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cond-mat0002078
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Axial Anomaly in Quasi-1D Chiral Superfluids
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"author": "J. Goryo"
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The axial anomaly in a quasi-one-dimensional (quasi-1D) chiral $p$-wave superfluid model, which has a $\varepsilon_{x} p_{x}+i \varepsilon_{y} p_{y}$-wave gap in 2D is studied. The anomaly causes an accumulation of the quasiparticle and a quantized chiral current density in an inhomogeneous magnetic field. These effects are related to the winding number of the gap. By varying the parameters $\varepsilon_{x}$ and $\varepsilon_{y}$, the model could be applicable to Sr$_{2}$RuO$_{4}$ near the second superconducting transition point, some quasi-1D organic superconductors and the fractional quantum Hall state at $\nu = 5 / 2$ Landau level filling factor.
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"name": "anomaly-q1cs.tex",
"string": "%\\documentstyle[12pt,epsf]{article}\n\\documentstyle[prl,aps,multicol,epsf]{revtex}\n%\\documentstyle[prb,aps,multicol,epsf]{revtex}\n%\\documentstyle[preprint,aps,epsf]{revtex}\n%\\setlength{\\textwidth}{170mm}\n%\\setlength{\\oddsidemargin}{-0mm}\n%%%%%%%%%%%%%%%%%%%%%%%%%\n% EPS.FIGURE FILES \n% ``fermi-cir.eps '' \n% ARE CONTAINED\n%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newcommand{\\Sla}[1]{{\\not \\! \\! {#1}}}\n\\newcommand{\\sla}[1]{{\\not \\! \\! \\: {#1}}}\n\\newcommand{\\slaD}{\\sla{D}}\n\\newcommand{\\slag}{\\Sla{\\! \\; \\gamma}}\n\\newcommand{\\slad}{\\Sla{\\, d}}\n\\begin{document}\n\\input epsf \n\\title{Axial Anomaly in Quasi-1D Chiral Superfluids}\n% \\draft command makes pacs numbers print\n\\draft\n\\author{J. Goryo} \n\\address{\\it Department of Physics, Hokkaido University, \nSapporo, 060-0810 Japan}\n\\date{\\today}\n\\maketitle\n\n \n\\begin{abstract}\nThe axial anomaly in a quasi-one-dimensional (quasi-1D) \nchiral $p$-wave superfluid model, which has a \n$\\varepsilon_{x} p_{x}+i \\varepsilon_{y} p_{y}$-wave gap in 2D is studied. \nThe anomaly causes an accumulation of the quasiparticle \nand a quantized chiral current density in an inhomogeneous magnetic field. \nThese effects are related to the winding number of the gap. \nBy varying the parameters $\\varepsilon_{x}$ and $\\varepsilon_{y}$, \nthe model could be applicable to Sr$_{2}$RuO$_{4}$ near the second \nsuperconducting transition point, some quasi-1D organic superconductors\nand the fractional quantum Hall state at $\\nu = 5 / 2$ \nLandau level filling factor. \n\n\\end{abstract}\n\n\\pacs{PACS numbers: 74.25.Ha, 03.70.+k, 73.40.Hm}\n\n\n\\begin{multicols}{2}\n\n%\\section{Introduction}\n\nThe chiral superfluidity is realized \nin the superfluid $^{3}$He-A \\cite{volovik-1}. \nRecently, the possibility of \nthe chiral superconductivity is argued\\cite{cpsc,cdsc}. \nIn such superfluids or superconductors, the ground state is the condensate of \nthe Cooper pairs which have orbital angular \nmomentum along a same direction. Therefore, time-reversal symmetry (T) and \nalso parity (P) in two-dimensional space (2D) are violating. \nWe investigate a quasi-1D chiral $p$-wave superfluid in 2D. \nIt is revealed that the axial anomaly causes \nP- and T-violating phenomena related to the quantized number. \n\nThe axial anomaly has been originally pointed out \nin the Dirac QED in 3D\\cite{a-b,b-j}. It is a phenomenon \nthat a symmetry under the phase transformation $e^{i \\gamma_{5} \\alpha}$ \nof the Dirac field in the action \nat the classical level is broken in the quantum theory. \nHere, $\\alpha$ is a constant and $\\gamma_{5}$ is a hermitian \nmatrix which anti-commutes with all of the Dirac matrices $\\gamma_{\\mu}$, \nwhere $\\mu$ is the spacetime index. \nThe Adler-Bardeen's theorem guarantees the absence of higher order \ncollections to the divergence of the axial current\\cite{a-b}. \nTherefore, the exact calculation of the two-photon decay rate \nof neutral $\\pi$ meson can be done.\nIt has been pointed out that the same results are \nobtained by using the path-integral formalism and has been clarified \nthe relation between the axial anomaly and topological \nquantized numbers through the Atiyah-Singer index theorem\\cite{fujikawa}. \n\nIt has been pointed out that the axial anomaly also plays important role \nin the quantum Hall effect (QHE) in the 2D massive Dirac QED. \n%although $\\gamma_{5}$ can be defined in the odd-dimensional space. \nIn 2D, the mass term of the Dirac Fermion violates P and T like \nthe magnetic field, and \nthe Hall effect may occur. \nIt was shown that the existence of the Hall current and its quantization \nare caused by the axial anomaly in 1D\\cite{red-nie-sem,ishikawa}. \nThe relation between the axial anomaly and QHE \nin 2D electron gas in the magnetic field was \nalso discussed\\cite{ishikawa,edge}, and the quantized Hall conductance is \nexpressed by the winding number of the fermionic propagator in \nthe momentum space\\cite{i-i-m-t}. \nOther applications of the axial anomaly \nto the condensed matter physics are studied in \nthe field of the superfluid $^{3}$He in 3D \nand in charge density waves in 1D conductors\\cite{volovik-1,he-3-cdw-anomaly}. \n\nThe phenomena caused by the axial anomaly are \nrelated to the topologically quantized numbers. \n%Therefore, it is important to search other \n%phenomena in which the axial anomaly is at work. \nOn the analogy of QHE, it is expected that the axial anomaly \nalso plays important role in other P- and T-violating 2D systems. \n In this letter, we investigate the axial anomaly in a quasi-1D \nchiral superfluid model in 2D, which has the spin-triplet \n$\\varepsilon_{x} p_{x} + i \\varepsilon_{y} p_{y}$-wave symmetry. \nP and T-violation occur whenever \nboth of $\\varepsilon_{x}$ and $\\varepsilon_{y}$ are non-zero. \nWe show that the axial anomaly in 1D causes an accumulation of \nthe mass density of the quasiparticle in an inhomogeneous magnetic field. \n%which is \n%proportional to the magnitude of \n%the gradient of the magnetic field. \n%with a coefficient \n%$- e \\mu N_{\\rm 1D}(0) \\times {\\rm sgn}(\\varepsilon_{x} \\varepsilon_{y}) $. \n%Here, $e$ and $\\mu$ is a mass and a magnetic moment of the quasiparticle, and \n%$N_{\\rm 1D}(0)$ is the density of state at the Fermi surface in 1D. \n%The axial anomaly \nThe axial anomaly also causes a chiral current density, which is \nperpendicular to the gradient of the magnetic field. \n%with \n%a quantized conductance $- e \\mu / 2 \\pi \\times \n%{\\rm sgn}(\\varepsilon_{x} \\varepsilon_{y})$. \nThese effects are related to the winding number of \nthe gap; ${\\rm sgn}(\\varepsilon_{x} \\varepsilon_{y})$\\cite{vol-g-i}. \nOur discussion would be valid for the superconductors \nby taking into account the Meissner effect. \nBy varying the parameters $\\varepsilon_{x}$ and $\\varepsilon_{y}$, \nthe model could be applicable to \nSr$_{2}$RuO$_{4}$ near the second superconducting transiton point\\cite{s-r-j}, \nand some quasi-1D organic superconductors\\cite{orgsc,lee,kohmoto}\nor the fractional quantum Hall (FQH) state at $\\nu = 5 / 2$ \nLandau level (LL) filling factor \\cite{maeda,5/2exp,pfaff}. \nWe use the 2+1-dimensional Euclidian spacetime and \nthe natural unit ($\\hbar=c=1$) in the present paper.\n\n\n\nLet us consider a quasi-1D chiral superfluid model. \nWe assume a linearized fermion spectrum and \na spin-triplet chiral $p$-wave gap near the Fermi surface \nin the normal state written as, \n\\begin{eqnarray}\n\\epsilon_{\\rm R,L}({\\bf p})&=&\\pm v_{\\rm F} (p_{x} \\mp p_{\\rm F}), \n\\label{udkin}\\\\ \n\\Delta({\\bf p}) &=& i \\sigma_{3} \\sigma_{2} \n\\frac{\\Delta}{|p_{F}|}(\\varepsilon_{x} p_{x} + i \\varepsilon_{y} p_{y}),\n\\label{c-p-w}\n\\end{eqnarray} \nwhere \n$v_{\\rm F}$ and $p_{\\rm F}$ are the Fermi velocity and the Fermi momentum, \nrespectively. $\\epsilon_{R} ({\\bf p})$ ($\\epsilon_{L} ({\\bf p})$) is \nthe kinetic energy for the right (left) mover. \n%$\\varepsilon_{x}$ and $\\varepsilon_{y}$ are parameters. \nWhen $\\varepsilon_{x}<<1$ and $\\varepsilon_{y}\\sim 1$, \nthe model describes the low energy excitations \n(the quasiparticle excitations around the tiny gap points) \nof Sr$_{2}$RuO$_{4}$ near the second superconducting transition point \nunder the uniaxial pressure in the $x-y$ plane (the \nbasal plane)\\cite{s-r-j}. For simplicity, \nwe assume a circular Fermi Surface in the normal state (Fig.1(a)). \nWhen $\\varepsilon_{x}\\sim1$ and $\\varepsilon_{y}<< 1$, \nthe model describes the excitations near $p_{x}=\\pm p_{\\rm F}$ \nwith the chiral $p$-wave gap, whose \nkinetic energy in the normal state is \n$\\epsilon({\\bf p})= - 2 t_{x} \\cos(p_{x} a) - 2 t_{y} \\sin(p_{y} b) \n- \\epsilon_{\\rm F}$ $(t_{y}<<t_{x}, \\epsilon_{\\rm F}$; the Fermi energy). \nThe model in this case is applicable to \nsome quasi-1D organic superconductors \nor the FQH state at $\\nu = 5 / 2$ LL \nfilling factor (Fig.1(b)). \nThe quasi-1D superconductivity \nhas been observed in organic conductors, such as (TMTSF)$_{2}$X\\cite{orgsc}. \nThe NMR knight shift study in Ref.\\cite{lee} is a evidence supporting \na spin-triplet pairing state in (TMTSF)$_{2}$PF$_{6}$. \nThe spin-triplet superconductivity in a quasi-1D system \nwith a nodeless gap is obtained theoretically \nwhen an electron-phonon coupling and \nantiferromagnetic fluctuations are taken into account\\cite{kohmoto}.\nOur discussion would be valid for such superconductors if they have \nthe chiral $p$-wave pairing symmetry.\nIt has been pointed out that the unidirectional charge density wave state, \nwhich has the belt-shaped Fermi sea like Fig. 1(b), seems to be \nthe most plausible compressible state at the half-filled Landau levels \nin the quantum Hall system\\cite{maeda}. \nRecently, the FQH effect has been observed at $\\nu=5 / 2$ \n\\cite{5/2exp}, and the effect could be described by \nthe chiral $p$-wave pairing state (the Pfaffian state)\\cite{pfaff}. \nTherefore, our model could be a candidate of \nthe $\\nu=5 / 2$ FQH state. \n \n\n%%%%%%%%%\n% Insert Figure 1 (a) and (b) \n%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\vspace{0cm}\n\\begin{figure}\n\\centerline{\n\\epsfysize=7cm\\epsffile{fermi-cir.eps}}\n\\vspace{-4cm}\nFig. 1. The Fermi sea in the normal state and \nthe momentum dependence of the gap functions \nfor (a) Sr$_{2}$RuO$_{4}$ near the second superconducting \nphase transition point, which corresponds to \n$\\varepsilon_{x} <<1$, $\\varepsilon_{y}\\sim 1$ in our model, \nand for (b) some quasi-1D organic superconductors with \nthe chiral $p$-wave pairing symmetry or \nthe FQH state at $\\nu=5 / 2$ LL\nfilling factor, which corresponds to \n$\\varepsilon_{x} \\sim 1$, $\\varepsilon_{y}<<1$. \nThe shadows show the Fermi sea, and the distance between \nouter lines and inner lines shows the magnitude of the gap. \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nThe Lagrangian of our model is written as \n\\begin{eqnarray} \n{\\cal {L}}&=&\\bar{\\Psi}_{\\bf p} \n\\left[\\{ i \\partial_{\\tau} + \\mu \\frac{d {\\rm B}_{z}}{d y} y \\sigma_{3} \\} \n\\otimes \\gamma_{\\tau} \\right. \n\\label{ucpsc-lag}\\\\\n&&\\left.+ \\frac{\\Delta}{|p_{\\rm F}|} \\sigma_{3} \\otimes\n( \\varepsilon_{x} p_{x} \\gamma_{x} + \\varepsilon_{y} p_{y} \\gamma_{y} ) \n- i v_{\\rm F} (p_{x} - p_{\\rm F}) \\right] \\Psi_{\\bf p}. \n\\nonumber\n\\end{eqnarray} \nHere, we use the Bogoliubov-Nambu representation with an isospin $\\alpha=1,2$\n$$\n\\Psi({\\bf x})=e^{i p_{\\rm F} x} \n\\left(\\begin{array}{c}\n\\psi(p_{\\rm F},{\\bf x}) \\\\ \ni \\sigma_{2} \\psi^{*}(-p_{\\rm F},{\\bf x}) \n\\end{array} \\right), \n$$ \nand $\\Psi_{\\bf p}$ is its Fourier transform. \n$\\psi(p_{\\rm F},{\\bf x})$ and $\\psi(-p_{\\rm F},{\\bf x})$ are\nthe slowly varying fields for the right mover and the left mover \nwith a real spin index $s=1,2$, \nrespectively. The matrices \n$\\gamma_{\\tau}=\\tau_{3}, \\gamma_{x}=-\\tau_{2}$ and $\\gamma_{y}=-\\tau_{1}$ \nare the $2\\times2$ Pauli matrices with isospin indices and \n$\\sigma_{i}~(i=1,2,3)$ is the $2\\times2$ Pauli matrices with spin indices. \nThe symbol $\\sigma_{i} \\otimes \\gamma_{\\tau,x,y}$ shows the direct product.\n$\\bar{\\Psi}$ is defined as $\\bar{\\Psi}= - i \\Psi^{\\dagger} \\gamma_{\\tau}$. \nWe assume a magnetic field, which is directed to the $z$-axis \n(the $c$-axis in the crystal) and \nhas a constant gradient in the $y$-direction, \ni.e. $B_{z}(y)=(d B_{z} / d y) y$, and $(d B_{z} / d y)=const.$ \nThe magnetic field couples with the Fermion through the Zeeman term \n$\\mu B_{z} \\bar{\\Psi} \\sigma_{3} \\otimes \\gamma_{\\tau} \\Psi$, \nwhere $\\mu$ is the magnetic moment of the Fermion. \nWe note that the Lagrangian is similar to that of \n2D Dirac QED in a background scalar potential, except for the last term \nand the existence of $\\sigma_{3}$. \nThe axial anomaly in such a system is discussed \nin Ref. \\cite{red-nie-sem,ishikawa}. \n\nLet us calculate the expectation value of the mass density \n\\begin{eqnarray}\n\\left<\\rho_{e}({\\bf x})\\right>\n&=& e \\left< \\bar{\\Psi} ({\\bf x}) \\Psi ({\\bf x}) \\right> \n\\label{j-exp}\\\\\n&=& {\\rm Tr}\\left[\\frac{e \\sigma_{3}}{\\slad + \n\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{x} p_{x} \\hat{1} \\otimes \\gamma_{x} \n- i v_{\\rm F} (p_{x} - p_{\\rm F}) \\sigma_{3} \\otimes \\hat{1}} \\right], \n\\nonumber\n\\end{eqnarray}\nwhere $e$ shows a mass of the quasiparticle. \nIt shows an electric charge when we consider a superconductor. \nA hermitian operator $\\slad$ in the $y$-direction is defined as \n\\begin{equation} \n\\slad = \\left\\{ i \\partial_{\\tau} \\sigma_{3} + \n\\mu \\frac{d B_{z}}{d y} y \\right\\} \\otimes \\gamma_{\\tau} \n\\nonumber\\\\\n+ \n\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{y} p_{y} \\hat{1} \\otimes \\gamma_{y}. \n\\end{equation} \nWe define $\\gamma_{5}$ as $\\gamma_{5}=i \\gamma_{\\tau} \\gamma_{y}=-\\gamma_{x}$,\nand it is anti-commute with $\\gamma_{\\tau}$ and $\\gamma_{y}$, \ntherefore, $\\gamma_{5}$ is a hermitian matrix and satisfies \n$\\{ \\gamma_{5}, \\slad \\}= 0$. \nThese facts suggest that \nif an eigenstate $u_{n}$ of $\\slad$ with a nonzero \neigenvalue $\\xi_{n}$ ($0<n$) exists \n(i.e. $\\slad u_{n} = \\xi_{n} u_{n}$), \n$\\gamma_{5} u_{n}$ should be another eigenstate with an eigenvalue $-\\xi_{n}$.\nIf zeromodes of $\\slad$ exist (i.e. $\\slad u_{0} = 0$ and \n$\\slad \\gamma_{5} u_{0}=0$), they are divided into two groups. \nOne of them is $u_{0}^{(+)}=(1/2)(1+\\gamma_{5})u_{0}$ \nwith an eigenvalue $\\gamma_{5}=+1$ and \nanother is $u_{0}^{(-)}=(1/2)(1-\\gamma_{5})u_{0}$ with \nan eigenvalue $\\gamma_{5}=-1$, since $\\gamma_{5}^{2}=1$. \n\n\nLet us research eigenmodes of $\\slad$. \nThe expectation value of $\\slad^{2}$ is \n\\begin{eqnarray} \n(u_{n}, \\slad^{2} u_{n})&=&|\\omega_{c}| \n(n + \\frac{1}{2}) \n+ \\frac{\\omega_{c}}{2} (u_{n}, \\gamma_{5} u_{n}), \n\\nonumber\\\\ \n\\omega_{c}&=&\\mu \\frac{d B_{z}}{d y} \\frac{2 \\Delta}{|p_{\\rm F}|} \n\\varepsilon_{y}, \n\\label{expvalue}\n\\end{eqnarray} \nwhere $u_{n}=u_{n}(y - y_{c}(p_{\\tau}, \\sigma_{3}))$ \nis the eigenfunction of \nthe harmonic oscillator with the frequency \n$\\omega_{c}$. The oscillator is centered at \n%\\begin{equation}\n$\ny_{c} (p_{\\tau},\\sigma_{3}) = \n- (d B_{z} / d y)^{-1} (p_{\\tau}/\\mu) \\sigma_{3}. \n$\n%\\end{equation}\nEq. (\\ref{expvalue}) indicates that \nonly zeromodes which belong to $u_{0}^{-}$ ($u_{0}^{+}$) exist \nwhen $0<\\omega_{c}$ ($\\omega_{c}<0$). \nIt suggests the nonconservation of the vacuum expectation value of \nthe axial charge which is defined in the second-quantized formalism as \n\\begin{eqnarray}\n\\left<Q_{5}\\right>&=&\\left<N_{+} - N_{-}\\right>, \n\\nonumber\\\\ \nN_{\\pm}&=&\\int dp_{y} \\hat{u}_{0}^{\\dagger (\\pm)} \\hat{u}_{0}^{(\\pm)}, \n\\end{eqnarray} \nwhile the classical 1D theory \n${\\cal{L}}_{\\rm 1D}=\\bar{\\Psi} \\slad \\Psi$ \nhas the axial symmetry $\\Psi \\rightarrow e^{i \\alpha \\gamma_{5}} \\Psi$, \ni.e. {\\it the axial anomaly occurs.} \nHere, $\\hat{u}_{0}^{\\pm}$ is a second-quantized fermionic field. \nThe anomaly comes from the spectral asymmetry \nof zeromodes as same as the discussions \nin Refs.\\cite{red-nie-sem,ishikawa,j-l-h}. \nIn the free system, the energy spectrum of $u_{0}^{(\\pm)}$ is \n$p_{0}= \\pm \\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{y} p_{y}$ \nin Minkowski spacetime, \nand all of the negative energy states are filled \nwhile all of the positive energy states are empty and $\\left<Q_{5}\\right>=0$. \nAfter we turn on the magnetic field adiabatically \n(for a while, we assume $0<\\omega_{c}$), \nthe energy spectrum of $u_{0}^{(+)}$ is lowered \nand $\\left<N_{+}\\right>$ decreases (i.e. empty negative energy states arise \non the spectrum of $u_{0}^{(+)}$), on the other hand, \nthe energy spectrum of $u_{0}^{(-)}$ is lifted \nand $\\left<N_{-}\\right>$ increases (i.e. filled positive energy states arise \non the spectrum of $u_{0}^{(-)}$), \ntherefore $\\left<{Q_{5}}\\right>$ does {\\it not} conserve. \nFinally $\\left<N_{+}\\right>=0$ and only $u_{0}^{(-)}$ exists. \nThe nonzero eigenvalues of $\\slad^{2}$ is \n$E_{n}=\\omega_{c} (n + 1/2)$, since \nthe inner product $(u_{n}, \\gamma_{5} u_{n})$ vanishes whenever \n$\\slad u_{n} \\neq 0$ because of the orthogonal relation between \nthe eigenfunctions of the hermitian operator. \n\n\nNext, we consider the eigenvalue problem of a 2D operator\n\\begin{equation}\n\\slaD = \\slad + \n\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{x} p_{x} \\hat{1} \\otimes \\gamma_{x} \n= \n\\slad - \n\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{x} p_{x} \\hat{1} \\otimes \\gamma_{5}.\n\\end{equation}\nLet \n\\begin{equation}\n\\varphi_{n}= (\\alpha_{n} u_{n} + \\beta_{n} \\gamma_{5} u_{n}) \ne^{i p_{x} x}\n\\end{equation}\nstands for an eigenfunction. \nWe use a representation for the $n$-th level such as ($\\nearrow$)\n\\begin{equation} \n\\slad= \n\\left( \\begin{array}{cc} \n\\xi_{n} & 0 \\\\\n0 & - \\xi_{n} \n\\end{array} \\right) , \nu_{n}=\n\\left( \\begin{array}{c} \n1 \\\\\n0\n\\end{array} \\right) , \n\\gamma_{5}= \n\\left( \\begin{array}{cc} \n0 & 1 \\\\\n1 & 0 \n\\end{array} \\right), \n\\end{equation} \nwhere, \n$$\n\\xi_{n}=\n\\left\\{\\begin{array}{cl}\n\\sqrt{|\\omega_{c}| (n + \\frac{1}{2})} \n& (n = 1,2,\\cdot\\cdot\\cdot), \n\\\\\n0 & (n=0). \n\\end{array}\\right.\n$$\nTherefore, the eigenvalue equation is written as \n\\begin{equation}\n\\left(\\begin{array}{cc}\n\\xi_{n} & -\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{x} p_{x} \\\\ \n-\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{x} p_{x} & -\\xi_{n} \n\\end{array} \\right)\n\\left(\\begin{array}{c}\n\\alpha_{n} \\\\ \n\\beta_{n} \n\\end{array} \\right)\n=\\zeta_{n}\n\\left(\\begin{array}{c}\n\\alpha_{n} \\\\ \n\\beta_{n} \n\\end{array} \\right). \n\\end{equation} \nThere are two eigenstates for an oscillator in the $n (\\neq 0)$-th level \nwritten as \n\\begin{eqnarray}\n&\\zeta_{n}^{(\\pm)}(p_{x})&\n=\\pm \\sqrt{\\xi_{n}^{2} + \n\\frac{\\Delta^{2}}{p_{\\rm F}^{2}} \\varepsilon_{x}^{2} p_{x}^{2}}, \n\\\\\n&\\left(\\begin{array}{c}\n\\alpha_{n}^{+} \\\\\n\\beta_{n}^{+} \n\\end{array} \\right)& \n=\\frac{1}{C_{+}} \n\\left(\\begin{array}{c} \n\\zeta_{n}^{(+)} + \\xi_{n} \\\\\n-\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{x} p_{x} \n\\end{array} \\right) , \n\\nonumber\\\\\n&\\left(\\begin{array}{c}\n\\alpha_{n}^{-} \\\\\n\\beta_{n}^{-} \n\\end{array} \\right)&\n=\\frac{1}{C_{-}} \n\\left(\\begin{array}{c} \n\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{x} p_{x} \\\\ \n-\\zeta_{n}^{(-)} + \\xi_{n} \n\\end{array} \\right), \n\\nonumber\n\\end{eqnarray}\nwhere $C_{\\pm}$ are normalization constants, but for $n=0$, there is only \none eigenstate \n\\begin{eqnarray} \n\\zeta_{0}(p_{x})&=& \\frac{\\omega_{c}}{|\\omega_{c}|} \n\\frac{\\Delta}{|p_{\\rm F}|} \\varepsilon_{x} p_{x} \n, \n\\nonumber\\\\\n\\left(\\begin{array}{c} \n\\alpha_{0} \\\\\n\\beta_{0} \n\\end{array} \\right) \n&=& \\frac{1}{2}\n\\left(\\begin{array}{c} \n1 \\\\\n- \\omega_{c}/|\\omega_{c}| \n\\end{array} \\right), \n\\end{eqnarray} \nbecause the solution should satisfy \n$\\gamma_{5} \\varphi_{0} = - (\\omega_{c}/|\\omega_{c}|)\\varphi_{0}$. \nThis condition comes from the axial anomaly in the $y$-direction. \n\nFinally, we show the accumulation of \nthe mass density from Eq. (\\ref{j-exp}), \nwhich is derived as\n\\end{multicols}\n\\begin{eqnarray} \n\\left<\\rho_{e}({\\bf x})\\right>&=& {\\rm Tr} \n\\left[ \n\\frac{e \\sigma_{3}} \n{\\slaD - i v_{\\rm F}(p_{x} - p_{\\rm F})\\sigma_{3} \\otimes \\hat{1}} \n\\right] \n%\\nonumber\\\\ \n%&=& \\sum_{n} \n=\\sum_{n}\n\\int_{-\\infty}^{\\infty} \\frac{d p_{\\tau}}{2 \\pi} \n\\int \\frac{d p_{x}}{2 \\pi} {\\rm tr} \\left[ \n\\frac{e \\sigma_{3} \n|u_{n}(y - y_{0}(p_{\\tau}, \\sigma_{3}))|^{2}} \n{\\zeta_{n}(p_{x}) - \ni v_{\\rm F} (p_{x} - p_{\\rm F}) \\sigma_{3}} \\right] \n\\nonumber\\\\\n&=& \\frac{-e \\mu}{2 \\pi}\\frac{d B_{z}}{d y} \\sum_{n \\neq 0} \n\\int \\frac{d p_{x}}{2 \\pi} {\\rm tr} \\left[ \n\\frac{1} \n{\\zeta_{n}^{(+)}(p_{x}) - \ni v_{\\rm F} (p_{x} - p_{\\rm F}) \\sigma_{3}} \n+\n\\frac{1} \n{\\zeta_{n}^{(-)}(p_{x}) - \ni v_{\\rm F} (p_{x} - p_{\\rm F}) \\sigma_{3}} \n\\right]\n\\nonumber\\\\\n&&\n- \\frac{e \\mu}{2 \\pi} \\frac{d B_{z}}{d y} \n\\int \\frac{d p_{x}}{2 \\pi} {\\rm tr} \\left[ \n\\frac{1} \n{\\zeta_{0}(p_{x}) - \ni v_{\\rm F} (p_{x} - p_{\\rm F})\\sigma_{3}} \\right]\n=- {\\rm sgn}(\\varepsilon_{x} \\varepsilon_{y}) \ne \\mu N_{\\rm 1D}(0) \n\\frac{d B_{z}}{d y}, \n\\end{eqnarray}\n\\begin{multicols}{2}\nwhere the symbol $tr$ means a trace on the real spin, \nand we use the normal-orthogonal relation $\\int dy |u_{n}|^{2}=1$. \n$\\int \\frac{d p_{x}}{2 \\pi}\n=\\int_{p_{\\rm F}-\\Lambda}^{p_{\\rm F}+\\Lambda}\n\\frac{d p_{x}}{2 \\pi}$ \n, and $\\Lambda$ is a momentum cutoff. \nWe assume a relation $|\\Delta|<<\\Lambda^{2} / 2 m<<\\epsilon_{F}$. \n$N_{\\rm 1D}(0)=(2 \\pi v_{\\rm F})^{-1}$ is the density of state at \nthe Fermi surface in 1D.\nAll of the $n \\neq 0$ parts are canceled out because of the \nco-existence of the eigenvalues $\\zeta_{n}^{(+)}$ and $\\zeta_{n}^{(-)}$. \n{\\it Only the $n=0$ part survives because of the axial anomaly in the \n$y$-direction}. \n\nWe can define a chiral transformation in the $x$-direction \nsuch as $\\psi(\\pm p_{\\rm F}, {\\bf x}) \\rightarrow \ne^{\\pm i \\alpha} \\psi(\\pm p_{\\rm F}, {\\bf x})$, therefore \n$\\Psi \\rightarrow e^{i \\alpha} \\Psi, \\bar{\\Psi} \\rightarrow \n\\bar{\\Psi}e^{i \\alpha}$. The expectation value of \nthe corresponding current density which is perpendicular to $d B_{z} / dy$ is \n\\begin{equation} \n\\left< j_{x}^{chi} ({\\bf x}) \\right> \n=e v_{\\rm F} \\left< \\bar{\\Psi}({\\bf x})\\Psi({\\bf x}) \\right> \n=- {\\rm sgn}(\\varepsilon_{x} \\varepsilon_{y}) \\frac{e \\mu}{2 \\pi} \n\\frac{d B_{z}}{d y}, \n\\end{equation} \nand we call it a chiral Hall current density. \n\nThese two effects are related to the winding number of \nthe gap Eq. (\\ref{c-p-w})\\cite{vol-g-i}, \n\\begin{eqnarray}\n{\\rm sgn}(\\varepsilon_{x} \\varepsilon_{y})\n&=&\\int \\frac{d^{2} p}{16 \\pi} \ntr[\\hat{{\\bf g}} \\cdot \n({\\bf \\nabla}\\hat{{\\bf g}} \\times {\\bf \\nabla} \\hat{{\\bf g}})] \n, \n\\\\ \n{\\bf g}({\\bf {p}})&=& \n\\left(\\begin{array}{c} \n{\\rm Re}[\\Delta({\\bf {p}})(-i \\sigma_{2})] \\\\ \n-{\\rm Im}[\\Delta({\\bf {p}})(-i \\sigma_{2})] \\\\\n({\\bf p}^{2} / 2 m) - \\epsilon_{\\rm F}\n\\end{array} \\right), \n\\nonumber\n\\end{eqnarray}\nwhere ${\\bf \\nabla}=\\partial / \\partial {\\bf p}$. \nIt suggests that these effects occur even if $\\varepsilon_{x}$ and/or \n$\\varepsilon_{y}$ are infinitesimally small, and that these effects \ncome from the P- and T-violation of the gap. \n\n\n \nThe accumulated mass density and the chiral Hall current density \nexist in the bulk region of the superfluid. \nIn the superconductors, the Meissner effect occurs \nand the magnetic field cannot penetrate into the bulk, therefore \nthe accumulated charge density and the chiral Hall \ncurrent density would exist near the edge \nof the superconductors\\cite{note} and also around the vortex core. \nAs we mentioned before, \nour discussion could be applicable to Sr$_{2}$RuO$_{4}$ near the \nsecond superconducting phase transition point, \n%\\cite{cpsc,s-r-j}, \nsome quasi-1D organic superconductors\n%\\cite{orgsc,lee,kohmoto} \nand the FQH state \nat $\\nu = 5 / 2$ LL filling factor\n%\\cite{maeda,5/2exp,pfaff} \nby varying the parameters $\\varepsilon_{x}$ and $\\varepsilon_{y}$. \n%(Fig. 1). \nRecently, the vortex in chiral superconductors has \nbeen discussed \\cite{goryo}, and such a vortex has a fractional charge and \na fractional angular momentum. \nInteresting phenomena related to these fractional quantum numbers \nand the present effects are expected to occur \naround the vortex core. \n\n\nThe axial anomaly also causes the spin quantum Hall \neffect (SQHE) in the chiral $d$-wave ($d_{x^{2}-y^{2}} + i d_{xy}$-wave) \nsuperconductors\\cite{fisher-sqhe}. \nThe low energy quasiparticles in a magnetic field \nwith a constant gradient can be mapped onto the massive Dirac Fermion in \na constant electric field, and the spin rotation around the $z$-axis \nfor the quasiparticle \ncorresponds to the $U(1)$ transformation for the Dirac Fermion. \nTherefore, according to the \ndiscussions in Ref.\\cite{red-nie-sem,ishikawa}, \nwe can see that the axial anomaly causes the quantized spin Hall current, \nwhich is perpendicular to the gradient of the \nmagnetic field. \n\nSQHE has been pointed out by Volovik and Yakovenko \nin superfluid $^{3}$He-A film, \nwhich is the chiral $p$-wave superfluid\\cite{volovik-1,vol-yak}. \nThey have described the effect by the Chern-Simons term. \nIt has been clarified the relation between the axial anomaly \nand the Chern-Simons term in 2D Dirac QED\\cite{ishikawa}. \nTherefore, SQHE in $^{3}$He-A could be related \nto the axial anomaly. According to Ref.\\cite{vol-yak}, \nSQHE also occurs at the edge or around the vortex core of\n the superconducting Sr$_{2}$RuO$_{4}$\\cite{cpsc} by \na magnetic field in the basal plane. \n\nThe author thanks K. Ishikawa and N. Maeda for useful discussions and \nencouragement. \nThis work was partially supported by the special Grant-in-Aid for \nPromotion of Education and Science in Hokkaido University provided by \nthe Ministry of Education, Science, Sports, and Culture, the Grant-in-Aid \nfor Scientific Research on Priority area \n(Physics of CP violation)\n(Grant No. 10140201), and the Grant-in-Aid for International Science Research \n(Joint Research 10044043) from the \nMinistry of Education, Science, Sports and Culture, Japan. \n\\begin{references}\n\n\\bibitem{volovik-1}\nSee, for example, G. E. Volovik, \n{\\it ``Exotic Property of Superfluid $^{3}$He''}, \nWorld Scientific, Singapole (1992). \n\n\\bibitem{cpsc}\nG. M. Luke, {\\it et. al.}, Nature {\\bf 394}, 558-561 (1998). \n\n\\bibitem{cdsc}\nK. Krishana {\\it et. al.}, Science {\\bf 277}, 83 (1997); \nR. B. Laughlin, Phys. Rev. Lett. {\\bf 80} 5188 (1998). \n\n\n\\bibitem{a-b}\nS. L. Adler, Phys. Rev. {\\bf 177} 2426 (1969); \nW. Bardeen, {\\it ibid.} {\\bf 184} 1848 (1969). \n\n\\bibitem{b-j}\nJ. S. Bell and R. Jackiw, Nuovo Cim. {\\bf 60A} 47 (1969). \n\n\\bibitem{fujikawa}\nK. Fujikawa, Phys. Rev. Lett. {\\bf 42} 1195 (1979); {\\it ibid.} \n{\\bf 44} 1733 (1980); \nM. Atiyah and I. Singer, Ann. Math. {\\bf 87},484 (1968). \n\n\\bibitem{red-nie-sem}\nA. Niemi and G. Semenoff, Phys. Rev. Lett. {\\bf 51}, 2077 (1983); \nA. Redlich, {\\it ibid.} {\\bf 52} 18 (1984). \n\n\\bibitem{ishikawa}\nK. Ishikawa, Phys. Rev. Lett. {\\bf 53} 1615 (1984); \nPhys. Rev. D {\\bf 31} 1432 (1985). \n\n\\bibitem{edge}\nX. G. Wen, Phys. Rev. Lett {\\bf 64} 2206 (1990); \nN. Maeda, Phys. Lett. B, {\\bf 376} 142 (1996), and references therein. \n\n\\bibitem{i-i-m-t}\nK. Ishikawa and T. Matsuyama, Z. Phys. C {\\bf 33}, 41 (1986); \nNucl. Phys. {\\bf B280}, 523 (1987); \nN. Imai, {\\it et. al.}, Phys. Rev. B{\\bf 42}, 10610 (1990). \n\n\\bibitem{he-3-cdw-anomaly}\nG. E. Volovik, Sov. Phys. JETP {\\bf 65}, 1193 (1987); \nM. Stone and F. Gaitan, Annals of Phys. {\\bf 178}, 89 (1987); \nB. Sakita and K. Shizuya, Phys. Rev. B, {\\bf 42} 5586 (1990). \n \n\\bibitem{vol-g-i}\nG.E. Volovik, Sov. Phys. JETP {\\bf 67}, 1804 (1988);\nJ. Goryo and K. Ishikawa, Phys. Lett. A {\\bf 260}, 294 (1999). \n\n\\bibitem{s-r-j} \nM. Sigrist, R. Joynt and T. M. Rice, Europhys. Lett., {\\bf 3} 629 (1987). \n\n\\bibitem{orgsc} \nSee, for reviews, \nT. Ishiguro and K. Yamaji, {\\it Organic superconductors} \n(Springer, 1990). \n\n\\bibitem{lee} \nI. J. Lee, {\\it et. al.}, cond-mat/0001332. \n\n\\bibitem{kohmoto}\nM. Kohmoto and M. Sato, cond-mat/0001331. \n\n\\bibitem{maeda} \nN. Maeda, Phys. Rev. B {\\bf 61} 4766 (2000). \n\n\\bibitem{5/2exp} \nW. Pan {\\it et. al.}, Phys. Rev. Lett. {\\bf 83} 3530 (1999). \n\n\\bibitem{pfaff} \nG. Moore and N. Read, Nucl. Phys. {\\bf B360} 362 (1991). \n\n\\bibitem{j-l-h} R. Jackiw, The Proceedings of \n``{\\it Lectures in Les Houches summer school, \nSession XL}'', 221 (1983). \n\n\\bibitem{note} Note that the axial anomaly discussed here \nis in the $y$-direction, which is perpendicular to the edge, \ntherefore it is different from that of the \nchiral edge mode\\cite{edge}. \n\n\\bibitem{goryo} \nJ. Goryo, Phys. Rev. B {\\bf 61} 4222 (2000). \n\n\\bibitem{fisher-sqhe}\nT. Senthil, J. B. Marston and M. P. A. Fisher, \ncond-mat/9902062. \n\n\\bibitem{vol-yak}\nG. E. Volovik and V. M. Yakovenko, \nJ. Phys. Condens. Matter {\\bf 1} 5263 (1989). \nSee, also N. Read and D. Green, cond-mat/9906453. \n\n\n\n\\end{references}\n\n\\end{multicols}\n\n\\end{document}\n\n\\newpage \n\n\\begin{center}\nFIGURE CAPTION \n\\end{center}\n\nFig. 1. The Fermi sea in the normal state \nand the momentum dependence of the gap functions \nfor (a) Sr$_{2}$RuO$_{4}$ near the second superconducting \nphase transition point, which corresponds to \n$\\varepsilon_{x} <<1$, $\\varepsilon_{y}\\sim 1$ in our model, \nand for (b) some quasi-1D organic superconductor with \nthe chiral $p$-wave pairing symmetry or \nthe FQH state at $\\nu=5 / 2$ LL \nfilling factor, which correspond to \n$\\varepsilon_{x} \\sim 1$, $\\varepsilon_{y}<<1$. \nThe shadows show the Fermi sea, and the distance between \nouter lines and inner lines shows the magnitude of the gap. \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\vspace{0cm}\n\\begin{figure}\n\\centerline{\n\\epsfysize=10cm\\epsffile{fermi-cir.eps}}\n\\vspace{-4cm}\nFig.1. \n\n\\vspace{1cm}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n%\\end{multicols}\n\n\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"
}
] |
[
{
"name": "cond-mat0002078.extracted_bib",
"string": "\\bibitem{volovik-1}\nSee, for example, G. E. Volovik, \n{\\it ``Exotic Property of Superfluid $^{3}$He''}, \nWorld Scientific, Singapole (1992). \n\n\n\\bibitem{cpsc}\nG. M. Luke, {\\it et. al.}, Nature {\\bf 394}, 558-561 (1998). \n\n\n\\bibitem{cdsc}\nK. Krishana {\\it et. al.}, Science {\\bf 277}, 83 (1997); \nR. B. Laughlin, Phys. Rev. Lett. {\\bf 80} 5188 (1998). \n\n\n\n\\bibitem{a-b}\nS. L. Adler, Phys. Rev. {\\bf 177} 2426 (1969); \nW. Bardeen, {\\it ibid.} {\\bf 184} 1848 (1969). \n\n\n\\bibitem{b-j}\nJ. S. Bell and R. Jackiw, Nuovo Cim. {\\bf 60A} 47 (1969). \n\n\n\\bibitem{fujikawa}\nK. Fujikawa, Phys. Rev. Lett. {\\bf 42} 1195 (1979); {\\it ibid.} \n{\\bf 44} 1733 (1980); \nM. Atiyah and I. Singer, Ann. Math. {\\bf 87},484 (1968). \n\n\n\\bibitem{red-nie-sem}\nA. Niemi and G. Semenoff, Phys. Rev. Lett. {\\bf 51}, 2077 (1983); \nA. Redlich, {\\it ibid.} {\\bf 52} 18 (1984). \n\n\n\\bibitem{ishikawa}\nK. Ishikawa, Phys. Rev. Lett. {\\bf 53} 1615 (1984); \nPhys. Rev. D {\\bf 31} 1432 (1985). \n\n\n\\bibitem{edge}\nX. G. Wen, Phys. Rev. Lett {\\bf 64} 2206 (1990); \nN. Maeda, Phys. Lett. B, {\\bf 376} 142 (1996), and references therein. \n\n\n\\bibitem{i-i-m-t}\nK. Ishikawa and T. Matsuyama, Z. Phys. C {\\bf 33}, 41 (1986); \nNucl. Phys. {\\bf B280}, 523 (1987); \nN. Imai, {\\it et. al.}, Phys. Rev. B{\\bf 42}, 10610 (1990). \n\n\n\\bibitem{he-3-cdw-anomaly}\nG. E. Volovik, Sov. Phys. JETP {\\bf 65}, 1193 (1987); \nM. Stone and F. Gaitan, Annals of Phys. {\\bf 178}, 89 (1987); \nB. Sakita and K. Shizuya, Phys. Rev. B, {\\bf 42} 5586 (1990). \n \n\n\\bibitem{vol-g-i}\nG.E. Volovik, Sov. Phys. JETP {\\bf 67}, 1804 (1988);\nJ. Goryo and K. Ishikawa, Phys. Lett. A {\\bf 260}, 294 (1999). \n\n\n\\bibitem{s-r-j} \nM. Sigrist, R. Joynt and T. M. Rice, Europhys. Lett., {\\bf 3} 629 (1987). \n\n\n\\bibitem{orgsc} \nSee, for reviews, \nT. Ishiguro and K. Yamaji, {\\it Organic superconductors} \n(Springer, 1990). \n\n\n\\bibitem{lee} \nI. J. Lee, {\\it et. al.}, cond-mat/0001332. \n\n\n\\bibitem{kohmoto}\nM. Kohmoto and M. Sato, cond-mat/0001331. \n\n\n\\bibitem{maeda} \nN. Maeda, Phys. Rev. B {\\bf 61} 4766 (2000). \n\n\n\\bibitem{5/2exp} \nW. Pan {\\it et. al.}, Phys. Rev. Lett. {\\bf 83} 3530 (1999). \n\n\n\\bibitem{pfaff} \nG. Moore and N. Read, Nucl. Phys. {\\bf B360} 362 (1991). \n\n\n\\bibitem{j-l-h} R. Jackiw, The Proceedings of \n``{\\it Lectures in Les Houches summer school, \nSession XL}'', 221 (1983). \n\n\n\\bibitem{note} Note that the axial anomaly discussed here \nis in the $y$-direction, which is perpendicular to the edge, \ntherefore it is different from that of the \nchiral edge mode\\cite{edge}. \n\n\n\\bibitem{goryo} \nJ. Goryo, Phys. Rev. B {\\bf 61} 4222 (2000). \n\n\n\\bibitem{fisher-sqhe}\nT. Senthil, J. B. Marston and M. P. A. Fisher, \ncond-mat/9902062. \n\n\n\\bibitem{vol-yak}\nG. E. Volovik and V. M. Yakovenko, \nJ. Phys. Condens. Matter {\\bf 1} 5263 (1989). \nSee, also N. Read and D. Green, cond-mat/9906453. \n\n\n\n"
}
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cond-mat0002079
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Trapped atomic condensates with anisotropic interactions
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[
{
"author": "S. Yi and L. You"
}
] |
We study the ground state properties of trapped atomic condensates with electric field induced dipole-dipole interactions. A rigorous method for constructing the pseudo potential in the spirit of ladder approximation is developed for general non-spherical (polarized) particles interacting anisotropically. We discuss interesting features not previously considered for currently available alkali condensates. In addition to provide a quantitative assessment for controlling atomic interactions with electric fields, our investigation may also shed new light into the macroscopic coherence properties of the Bose-Einstein condensation (BEC) of dilute interacting atoms.
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[
{
"name": "nabec.tex",
"string": "%\\documentstyle[preprint,aps,eqsecnum]{revtex}\n%\\documentstyle[preprint,aps]{revtex}\n%\\documentstyle[twocolumn,aps]{revtex}\n%\\setlength{\\oddsidemargin}{0in}\n%macros for journals\n\n\\documentstyle[aps]{revtex}\n%\\input{psfig.sty}\n\n\\begin{document}\n\\draft\n\\title{Trapped atomic condensates with anisotropic interactions}\n\\author{S. Yi and L. You}\n\\address{School of Physics, Georgia Institute of Technology,\nAtlanta, GA 30332-0430}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nWe study the ground state properties of trapped atomic condensates\nwith electric field induced dipole-dipole interactions.\nA rigorous method for constructing the pseudo potential in the spirit\nof ladder approximation is developed for general non-spherical (polarized)\nparticles interacting anisotropically.\nWe discuss interesting features not previously considered for\ncurrently available alkali condensates. In addition\nto provide a quantitative assessment for controlling atomic\ninteractions with electric fields, our investigation may also\nshed new light into the macroscopic coherence properties of the\nBose-Einstein condensation (BEC) of dilute interacting atoms.\n\\end{abstract}\n\n\\pacs{03.75.Fi,34.10.+x,32.80.Cy}\n\n\\narrowtext\n\nThe success of atomic Bose-Einstein condensation (BEC)\n\\cite{bec,mit,rice,Edwards} has stimulated great interest\nin the properties of trapped quantum gases.\nIn standard treatments of interacting quantum gases,\nrealistic inter-atomic potentials $V(\\vec R)$ are replaced by\ncontact forms $u_0\\delta(\\vec R)$ in the so-called\nshape independent approximation (SIA) \\cite{yang}.\nSuch an approximation results in tremendous simplification.\nTo date, the SIA has worked remarkably well as recent\ntheoretical investigations \\cite{Edwards} have successfully\naccounted for almost all experimental observations \\cite{glauber,esry}.\n\nCurrently available degenerate quantum gases are cold and dilute,\nwith interactions dominated by low energy binary collisions.\nWhen realistic interatomic potentials are\nassumed to be isotropic and short ranged, i.e. decreasing faster\nthan $-1/R^3$ asymptotically for large interatomic separations $R$,\nthe properties of\na complete two body collision is described by just one\natomic parameter: $a_{\\rm sc}$,\nthe s-wave {\\it scattering length}.\nThe scattering amplitude is isotropic\nand energy-independent:\n$f(\\vec k,\\vec k')=-4\\pi a_{\\rm sc}$ for collisions involving\nincident momentum $\\vec k$ scattering into $\\vec k'$.\nEffective physical mechanisms exist for control of\nthe atom scattering lengths \\cite{gora,verhaar,eite}.\nIf implemented, these control\n`knobs' allow for unprecedented comparison between\ntheory and experiment over a wide range of interaction strength.\nIndeed, very recently several groups have successfully\nimplemented {\\it Feshbach resonance} \\cite{fesh},\nthus enabling a control knob on $a_{\\rm sc}$ through the\nchanging of an external magnetic field.\nOther physical mechanisms also exist for modifying\natom-atom interactions, e.g. the {\\it shape resonance}\ndue to anisotropic dipole interactions\ninside an external electric field \\cite{mm}.\n\nAlthough fermions with anisotropic interactions are\nwell studied within the context of $^3$He fluid \\cite{leggett}\nand in d-wave high $T_c$ superconductors, anisotropically\ninteracting bosons have not been studied in great detail.\nIn particular, we are not aware of any systematic approach\nfor constructing an anistropic pseudo potential \\cite{yang}.\n\nIn this paper, we study the ground state properties of\ntrapped condensates with dipole interactions.\nA rigorous method is developed for constructing the\nanisotropic pseudo potential that can also be applied\nto future polar molecular BEC \\cite{doyle,stwalley}.\nThis Letter is\norganized as follows. First we briefly review the\nSIA pseudo-potential approximation.\nWe then construct an analogous effective low energy\nanisotropic pseudo-potential.\nNumerical results are then discussed for\n$^{87}$Rb \\cite{bec} inside the external E-field in\nthe JILA TOP trap. We conclude with a brief discussion\nof prospects for realistic experiments.\n\nFor $N$ trapped spinless bosonic atoms in a potential $V_t(\\vec r)$,\nthe second quantized Hamiltonian is given by\n\\begin{eqnarray}\n{\\cal H}&=& \\int\\! d\\vec r\\,\\hat\\Psi^{\\dag}(\\vec r)\n\\left[-\\frac{\\hbar^2}{2M}\\nabla^2+V_{t}(\\vec r)\n-\\mu\\right]\\hat\\Psi(\\vec r) \\nonumber\\\\\n&\\ &+\\frac{1}{2}\\int\\! d\\vec r\\int\\! d\\vec r'\n\\hat\\Psi^{\\dag}(\\vec r)\\hat\\Psi^{\\dag}(\\vec r')\nV(\\vec r-\\vec r')\\hat\\Psi(\\vec r')\\hat\\Psi(\\vec r),\n\\label{h}\n\\end {eqnarray}\nwhere $\\hat\\Psi(\\vec r)$ and $\\hat\\Psi^{\\dag}(\\vec r)$ are atomic\n(bosonic) annihilation and creation fields.\nThe chemical potential $\\mu$ guarantees the\natomic number $\\hat N=\n\\int d\\vec r\\,\\hat\\Psi^{\\dag}(\\vec r)\\hat\\Psi(\\vec r)$\nconservation.\n\nThe bare potential $V(\\vec R)$ in (\\ref{h})\nneeds to be renormalized for a meaningful\nperturbation calculation. For bosons, the\nusual treatment is based on field theory and\nis rather involved \\cite{yang,Galits,Beliaev,Brueckner}.\nPhysically the SIA can be viewed as a valid low energy and low density\nrenormalization scheme. The physics involved is rather simple:\none simply replaces the bare potential\n$V(\\vec R)$ by the pseudo potential\n$u_0\\delta(\\vec R)$ such that whose\nfirst order Born scattering amplitude\nreproduces the complete scattering amplitude\n($-a_{\\rm sc}$). This requires\n$u_0=4\\pi\\hbar^2 a_{\\rm sc}/M$.\n\nWhen an electric field is introduced along the positive z\naxis, an additional dipole interaction\n\\begin{eqnarray}\nV_{E}(\\vec R)&&=-u_2{Y_{20}(\\hat R)\\over R^3},\n\\label{ve}\n\\end{eqnarray}\nappears, where $u_2=4\\sqrt{(\\pi/5)}\\,\\alpha(0)\\alpha^*(0){\\cal E}^2$,\nwith $\\alpha(0)$ being the polarizability,\nand ${\\cal E}$ the electric field strength.\nAs was shown in Ref. \\cite{mm},\nthis modification\nresults in a completely new low-energy scattering\namplitude\n\\begin{eqnarray}\nf(\\vec k,\\vec k')\\Big|_{k=k'\\to 0}=4\\pi\\sum_{lm,l'm'}\nt_{lm}^{l'm'}({\\cal E})Y_{lm}^*({\\hat k})Y_{l'm'}({\\hat k'}),\n\\label{cf}\n\\end{eqnarray}\nwith $t_{lm}^{l'm'}({\\cal E})$ the reduced T-matrix\nelements. They are all energy independent\nand act as generalized scattering lengths.\nThe anisotropic $V_{E}$ causes the dependence on both\nincident and scattered directions: $\\hat k$ and $\\hat k'=\\hat R$.\n\nA general anisotropic pseudo potential can be\nconstructed according to\n\\begin{eqnarray}\nV_{\\rm eff}(\\vec R)=u_0\\delta(\\vec R)\n+\\sum_{l_1>0, m_1} \\gamma_{l_1m_1}\n{Y_{l_1m_1}(\\hat R)\\over R^3},\n\\label{veff}\n\\end{eqnarray}\nwhose first Born amplitude is then given by\n\\begin{eqnarray}\nf_{\\rm Born}(\\vec k,\\vec k')\n=-(4\\pi)^2 a_{\\rm sc}Y_{00}^*(\\hat k)Y_{00}(\\hat k')\n-{M\\over 4\\pi\\hbar^2}\\sum_{l_1 m_1} \\gamma_{l_1m_1} (4\\pi)^2 \\sum_{lm}\n\\sum_{l'm'}{\\cal T}_{lm}^{l'm'}(l_1,m_1)\nY_{lm}^*(\\hat k)Y_{l'm'}(\\hat k'),\n\\label{bf}\n\\end{eqnarray}\nwith $\n{\\cal T}_{lm}^{l'm'}(l_1,m_1)=(i)^{l+l'}{\\cal R}_{l}^{l'}\nI_{lm}^{l'm'}(l_1,m_1).\n$\nBoth\n\\begin{eqnarray}\nI_{lm}^{l'm'}(l_1m_1)&&=\\langle\nY_{l'm'}|Y_{l_1m_1}|Y_{lm}\\rangle, {\\hskip 24pt \\rm and}\\nonumber\\\\\n{\\cal R}_{l}^{l'} &&=\\int_0^{\\infty} d R\\,{1\\over\nR}j_l(kR)j_{l'}(k'R),\\nonumber\n\\end{eqnarray}\ncan be computed analytically \\cite{su}. The $1/R^3$\nform in Eq. (\\ref{veff}) assures all ${\\cal R}_{l}^{l'}$ to be\n$k=k'$ independent [by a change of variable to\n$x=kR$ in the integral]. Putting\n\\begin{eqnarray}\nf_{\\rm Born}(\\vec k,\\vec k')=f(\\vec k,\\vec k'),\n\\end{eqnarray}\none can solve for the $\\gamma_{l_1m_1}({\\cal E})$\nas $t_{lm}^{l'm'}({\\cal E})$ are known numerically \\cite{mm,mm2}.\nThis reduces to the linear equations\n\\begin{eqnarray}\n-{M\\over 4\\pi \\hbar^2}\\sum_{l_1 m_1} \\gamma_{l_1m_1} (4\\pi)\n{\\cal T}_{lm}^{l'm'}(l_1,m_1)\\equiv\nt_{lm}^{l'm'},\n\\end{eqnarray}\nfor all ($lm$) and ($l'm'$) with $l,l'\\ne 0$,\nand separately $a_{\\rm sc}({\\cal E})=-t_{00}^{00}({\\cal E})$.\nThe problem\nsimplifies further for Bosons (fermions) as only\neven (odd) $(l,l')$ terms are needed to match.\nFigure \\ref{fig1} displays result of\n$a_{\\rm sc}({\\cal E})$ for the triplet state of $^{87}$Rb.\nThe Born amplitude for the dipole term $V_E$ is\n\\begin{eqnarray}\nf_{\\rm Born}(\\vec k,\\vec k') &&=u_2{M\\over 4\\pi \\hbar^2}\n(4\\pi)^2{\\cal T}_{00}^{20}\n\\sum_{lm,l'm'}\\overline{\\cal T}_{lm}^{l'm'}\nY_{lm}^*(\\hat k)Y_{l'm'}(\\hat k'),\n\\end{eqnarray}\nwith\n${\\cal T}_{00}^{20}=-0.023508$.\n$\\overline {\\cal T}_{lm}^{l'm'}={\\cal T}_{lm}^{l'm'}(2,0)/{\\cal T}_{00}^{20}$\nare tabulated below for small $(l,l')$.\n\\begin{table}\n\\caption{$\\bar{\\cal T}_{lm}^{l'm'}$}\n\\begin{tabular}{c|ccccc}\n$(lm),(l'm')$&(00)&(20)&(40)&(60)&(80)\\\\ \\hline (00)&0&1&0&0&0\\\\\n(20)&1&-0.63889&0.14287&0&0\\\\(40)&0&0.14287&-0.17420&0.05637&0\\\\\n(60)&0&0&0.05637&-0.08131&0.03008\\\\(80)&0&0&0&0.03008&-0.04707\n\\end{tabular}\n\\label{table1}\n\\end{table}\nWe found that away from {\\it shape resonances},\nTable (\\ref{table1}) agrees ($\\sim$\na few per cent) with the same ratios\n$t_{lm}^{l'm'}({\\cal E})/t_{00}^{20}({\\cal E})$\nfrom the numerical multi-channel calculations \\cite{mm}.\nThis interesting observation applies for all\nbosonic alkali triplet states we computed:\n$^7$Li, $^{39,41}$K, and $^{85,87}$Rb,\nfor up to a field strength of $3\\times 10^6$ (V/cm) \\cite{mm,mm2}.\nPhysically, this implies that effect of $V_E$\nis perturbative as ${\\cal E}$\nremains small in atomic units.\nWhat is remarkable is that\n${\\cal T}_{00}^{20}({\\cal E})$ and $t_{00}^{20}({\\cal E})$\nalso agree in absolute values \\cite{mm}.\nFor $^{87}$Rb, we found\n\\begin{eqnarray}\nu_2{M\\over 4\\pi\\hbar^2}\n(4\\pi)^2{\\cal T}_{00}^{20}\n=-1.495\\times 10^{10} {\\overline {\\cal E}}^2 (a_0),\n\\end{eqnarray}\nwith ${\\overline {\\cal E}}$ in\natomic units ($5.142\\times 10^9$ V/cm).\n$a_0$ is the Bohr radius. While multi-channel scattering gives \\cite{mm2}\n\\begin{eqnarray}\n(4\\pi)t_{00}^{20} =-1.512\\times 10^{10} {\\overline {\\cal E}}^2 (a_0).\n\\end{eqnarray}\nThe cause of this slight difference (1\\%)\nis not entirely clear and but is within numerical error.\n\nWe can thus approximate Eq. (\\ref{veff}) by keeping only\nthe $l_1=2,m_1=0$ term in the sum\n\\begin{eqnarray}\nV_{\\rm eff}(\\vec R)=u_0\\delta(\\vec R)-u_2 Y_{20}(\\hat R)/R^3,\n\\end{eqnarray}\naway from the {\\it shape resonance}.\nAt zero temperature the condensate wave function\n$\\psi(\\vec r,t)=\\langle\\hat\\Psi(\\vec r,t)\\rangle$ then obeys the\nfollowing nonlinear Schrodinger equation\n\\begin{eqnarray}\ni\\hbar {d\\over dt}\\psi(\\vec r,t)\n&&=\\left[-\\frac{\\hbar^2}{2M}\\nabla^2+V_{t}(\\vec r)\n-\\mu+u_0|\\psi(\\vec r,t)|^2\\right.\\nonumber\\\\\n&&\\left.-u_2\\int d\\vec r' {Y_{20}(\\hat R)\\over R^3}\n|\\psi(\\vec r',t)|^2\\right]\\psi(\\vec r,t),\n\\label{nlse}\n\\end{eqnarray}\nwith $\\psi(\\vec r,t)$ normalized to $N$. The ground state\nis found by steepest descent through propagation\nof Eq. (\\ref{nlse}) in imaginary time $(it)$.\nFor a cylindrical symmetric trap\n$V_{\\rm t}(\\vec r)=M(\\omega _\\perp^2x^2+\\omega_\\perp^2y^2+\\omega _z^2z^2)/2$,\nthe ground state also possesses azimuthal symmetry. Therefore\nthe non-local term simplifies to\n\\begin{eqnarray}\n\\int d\\vec r'|\\psi(\\rho',z')|^2 {Y_{20}(\\hat R)\\over\nR^3}=\\int dz' d\\rho' {\\cal K}(.,.;.)\n|\\psi(\\rho',z')|^2,\\nonumber\n\\end{eqnarray}\nwith the kernel ${\\cal K}(\\rho,\\rho';z-z')$\nexpressed in terms of the standard Elliptical\nintegrals ${\\rm E}[.]$ and ${\\rm K}[.]$. The kernel is\ndivergent at $\\vec r=\\vec r'$, so a cut-off radius $R_c$\nis chosen such that ${\\cal K}(\\rho',\\rho,z'-z)=0$\nwhenever $|\\vec r-\\vec r'|<R_c$. We typically\n$R_c\\sim 50 (a_0)$, much smaller than the grid size,\nto minimize numerical errors.\nTechnical details for numerical computations and\nfor handling the singular\nrapid variation of the kernel over\nsmall length scale will be discussed elsewhere \\cite{su}.\n\nFigure \\ref{fig2} presents $\\psi(\\rho,z)$ along\n$\\rho=0$ (a) and $z=0$ (b) cuts respectively for $^{87}$Rb\n($a_{\\rm sc}=5.4$ nm) at several different ${\\cal E}$. We note\nthe condensate shrinks radially while stretches along z-axis\n to minimize the\ndipole interaction $V_E$. The top-right corner inserts shows\nelectric field polarized atoms in (radially) repulsive (a) and\n(longitudinally) attractive (b) configurations.\nAn elongated condensate along the z-axis\nreduces the total energy. The same mechanism could cause\nspontaneous alignment of polar molecular condensates inside\nisotropic traps \\cite{doyle}.\nFor better insights we try a variation ansatz\n\\begin{eqnarray}\n\\psi_T(\\rho,z)={\\kappa^{1/2}\\over \\pi^{3/4}d^{3/2}} \\exp\\left[\n-{1\\over 2d^2}(\\rho^2+\\kappa^2 z^2)\\right],\n\\end{eqnarray}\nwith parameters $d$ and $\\kappa$. In\ndimensionless units for length\n($a_{\\perp}=\\sqrt{\\hbar/M\\omega_\\perp}$), energy\n($\\hbar\\omega_\\perp$), and $\\lambda=\\omega_z/\\omega_\\perp$, we\nobtain\n\\begin{eqnarray}\nE[\\psi_T]&&=(1+{\\lambda^2\\over 2\\kappa^2})d^2+(1+{\\kappa^2\\over\n2}){1\\over d^2} +{4N\\kappa\\over\\sqrt{2\\pi}} {a_{\\rm sc}^{\\rm\neff}\\over a_{\\perp}}{1\\over d^3}, \\label{energy}\n\\end{eqnarray}\nwith the effective scattering length\n$a_{\\rm sc}^{\\rm eff}=a_{\\rm sc}[1-b(\\kappa){u_2/\nu_0}]$, and\n\\begin{eqnarray}\nb(\\kappa)={\\sqrt{5\\pi}\\over\n3(\\kappa^2-1)}\\left(-2\\kappa^2-1+{3\\kappa^2\\tanh^{-1}\\sqrt\n{1-\\kappa^2}\\over \\sqrt {1-\\kappa^2} }\\right).\\nonumber\n\\end{eqnarray}\nThe $b(\\kappa)$ is monotonically decreasing, and bounded between\n$b(0)=\\sqrt{5\\pi}/3$ and $b(\\infty)=-2\\sqrt{5\\pi}/3$.\n$a_{\\rm sc}^{\\rm eff}$ is shown in\nFig. \\ref{fig3} as a function of ${\\cal E}$\nfor several different values of $\\kappa$.\nFor increasing electric field ${\\cal E}$, variational calculation\nresults in decreasing $\\kappa$, eventually $\\kappa$ becomes less\nthan one, i.e. the condensate changes from oblate (pancake) shaped\nat zero field (for the TOP trap) to prolate (cigar) shaped. We\nalso note that $b(\\kappa)\\ge 0$ for $\\kappa\\le 1$, therefore\n$a_{\\rm sc}^{\\rm eff}$ becomes negative at certain field value\n${\\cal E}_c$ in the case of a positive $a_{\\rm sc}({\\cal E}=0)$,\ncausing the collapse of the condensate. This is indeed what we\nfound as illustrated in Fig. \\ref{fig4}. A detailed discussion of\nthe collapse and other interesting\nfeatures will be given elsewhere \\cite{su}.\n\nWe note the energy of dipole alignment\n\\begin{eqnarray}\nE_P &&\\sim -(2\\pi) 1\\times10^{18}\n\\times {\\overline{\\cal E}}^2 {\\rm (Hz)},\n\\end{eqnarray}\nbecomes much larger than the trap depth\nat the proposed ${\\cal E}$ values for $^{87}$Rb.\nTherefore spatial homogeneity for ${\\cal E}(\\vec r)$ is required.\nAt ${\\cal E}\\sim 5\\times 10^5$ (V/cm) [${\\overline{\\cal E}}\\sim 10^{-4}$]\nwith a spatial gradient\n$<10^{-4}$/cm$^3$, the corresponding force\nis smaller than the magnetic trapping force\nfor typical traps at $\\sim 100$ (Hz).\nFor comparison, the magnetic field gradient is\n$\\sim 10^{-6}$/cm$^3$ inside the Penning trap magnets.\nAlthough the proposed electric field [$10^5$ (V/cm) $< {\\cal E}<10^6$ (V/cm)]\nis large, it can be created through careful laboratory techniques as breakup is\nfundamentally limited by field ionization, which\ntypically occurs at ${\\cal E}>10^7$ (V/cm) \\cite{latham}.\nRecently a ${\\cal E}$ field of upto $1.25\\times 10^5$ (V/cm)\nwas used successfully to decelerate a molecular beam \\cite{meijer}.\n\nIn conclusion, we have developed a general scheme for\nconstructing effective pseudo-potentials for anisotropic\ninteractions. Our scheme guarantees that the first order\nBorn scattering amplitude from the pseudo-potential\nreproduces the complete\nscattering amplitude obtained from a multi-channel\ncomputation including the anisotropic dipole-interaction,\nthus contains no energy dependence at low temperatures\nof the trapped atomic gases \\cite{mm}. Our scheme is\nthus more pleasing than the standard\nSkyrme type velocity dependent effective potentials\ncommonly adopted in nuclear physics \\cite{ring}.\nWe also presented results for both the electric field\nmodified atomic scattering parameters and the\ninduced changes to the condensate\nfor $^{87}$Rb in the JILA TOP trap. Our theory can be\ndirectly extended to systems involving magnetic dipole interaction\nof atoms/molecules in a static magnetic trap and systems\nof trapped molecules with permanent electric dipoles \\cite{doyle}.\nFor alkali atoms, typical magnetic dipole interaction is weak\nsince a Bohr magneton ($\\mu_B=e\\hbar/2mc$) only corresponds to\nan electric dipole of $\\sim (1/2\\alpha_f)(e a_0)$ (fine structure constant\n$\\alpha_f\\approx 1/137$), which is equivalent to the\ninduced electric dipole at ${\\cal E}=6\\times 10^4$ (V/cm)\nfor $^{87}$Rb. Other atoms with\nlarger magnetic dipole moments \\cite{doyle2} will\ndisplay clearer anisotropic effects. Typical\nhetero-nuclear diatomic molecules have a permanent\nelectric dipole moment of $\\sim (e a_0)$, corresponding to\nan induced moment in $^{87}$Rb at ${\\cal E}=1.6\\times 10^7$ (V/cm)\n\\cite{doyle}. Trapped molecules with aligned permanent\nelectric dipoles (by an external E-field) would\ngive similar results. However, magnetic trapped molecules \\cite{doyle}\nwith unaligned electric dipoles interacting with the spin axis\nrepresents an interesting\nextension that requires further investigation.\n\nWe thank Dr. M. Marinescu for helpful discussions during the\nearly stages of this work.\nThis work is supported by the U.S. Office of Naval Research\ngrant No. 14-97-1-0633 and by the NSF grant No. PHY-9722410.\n\n\\begin{references}\n\\bibitem{bec}M. H. Anderson {\\it et al.}, Science {\\bf 269}, 198 (1995).\n\n\\bibitem{mit}K. B. Davis {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 3969\n(1995).\n\n\\bibitem{rice}C. C. Bradley {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 1687 (1995);\n{\\it ibid} {\\bf 79}, 1170 (1997).\n\n\\bibitem{Edwards}Extentive list of references exists at:\nhttp://amo.phy.gasou.edu/bec.html/bibliography.html .\n\n\\bibitem{yang}Kerson Huang and C. N. Yang,\nPhys. Rev. {\\bf 105}, 767 (1957).\n\n\\bibitem{glauber}M. Naraschewski and R.J. Glauber, Phys. Rev. A {\\bf 59}, 4595 (1999).\n\n\\bibitem{esry}B. Esry and C. Greene, Phys. Rev. A {\\bf 60}, 1451 (1999).\n\n\\bibitem{gora}P. O. Fedichev {\\it et al.}, Phys. Rev. Lett. {\\bf 77}, 2913 (1996);\nJ. L. Bohn and P. S. Julienne, Phys. Rev. A {\\bf 56},\n1486 (1997).\n\n\\bibitem{verhaar}A. J. Moerdijk {\\it et al.}, Phys. Rev. A {\\bf 53}, 4343 (1996).\n\n\\bibitem{eite}E. Tiesinga {\\it et al.}, Phys. Rev. A {\\bf 46}, R1167 (1992).\n\n\\bibitem{fesh}S. Inouye {\\it et. al}, Nature (London) {\\bf 392}, 151 (1998);\nPh. Courteille {\\it et. al}, \\prl {\\bf 81}, 69 (1998);\nJ. L. Roberts {\\it et. al}, {\\it ibid} {\\bf 81}, 5179 (1998);\nVladan Vuletic {\\it et. al},\n {\\it ibid} {\\bf 82}, 1406 (1999).\n\n\\bibitem{mm}M. Marinescu and L. You, Phys. Rev. Lett. {\\bf 81},\n4596 (1998).\n\n\\bibitem{leggett}A. J. Leggett, Rev. Mod. Phys. {\\bf 47}, 331 (1975).\n\n\\bibitem{doyle}J. D. Weinstein {\\it et. al}, Nature {\\bf 395}, 148 (1998).\n\n\\bibitem{stwalley}J. T. Bahns {\\it et al.},(preprint, 1999).\n\n\\bibitem{Galits}V. Galitskii, Sov. Phys. JETP {\\bf 34}, 104 (1958)\n[Zh. Eksp. Teor. Fiz. {\\bf 34}, 151 (1958)].\n\n\\bibitem{Beliaev}S. T. Beliaev, Sov. Phys. JETP {\\bf 7}, 289 (1958)\n[Zh. Eksp. Teor. Fiz. {\\bf 34}, 417 (1958)]; N. M. Hugenholtz\n and D. Pines, Phys. Rev. {\\bf 116}, 489 (1959).\n\n\\bibitem{Brueckner}K. A. Brueckner\nand K. Sawada, Phys. Rev. {\\bf 106}, 1117 (1957).\n\n\\bibitem{mm2}M. Marinescu and L. You, (unpublished).\n\n\\bibitem{su}S. Yi and L. You, (to be published).\n\n\\bibitem{nlse}E. P. Gross, Nuovo Cimento {\\bf 20}, 454 (1961);\nL. Pitaevskii, Sov. Phys. JETP {\\bf 13}, 451 (1961).\n\n\\bibitem{var}G. Baym and C. J. Pethick, \\prl {\\bf 76}, 6 (1996);\nVictor M. Perez-Garcia {\\it et al.}, {\\it ibid} {77}, 5320 (1996).\n\n\\bibitem{latham}R. V. Latham, {\\it High Voltage Vacuum Insulation:\nThe physical basis}, (Academic Press, New York, 1981);\nT. B. Mitchell, AIP Proceedings {\\bf 331}, 115 (1995);\nJ. C. Davis, (private communications).\n\n\\bibitem{meijer}H. L. Bethlem {\\it et al.}, \\prl {\\bf 83}, 1558 (1999).\n\n\\bibitem{ring}P. Ring and P. Schuck,\n{\\it The Nuclear Many-Body Problem},\np. 172, (Springer-Verlag, New York, 1980).\n\n\\bibitem{doyle2}J. Weinstein {\\it et al.}, Phys. Rev. A {\\bf 57}, R3173 (1998).\n\n\\end{references}\n\n\\begin{figure}\n\\caption{The field dependent value for $a_{\\rm sc}$. Note the\nshape resonance for ${\\cal E}$ around $8.3\\times 10^5$ (V/cm).}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\caption{(a) $\\psi(\\rho,0)$ for $^{87}$Rb with\n$\\omega_\\perp=(2\\pi) 70$ (Hz), $\\omega_z=\\sqrt{8}\\,\\omega_\\perp$,\nand $N=5000$ atoms. Solid, dashed-dot, dashed, and dotted\nlines are for\n${\\cal E}=0, 4.0\\times 10^5, 5.7\\times 10^5$, and\n$5.88\\times 10^5$ (V/cm) respectively.\n$a_{\\perp}$ is the radial trap width.\n \\\\ (b) same as in\nFig. 2 (a), but for $\\psi(0,z)$.} \\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Typical behavior of the effective scattering length\n$a_{\\rm sc}^{\\rm eff}.$ Lines corresponds to\n$\\kappa=5.1,1.7,1.02,0.34,0.017,$ in descending order of\n$a_{\\rm sc}^{\\rm eff}$.}\\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Electric field dependence of the\nwidth aspect ratio for parameters of Fig. 2. The\nsolid line is the result of our variational calculation\nwhile circles denote exact numerical results.\nThe dashed line corresponds to\n$\\sqrt{\\sqrt{8}}$, the result for a non-interacting\ngas in the TOP trap.}\\label{fig4}\n\\end{figure}\n\n\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002079.extracted_bib",
"string": "\\bibitem{bec}M. H. Anderson {\\it et al.}, Science {\\bf 269}, 198 (1995).\n\n\n\\bibitem{mit}K. B. Davis {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 3969\n(1995).\n\n\n\\bibitem{rice}C. C. Bradley {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 1687 (1995);\n{\\it ibid} {\\bf 79}, 1170 (1997).\n\n\n\\bibitem{Edwards}Extentive list of references exists at:\nhttp://amo.phy.gasou.edu/bec.html/bibliography.html .\n\n\n\\bibitem{yang}Kerson Huang and C. N. Yang,\nPhys. Rev. {\\bf 105}, 767 (1957).\n\n\n\\bibitem{glauber}M. Naraschewski and R.J. Glauber, Phys. Rev. A {\\bf 59}, 4595 (1999).\n\n\n\\bibitem{esry}B. Esry and C. Greene, Phys. Rev. A {\\bf 60}, 1451 (1999).\n\n\n\\bibitem{gora}P. O. Fedichev {\\it et al.}, Phys. Rev. Lett. {\\bf 77}, 2913 (1996);\nJ. L. Bohn and P. S. Julienne, Phys. Rev. A {\\bf 56},\n1486 (1997).\n\n\n\\bibitem{verhaar}A. J. Moerdijk {\\it et al.}, Phys. Rev. A {\\bf 53}, 4343 (1996).\n\n\n\\bibitem{eite}E. Tiesinga {\\it et al.}, Phys. Rev. A {\\bf 46}, R1167 (1992).\n\n\n\\bibitem{fesh}S. Inouye {\\it et. al}, Nature (London) {\\bf 392}, 151 (1998);\nPh. Courteille {\\it et. al}, \\prl {\\bf 81}, 69 (1998);\nJ. L. Roberts {\\it et. al}, {\\it ibid} {\\bf 81}, 5179 (1998);\nVladan Vuletic {\\it et. al},\n {\\it ibid} {\\bf 82}, 1406 (1999).\n\n\n\\bibitem{mm}M. Marinescu and L. You, Phys. Rev. Lett. {\\bf 81},\n4596 (1998).\n\n\n\\bibitem{leggett}A. J. Leggett, Rev. Mod. Phys. {\\bf 47}, 331 (1975).\n\n\n\\bibitem{doyle}J. D. Weinstein {\\it et. al}, Nature {\\bf 395}, 148 (1998).\n\n\n\\bibitem{stwalley}J. T. Bahns {\\it et al.},(preprint, 1999).\n\n\n\\bibitem{Galits}V. Galitskii, Sov. Phys. JETP {\\bf 34}, 104 (1958)\n[Zh. Eksp. Teor. Fiz. {\\bf 34}, 151 (1958)].\n\n\n\\bibitem{Beliaev}S. T. Beliaev, Sov. Phys. JETP {\\bf 7}, 289 (1958)\n[Zh. Eksp. Teor. Fiz. {\\bf 34}, 417 (1958)]; N. M. Hugenholtz\n and D. Pines, Phys. Rev. {\\bf 116}, 489 (1959).\n\n\n\\bibitem{Brueckner}K. A. Brueckner\nand K. Sawada, Phys. Rev. {\\bf 106}, 1117 (1957).\n\n\n\\bibitem{mm2}M. Marinescu and L. You, (unpublished).\n\n\n\\bibitem{su}S. Yi and L. You, (to be published).\n\n\n\\bibitem{nlse}E. P. Gross, Nuovo Cimento {\\bf 20}, 454 (1961);\nL. Pitaevskii, Sov. Phys. JETP {\\bf 13}, 451 (1961).\n\n\n\\bibitem{var}G. Baym and C. J. Pethick, \\prl {\\bf 76}, 6 (1996);\nVictor M. Perez-Garcia {\\it et al.}, {\\it ibid} {77}, 5320 (1996).\n\n\n\\bibitem{latham}R. V. Latham, {\\it High Voltage Vacuum Insulation:\nThe physical basis}, (Academic Press, New York, 1981);\nT. B. Mitchell, AIP Proceedings {\\bf 331}, 115 (1995);\nJ. C. Davis, (private communications).\n\n\n\\bibitem{meijer}H. L. Bethlem {\\it et al.}, \\prl {\\bf 83}, 1558 (1999).\n\n\n\\bibitem{ring}P. Ring and P. Schuck,\n{\\it The Nuclear Many-Body Problem},\np. 172, (Springer-Verlag, New York, 1980).\n\n\n\\bibitem{doyle2}J. Weinstein {\\it et al.}, Phys. Rev. A {\\bf 57}, R3173 (1998).\n\n"
}
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cond-mat0002080
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Double-dot charge transport in Si single electron/hole transistors
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[
{
"author": "L.~P. Rokhinson"
},
{
"author": "L.~J. Guo$^{a)}$"
},
{
"author": "S.~Y. Chou and D.~C. Tsui"
}
] |
We studied transport through ultra-small Si quantum dot transistors fabricated from silicon-on-insulator wafers. At high temperatures, 4 K $<T<$ 100 K, the devices show single-electron or single-hole transport through the lithographically defined dot. At $T<4$ K, current through the devices is characterized by multidot transport. From the analysis of the transport in samples with double-dot characteristics, we conclude that extra dots are formed inside the thermally grown gate oxide which surrounds the lithographically defined dot.
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[
{
"name": "mqd-cond-mat.tex",
"string": "\\documentstyle[twocolumn,floats,aps,prl,epsfig]{revtex}\n\n\\begin{document}\n\\draft\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname%\n\n\\title{Double-dot charge transport in Si single electron/hole\ntransistors}\n\n\\author{L.~P. Rokhinson, L.~J. Guo$^{a)}$, S.~Y. Chou and D.~C. Tsui}\n\n\\address{Department of Electrical Engineering, Princeton University,\nPrinceton, NJ 08544}\n\n\\date{To appear in \\apl on March 20, 2000}\n\n\\maketitle\n\n\\begin{abstract}\nWe studied transport through ultra-small Si quantum dot transistors\nfabricated from silicon-on-insulator wafers. At high temperatures, 4\nK $<T<$ 100 K, the devices show single-electron or single-hole\ntransport through the lithographically defined dot. At $T<4$ K,\ncurrent through the devices is characterized by multidot transport.\nFrom the analysis of the transport in samples with double-dot\ncharacteristics, we conclude that extra dots are formed inside the\nthermally grown gate oxide which surrounds the lithographically\ndefined dot.\n\\end{abstract}\n\n\\pacs{\\\\PACS numbers: 73.23.Hk, 85.30.Wx, 85.30.Vw, 85.30.Tv}\n\\vskip2pc]\n\nRecent advances in miniaturization of Si metal-oxide-semiconductor\nfield-effect transistors (MOSFETs) brought to light several issues\nrelated to the electrical transport in Si nanostructures. At low\ntemperatures and low source-drain bias Si nanostructures do not\nfollow regular MOSFET transconductance characteristics but show\nrather complex behavior, suggesting transport through\nmultiply-connected dots. Even in devices with no intentionally\ndefined dots (like Si quantum\nwires\\cite{nakajima95,ishikuro96,hiramoto97,smith97} or point\ncontacts\\cite{ishikuro97}) Coulomb blockade oscillations were\nreported. In the case of quantum wires, formation of tunneling\nbarriers is usually attributed to fluctuations of the thickness of\nthe wire or of the gate oxide. However, formation of a dot in point\ncontact samples is not quite consistent with such explanation.\nRecently in an elegant experiment with both $n^+$ and $p^+$\nsource/drain connected to the same Si point contact Ishikuro and\nHiramoto\\cite{ishikuro99} have shown that the confining potential in\nunintentionally created dots is similar for both holes and electrons.\nHowever, there is no clear picture where and how these dots are\nformed.\n\n\n\n\nIn this work we analyze the low temperature transport through an\nultra-small lithographically defined Si quantum dots. While at high\ntemperature 4 K $<T<$ 100 K we observe single-electron tunneling\nthrough the lithographically defined dot, at $T<4$ K transport is\nfound to be typical for a multi-dot system. We restrict ourselves to\nthe analysis of samples with double-dot transport characteristics.\nFrom the data we extract electrostatic characteristics of both the\nlithographically defined and the extra dots. Remarkably, transport\nin some samples cannot be described by tunneling through two dots\nconnected in sequence but rather reflects tunneling through dots\nconnected in parallel to both source and drain. Taking into account\nthe geometry of the samples we conclude that extra dots should be\nformed within the gate oxide. Transport in p- and n-type samples are\nsimilar, suggesting that the origin of the confining potential for\nelectrons and holes in these extra dots is the same.\n\n%\n%-----figure1--------------------------------------------------\n%\n\\begin{figure}[tb]\n\\epsfig{file=fig1.eps,width=3.25in}\n\\vspace{-1.8in}\n\\caption{(a) Schematic of the device structure, (b) SEM micrograph of\na device, and (c) schematic view of two dots D$_1$ and D$_2$\nconnected to source and drain contacts L and R. G represents a gate\nelectrode and C$_{g1}$ and C$_{g2}$ are gate capacitances. Dashed\nlines represent possible tunneling barriers.}\n\\label{scheme}\n\\end{figure}\n\n\nThe samples are metal-oxide-semiconductor field-effect transistors\n(MOSFETs) fabricated from a silicon-on-insulator (SOI) wafer. The top\nsilicon layer is patterned by an electron-beam lithography to form a\nsmall dot connected to wide source and drain regions, see schematic\nin Fig.~\\ref{scheme}a. Next, the buried oxide is etched beneath the\ndot transforming it into a free-standing bridge. Subsequently, 40 or\n50 nm of oxide is thermally grown which further reduces the size of\nthe dot. Poly-silicon gate is deposited over the bridge with the dot\nas well as over the adjacent regions of the source and drain. It is\nimportant to note that in this type of devices the gate not only\ncontrols the potential of the dot but also changes the dot-source and\ndot-drain barriers. Finally, the uncovered regions of the source and\ndrain are n--type or p--type doped. More details on samples\npreparation can be found in Ref.~\\cite{leobandung95}. Totally, about\n30 hole and electron samples have been studied. Here we present data\nfrom two samples with hole (H5A) and electron (E5-7) field-induced\nchannels.\n\n\n\nAn SEM investigation of test samples, Fig.~\\ref{scheme}b, reveals\nthat the lithographically defined dot in the Si bridge is 10-40 nm in\ndiameter and the distance between narrow regions of the bridge is\n$\\sim$70 nm. Taking into account the oxide thickness we estimate the\ngate capacitance to be 0.8--1.5 aF.\n\n%\n%-----figure2--------------------------------------------------\n%\n\\begin{figure}[tb]\n\\psfig{file=fig2.eps,width=3.25in}\n\\vspace{-1.5in}\n\\caption{(a) Differential conductance in the hole quantum dot sample\nH5A is shown as a function of the gate voltage $V_g$ for $T=31$, 22,\n10, 4.2 and 0.3 K (from top to bottom). The trace at the lowest\ntemperature 0.3 K has been taken in a separate cooldown. In the inset\npeak width $w$ vs $T$ is plotted for peaks between $-3.0<V_g<-2.2$ V\n($w$ is defined in the text). (b)-(d) Modeling of the total\nconductance at $T=0.3$ K assuming that the two dots are connected (b)\nin series, (c) in parallel, or (d) mixed.}\n\\label{h5a-t-dep}\n\\end{figure}\n\nIn most of our samples (with both n-- and p--channel) we see clear\nCoulomb blockade oscillations with a period $\\Delta V_{g1}=100-160$\nmV up to $\\sim$100 K. A typical charge addition spectra is plotted in\nFig.~\\ref{h5a-t-dep} and Fig~\\ref{seq1} for samples H5A and E5-7. In\nH5A the spectrum is almost periodic as a function of the gate voltage\n$V_g$ at $T>4$ K with the period $\\Delta V_{g1}\\approx 130$ mV.\nAssuming that each peak corresponds to an addition of one hole into\nthe dot we calculate the gate capacitance $C_{g1}=e/\\Delta\nV_{g1}=1.2$ aF, which is within the error bars for the capacitance\nestimated from the sample geometry. The lineshape of an individual\npeak can be described\\cite{kulik75,beenakker91} by $G\\propto\n\\cosh^{-2}[(V_g-V_g^i)/2.5\\alpha k_B T]$, where $V_g^i$ is the peak\nposition and coefficient $\\alpha=C_{total}/eC_g$ relates the change\nin the $V_g$ to the shift of the energy levels in the dot relative to\nthe Fermi energy in the contacts. This expression is valid if both\ncoupling to the leads $\\Gamma$ and single-particle level spacing\n$\\Delta E$ are small: $\\Gamma,\\Delta E \\ll k_B T \\ll e^2/C_{total}$.\nWe fit the data for H5A with $\\sum_{i} \\cosh^{-2}[(V_g-V_g^i)/w]$ in\nthe range -3.0 V $<V_g<-2.2$ V and the extracted $w$ is plotted in\nthe inset in Fig.~\\ref{h5a-t-dep}. From the linear fit $w=11.3+2.2T$\n[mV] we find the coefficient $\\alpha=10$ [mV/meV], thus the Coulomb\nenergy is $\\approx 13$ meV and the total capacitance $C_{total}=12.3$\naF. The main contribution to $C_{total}$ comes from dot-to-lead\ncapacitances (an estimated self-capacitance is a few aF). The\nextrapolated value of $w$ at zero temperature provides an estimate\nfor the level broadening $\\Gamma\\approx1$ meV.\n\nAt $T<4$ K oscillations with another period, much smaller than\n$\\Delta V_{g1}$, appear as a function of $V_g$. The small period is\nin the range $\\Delta V_{g2}=8-25$ mV in different devices ($\\Delta\nV_{g2}=11.8$ mV for the sample in Fig.~\\ref{h5a-t-dep}). This small\nperiod is due to a single-hole tunneling through a second dot and the\ncorresponding gate capacitance $C_{g2}=e/\\Delta V_{g2}=6-20$ aF.\nHowever, there is no intentionally defined second dot in our devices.\nBelow we first analyze the experimental results and then discuss\nwhere the second dot can be formed.\n\n%\n%-----figure3--------------------------------------------------\n%\n\\begin{figure}[tb]\n\\epsfig{file=fig3.eps,width=3.25in}\n\\vspace{-3.1in}\n\\caption{Differential conductance in an electron quantum dot sample\nE5-7 is plotted as a function of the gate voltage $V_g$ for (a)\ndifferent dc source-drain bias $V_b$ and (b) different temperatures.\nIn (a) each curve is measured at different $V_b$ from -20 mV (bottom\ncurve) to 20 mV (top curve) at $T=1.5$ K. Arrows indicate the curve\nwith $V_b=0$. All curves are offset by 0.5 $\\mu$S. Data in (b) is\ntaken at zero bias. The excitation voltage is 100 $\\mu$V.}\n\\label{seq1}\n\\end{figure}\n\n\n\n\nAt low temperatures and small gate voltages (close to the turn-on of\nthe device at high temperatures) current is either totally\nsuppressed, as in E5-7 at $V_g<3.5$ V, Fig.~\\ref{seq1}a, or there\nare sharp peaks with no apparent periodicity, as in H5A at $V_g>-2.3$\nV, Fig.~\\ref{h5a-t-dep}. Both suppression of the current and\n``stochastic Coulomb blockade''\\cite{ruzin92} are typical signatures\nof tunneling through two sequentially connected dots. The non-zero\nconductance can be restored either by raising the temperature\n(Fig.~\\ref{h5a-t-dep}) or by increasing the source-drain bias $V_b$\n(Fig.~\\ref{seq1}a). In both cases, $G$ is modulated with $\\Delta\nV_{g1}$ and $\\Delta V_{g2}$, consistent with sequential tunneling. We\nconclude that in these regime the two dots are connected in series\nL-D$_1$-D$_2$-R (see schematic in Fig.~\\ref{scheme}c).\n\nAt larger gate voltages ($V_g>6$ V for E5-7 and $V_g<-2.3$ V for H5A)\ncurrent is not suppressed even at the lowest temperatures. However,\nthe $G$ pattern is different in the H5A and E5-7 samples. In H5A, the\noscillations with $\\Delta V_{g2}$ have approximately the same\namplitude (except for the sharp peaks which are separated by\napproximately $\\Delta V_{g1}$), while in E5-7 the amplitude of the\nfast oscillations is modulated by $\\Delta V_{g1}$. Also, the\ndependence of the amplitude of the fast modulations on the average\nconductance $<G>$ is different: in H5A the amplitude is almost\n$<G>$-independent, while in E5-7 it is larger for larger $<G>$.\n\n\nNon-vanishing periodic conductance at low temperatures requires that\nthe transport is governed by the Coulomb blockade through only one\ndot D$_2$. That can be achieved either if both barriers between the\ncontacts and the D$_2$ become transparent enough to allow substantial\ntunneling or if the strong coupling between the main dot D$_1$ and\none of the leads results in a non-vanishing density of states in the\ndot at $T=0$. If we neglect coupling between the dots, in the former\ncase the total conductance is approximately the sum of two\nconductances, $G_{parallel}\\approx G_1+G_2$, where $G_1$ is\nconductance through the main dot L-D$_1$-R and $G_2$ is conductance\nthrough the second dot L-D$_2$-R. This case is modeled in\nFig~\\ref{h5a-t-dep}c using experimentally determined parameters of\nsample H5A. From the analysis of high-temperature transport we found\nthat the zero-temperature broadening of D$_1$ peaks\n$\\alpha\\Gamma\\approx10$ mV $\\approx\\Delta V_{g2}\\ll \\Delta\nV_{g1}=130$ mV and that $G$ should be exponentially suppressed\nbetween D$_1$ peaks at $T=0.3$ K if the dots are connected in series\nL-D$_1$-D$_2$-R, Fig~\\ref{h5a-t-dep}b. The best description of the\nlow temperature transport at -3.0 V $ <V_g<-2.3$ V in H5A is achieved\nif we assume that there are two conducting paths in parallel: through\nthe extra dot L-D$_2$-R and through both dots together\nL-D$_1$-D$_2$-R, Fig~\\ref{h5a-t-dep}d.\n\nIn the latter case, the dots are connected in series L-D$_1$-D$_2$-R.\nAt high $V_g$ the barrier between L and D$_1$ is reduced giving rise\nto a large level broadening $\\Gamma$. The total conductance is\n$G_{series}\\approx G_{BW} G_2/(G_{BW}+G_2)$, where $G_2$ is the\nCoulomb blockade conductance through D$_2$ alone and\n$G_{BW}={{2e^2}\\over{h}} \\Gamma^2/(\\Gamma^2+\\delta E^2)$ is the\nBreit-Wigner conductance through D$_1$ and $\\delta\nE=(V_g-V_g^i)/\\alpha$. In this case $G_{series}$ is following\n$G_{BW}$ and is modulated by $G_2$. Moreover, if we assume that the\namplitude of $G_2$ is not a strong function of $V_g$, the amplitude\nof $G_{series}$ modulation will be a function of $G_{BW}$, namely the\nlarger $G_{BW}$ the larger the amplitude of the modulation of the\ntotal conductance. This model of two dots in series with one being\nstrongly coupled to the leads is in qualitative agreement with the\ndata from sample E5-7.\n\n\n\nNon-equilibrium transport through E5-7 is shown in\nFig.~\\ref{E57-gray} with a single $G$ vs. $V_b$ trace at a fixed\n$V_g$ shown at the top of the figure. White diamond-shaped Coulomb\nblockade regions are clearly seen on the gray-scale plot. Peaks in\n$G$ at positive bias are due to asymmetry in the tunneling\nbarriers\\cite{su92}: at negative biases tunneling to the dot is\nslower than tunneling off the dot and only one extra electron\noccupies the dot at any given time, thus only one peak, corresponding\nto the onset of the current, is observed (we have not seen any\nfeatures due to the size quantization, which is not surprising if we\ntake into account the large number of electrons in this dot). At\npositive biases current is limited by the time the electron spends in\nthe dot before it tunnels out. In this regime an extra step in the\nI-V characteristic (and a corresponding peak in its derivative $G$)\nis observed every time one more electron can tunnel into the dot.\nThese peaks, marked with arrows, are separated by the charging energy\n$U_c=e\\Delta V_b=8$ meV.\n\n\n%\n%-----figure4--------------------------------------------------\n%\n\\begin{figure}[tb]\n\\epsfig{file=fig4.eps,width=3.25in}\n\\vspace{0in}\n\\caption{Differential conductance on a gray scale as a function of\nboth $V_g$ and $V_b$. A single trace at $V_g=7.922$ is shown at the\ntop. Arrows indicate onset of the tunneling of 1,2 and 3 electrons\nsimultaneously, as discussed in the text.}\n\\label{E57-gray}\n\\end{figure}\n%\n%-----figure--------------------------------------------------\n%\n\n\n\nElectrostatic parameters of the D$_2$ dot can be readily extracted\nfrom Fig.~\\ref{E57-gray}. The source, drain and gate capacitances are\n8.5, 2.7 and 6.4 aF and the corresponding charging energy is\n$\\approx9$ meV. The charging energy of $\\approx11$ meV is obtained by\nanalyzing Fermi-Dirac broadening of the conductance peaks as a\nfunction of temperature and the period of oscillations. The fact that\nit requires the application of $V_b=10$ mV to lift the Coulomb\nblockade means that in the Coulomb blockade regime all the bias is\napplied across the second dot, consistent with large conductance\nthrough D$_1$.\n\nWhere does the second dot reside? One possibility is that the silicon\nbridge, containing the lithographically defined dot, breaks up at low\ntemperatures as a result of the depletion due to variations of the\nbridge thickness and fluctuations in the thickness of the gate oxide,\nor due to the field induced by ionized impurities. However, in this\ncase $C_{g2}$ should be less than $C_{g1}$. In fact, if we assume\nthat the thickness of the thermally grown oxide is uniform, the gate\ncapacitance of the largest possible dot in the channel cannot be\nlarger than 1.5 aF. Also, if at low temperatures the main dot would\nsplit into two or more dots we should see the change in the period of\nthe large oscillations\\cite{waugh95}, inconsistent with our\nobservations.\n\nAnother possibility is that the dot is formed in the contact region\nadjacent to the bridge. Given that the oxide thickness is 40 nm, the\nsecond dot diameter should be $\\approx 100$ nm. We measured two\ndevices which have 30 nm wide and 500 nm long channels, fabricated\nusing the same technique as the dot devices. Both samples show\nregular MOSFET characteristics down to 50 mK. Thus, it is unlikely\nthat a dot is formed in the wide contact regions of the device. Even\nif such a dot was formed occasionally in some device by, for example,\nrandomly distributed impurities, it is unlikely that dots of\napproximately the same size would be formed in all samples. Another\nargument against such a scenario is that if the second dot is formed\ninside one of the contact regions, it cannot be coupled to the other\ncontact to provide a parallel conduction channel, as in sample H5A.\n\nThus, the second dot should reside within the gate oxide, which\nsurrounds the lithographically defined dot. Some traps can create\nconfining potential in both conduction and valence bands, for example\nP$_b$ center has energy levels at $E_c-0.3$ eV and $E_v+0.3$ eV.\nSeveral samples show a hysteresis during large gate voltage scans\naccompanied by sudden switching. This behavior can be attributed to\nthe charging-discharging of traps in the oxide. If such a trap\nhappens to be in a tunneling distance from both the lithographically\ndefined dot and a contact, or the trap is extended from one contact\nto the other, it may appear as a second dot in the conductance.\n\nTo summarize our results, we performed an extensive study of a large\nnumber of Si quantum dots. We found that all devices show multi-dot\ntransport characteristics at low temperatures. From the data analysis\nwe arrived at the conclusion that at least double-dot behavior is\ncaused not by the depletion of the silicon channel but by additional\ntransport through traps within the oxide.\n\nWe acknowledge the support from ARO, ONR and DARPA.\n\n\n\\begin{references}\n\n\\bibitem[a)]{addr88}\nCurrent address: Department of Electrical Engineering and\nComputer Science, University of Michigan, Ann Arbor, MI 48109.\n\n\\bibitem{nakajima95}\nY. Nakajima, Y. Takahashi, S. Horiguchi, K. Iwadate, H. Namatsu, K.\nKurihara,\n and M. Tabe, Jpn.~J.~Appl.~Phys. {\\bf 34}, 1309 (1995).\n\n\\bibitem{ishikuro96}\nH. Ishikuro, T. Fujii, T. Saraya, G. Hashiguchi, T. Hiramoto, and T.\nIkoma,\n \\apl {\\bf 68}, 3585 (1996).\n\n\\bibitem{hiramoto97}\nT. Hiramoto, H. Ishikuro, T. Fujii, G. Hashiguchi, and T. Ikoma,\nJpn.~J.\n ~Appl.~Phys. {\\bf 36}, 4139 (1997).\n\n\\bibitem{smith97}\nR.~A. Smith Abd~H. Ahmed, \\apl {\\bf 71}, 3838 (1997).\n\n\\bibitem{ishikuro97}\nH. Ishikuro and T. Hiramoto, \\apl {\\bf 71}, 3691 (1997).\n\n\\bibitem{ishikuro99}\nH. Ishikuro and T. Hiramoto, \\apl {\\bf 74}, 1126 (1999).\n\n\\bibitem{leobandung95}\nE. Leobandung, L. Guo, Y. Wang, and S.~Y. Chou, \\apl {\\bf 67}, 938\n(1995).\n\n\\bibitem{kulik75}\nI.~O. Kulik and R.~I. Shekhter, Zh. Eksp. Teor. Fiz. {\\bf 68}, 623\n(1975),\n [Sov.~Phys.~JETP {\\bf 41}, 308 (1975)].\n\n\\bibitem{beenakker91}\nC.~W.~J. Beenakker, \\prb {\\bf 44}, 1646 (1991).\n\n\\bibitem{ruzin92}\nI.~M. Ruzin, V. Chandrasekhar, E.~I. Levin, and L.~I. Glazman, \\prb\n{\\bf 45},\n 13469 (1992).\n\n\\bibitem{su92}\nB. Su, V.~J. Goldman, and J.~E. Cunningham, \\prb {\\bf 46}, 7644\n(1992).\n\n\\bibitem{waugh95}\nF.~R. Waugh, M.~J. Mar, R.~M. Westervelt, K.~L. Campman, and A.~C.\nGossard,\n \\prl {\\bf 75}, 705 (1995).\n\n\\end{references}\n\n\\end{document}\n"
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[
{
"name": "cond-mat0002080.extracted_bib",
"string": "\\bibitem[a)]{addr88}\nCurrent address: Department of Electrical Engineering and\nComputer Science, University of Michigan, Ann Arbor, MI 48109.\n\n\n\\bibitem{nakajima95}\nY. Nakajima, Y. Takahashi, S. Horiguchi, K. Iwadate, H. Namatsu, K.\nKurihara,\n and M. Tabe, Jpn.~J.~Appl.~Phys. {\\bf 34}, 1309 (1995).\n\n\n\\bibitem{ishikuro96}\nH. Ishikuro, T. Fujii, T. Saraya, G. Hashiguchi, T. Hiramoto, and T.\nIkoma,\n \\apl {\\bf 68}, 3585 (1996).\n\n\n\\bibitem{hiramoto97}\nT. Hiramoto, H. Ishikuro, T. Fujii, G. Hashiguchi, and T. Ikoma,\nJpn.~J.\n ~Appl.~Phys. {\\bf 36}, 4139 (1997).\n\n\n\\bibitem{smith97}\nR.~A. Smith Abd~H. Ahmed, \\apl {\\bf 71}, 3838 (1997).\n\n\n\\bibitem{ishikuro97}\nH. Ishikuro and T. Hiramoto, \\apl {\\bf 71}, 3691 (1997).\n\n\n\\bibitem{ishikuro99}\nH. Ishikuro and T. Hiramoto, \\apl {\\bf 74}, 1126 (1999).\n\n\n\\bibitem{leobandung95}\nE. Leobandung, L. Guo, Y. Wang, and S.~Y. Chou, \\apl {\\bf 67}, 938\n(1995).\n\n\n\\bibitem{kulik75}\nI.~O. Kulik and R.~I. Shekhter, Zh. Eksp. Teor. Fiz. {\\bf 68}, 623\n(1975),\n [Sov.~Phys.~JETP {\\bf 41}, 308 (1975)].\n\n\n\\bibitem{beenakker91}\nC.~W.~J. Beenakker, \\prb {\\bf 44}, 1646 (1991).\n\n\n\\bibitem{ruzin92}\nI.~M. Ruzin, V. Chandrasekhar, E.~I. Levin, and L.~I. Glazman, \\prb\n{\\bf 45},\n 13469 (1992).\n\n\n\\bibitem{su92}\nB. Su, V.~J. Goldman, and J.~E. Cunningham, \\prb {\\bf 46}, 7644\n(1992).\n\n\n\\bibitem{waugh95}\nF.~R. Waugh, M.~J. Mar, R.~M. Westervelt, K.~L. Campman, and A.~C.\nGossard,\n \\prl {\\bf 75}, 705 (1995).\n\n"
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cond-mat0002081
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Free Energy Landscape Of Simple Liquids Near The Glass Transition
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"author": "Chandan Dasgupta\\dag \\footnote[3]{Also at the Condensed Matter Theory Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India.} and Oriol T. Valls\\ddag"
}
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% insert abstract here Properties of the free energy landscape in phase space of a dense hard sphere system characterized by a discretized free energy functional of the Ramakrishnan-Yussouff form are investigated numerically. A considerable number of glassy local minima of the free energy are located and the distribution of an appropriately defined ``overlap'' between minima is calculated. The process of transition from the basin of attraction of a minimum to that of another one is studied using a new ``microcanonical'' Monte Carlo procedure, leading to a determination of the effective height of free energy barriers that separate different glassy minima. The general appearance of the free energy landscape resembles that of a putting green: deep minima separated by a fairly flat structure. The growth of the effective free-energy barriers with increasing density is consistent with the Vogel-Fulcher law, and this growth is primarily driven by an entropic mechanism.
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"name": "trms.tex",
"string": "\\documentclass[10pt]{iopart}\n\\input epsf\n\\begin{document}\n\\jl{3}\n\\title[Free Energy Landscape Near The Glass Transition]\n{Free Energy Landscape Of Simple Liquids Near The Glass Transition}\n\\author{Chandan Dasgupta\\dag \\footnote[3]{Also at the Condensed \nMatter Theory Unit, Jawaharlal \nNehru Centre for Advanced Scientific Research, Bangalore 560064,\nIndia.} and Oriol T. Valls\\ddag}\n\\address{\\dag\\ Centre for Condensed Matter Theory, \nDepartment of Physics, Indian Institute of Science, Bangalore \n560012, India}\n\\address{\\ddag\\ School of Physics and Astronomy and Minnesota \nSupercomputer Institute, \\\\ University of Minnesota, \nMinneapolis, Minnesota 55455, USA}\n%\\maketitle\n\\begin{abstract}\n% insert abstract here\nProperties of the free energy landscape in\nphase space of a dense hard sphere system characterized by a discretized\nfree energy functional of the Ramakrishnan-Yussouff form are\ninvestigated numerically. A considerable number of glassy local minima \nof the free energy are located and the distribution of\nan appropriately defined ``overlap'' between minima is calculated.\nThe process of transition from the basin of attraction of \na minimum to that of another one is studied using a new\n``microcanonical'' Monte Carlo procedure, leading to a determination\nof the effective height of free energy\nbarriers that separate different glassy minima. \nThe general appearance of the free energy landscape resembles \nthat of a putting green: deep minima separated by\na fairly flat structure. The growth of the effective\nfree-energy barriers with increasing density is consistent with \nthe Vogel-Fulcher law, and this growth is primarily driven by an\nentropic mechanism.\n\\end{abstract}\n% insert suggested PACS numbers in braces on next line\n\\pacs{64.70.Pf, 64.60.Ak, 64.60.Cn}\n\\submitted{{\\noindent \\it }}\n\n% body of paper here\n\\section{Introduction}\n\\label{sec:intro}\nA liquid quickly cooled to temperatures below its freezing\npoint enters a metastable supercooled state. At lower \ntemperatures, the supercooled liquid undergoes a\nglass transition to a state in which it resembles\na disordered solid. The dynamics of supercooled liquids near\nthe glass transition exhibits~\\cite{rev1,rev2} multi-stage, non-exponential \ndecay of fluctuations and a rapid growth of relaxation times,\nfeatures which are not fully understood.\n\nAn intuitive description that\nis often used~\\cite{pwa,pgw} for a qualitative understanding of the\nobserved behavior near the glass transition is based on the\n``free energy landscape'' paradigm. \nThis description starts from a functional that expresses the free\nenergy of a liquid in terms of the time-averaged local number\ndensity. At high temperatures (or at low densities in systems, such as\nhard spheres, where the\ndensity is the control parameter), this functional is\nbelieved to have only one minimum, that representing the uniform liquid\nstate. As the temperature is decreased to near the \ncrystallization point, a new minimum representing the crystal,\nwith a periodic modulation of the local density, \nshould also develop. In the ``free energy landscape'' picture, \na large number of ``glassy'' local minima\nof the free energy, characterized by inhomogeneous, aperiodic\ndensity distributions, also appear at temperatures below\nthe equilibrium freezing point. If the system gets \ntrapped in one of these glassy local minima as it is cooled\nrapidly, crystallization can not occur and\nthe subsequent dynamics is governed by\nthermally activated transitions among some of the many \nmetastable glassy minima. If the system visits many of these minima\nduring its evolution over a certain observation time, \nit behaves like a liquid\nover such time scales: the time-averaged local density\nremains uniform. However, the dynamics in this regime, \ngoverned by thermally activated transitions, is \nslow and complex. In this picture,\nthe glass transition occurs when the time scale of transitions among \nthe glassy minima becomes so long that the system is confined in\na single ``valley'' of the landscape over experimentally accessible \ntime scales. The features of such a free energy landscape would be\nvery similar to those found~\\cite{kw,kt,ktw} in\ncertain generalized spin glass models with infinite-range interactions,\nand in spin models~\\cite{bm,parisi} with \ncomplicated infinite-range interactions, but no quenched\ndisorder. The behavior of these mean-field models exhibits a remarkable\nsimilarity with the phenomenology of the glass transition.\nThese results suggest that the free energy landscape paradigm \nindeed provides a good framework for the \nunderstanding of the properties of supercooled liquids near the glass\ntransition. Such a description requires numerically\nobtained information about the topography of the free energy\nlandscape of these liquids. \n\nWe have carried out several numerical studies of a dense hard-sphere \nsystem, using a model free\nenergy functional proposed by Ramakrishnan and Yussouff (RY)~\\cite{RY},\na discretized version of which exhibits~\\cite{cd1,cd2}\na large number of glassy local minima at densities\nhigher than the value at which equilibrium crystallization \noccurs. [The control parameter for a hard-sphere system is the dimensionless \ndensity $n^* \\equiv \\rho_0 \\sigma^3$, where $\\rho_0$ is the average\nnumber density in the fluid phase and $\\sigma$ is the hard-sphere\ndiameter; increasing \n(decreasing) $n^*$ has the same effect as decreasing (increasing) \nthe temperature where the temperature is the \ncontrol parameter.] From numerical studies~\\cite{lvd,dv94,vd95}\nof the Langevin equations for this system, we found that the \ndynamics changes qualitatively \nat a ``crossover'' density near $n^*_x = 0.95$. The dynamics of a\nsystem initially in the uniform liquid state remains\ngoverned by small fluctuations near the liquid free energy minimum \nwhen the density is lower than $n^*_x$. \nFor higher values of $n^*$, the dynamics\nis governed by transitions among the glassy minima. \nThe time scales for such transitions were estimated \nfrom a Monte Carlo (MC) method in Ref.~\\cite{dv96} and found to \nincrease rapidly with density. \n\nHere we report results of additional numerical studies in\nwhich a new approach to the free energy landscape is used. \nWe have developed and\nused a new MC procedure that enables us\nto study transitions between different glassy\nminima and thus investigate\nthe topography of the free energy surface in phase space. We have \nlocated a large number of glassy minima of the free energy so\nas to yield their statistical properties as a function\nof density. The total number of glassy minima is found to remain \nnearly constant as the density\nis varied in the range $0.94 \\le n^* \\le 1.06$. The free energies of the\nglassy minima are distributed over a wide range between the free\nenergy of the uniform liquid and that of the crystal. An appropriately\ndefined ``overlap'' between different glassy minima is also found to\nexhibit a broad distribution. We have found pairs of glassy minima that differ\nfrom each other in the rearrangement of a very small number of\nparticles. The height of the free energy barrier that separates two\nsuch minima is quite small. Such pairs\nmay be identified as ``two-level systems'' which are believed~\\cite{tls}\nto exist in all glassy systems. \n\nOur computation of the probability of transition from a glassy minimum\nto the others as a function of the free energy increment (see below) \nand the MC\n``time'' $t$ leads us to define an effective barrier height that\ndepends weakly on $t$. The growth of this \neffective barrier height with density is consistent with a \nVogel-Fulcher form~\\cite{vf} for a hard-sphere system~\\cite{wa81}. \nThe dependence of the effective barrier height on $t$ and the\ndensity indicates that the growth of \nthe barrier height (and the consequent growth of the relaxation time)\nis primarily due to entropic effects arising from an increase in the \ndifficulty of finding low free-energy paths (saddle points) that \nconnect one glassy local minimum with the others. \n\n\\section{Model and Methods}\n\\label{section:m&m}\n\nWe characterize our system \nby a free energy functional $F[\\rho]$ of the\nform~\\cite{RY}:\n\n\\begin{eqnarray}\nF[\\rho] &=& F_l(\\rho_0)+ k_B T \\left[ \\int{d {\\bf r}\\{\\rho({\\bf r})\n\\ln (\\rho({\\bf r})/\\rho_0)-\\delta\\rho({\\bf r})\\} } \\right. \\nonumber \\\\\n&-& \\left. (1/2)\\int{d {\\bf r} \\int {d{\\bf r}^\\prime\nC({|\\bf r}-{\\bf r^\\prime|}) \\delta \\rho ({\\bf r}) \\delta\n\\rho({\\bf r}^\\prime)}} \\right],\n\\label{ryfe}\n\\end{eqnarray}\nwhere $F_l(\\rho_0)$ is the free energy of the uniform liquid at density\n$\\rho_0$, and $\\delta \\rho ({\\bf r})\\equiv \\rho({\\bf r})-\\rho_0$ is the\ndeviation of the density $\\rho$ at point ${\\bf r}$ from $\\rho_0$.\nWe set $F_l(\\rho_0)=0$. In Eq.(\\ref{ryfe}),\n$T$ is the temperature and $C(r)$ the direct pair correlation\nfunction~\\cite{hm86} of the uniform liquid at density $\\rho_0$, which\nwe express in terms of \n$n^*\\equiv \\rho_0 \\sigma^3$ ($\\sigma$ is the hard-sphere diameter) \nby making use of the Percus-Yevick~\\cite{hm86} approximation. \nThe direct pair correlation function~\\cite{hm86} of simple model\nliquids characterized by an isotropic, short-range pair-potential with \na strongly repulsive core (such as the Lennard-Jones potential) is very\nsimilar to that of the hard-sphere system at high densities. Therefore,\nwe expect our results to apply, at least\nqualitatively, to such dense liquids.\n\nWe discretize our system\nby introducing a cubic lattice of size $L^3$ and mesh constant\n$h$ in which the variables \n$\\rho_i,\\, i= 1, L^3$, are defined as\n$\\rho_i \\equiv \\rho({\\bf r}_i) h^3$, where $\\rho({\\bf r}_i)$ is the\ndensity at mesh point $i$.\nThe dimensionless free energy per particle $f[\\rho]$ is\n$f[\\rho]= \\beta F[\\rho]/N$\nwhere $N = \\rho_0 (Lh)^3 = n^* L^3 a^3$ is the total number of\nparticles in the simulation box, $\\beta \\equiv 1/(k_BT)$ and \n$a$ is the ratio $h/\\sigma$.\n\nIdeally, one would like to start\nthe system in a known glassy local minimum\nof the free energy, and \ninvestigate the topography of the free energy surface near the starting\npoint by allowing the system to evolve, and finding out which\nconfigurations it subsequently visits and where it ends up. \nA conventional Metropolis algorithm MC\nprocedure~\\cite{dv96} is inefficient at doing \nthis because at the relatively high\ndensities studied here, it would take a very long time for the\nsystem to move out of the basin of attraction of the initial minimum.\nTo obviate this difficulty, \nwe have devised what we call a ``microcanonical'' MC\nmethod. The algorithm is as\nfollows: we choose a trial value of what we call the free energy increment,\n$\\Delta F$, or $\\Delta f$ if we are \ndealing with the dimensionless version of $F$. Then, starting with\ninitial conditions which correspond to a\nlocal free energy minimum, we sweep the sites $i$ of the lattice sequentially.\nAt each step and site, we pick another site $j$ at random from the ones\nwithin a distance $\\sigma$ from $i$. \nWe then attempt to change the values\nof $\\rho_i$ and $\\rho_j$ to $p(\\rho_i+\\rho_j)$ and $(1-p)(\\rho_i+\\rho_j)$,\nwhere $p$ is a random number distributed uniformly in $[0,1]$. \nThe attempted change\nis accepted, and this is the crucial point, if and only if the free energy\nafter the change is less than $F_{max}\\equiv F_0 +\\Delta F$ where\n$F_0$ is the value of the free energy at\nthe minimum where we start the computation.\nThe simulation proceeds up to a maximum ``time'', $t_{m}$,\nmeasured in MC steps per site (MCS). We\nperform a sweep over a range of values of $\\Delta F$, with the same\ninitial conditions. If $\\Delta F$ is smaller than the height of \nthe lowest free energy barrier between the starting minimum\nand any other ``nearby'' minima, the system will\nremain in the basin of attraction of the starting minimum. As we\nincrease $\\Delta F$, there will eventually be \none or more minima that the system can find within \na ``time'' $t<t_m$. These minima\nare separated from the initial minimum by free energy barriers of\nheight less than $\\Delta F$.\nAs $\\Delta F$ is further increased, additional minima\nwill be made accessible, and since additional paths will become\navailable between the initial minimum and the minima already accessible at\nsmaller values of $\\Delta F$, these minima may be reached in fewer MC steps.\nClearly, if one obtains the information of near which minimum the\nsystem is, and how long it takes to get there, one can begin to map out\nthe free energy landscape.\n\nTo find out which basin of attraction the system is in at\ntime $t$, we save the values of the \nvariables $\\rho_i$ at relatively\nfrequent time intervals $\\Delta t$. These configurations are then used\nas the inputs in a minimization procedure~\\cite{cd1} that determines which\nbasin of attraction the system is in. \nThe entire procedure is repeated\na number of times (the ``number of runs'') and averaged over.\nWe have carried out this procedure at \ndensities in the range $0.94 \\le n^* \\le 1.06$. We did not consider\ndensities lower than 0.94 because previous studies~\\cite{dv94,vd95} \nshow that the dynamics of the system is governed by transitions among\nglassy local minima only at higher densities. Since\nthe Percus-Yevick approximation becomes \nless accurate at relatively high densities~\\cite{hm86}, \nvalues of $n^* > 1.06$ were not considered.\n\nWe used two different sets of the sample size $L$ and \nthe mesh size $h$, in one case\ncommensurate with a close-packed lattice, and in the other incommensurate.\nThe computationally more intensive part of our \nsimulations was carried out for systems of size $L = 15$ \nwith periodic boundary conditions and mesh size $h = \\sigma/4.6$. \nNo crystalline\nminimum was found for these incommensurate values. \nThe other portion of the computations was performed for systems\nwith $L = 12$\nand $h = 0.25 \\sigma$. These values are commensurate with a\nfcc structure and a crystalline minimum is found at\nsufficiently high densities. Because of the smaller size of these\nsamples, we were\nable to explore more extensively several\naspects of the problem under consideration.\nOur computations for the $L=15$ sample were carried out for \n$t_m=15000$ MCS and $\\Delta t=5000$ MCS, whereas computations for \nthe system with $L=12$ were carried out to $t_m=8000$ MCS with \n$\\Delta t=2000$ MCS. A detailed discussion of how the glassy minima\nused in our study were chosen and the structure of these glassy minima\nmay be found in Refs.~\\cite{dv96,dv98,dv99}.\n\n\\section{Results}\n\\label{section:r&d}\n\nDuring the evolution\nof the system, we monitor $\\beta F$ and\nthe maximum and minimum values\nof the variables $\\rho_i, i = 1,L^3$. If the system\nfluctuates near one of the inhomogeneous minima,\nthen the maximum value of $\\rho_i$ would be much higher than the value\n(close to $\\rho_0 h^3$) it would have in the vicinity\nof the uniform liquid minimum. The system does not move to\nthe neighborhood of the liquid minimum for the values of $\\Delta F$\nconsidered here. The total free energy remains nearly\nconstant at a value slightly lower than the maximum allowed value, \n$F_{max} = F_0 + \\Delta F$. \n\nIn our analysis of the process of transition of the system \nfrom the initial glassy minimum to the basins\nof other minima, we define a ``critical'' value, $\\Delta f_c(t)$ (or\n$\\Delta F_c(t)$), of the free-energy increment $\\Delta f$ (or $\\Delta\nF$) as follows:\nat every time investigated (i.e. times 5000, 10000 and 15000 MCS \nfor $L=15$\nand times 2000, 4000, 6000 and 8000 MCS for the $L=12$ samples),\nwe test, for increasing\nvalues of $\\Delta f$, what is the probability,\n$P(\\Delta f,t)$, that the\nsystem has moved to the basin of attraction of a free energy minimum\ndistinct from the starting one. This probability, which \nwe obtain by averaging over a sufficient number (ten\nto fifteen) of runs, is (at constant time) zero\nfor very small $\\Delta f$ and rises toward unity as $\\Delta f$ \nincreases. At a constant $\\Delta f$, it increases somewhat with \nMC time, as the system explores further regions\nof phase space. We define $\\Delta f_c(t)$ as the value of $\\Delta \nf$ for which, at that time, the switching probability reaches $1/2$. \nOf course, $P$ and $\\Delta f_c$ are also functions\nof $n^*$. The minima to which the system moves for values of \n$\\Delta f$ close to\nor higher than $\\Delta f_c$ are, in general, different for\ndifferent runs. This suggests that $\\Delta f_c$ represents\na measure of the free energy increment for which a \nrelatively large region of\nphase space becomes accessible to the system. \nThe system almost never returns to\nthe basin of attraction of the initial minimum: after having\nleft the initial minimum, the system cannot find\nits way back. \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode \n \\hspace*{1.8cm}\n \\epsfysize=5.5cm\\epsfbox{dasg1a.eps}\n \\epsfysize=5.5cm\\epsfbox{dasg1b.eps}\n \\end{center}\n \\caption{ (a) Example of the determination of the \n``critical'' value $\\Delta f_c$, defined as the value of the free\nenergy increment $\\Delta f$ at which the transition probability $P$ is 1/2.\nThe black dots mark the intersections of the plots with\nthe line $P=0.5$ (see text for a complete discussion). The data shown\nare for a sample of size $L = 15$, and three values (5000, 10000 and\n15000 MCS) of the Monte Carlo time $t$.\n(b) Results for $\\Delta f_c$ as a function of $t$ at four \ndifferent densities for a $L = 15$ minimum. Results for\nother times and densities studied interpolate smoothly with the\nresults shown. } \n\\label{fig1}\n\\end{figure}\n\nThe procedure for determining $\\Delta f_c$ is illustrated\nin Fig.\\ \\ref{fig1}a, where we have \nshown the results for the \ntransition probability $P$ as a function of the free energy increment\n$\\Delta f$ for a \n$L = 15$ minimum at $n^* = 0.99$. \nIt is clear from our data \nthat the uncertainty in the estimated values of $\\Delta\nf_c$ is $\\sim 0.05$, the spacing between successive values of \n$\\Delta f$ in the simulation. Typical results for $\\Delta f_c$\nare shown in Fig.\\ \\ref{fig1}b for the same $L = 15$ minimum and four \nvalues of $n^*$. Clearly, $\\Delta f_c$ is a\nweak function of $t$, and a stronger function of $n^*$.\nThe dependence of $\\Delta f_c$ on $t$ for fixed $n^*$\nbecomes more pronounced as $n^*$ is increased. These dependences are\nanalyzed later in this section. \n\nThe large number of minimization runs we carried out locate a \nlarge fraction of the full collection of\nglassy minima of the free energy. \nFor the ``incommensurate'' $L = 15$ sample used in our work, the\nnumber of minima we\nhave located at each density is in the range of four to six.\nThe ``commensurate'' $L = 12$ sample exhibits a substantially larger\nnumber of minima, one of which is crystalline (fcc). \nA similar sensitivity of the number of local minima to the sample size and\nboundary conditions has been found in numerical studies~\\cite{ah97,dpvr} of \nthe potential energy landscape of model liquids described \nby simple Hamiltonians. In the following discussion of the statistical\nproperties of the collection of glassy minima, we consider chiefly the \nresults obtained for $L = 12$, for which we can produce significant \nstatistics.\n\nThe number of glassy local minima of the $L = 12$ system remains nearly\nconstant as the density is varied in the range $0.96 \\le n^* \\le 1.06$.\nThis number is close to 25. There is no systematic trend in the\ndependence of this number on the density. \nThe free energies of these minima are distributed in a band that lies\nbetween the free energy of the uniform liquid (zero)\nand that of the crystal. The\nwidth of this band increases with $n^*$. Since the number of\nminima is approximately independent of the density, this implies that\nthe ``density of states'' of the glassy minima decreases as $n^*$ is\nincreased. Let $p(\\beta F) \\delta $ be the probability of\nfinding a glassy minimum with dimensionless free energy between $\\beta\nF-\\delta/2$ and $\\beta F + \\delta/2$. \nWe have calculated this quantity \nat different values of $n^*$. Representative results at two\ndensities, $n^*=0.96$ and $n^* = 1.02$, are shown in Fig.\\ \\ref{fig2}a.\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode \n \\hspace*{1.8cm}\n \\epsfysize=5.5cm\\epsfbox{dasg2a.eps}\n \\epsfysize=5.5cm\\epsfbox{dasg2b.eps}\n \\end{center}\n \\caption{(a) The ``density of states'' for glassy free energy minima, defined\nas the probability of finding a glassy minimum with free energy in\na given range (see text). Results for $L = 12$ samples at two densities\nare shown.\n(b) The distribution $P(q)$ of the overlap $q$ defined in\nEq.(\\protect\\ref{ovlp}). The distributions of the self-overlap and the\noverlap between different minima are shown separately for $L=12$\nsamples at density $n^*=1.02$.}\n\\label{fig2}\n\\end{figure}\nThe values of $\\delta$ used are 4.0 and 8.0 for $n^*=0.96$ and $n^* =\n1.02$, respectively. The\nrange of $\\beta F$ over which $p(\\beta F)$ is nonzero is clearly wider\nat the higher density. The consequent decrease in the values of\n$p(\\beta F)$ with increasing density is also clearly seen. Both\ndistributions show peaks near the upper end, and tails extending to\nsubstantially lower values. However, the lowest free energy of the\nglassy minima is substantially higher than the free energy of the\ncrystalline minimum. If the probability of\nfinding the system in a glassy minimum is assumed to be proportional to\nthe Boltzmann factor $e^{-\\beta F}$, then only those minima with free\nenergies lying near the lower end of the band would be relevant in\ndetermining the equilibrium and dynamic properties of the system. Our\nresults indicate that the number of such ``relevant'' minima decreases\nwith increasing $n^*$. We find correlations between the free\nenergy of a glassy minimum and its structure, similar to those\nfound in Ref.\\cite{dv94}. Minima with\nlower free energies have more ``structure'' (as indicated by e.g. the\nheights of the first and second peaks of the two-point correlation \nfunction of the local density) and higher average density than those with\nhigher free energies.\n\nWe have also studied how the distributions of the local density \nvariables in two distinct glassy minima differ from one another. \nThe degree of similarity between two minima may be quantified in terms\nof their ``overlap''~\\cite{ktw}. For the discretized system\nconsidered here, the overlap $q(1,2)$ between two minima labeled ``1''\nand ``2'' may be defined in the following way:\n\\begin{equation}\nq(1,2) = \\frac{1}{\\rho_{av}L^3}\\,max\\{{\\bf R}\\}\n\\sum_i[\\rho_i^{(1)} - \\rho_{av}][\\rho_{{\\bf R}(i)}^{(2)}-\\rho_{av}].\n\\label{ovlp}\n\\end{equation}\nHere, $\\rho_i^{(1)}$ and $\\rho_i^{(2)}$ are the discretized densities\nat the two minima and $\\rho_{av}$ is the average value of the $\\rho_i$, which\nis assumed to be the same for the two minima.\n$\\bf R$ represents one of the 48 symmetry operations\nof the cubic mesh, plus all translations taking into account periodic\nboundary conditions,\n${\\bf R}(i)$ is the mesh point to which mesh point $i$\nis transformed under $\\bf R$, and $max\\{{\\bf R}\\}$ means that the $\\bf\nR$ that maximizes the quantity on the right is to be taken.\nIn Fig.\\ \\ref{fig2}b, we display\nthe results for the distribution $P(q)$ of\n$q$ at $n^*=1.02$. The distribution of the self-overlap, defined as\n$q(i,i)$ for the $i$th minimum, is also shown. As expected, the\ndistribution of the self-overlap exhibits a sharp peak at a large\nvalue of $q$. The overlap between different minima exhibits a broad\ndistribution with a peak at a small value of $q$, indicating that\nmost of the glassy minima are rather different\nfrom one another. This distribution, however, extends to values of $q$ as\nlarge as 0.6, indicating that there are a few pairs of \nglassy minima which are very similar to each other. For each value of \n$n^*$, we find a small number (3-5) of such pairs of minima.\nThe main difference between their structures comes from small\ndisplacements of just 2-3 particles. \nThese pairs of minima are examples of ``two-level systems'' whose existence\nin glassy materials was postulated~\\cite{tls} many years ago. \n\nWe have also looked at how the quantity $ F_c = F_0 +\\Delta F_c$,\nvaries from one minimum to another. While the free energy $F_0$ of a glassy\nminimum varies over a wide range (see Fig.~\\ref{fig2}a), the value of $F_c$\nis nearly constant for each value of $n^*$. This suggests a \n``putting green like'' free energy landscape in which the local minima\nare like ``holes'' of varying depth in a nearly flat background. This\nstructure also implies that there is a strong correlation between the\ndepth of a minimum and the height of the barriers that separate it from\nthe other minima: the barriers are higher for deeper minima. \n\nWe now discuss the dependence of $\\Delta f_c$ on the density $n^*$ and\nMC time $t$. Since the transition probability $P$ is an \nincreasing function of both $\\Delta f$ and $t$,\n$\\Delta f_c(n^*,t)$ decreases as $t$ is increased (see Fig.~\\ref{fig1}). \nIn agreement with the previously observed~\\cite{dv96}\ngrowth of the barrier-crossing time scale with $n^*$, we find that \n$\\Delta f_c$ is an increasing function of $n^*$. \nThe $t$-dependence of $\\Delta f_c$ becomes {\\em stronger} as $n^*$\nis increased. The $t$-dependence of $\\Delta f_c$ is\nclosely related to the probability of finding a path \n(``saddle point'') that connects\nthe starting minimum to a different one. \nIf such paths were relatively easy to find, then the transition \nprobability would be insensitive to the value of $t$ as long as\nit is not very short. If, however, paths to other minima are few,\na large number of configurations have to be explored \nbefore one of them is found. The $t$-dependence \nof $\\Delta f_c$ would then be more pronounced\nand extend to larger values of $t$. To make the idea more concrete, \nwe ignore the short-range time correlations among \nthe configurations generated in a\nMC run and assume that they represent $t$ independent \nsamplings of configurations with\nfree energy less than $F_0+\\Delta F$. \nNeglecting the rare return to the basin of attraction of \nthe starting minimum after \na transition to a different basin of attraction,\nthe transition probability may then be estimated\nas $P(n^*,\\Delta f,t) = 1- [1-p(n^*,\\Delta f)]^{t} \\simeq 1 -\n\\exp(-tp)$, where \n$p(n^*, \\Delta f) \\ll 1$ is the probability that a randomly \nchosen configuration with\n$\\beta F \\le \\beta F_0 + N \\Delta f$ belongs in the basin of attraction of\na different minimum. One expects $p$ to be zero if $\\Delta f \\le\n\\Delta f_0(n^*)$ where $N k_B T \\Delta f_0$ is the height of the lowest \nfree energy barrier, and\n$p = g(n^*,\\Delta f - \\Delta f_0)$ for $\\Delta f > \\Delta f_0$ where \n$g(n^*,x)$ grows \ncontinuously from zero as $x$ is increased from zero.\nCombining this with the definition of $\\Delta f_c$, we obtain the relation \n$g(n^*,\\Delta f_c(n^*,t)-\\Delta f_0(n^*)) = \\ln 2/t$. \nSince $\\Delta f_c(n^*,t_1) - \\Delta f_c(n^*, t_2)$ for fixed $t_1 < t_2$ \n{\\em increases} with $n^*$, \nthe function $g(n^*,x)$\n{\\em decreases} (i.e. the difficulty of finding \npaths to other minima increases) as\n$n^*$ is increased at fixed $x$.\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode \n \\hspace*{1.8cm}\n \\epsfysize=5.5cm\\epsfbox{dasg3a.eps}\n \\epsfysize=5.5cm\\epsfbox{dasg3b.eps}\n \\end{center}\n% \\hbox to \\textwidth{\\vspace*{5mm}\\hfil \\hfil (a)\\hfil\\hfil \n% (b)\\hfil}\n \\caption{(a) Plots of $\\Delta f_c$, obtained for a $L$ = 15 minimum, against \n$(t/1000)^{-0.35}$ for four values of $n^*$. The dashed lines are the\nbest straight-line fits.\n(b) The dependence of $\\Delta f_c(n^*, t=15000)$ on $n^*$ for $L$ = 15. \nThe dashed line shows the best fit to a Vogel-Fulcher form (see text).} \n\\label{fig3}\n\\end{figure}\n\nThe observed $t$-dependence of $\\Delta f_c$ for \nall values of $n^*$\nand all the minima in our study is well-represented by \n\\begin{equation}\n\\Delta f_c(n^*,t) = \\Delta f_0(n^*) + c(n^*) t^{-\\alpha},\n\\label{fit}\n\\end{equation}\nwith $\\alpha$ in the range $0.25-0.40$. \nFits to this form with $\\alpha = 0.35$\nfor a minimum with $L$ = 15 are shown in Fig.~\\ref{fig3}a. \nThe values of $\\Delta f_0$ obtained from such\nfits with a fixed value of $\\alpha$ are nearly independent of $n^*$, \nbut exhibit a\ndependence on the value of $\\alpha$, varying between 0 and 0.5\nfor the $L$ = 15 minimum. Similar results are obtained for $L=12$, with\nvalues of $\\Delta f_0$ between 1.3 and 1.5.\nThe quantity $c(n^*)$ increases with $n^*$. \nThese results correspond to \n$g(n^*,x) \\sim A(n^*)x^{1/\\alpha}$ with $A(n^*)$ \ndecreasing with increasing $n^*$. \nWe conclude that the \ngrowth of the effective barrier height\nwith increasing $n^*$ is primarily due to an \nentropic mechanism associated with an increase\nof the difficulty in finding low-lying saddle points that connect different \nglassy local minimum of the free energy. \n\nThe dependence of $\\Delta f_c$ on $n^*$ \nis consistent with the Vogel-Fulcher\nlaw~\\cite{vf} which assumes the following form~\\cite{wa81} for our system:\n\\begin{equation}\n\\Delta f_c(n^*) = a + b/(n^*_c - n^*),\n\\label{vfeq}\n\\end{equation}\nwhere $a$, $b$ and $n^*_c$ are constants. \nThe value of $n^*_c$ obtained\nfrom fits of our data for $\\Delta f_c(n^*, t)$ to Eq.(\\ref{vfeq}) \nwith fixed $a$ is\nnearly independent of $t$. This is\nconsistent with the form of Eq.(\\ref{fit}) if \n$a = \\Delta f_0$ , $b \\propto t^{-\\alpha}$, and $c \\propto 1/(n^*_c-n^*)$. \n$\\Delta f_0$ is indeed nearly independent of\n$n^*$, and we find that\nthe $t$-dependence of $b$ and the $n^*$-dependence of $c$ \nare in agreement\nwith the other two conditions. For the $L$ = 15 case,\nwe can fit the data for $\\Delta f_c$ at $t$ = 15,000 to \nthe form of Eq.(\\ref{vfeq}) with\n$a$ = 0 ($\\Delta f_0=0)$. \nThe best fit, shown in Fig.~\\ref{fig3}b, \ncorresponds to $n^*_c = 1.225$, very\nclose to the expected random close packing density \n$n^*_{rcp} \\simeq 1.23$. \nThe best fit to the $L$ = 12 data with $a \\simeq 1.0$ \nalso yields a similar value of $n^*_c$. \nWe conclude that the observed growth of\nthe effective barrier height is consistent with the Vogel-Fulcher form. \nThe increase in the effective barrier height as $n^*$ is increased from\n0.96 to 1.06 is about 25$k_BT$, corresponding to a growth of the \ncharacteristic time\nscale of about ten orders of magnitude. Thus, the range of time \nscales covered in\nour study is comparable to that used in Vogel-Fulcher fits \nof experimental data, and\nmuch wider than what can be achieved in standard MC or \nmolecular dynamics simulations.\n\n\\section{Discussion}\n\\label{s&d}\n\nWe close with a discussion of the connections between our\nresults and those of \nspin-glass-like theories~\\cite{kt,ktw,gp1} of the structural \nglass transition (see Ref.~\\cite{dv99} for a discussion of the relation\nof our work with other recent studies of the behavior of simple liquids\nnear the glass transition). These theories are based on the similarity \nbetween the phenomenology of the structural glass transition in \nso-called ``fragile''~\\cite{rev2} liquids and the behavior found in a class\nof generalized mean-field spin glass models~\\cite{kw,ktn} with \ninfinite-range interactions.\nAt high temperatures, the free energy of these mean-field models,\nexpressed as a function of the single-site magnetizations, \nexhibits only one, ``paramagnetic'', minimum. As\n$T$ is lowered, an exponentially large number of\nnon-trivial local minima come into existence at a\ntemperature $T_d$, where a ``dynamic transition'', \ncharacterized by a breaking of ergodicity, occurs. \nThis ``dynamic transition'' does not have any signature in\nthe equilibrium behavior of the system. A thermodynamic phase\ntransition occurs at a lower temperature $T_c$. \nIn the suggested analogy between these models and the structural glass\ntransition, the paramagnetic minimum of the free energy is identified\nwith that corresponding to the uniform liquid, and the role of the \nnon-trivial local minima of the free energy is played by the glassy\nlocal minima. The analogue of the ``dynamic transition'' at $T_d$ \nis thought to be smeared out in liquids. It has been \nsuggested~\\cite{kt,ktw,gp1} that $T_d$ should be identified with the\n``ideal glass transition'' temperature of mode-coupling\ntheories~\\cite{mct}. \nThe temperature $T_c$ is interpreted as the ``Kauzmann \ntemperature''~\\cite{kauz} at which the difference in entropy \nbetween the supercooled \nliquid and the crystalline solid extrapolates to zero. The relaxation\ntime of the supercooled liquid is supposed to diverge at this\ntemperature. Heuristic arguments suggest that this divergence is\nof the Vogel-Fulcher form~\\cite{ktw,gp1}.\n\nOur results qualitatively support this scenario. We find a\ncharacteristic density at which a large number of glassy\nminima of the free energy appear. We do not \nknow whether the number of glassy minima depends exponentially on the\nsample volume. The configurational entropy associated with these\nminima decreases with increasing density because the width of the band\nover which the free energy of these minima is distributed increases\nwith density. We have also found evidence for a\nVogel-Fulcher-type growth of relaxation times driven by an entropic \nmechanism. \n\nThere are, however, certain differences between our\nfindings and the predictions of spin-glass-like theories. In our\nearlier work \\cite{dv94,vd95}, we found that the free energy of a typical\nglassy minimum becomes lower than that of the uniform liquid as the\ndensity is increased slightly above the value ($n^* \\approx$ 0.8) at \nwhich the minimum comes into existence. In particular, the free\nenergies of the glassy minima are substantially lower than that of the\nuniform liquid one for $n^*$ near $n^*_x \\simeq 0.95$, the crossover \ndensity~\\cite{dv94,vd95} above which the dynamics is governed by \ntransitions among glassy minima. This is different from the\nbehavior found in the spin glass models. Our results for the\ndistribution of the overlap between different minima are also somewhat\ndifferent from those for the spin glass models. Some of these\ndifferences may be due to finite-size effects which are probably \nsignificant\nfor the small samples considered here. Also, fluctuation effects, which\nare unimportant in mean-field models, may play an important role in our\nsystem. A careful investigation of these issues would be interesting.\n\n\\section*{References}\n\\begin{thebibliography}{999}\n\\bibitem{rev1} J\\\"{a}ckle J 1986 {\\it Rep. Prog. Phys.} {\\bf 49} 171.\n\\bibitem{rev2} Angell C A 1988 {\\it J. Phys. Chem. Solids} {\\bf 49} 863.\n\\bibitem{pwa} Anderson P W 1979 in {\\it Ill Condensed Matter, Lecture\nNotes of the Les Houches Summer School}, ed R Balian, R Maynard and G \nToulouse (Amsterdam: North Holland). \n\\bibitem{pgw} Wolynes P G 1988 in {\\it Proceedings of International\nSymposium on Frontiers in Science, (AIP Conf. Proc. No. 180)}, ed\nS S Chen and P G Debrunner (New York: American Institute of Physics).\n\\bibitem{kw} Kirkpatrick T R and Wolynes P G 1987 {\\it Phys. Rev. A}\n{\\bf 35} 3072; {\\it Phys. Rev B} {\\bf 36} 8552.\n\\bibitem{kt} Kirkpatrick T R and Thirumalai D 1989 {\\it J. Phys. A}\n{\\bf 22} L149.\n\\bibitem{ktw} Kirkpatrick T R, Thirumalai D and Wolynes P G 1989\n{\\it Phys. Rev. A} {\\bf 40} 1045.\n\\bibitem{bm} Bouchaud J -P and Mezard M 1994 {\\it J. Phys. I (France)}\n{\\bf 4} 1109.\n\\bibitem{parisi} Cugliandolo L F, Kurchan J, Parisi G and \nRitort F 1995 {\\it Phys. Rev. Lett.} {\\bf 74} 1012.\n\\bibitem{RY} Ramakrishnan T V and Yussouff M 1979 {\\it Phys. Rev. B}\n{\\bf 19} 2775.\n\\bibitem{cd1} Dasgupta C 1992 {\\it Europhys. Lett.} {\\bf 20} 131.\n\\bibitem{cd2} Dasgupta C and Ramaswamy S 1992 {\\it Physica A} {\\bf 186}\n314.\n\\bibitem{lvd} Lust L M, Valls O T, and Dasgupta C 1993 {\\it Phys. Rev.\nE} {\\bf 48} 1787.\n\\bibitem{dv94} Dasgupta C and Valls O T 1994 {\\it Phys Rev. E} {\\bf 50}\n3916.\n\\bibitem{vd95} Valls O T and Dasgupta C 1995 {\\it Transport Theory\nand Stat. Physics} {\\bf 24} 1199.\n\\bibitem{dv96} Dasgupta C and Valls O T 1996 {\\it Phys. Rev. E} {\\bf\n53} 2603.\n\\bibitem{tls} Anderson P W, Halperin B I and Varma C M 1972 {\\it\nPhilos. Mag.} {\\bf 25} 1; Phillips W A 1972 \n{\\it J. Low. Temp. Phys.} {\\bf 7} 351.\n\\bibitem{vf} Vogel H 1921 {\\it Z. Phys.} {\\bf 22} 645; Fulcher G S\n1925 {\\it J. Amer. Ceram. Soc.} {\\bf 8} 339.\n\\bibitem{wa81} Woodcock L V and Angell C A 1981 {\\it Phys. Rev. Lett.} \n{\\bf 47} 1129.\n\\bibitem{hm86} Hansen J P and McDonald I R 1986 {\\it Theory of Simple \nLiquids} (London: Academic).\n\\bibitem{dv98} Dasgupta C and Valls O T 1998 {\\it Phys. Rev. E} {\\bf\n58} 801.\n\\bibitem{dv99} Dasgupta C and Valls O T 1998 {\\it Phys. Rev. E} {\\bf\n59} 3123.\n\\bibitem{ah97} Heuer A 1997 {\\it Phys. Rev. Lett.} {\\bf 78} 4051.\n\\bibitem{dpvr} Daldoss G, Pilla O, Villani G, and Ruocco G, 1998\npreprint (cond-mat/9804113).\n\\bibitem{gp1} Parisi G 1997 in {\\it Complex Behaviour of Glassy Systems:\nProceedings of the XIV Sitges Conference}, ed M Rubi and C\nPerez-Vicente (Berlin: Springer).\n\\bibitem{ktn} Kirkpatrick T R and Thirumalai D 1987 {\\it Phys. Rev. B} \n{\\bf 36} 5388.\n\\bibitem{mct} G\\\"{o}tze W 1991 in {\\it Liquids, Freezing and the Glass\nTransition} ed J P Hansen, D Levesque and J Zinn-Justin (New York: Elsevier).\n\\bibitem{kauz} Kauzmann W 1948 {\\it Chem. Rev.} {\\bf 48} 219.\n\n\\end{thebibliography}\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002081.extracted_bib",
"string": "\\begin{thebibliography}{999}\n\\bibitem{rev1} J\\\"{a}ckle J 1986 {\\it Rep. Prog. Phys.} {\\bf 49} 171.\n\\bibitem{rev2} Angell C A 1988 {\\it J. Phys. Chem. Solids} {\\bf 49} 863.\n\\bibitem{pwa} Anderson P W 1979 in {\\it Ill Condensed Matter, Lecture\nNotes of the Les Houches Summer School}, ed R Balian, R Maynard and G \nToulouse (Amsterdam: North Holland). \n\\bibitem{pgw} Wolynes P G 1988 in {\\it Proceedings of International\nSymposium on Frontiers in Science, (AIP Conf. Proc. No. 180)}, ed\nS S Chen and P G Debrunner (New York: American Institute of Physics).\n\\bibitem{kw} Kirkpatrick T R and Wolynes P G 1987 {\\it Phys. Rev. A}\n{\\bf 35} 3072; {\\it Phys. Rev B} {\\bf 36} 8552.\n\\bibitem{kt} Kirkpatrick T R and Thirumalai D 1989 {\\it J. Phys. A}\n{\\bf 22} L149.\n\\bibitem{ktw} Kirkpatrick T R, Thirumalai D and Wolynes P G 1989\n{\\it Phys. Rev. A} {\\bf 40} 1045.\n\\bibitem{bm} Bouchaud J -P and Mezard M 1994 {\\it J. Phys. I (France)}\n{\\bf 4} 1109.\n\\bibitem{parisi} Cugliandolo L F, Kurchan J, Parisi G and \nRitort F 1995 {\\it Phys. Rev. Lett.} {\\bf 74} 1012.\n\\bibitem{RY} Ramakrishnan T V and Yussouff M 1979 {\\it Phys. Rev. B}\n{\\bf 19} 2775.\n\\bibitem{cd1} Dasgupta C 1992 {\\it Europhys. Lett.} {\\bf 20} 131.\n\\bibitem{cd2} Dasgupta C and Ramaswamy S 1992 {\\it Physica A} {\\bf 186}\n314.\n\\bibitem{lvd} Lust L M, Valls O T, and Dasgupta C 1993 {\\it Phys. Rev.\nE} {\\bf 48} 1787.\n\\bibitem{dv94} Dasgupta C and Valls O T 1994 {\\it Phys Rev. E} {\\bf 50}\n3916.\n\\bibitem{vd95} Valls O T and Dasgupta C 1995 {\\it Transport Theory\nand Stat. Physics} {\\bf 24} 1199.\n\\bibitem{dv96} Dasgupta C and Valls O T 1996 {\\it Phys. Rev. E} {\\bf\n53} 2603.\n\\bibitem{tls} Anderson P W, Halperin B I and Varma C M 1972 {\\it\nPhilos. Mag.} {\\bf 25} 1; Phillips W A 1972 \n{\\it J. Low. Temp. Phys.} {\\bf 7} 351.\n\\bibitem{vf} Vogel H 1921 {\\it Z. Phys.} {\\bf 22} 645; Fulcher G S\n1925 {\\it J. Amer. Ceram. Soc.} {\\bf 8} 339.\n\\bibitem{wa81} Woodcock L V and Angell C A 1981 {\\it Phys. Rev. Lett.} \n{\\bf 47} 1129.\n\\bibitem{hm86} Hansen J P and McDonald I R 1986 {\\it Theory of Simple \nLiquids} (London: Academic).\n\\bibitem{dv98} Dasgupta C and Valls O T 1998 {\\it Phys. Rev. E} {\\bf\n58} 801.\n\\bibitem{dv99} Dasgupta C and Valls O T 1998 {\\it Phys. Rev. E} {\\bf\n59} 3123.\n\\bibitem{ah97} Heuer A 1997 {\\it Phys. Rev. Lett.} {\\bf 78} 4051.\n\\bibitem{dpvr} Daldoss G, Pilla O, Villani G, and Ruocco G, 1998\npreprint (cond-mat/9804113).\n\\bibitem{gp1} Parisi G 1997 in {\\it Complex Behaviour of Glassy Systems:\nProceedings of the XIV Sitges Conference}, ed M Rubi and C\nPerez-Vicente (Berlin: Springer).\n\\bibitem{ktn} Kirkpatrick T R and Thirumalai D 1987 {\\it Phys. Rev. B} \n{\\bf 36} 5388.\n\\bibitem{mct} G\\\"{o}tze W 1991 in {\\it Liquids, Freezing and the Glass\nTransition} ed J P Hansen, D Levesque and J Zinn-Justin (New York: Elsevier).\n\\bibitem{kauz} Kauzmann W 1948 {\\it Chem. Rev.} {\\bf 48} 219.\n\n\\end{thebibliography}"
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cond-mat0002082
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Local magnetic structures induced by inhomogeneities of the lattice in $S=1/2$ bond-alternating chains and response to time-dependent magnetic field with a random noise
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[
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"author": "Masamichi Nishino"
}
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We study effect of inhomogeneities of the lattice in the $S=1/2$ bond-alternating chain by using a quantum Monte Carlo method and an exact diagonalization method. We adopt a defect in the alternating order as the inhomogeneity and we call it {\lq}{\lq}bond impurity{\rq}{\rq}. Local magnetic structures induced by the bond impurities are investigated both in the ground state and at very low temperatures. %The bond-alternating system has the gapful nature and %the correlation length is finite, which is similar to the case of $S=1$ %Haldane systems. The local magnetic structure can be looked on as an effective $S=1/2$ spin and the weakness of the interaction between the local structures causes the quasi-degenerate states in the low energy. We also investigate the force acting between bond impurities and find that the force is generally attractive. We also study the dynamical property of the local magnetic structure. While the local magnetic structure behaves as an isolated $S=1/2$ spin in the response to a time-dependent uniform field, it is found to be robust against the effect of a random noise applied at each site individually in the sweeping filed.
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"name": "nishino.tex",
"string": "\\documentstyle[aps,twocolumn]{revtex}\n\\tightenlines\n\n%\\documentstyle[aps,preprint]{revtex}\n\n\\begin{document}\n%\\wideabs{\n\\title{Local magnetic structures induced by inhomogeneities of the lattice\nin $S=1/2$ bond-alternating chains and response to time-dependent magnetic\nfield with a \nrandom noise}\n\n\\author{Masamichi Nishino}\n\n\\address{Department of Chemistry, Graduate School of Science,\\\\\nOsaka University, Toyonaka, Osaka 560, Japan\\\\} %\n\n\\author{Hiroaki Onishi}\n\\address{Department of Earth and Space Science, Graduate School of Science,\\\\\nOsaka University, Toyonaka, Osaka 560, Japan}%\n\n\\author{Kizashi Yamaguchi}\n\n\\address{Department of Chemistry, Graduate School of Science,\\\\\nOsaka University, Toyonaka, Osaka 560, Japan\\\\} %\n\n\n\\author{Seiji Miyashita}\n\\address{Department of Applied Physics, University of Tokyo\\\\\nBunkyoku, Tokyo, Japan}\n\n\\maketitle\n\n % DON'T CHANGE THIS LINE\n\\begin{abstract}\n\nWe study effect of inhomogeneities of the lattice in the $S=1/2$\nbond-alternating chain by using a quantum Monte Carlo method and an exact\ndiagonalization method.\nWe adopt a defect in the alternating order as the inhomogeneity and we call it \n{\\lq}{\\lq}bond impurity{\\rq}{\\rq}. \nLocal magnetic structures induced by the bond impurities\nare investigated both in the ground state and at very low temperatures.\n%The bond-alternating system has the gapful nature and \n%the correlation length is finite, which is similar to the case of $S=1$ \n%Haldane systems. \nThe local magnetic structure can be looked on as an effective $S=1/2$ spin\nand the weakness of the interaction between the local structures causes the\nquasi-degenerate states in the low energy.\nWe also investigate the force acting between bond impurities and \nfind that the force is generally attractive.\nWe also study the dynamical property of the local magnetic structure. \nWhile the local magnetic structure behaves as an isolated $S=1/2$ spin in the \nresponse to a time-dependent uniform field, it is found to be robust\nagainst the effect of a random noise applied at each site individually in\nthe sweeping filed.\n\n\n\\end{abstract}\n\\pacs{75.10.Jm 75.30.Hx 05.70.Ce 75.50.Ee 75.40.Gb}\n%}\n\n\\section{Introduction}\n\nIn the low dimensional quantum spin systems, it has been turned out that \nthe quantum mechanical \neffect, in particular the mechanism of the singlet pair formation, \nplays an important role for the ground state spin configuration~\\cite{Haldane,MG,AKLT}.\nThere have been found a wide variety of peculiar arrangements of the \nsinglet pairs in the ground state which brings new types of singlet \n(or nonmagnetic) ground state phases such as VBS~\\cite{AKLT}, RVB~\\cite{Anderson,Fazekas}, etc.\nAs general characteristics of these systems, \na finite energy gap exists between the ground\nstate and excitation states and the correlation length in the ground state is\nfinite. If a lattice has some inhomogeneities such as impurity sites, \ndefects of periodicity, and edge points, etc., then local magnetic structures are induced around them~\\cite{Sorensen,Pascal,Kennedy,Miyashita}.\nProperty of such induced moments is one of the most interesting\ncurrent topics. \n\nIn this paper we will study the interaction between the\ninduced moments and also dynamical properties of such moments in the \nbond-alternating Heisenberg antiferromagnetic (HAF) chain.\nThis model is one of the simplest systems of the singlet ground state\nwhich consists of singlet states at strong bonds. \nThe magnetic susceptibility and specific-heat of this model \nwas studied in detail as a function of the ratio of the alternation coupling\nconstants by Duffy and Barr~\\cite{Duffy_theory}.\nVarious materials corresponding to this model have been studied experimentally,\ni.e., aromatic free-radical compounds~\\cite{Duffy_exp},\nCu(NO$_3$)$_2\\cdot$2.5H$_2$O~\\cite{Diedrix}, and\n(VO)$_2$P$_2$O$_7$~\\cite{Garrett}, etc.\nThe ratio of the alternation coupling constants for theses materials \nhas been estimated comparing with the theoretical results~\\cite{Duffy_exp}.\nThis system has been also extensively studied concerning with the\nspin-Peirels transitions~\\cite{Bray,Uchinokura,Nishi}. \n\nHere we consider a defect of the alternating order of the bonds, such as\n$\\cdots$ABAB\\underline{AA}BAB$\\cdots$ or $\\cdots$ABA\\underline{BB}ABAB$\\cdots$,\nwhere A or B denotes the strength of bonds (the magnitude of exchange constant). \nAt this defect the singlet dimer state cannot be formed there.\nFirst we study how a magnetic structure appears at these positions. \nIn this paper we call such a defect of the alternating order \n{\\lq}{\\lq}bond impurity{\\rq}{\\rq}. \nFurthermore, the induced magnetic structure depends on whether \nthe edge bond is A or B.\nBond impurity effects in the uniform HAF have been investigated \nin our previous paper~\\cite{Nishino} and we compare the role of \nthe bond impurity in the present model with them.\n\nNext, we focus on what kind of interactions act between these locally induced\nmagnetic structures. We study the force by\ninvestigating the dependence of the ground state energy on the distance\nbetween the impurities for various types of configurations of defects. \nThe force between the bond impurities has been also studied for the\nuniform HAF~\\cite{Nishino}. There we found that the attractive force\nacts. In the present model we again find only attractive force \nregardless of the types of configuration.\n\nDue to the finite correlation length of the present model, the induced \nmoments behave almost independently.\nThus we can regard such a local moment as a microscopic moment.\nBecause recently the technology in microscopic processing makes remarkable\nprogress and the analyses in microscale or nanoscale phenomena have become\npossible, the quantum phenomena in microscale or nanoscale have received\nmuch attention. For example, the resonant tunneling phenomena \nhave been observed in recent experiments on high-spin molecules\n(Mn$_{12}$)~\\cite{Friedman,Thomas,Hernandez} and the concept of quantum\ntunneling of the magnetization (QTM) has become a topic of interest. \nRecently the adiabatic motion of spin $1/2$ was observed in V$_{15}$\nand the effects of thermal disturbance on the dynamics has been discussed~\\cite{Chiorescu}.\nWe have studied the tunneling dynamics from viewpoint of the nonadiabatic\ntransition~\\cite{Miya1,DeRaedt,Miya2,TFE}.\n\nUnder a time-dependent magnetic field, how does that local magnetic structure \nbehave�H We study the response of the\nmagnetization to a sweeping magnetic field. \nAlthough the induced moment consists of several spins, the total\nmagnetization behaves as a single spin when a uniform field is applied.\nUnder a sweeping magnetic field the behavior of magnetization is described\nby the Landau-Zener-St$\\rm{\\ddot{u}}$ckelberg (LZS)\nformula~\\cite{Landau,Zener,Stuckel}. \nIn realistic situations, noise disturbs the simple behavior of LZS \nformula~\\cite{Kaya1,Kaya2}.\nFurthermore, if the noise acts independently at each site, the behavior\nof induced moment (a cluster of spins) is expected to be different from\nthat of isolated single spin. We investigate effects of such individual \nnoise at each site on the dynamical properties.\n\n\nThis paper is organized as follows.\nIn the next section, we explain briefly the method used in this study.\nIn Sect. \\ref{alternate}, effects of bond impurities on the magnetic \nstructure in bond-alternating chains are studied. \nIn Sect. \\ref{force}, we study the force between bond impurities.\nIn Sect. \\ref{response}, we investigate the response of the magnetization\nto a sweeping field.\nSect. \\ref{sec:summary} is devoted to the summary and discussion.\n\n\n\n\\section{Model and Method}\nThe Hamiltonian treated in the present paper is given by\n\\begin{equation}\n{\\cal H}=\\sum_{i} J_{i,i+1}{\\bf S}_{i}\\cdot{\\bf S}_{i+1}, \n\\label{model}\n\\end{equation}\n%\\begin{equation}\n%{\\cal H}=\\sum_{i} J_{i,i+1}{\\bf S}_{i}\\cdot{\\bf S}_{i+1}-\\sum_{i}\n%h_i S_i^z, \n%\\end{equation}\nwhere ${\\bf S}_{i}=(S_i^x, S_i^y, S_i^z)$ are the $S=1/2$ spin operators at\nthe site $i$.\nWe study the bond-alternating chain where $J_{i,i+1}$ changes alternately \namong $J_1$ (a strong bond) and $J_2$ (a weak bond).\nHere we consider defects of the alternation which cause\ninhomogeneities of the lattice (bond impurities).\nTo study low temperature properties of the model, \nwe mainly use the loop algorithm of the continuous time quantum \nMonte Carlo method (LCQMC)~\\cite{loop,cont} with a method of\nspecification of the magnetization $M_z$~\\cite{Pascal,Nishino}.\nThis method overcomes the problem of long autocorrelation in Monte \nCarlo update and allows us to study systems at very low\ntemperatures. \nIn the present work, we performed $10^5$ Monte Carlo steps (MCS) \nfor getting equilibrium of the system and $10^6$ MCS to obtain quantities\nin the equilibrium state. Here a MCS means an update of the whole \nspins.\n\nIn order to study the dynamical response of spins under \na time-dependent external field, we investigate the following system:\n\\begin{equation}\n{\\cal H}=\\sum_{i} J_{i,i+1}{\\bf S}_{i}\\cdot{\\bf S}_{i+1}+\n2\\Gamma\\sum_{i}S_i^x-\\sum_{i}(H(t)+h_i(t))S_i^z, \n\\label{eq_noise}\n\\end{equation}\nwhere \n$H(t)$ is a time-dependent external \nfield and $h_i(t)$ is a random noise applied to each site individually.\n$\\Gamma$ is the transverse field which represents terms for \nquantum fluctuation of $M_z$. \nThe time evolution of the state is obtained by the time-dependent\nSchr$\\rm{\\ddot{o}}$dinger equation \n(TDSE)\n\\begin{equation}\ni\\hbar\\frac{\\partial}{\\partial t}|\\Psi (t) \\rangle={\\cal H}|\\Psi (t)\\rangle, \n\\label{eq:sch}\n\\end{equation}\nwhere $|\\Psi (t)\\rangle$ denotes the wave function of the spin system at\ntime $t$.\nWe set $\\hbar=1$.\nEquation (\\ref{eq:sch}) is solved using the fourth-order fractal\ndecomposition~\\cite{Suzuki}, \n\\begin{equation}\ne^{-it \\cal{H}}=[S_2(-it p_2)]^2 S_2(-it (1-4 p_2)) [S_2(-it p_2)]^2, \n\\end{equation}\nwhere $S_2(x)=e^{x {\\cal H}_1/2}e^{x {\\cal H}_2}e^{x {\\cal H}_1/2}$ \nwith $p_2=(4-4^{\\frac{1}{3}})^{-1}$,\nputting\n\\begin{eqnarray*} \n{\\cal H}_1&=&\\sum_{i} J_{i,i+1}{\\bf S}_{i}\\cdot{\\bf S}_{i+1}+\n2\\Gamma\\sum_{i}S_i^x, \\\\\n{\\cal H}_2&=&-\\sum_{i}(H(t)+h_i(t))S_i^z.\n\\end{eqnarray*}\n\nAs the initial state, \nwe set the applied field to its minimum value $H(t=0)=-H_0<0$, and put\nthe system to be the ground state for this field.\nThen we sweep the filed as \n\\begin{equation}\nH(t)=-H_0+ct. \n\\end{equation}\nBeside this sweeping field, we provide a random noise\nwith an exponential-decaying autocorrelation function \n$\\{h_i(t)\\}$ by a Langevin equation\n(Ornstein-Ulenbeck process),\n\\begin{equation}\n\\dot{h}(t)=-\\gamma h(t) + \\eta(t).\n\\label{Langevin}\n\\end{equation}\nHere $\\eta(t)$ is a white gaussian noise, \n\\begin{equation}\n\\langle \\eta(t) \\rangle=0 \\;\\;\\;{\\rm and}\\;\\;\\;\n\\langle \\eta(0)\\eta(t) \\rangle=A^2\\delta(t),\n\\label{white}\n\\end{equation}\nwhere $A$ is amplitude, $\\gamma$ is damping factor.\nThus obtained random process $h(t)$ has the properties\n\\begin{equation}\n\\langle h(t) \\rangle=0 \\;\\;\\;{\\rm and}\\;\\;\\;\n\\langle h(0)h(t) \\rangle=\\frac{A^2}{2\\gamma}\n{\\rm exp}(-\\frac{t}{\\tau}), \n\\label{rand_noise}\n\\end{equation}\nwhere $\\tau=1/\\gamma$.\nThe dynamics of magnetization is obtained as \n\\begin{equation}\nM(t)=\\langle \\Psi (t)| \\sum_i S_i^z |\\Psi (t) \\rangle.\n\\label{Eq_Mz}\n\\end{equation}\nIf the magnetic field changes rather slowly, i.e., the sweep rate is rather\nsmall, $|\\Psi (t)\\rangle$ changes adiabatically. In this case the state\nstays in the ground \nstate of the system for the current field $H(t)$, \nand the magnetization follows \nthe value of the ground state. When the sweeping rate becomes \nlarge the system cannot follow the change of the field completely,\nand the nonadiabatic transition occurs. \nThe probability to stay in the ground state was given by \nLZS formula as\n\\begin{equation}\np=1-{\\rm exp}(\\frac{-2\\pi \\Gamma^2}{c}).\n\\label{prob}\n\\end{equation}\nWhen the effect of noise does not become negligible, \nthe noise disturbs the quantum process and this probability\nchanges~\\cite{Kaya1}.\n\n\\section{bond impurity in bond-alternating chains}\n\\label{alternate}\n\nWe investigate magnetic structures in the system (\\ref{model}) with a\nbond alternation \n$\\cdots J_{1} J_{2} J_{1} J_{2} \\cdots$, where $J_{1} > J_{2}$. \nWe study effects of a defect in the alternation, such as \n$\\cdots J_{1} J_{2} J_{1} J_{2} \\underline{J_{1} J_{1}} J_{2} J_{1} J_{2}\nJ_{1} \\cdots $.\nBeside this defect, magnetic structures can be induced at the edges\nas naturally understood from the VBS picture, i.e., an unpaired spin\ncauses an induced magnetization (see the figures).\nThus we study the following four systems of\nthe bond configurations:\\\\\n(a) a chain of 63 sites where \nthe two strong bonds are at the center and both edges terminate \nwith a strong bond,\\\\\n($J_{1} J_{2} \\cdots J_{1} J_{2} \\underline{ J_{1} J_{1}}\n J_{2} J_{1} \\cdots J_{2} J_{1} $)\\\\\n(b) a chain of 65 sites where \nthe two strong bonds are at the center and both edges terminate \nwith a weak bond,\\\\\n($J_{2} J_{1} \\cdots J_{1} J_{2} \\underline{ J_{1} J_{1}}\n J_{2} J_{1} \\cdots J_{1} J_{2} $)\\\\\n(c) a chain of 63 sites where \nthe two weak bonds are at the center and both edges terminate \nwith a weak bond, \\\\\n($J_{2} J_{1} \\cdots J_{2} J_{1} \\underline{ J_{2} J_{2}}\n J_{1} J_{2} \\cdots J_{1} J_{2} $)\\\\\nand \\\\\n (d) a chain of 65 sites where \nthe two weak bonds are at the center and the edges terminate \nwith a strong bond\\\\\n($J_{1} J_{2} \\cdots J_{2} J_{1} \\underline{ J_{2} J_{2}}\n J_{1} J_{2} \\cdots J_{2} J_{1}$).\\\\\nHere we take the strong bond to be $J_{1}=1.3$ and the weak bond to be\n$J_{2}=0.7$. For this set of bonds,\nthe correlation length is estimated as $\\xi \\simeq0.82$ (see Appendix).\nHere we take $k_{\\rm B}$ as a unit of energy ($k_{\\rm B}=1$).\nThese four models have odd number of spins and their ground state is\ndoublet according to the Lieb-Mattis theorem~\\cite{Liep}.\nWe performed simulations at $T=0.01$ in the $M_z=1/2$ space to study\nmagnetic structures in the low energy state. \nThe magnetization profiles $\\{m_i\\}$ of (a)-(d) are drawn in Fig.\n\\ref{fig_bondalt_mag}, \nwhere $m_i=\\langle S_i^z\\rangle$ and $\\langle \\; \\rangle$ denotes the\ncanonical average at a given temperature.\nA magnetization is induced locally around the impurity. \nFirst we consider the cases where the bonds at edges are strong, i.e., \nthe model (a) and (d).\nIf we allocate a singlet pair at each strong bond, neighboring two strong bonds remain at the center in the model (a), while\none site remains in the model (d). The magnetization of $M_z=1/2$ is\nassigned in the remaining part at the center of the lattices to induce a\nlocal magnetic structure.\nBecause the edge bonds are strong, no magnetization is induced at the edges. \nFigures \\ref{fig_bondalt_mag}(a) and (d) are considered to describe well the \nmagnetization profiles of the ground state since these are gapful systems\nwithout quasi-degenerate states within a subspace with a fixed\nmagnetization (i.e., $M_z=1/2$). \nWe find that $M_z$ is distributed only into the $\\pm1/2$ space in the simulations at this temperature.\n\nIn the models (b) and (c), \nwhen we allocate singlet state at the strong bond, there are three\npositions for \nmagnetic moments, i.e., the center and both edges.\nIndeed in the both models (b) and (c), magnetizations are \ninduced around the impurity and both edges as shown in Fig. 1. \nIn order to study the distribution of magnetization in the lattice, we\nintroduce \nthe summation of the magnetization per site from the left edge site\n\\begin{equation}\n{\\cal M}_z(j)=\\sum_{i=1}^j m_i.\n\\label{tmag}\n\\end{equation}\nWe show this quantity for the model (b) in Fig. \\ref{fig_bondalt_sum}. \nThere the values of the left plateau and right plateau are 0.166 and 0.333,\nrespectively. \nFrom this figure we find a spin 1/6 locates at each local structure. This\ndeceptive fractional magnetization is considered to come\nfrom mixing of states.\n\nLooking on the local magnetic structure as an effective $S=1/2$ spin\ninteracting by an effective exchange $\\tilde{J}$, \nthis system is modeled by a \nthree-site Heisenberg model ${\\cal H}=\\tilde{J}{\\bf S}_{1}\\cdot{\\bf S}_{2}+\n\\tilde{J}{\\bf S}_{2}\\cdot{\\bf S}_{3}. $\nIn the $M_z=1/2$ space the eigenvalues are \n\\begin{equation}\nE_{1}=-\\tilde{J}, E_{2}=0$, and $E_{3}=\\tilde{J}/2.\n\\label{ground eq}\n\\end{equation}\nThe corresponding eigenvectors are denoted by $|\\phi_{i}\\rangle, (i=1,2, \n{\\rm and} 3)$.\nThe expectation values of magnetization of spins in each state are\n\\begin{eqnarray}\n&& \\langle \\phi_{1}|S_{1}^z|\\phi_{1}\\rangle=\\langle \\phi_{1}|\nS_{3}^z| \\phi_{1}\\rangle=1/3 \\;\\; {\\rm and} \\;\\; \\langle\n\\phi_{1}|S_{2}^z|\\phi_{1}\\rangle=-1/6, \\nonumber \\\\\n&& \\langle \\phi_{2}|S_{1}^z| \\phi_{2}\\rangle=\\langle \\phi_{2}|S_{3}^z| \n\\phi_{2}\\rangle=0 \\:\\:\\;\\;\\;\\; {\\rm and} \\;\\; \\langle\n\\phi_{2}|S_{2}^z|\\phi_{2}\\rangle=1/2, \\\\\n&& \\langle \\phi_{3}|S_{1}^z| \\phi_{3}\\rangle=\\langle \\phi_{3}|\nS_{2}^z| \\phi_{3}\\rangle=\\langle \\phi_{3}|S_{3}^z| \\phi_{3}\\rangle=1/\n6. \\nonumber \n\\end{eqnarray}\nIn the gapped spin system the correlation function decays exponentially\nand the effective coupling between the induced moments is expected to be\nvery small\\cite{Pascal,Kennedy}, i.e., $\\tilde{J} \\ll1$.\nThus these states are considered to\nbe almost degenerate even at this temperature ($T=0.01$),\nalthough there are energy gaps of order $O(\\tilde{J})$ \nbetween the state $|\\phi_{1}\\rangle$, \n$|\\phi_{2}\\rangle$, and $|\\phi_{3}\\rangle$. \nIn such a case the three states appear in the equal\nprobability, and the expectation value of magnetization is given by an equal-weight average in the three state. That is, $\\langle S_{1}^z \\rangle$, $\\langle S_{2}^z\n\\rangle$, and \n$\\langle S_{3}^z \\rangle$ are given \nby $(1/3+0+1/6)/3=1/6$, $(-1/6+1/2+1/6)/3=1/6$, and $(1/3+0+1/6)/3=1/6$,\nrespectively. \nThese values correspond to the observed deceptive fractional magnetization. \n\nIn order to confirm the above modeling we perform the following two\ninvestigations. \nFirst, we investigate a short chain of $L=21$ by an exact diagonalization method\nin order to check that the ground state of the type (b) is represented \nby $| \\phi_{1}\\rangle$ of the three-spin model.\nHere we choose a chain of a shorter \ncorrelation length because the length of the chain is short. \nNamely, we set $J_{1}=2$ and $J_{2}=0.5$, where $\\xi \\ll 1$.\n%Figure \\ref{fig_bondalt_diagmag} (a) shows the magnetization profile in the\n%ground state of this model, and \nFig. \\ref{fig_bondalt_diagmag} shows the\nsummation of the magnetization from the left edge site (Eq. (\\ref{tmag}))\nfor this model.\nThe local magnetization around the left edge is about 1/3 which corresponds to $\\langle \\phi_{1}|S_{1}^z| \\phi_{1}\\rangle$. This causes the left plateau.\nThe local magnetization around the impurity at the center is about $-$1/6 ($\\langle\n\\phi_{1}|S_{2}^z|\\phi_{1}\\rangle$). Therefore the right plateau is 1/6 ($\\langle \\phi_{1}|S_{1}^z| \\phi_{1}\\rangle$ + $\\langle\\phi_{1}|S_{2}^z|\\phi_{1}\\rangle$). \nFinally adding the local magnetization around the right edge (1/3), ${\\cal M}_z(21)$ \nterminates at 1/2.\nThus the ground state $| \\phi_{1}\\rangle$ represents well that of\nthe type (b) model.\n\nSecond, to confirm the quasi-degeneracy in the model (b), we check the \n distribution of $M_z$ in the Monte Carlo simulation, which is shown in\nFig. \\ref{fig_bondalt_prob}. \nHere the distribution for $M_z=3/2$ is about 12.5 $\\%$. \nBecause there are three states in the $M_z=1/2$ space and one state in the\n$M_z=3/2$ space in the three-spin model, this distribution indicates that\nthese four states are equally populated and that $T=0.01$ is much higher\nthan the energy gaps between these four states. \nIn the three-spin model the eigenvalue of the state of $M_z=3/2$ is\n$\\tilde{J}/2$, while the eigenvalues of the state of $M_z=1/2$ are\n$-\\tilde{J}$, 0, and $\\tilde{J}/2$ (Eq. (\\ref{ground eq})).\nIn the $M_z=3/2$ space we observe $S=1/2$ moment at each local structure \n%as shown in Figs. \\ref{fig_mag_Mz3/2}(a) and (b), \nwhich represents the state \n$|+++\\rangle$ in the effective three-site model. The values $\\{m_i\\}$ are\nalmost three times as large as those of Fig. \\ref{fig_bondalt_mag} (b).\nIn principle we can obtain the energy gap from the temperature \ndependence of the distribution~\\cite{Pascal}. \nHowever it is too small to detect here.\n\nWe find a similar scenario for the model (c).\nThus we conclude that in bond-alternating systems, \nlocal magnetic structures are induced by a bond impurity or weak edge \nbonds as an effective $S=1/2$ spin and they behave almost\nindependently.\n\nNow let us examine more detailed structures of the local magnetic structures.\n In the case (a) a negative magnetization appears at the middle site, while\nin the case (d) a positive magnetization appears there.\nThe interaction of the three spins at the center of the model (a) is \napproximately represented by the three-spin model with the strong bonds. \nHere the magnetizations at the center of the model (a) are distributed as\nabout (1/3, $-1/6$,1/3). \nOn the other hand in the model (d) a spin at the center is isolated from\nthe others.\n\n%Finally we consider the quasi-degenerate structure of the model (b)\n%(similarly (c)).\n%We found that the impurity-induced local structures in the bond-alternating\n%system can be well described by the effective spin model with small\n%$\\tilde{J}$.\n%In order to break a singlet pair a finite energy is necessary which causes the \n%gapful nature of the model. The degree of freedom of induced structures \n%without breaking singlet pairs causes the quasi-degenerate states.\n%In the Haldnae system in the open boundary condition, \n%the interaction between the edge magnetic moments brings the singlet and\n%triplet states which is called Kennedy triplet at the low\n%energy~\\cite{Kennedy,Miyashita}.\n%In the $S=1$ Haldane systems with $S=1/2$ impurities, doping impurities\n%causes a quasi degenerate low energy strucuture~\\cite{Pascal}.\n%Similar situation exists in the models (b) and (c).\n%In these cases, according to the above mentioned three spin model, two\n%doublets (S=1/2) and one quartet (S=3/2) state lie at the low energy as\n%shown in Fig. \\ref{fig_low_energy}. \n\nThe local magnetic structures in the present model are well isolated. Therefore \nwe can locate such magnetic structures as we desire.\nIn Fig. \\ref{five_moment} we show a magnetization profile which\nhas five positions of induced structures in a chain of $L=101$.\nThis system has three $J_2J_2$ defects and weak edge bonds.\nThis configuration is obtained at a very low temperature ($T=0.01$) in the space of\nthe total magnetization $M_z=1/2$. Each moment almost behaves\nindependently. Thus magnetization at each location \nis given by an average over all the possible states. \nIn the present case, each local moment has a deceptive magnetization $1/10$. \n%$\\langle S_i \\rangle$=1/10 ($i=1-5$).\nFurthermore we confirmed that the distribution of magnetization $P(M_z)$ is\ngiven \nby a binomial distribution \n\\begin{equation}\nP(M_z) = \\frac{1}{2^5} \\frac{5\\: !}{(5-2|M_z|) \\: ! \\: (2|M_z|)\\: ! }.\n\\end{equation}\n\n%In conclusion, quasi-degenerate states appears at the low energy in the bond-\n%alternating chain due to the interaction between the local magnetic structures\n%induced by the bond impurity and the edge structures.\n\n\\section{Force between two defects in an alternating chain}\n\\label{force}\n\nWhen a bond-alternating system has impurities, what kind of interaction\ndoes exist between them, attractive or repulsive? We study the force between the bond impurities in this section.\nWe estimate the ground state energies of the spin system in various fixed\nconfigurations of bonds and \ncompare the ground state energies as a function of the distance between the\nbond impurities. \n%Namely, we study effective interaction between the impurities \n%due to the spin interaction.\n\nIn the alternating chain ($ \\cdots J_{1} J_{2} J_{1} J_{2} \\cdots$ ), \nthe system may have a pair of defects by shifting a position \nof a strong bond by one. \n\\begin{equation}\n\\cdots J_{1} J_{2} \\underline{J_{1} J_{1} J_{2} J_{2} } J_{1} J_{2} \n\\cdots .\n\\label{ichi}\n\\end{equation}\nIf we shift the position furthermore, the system has a configuration \n\\begin{equation}\n\\cdots \\underline{J_{1} J_{1}} J_{2} J_{1} \\underline{ J_{2} J_{2} } \n\\cdots , \n\\label{ni}\n\\end{equation}\netc. We study dependence of the energy on the distance ($\\Delta_a$) between\nthe positions of $J_{1} J_{1}$ and $J_{2} J_{2}$. \n We define $\\Delta_a=0$ for the case of no defect, $\\Delta_a=1$ for the\nconfiguration (\\ref{ichi}), $\\Delta_a=2$ for the configuration (\\ref{ni})\nand so on. \nIn Fig. \\ref{fig_force_alt_diag} (a)\nwe plot the ground state energy as a function of $\\Delta_a$\nobtained by an exact diagonalization for $L=24$ with $J_1=2$ and \n$J_2=1$ in the periodic boundary condition (PBC). \nThe ground state energy becomes larger as $\\Delta_a$ becomes larger.\nTherefore we find an attractive force between the impurities ($J_{1} J_{1}$\nand $J_{2} J_{2}$).\n\nNext another situation is considered.\nIf one $J_2$ is exchanged by $J_1$ in the alternating chain, \nthe system has a configuration\n\\begin{equation}\n\\cdots J_{1} J_{2} \\underline{J_{1} J_{1} J_{1}} J_{2} J_{1} \n\\cdots .\n\\label{stand}\n\\end{equation}\nWe define $\\Delta_b= 0$ for this configuration.% (\\ref{stand}).\n Shifting a position of a $J_{1}$ by one, the system has \ntwo $J_{1} J_{1}$ pairs and has a configuration, \n\\begin{equation}\n\\cdots\\underline{J_{1} J_{1} } J_{2} \\underline{J_{1} J_{1} }\n\\cdots .\n\\label{one_shf}\n\\end{equation}\nWe define this distance between $J_{1} J_{1}$ and $J_{1} J_{1}$\nas $\\Delta_b= 1$.\nFurthermore, $\\Delta_b=2$ is defined for the configuration (\\ref{two_shf})\nand etc.\n\\begin{equation}\n\\cdots\\underline{J_{1} J_{1} } J_{2} J_{1} J_{2}\\underline{J_{1} J_{1} }\n\\cdots .\n\\label{two_shf}\n\\end{equation}\nThe ground state energies for a PBC chain of $L=24$ are shown as a function of $\\Delta_b$ in \nFig. \\ref{fig_force_alt_diag} (b). \nWe also study the interaction \nbetween $J_{2} J_{2}$ and $J_{2} J_{2}$, where we define the distance between \nthe impurities in the same way.\nWe again found \n%We plot the ground state energies as a function of $\\Delta_c$ in Fig.\n%\\ref{fig_force_alt_diag} (c). \n%In all cases, we observe \nthat the ground state energy increases as the distance becomes larger.\nThus it has been found that \nan attractive force acts between the impurities regardless of their type.\n\nIf we allow positions of impurities to move, the distance between\nimpurities would be distributed in the canonical distribution \naccording to the interaction between the impurities in the thermal equilibrium \nat a given temperature as has demonstrated in the previous paper~\\cite{Nishino}.\n\n\n\n\\section{responce of the local magnetic structure to the dynamical field}\n\\label{response}\n\nIn this section we investigate dynamical response of the\nlocal magnetic structure (effective $S=1/2$ spin) induced by a bond\nimpurity to a time-dependent magnetic field with a random noise\nin the dynamical model (\\ref{eq_noise}).\n\nFirst let us consider the response of the magnetization of a free $S=1/2$ spin to \na sweeping field.\nThe Hamiltonian is given by\n\\begin{equation}\n{\\cal H}(t)=2\\Gamma S_x-(-H_0+ct) S_z.\n\\label{free_spin}\n\\end{equation}\nThis system is two-level system whose energy levels are shown in\nFig.~\\ref{level_cross} as a function of $H$, where the avoided level\ncrossing occurs near $H=0$.\nIf $H_0 \\gg \\Gamma >0$, the ground state consists primarily of the\n$M_z=-1/2$ state at $t=0$.\nThe probability for the system to end up in the \n$M_z=1/2$ state (i.e., the probability to change its magnetization) at\n$t=\\infty$ is \ngiven by Eq. (\\ref{prob})\n\nNext we consider the case of a local magnetic structure (effective $S=1/2$)\non a lattice.\nWhen the noise $\\{h_i(t)\\}$ does not exist, dynamics of the system is\ngenerally the same as that of a free $S=1/2$ spin as far as we concern the\nstates in an $S=1/2$ space. \nIt is easily understood as follows.\nAdopting $\\{ |+\\rangle, |-\\rangle\\}$ as the basis set, the matrix\nrepresentation of Eq. (\\ref{free_spin}) is \n\\begin{equation}\n{\\cal H} = \\pmatrix{\n -\\frac{H}{2} & \\Gamma \\cr\n \\Gamma & \\frac{H}{2} \\cr\n }.\n\\end{equation}\nNoting that $\\sum_{i} J_{i,i+1}{\\bf S}_{i}\\cdot{\\bf S}_{i+1}$ and\n$2\\Gamma\\sum_{i}S_i^x-H(t)\\sum_{i}S_i^z$ commute, the matrix representation\nof Eq. (\\ref{eq_noise}) in the total $S=1/2$ space is given by\n\\begin{equation}\n{\\cal H} = \\pmatrix{\n -\\frac{H}{2}+{\\rm const.} & \\Gamma \\cr\n \\Gamma & \\frac{H}{2}+{\\rm const.} \\cr \n },\n\\label{lowest_two}\n\\end{equation}\nadopting $\\{ |S^{\\rm tot}=1/2, M_z=1/2 \\rangle, |S^{\\rm tot}=1/2,M_z=-1/2\n\\rangle\\}$ as the basis set.\nThe constant in Eq. (\\ref{lowest_two}) does not depend on $H(t)$. \nThus the dynamics is independent of the number of sites and the combination\nof $\\{J_{ij}\\}$.\n\nIn the experimental situation, however, noise usually has an influence on\nthe system. \nIf a random noise is applied to each site individually, the term\n$-\\sum_{i}(H(t)+h_i(t))S_i^z$ does not commute with $({\\bf S}^{\\rm\ntot})^2=(\\sum {\\bf S}_i)^2$, and therefore the dynamics would be changed\nwhen the interaction $\\{J_{ij}\\}$ and the system size are varied.\n\nHere for the study of dynamical properties we adopt the minimum model of the type (d) because we can treat a limited number of spins in the scheme of Eq.\n(\\ref{eq:sch}). This model consists of five sites and we take $J_1=1.0$ and\n$J_2=0.5$. This is the minimum bond-alternating model with a\n$J_2J_2$ bond impurity. \nThis system is looked on as an effective $S=1/2$ spin. \n\nIt would be expected that\nthe effect of noise is reduced as the system size becomes large and \na local magnetic structure (effective $S=1/2$ spin) consisting of several\nnumber of spins is less sensitive to such a random noise \nthan a free $S=1/2$ spin.\nIn order to study effects of the noise we investigate the following properties.\n\nFirst we study the broadening of the level due to the noise. \nA energy gap and a sweeping rate at the level crossing point are important\nfactors to determine the transition probability under a sweeping uniform field\n(see Eq. (\\ref{prob})).\nIt would be useful to investigate how the noise influences the energy gap at the\nlevel crossing point.\nFor a single spin the energy gap is given by\n\\begin{equation}\n\\Delta E=\\sqrt{(2\\Gamma)^2 \\; + \\; (h(t))^2}\n\\end{equation}\nfor $H(t)=0$.\nThus using the distribution of $h(t)$ (a gaussian distribution with the\nvariance \n$A^2/2\\gamma$ in Eq. (\\ref{rand_noise})), we can calculate the mean\n$\\langle \\Delta E \\rangle$ and the width $\\delta E=\\sqrt{ \\langle (\\Delta\nE)^2 \\rangle- \\langle \\Delta E \\rangle^2}$, \nwhich are listed in Table \\ref{Delta_E}.\nFor the local magnetic structure, we calculate the energy gap in a noise \nby a perturbation method. That gives \n\\begin{eqnarray}\n\\Delta E &=&2\\Gamma + \\frac{2 |\\langle \\phi_1^{(0)} | V_1 | \\phi_2^{(0)}\n\\rangle|^2}{E_2^{(0)}-E_1^{(0)}} \\\\\n&+& {\\rm contributions \\; from \\; higher \\; levels}, \\nonumber\n\\end{eqnarray}\nwhere $E_1^{(0)}$ and $\\phi_1^{(0)}$ ($E_2^{(0)}$ and $\\phi_2^{(0)}$) are the \neigenstate of the ground state (the first exited state) of the non \nperturbed hamiltonian. The term $V_1$ is $\\sum_i h_i S_i^z$.\nNoting that \n\\begin{equation}\n\\langle \\; |\\langle \\phi_1^{(0)} | V_1 | \\phi_2^{(0)} \\rangle|^2 \\rangle = \n\\sum_i \\langle h_i^2 \\rangle |\\langle \\phi_1^{(0)} | S_i^z | \\phi_2^{(0)}\n\\rangle |^2,\n\\end{equation}\nwe can calculate $\\langle \\Delta E \\rangle$ from the \ndistribution of $\\{h_i\\}$, which are also listed in Table \\ref{Delta_E} for \n$L=5$ and $L=9$.\nFor large values of $A$, we obtain the energies by diagonalizing the \nhamiltonian Eq. (\\ref{eq_noise}) and obtained $\\langle \\Delta E \\rangle$\nand $\\delta E$ numerically from 500 samples of $\\{h_i\\}$, which are also\nlisted in Table \\ref{Delta_E}.\nThus we find the noise causes almost the same effect on the broadening of the \nenergy levels at $H(t)=0$ in both systems, i.e., a \nsingle spin and local magnetic structures, which is not in accordance with\nthe above expectation.\n\nSecond, we investigate the dynamical properties of the both systems. \nIn particular, we study the time evolution of magnetization under the \nsweeping field.\nWe set parameters\n$\\Gamma=0.02$, $H_0=0.5$, $c=0.0005$, $t_{\\rm max}$=2000, and \n$dt=0.01$.\nIn this parameter set the probability $p$ in Eq. (\\ref{prob}) is nearly 1.\nAs was mentioned, the responses of the magnetization \nare the same in both systems when a random noise is not applied. \nTime evolution without noise is shown by thin dotted lines in Figs.\n\\ref{comp_mag} \n(a)-(c). \n\nNext we investigate the dynamics with the random field (Eq. (\\ref{Langevin})).\nWe observe the two cases changing the amplitude $A$ of noise (Eq.\n(\\ref{white})); (a) $A=0.01$ and (b) $A=0.02$. The value of\n$\\gamma$ is fixed at 0.1.\nThe lowest two energy levels of the system with a\nrandom noise $\\{h_i(t)\\}$ for case (b) are illustrated as a function of\ntime in Fig. \\ref{level_noise}.\n% It should be noted that under a random noise the dynamics does not obey simple LZS transition.\n\nFigure \\ref{comp_mag} (a) shows $M(t)$ in the noise of a small amplitude\n$(A=0.01)$. \nThe transition probabilities for \n a local magnetic structure (effective $S=1/2$ spin) and an $S=1/2$ free \nspin are very close to each other at this small amplitude and the\nprobabilities are reduced \nby a little amount from the probability of the pure system (i.e.,\n$h_i(t)=0$). Increasing the amplitude to $A=0.02$ (Fig. \\ref{comp_mag}\n(b)), reduction of $M(t)$s increases. \nFurthermore we find that $M(t)$ for the local magnetic structure \nremains larger than that of a free spin. \nThe errorbar shows the first standard deviation of the distribution of $\\{M(t)\\}$.\nHere we find that the distribution is rather wide but the mean of the distribution \nis definitely different.\nThe reduction of $M(t)$ by field fluctuation corresponds to the reduction\nof $p$ in Eq. (\\ref{prob}).\nThe reduction of $p$ has already pointed out for a single spin\nby Y. Kayanuma and H. Nakayama~\\cite{Kaya1}. \nOur results are qualitatively consistent with their results. \nFurthermore, the present observation indicates that a local magnetic structure induced by an\nimpurity is less sensitive to the noise. \n%We conclude that \n%the transition probability $p$ is different between a local magnetic \n%structure (effective $S=1/2$ spin) and a free $S=1/2$ spin under a sweeping\n%field with a random noise in the individual site. \nThis feature may be due to this bulky structure of the \nlocal magnetic structure. From this observation we may expect that local\nmagnetizations are easier to manipulate than single spins in a field with\nnoise. \nDetail analysis for the noise dependence will be reported elsewhere.\n\n\\section{Summary and Discussion}\n\\label{sec:summary}\n\nIn the bond-alternating ($S=1/2$) Heisenberg antiferromagnetic chain,\nwe studied properties of \nlocally induced magnetic structures by inhomogeneities of the lattice\nsuch as the defect of alternation or the edge of the lattice.\nBecause of the gapful nature, the role of the inhomogeneities is\nsimilar to that in the Haldane systems and is very different from that in the\nuniform chain. \nIn the uniform chain we found that the bond impurities divide the system into \ndomains.\nWe showed that a local magnetic structure induced by a bond impurity can be\nlooked on as an effective $S=1/2$ spin. \nThe interaction between the local magnetic moments \ndecays exponentially in the present model and \nthe local magnetic structures behave\nalmost independently even at very low temperatures.\n\nWe also studied the force acting between these defects of alternation. \nThere are several configurations of defects. We investigated the forces between $J_1J_1$ and $J_1J_1$, between $J_2J_2$ \nand $J_2J_2$, and between $J_1J_1$ and $J_2J_2$.\nFurthermore, there are two types of separations of defects, i.e.,\nthe type of (\\ref{ichi}) and (\\ref{stand}). \nIt turned out that the forces are attractive for all cases.\n\nWe also studied the dynamical response of the magnetization to \na sweeping field. The induced local moment behaves as a single spin\neven in dynamical property as far as the uniform field is applied. \nWe considered what property is different between the local magnetic\nstructure (effective $S=1/2$ spin) induced by an impurity and a free\n$S=1/2$ spin. In realistic situation there exists noise which is not\nnecessarily uniform but different from site to site.\nIt has been found that even when noise is applied individually at each site,\nthe total effect of the noise on the\nbroadening of the energy levels is almost the same as that of the \nsingle spin.\nHowever, the dynamical response of the magnetization to a\ntime-dependent magnetic field with a random noise is found to be different\nfrom that of a single spin.\nThat is, the local magnetic structure is found to be more robust against noise \napplied at each site individually.\n\nThe present study is the first attempt treating the dynamics of such a local\nmagnetic structure. We hope that this work \nprovides a basic information for the manipulation \nof microscopic or nanoscale magnetic devices in the future. \nWe also hope that the present study helps to analyze the magnetic properties \nat low temperatures such as observed by NMR measurement.\n\n\\acknowledgements\n\nThe present authors would like to thank Professor Jean-Paul Boucher\nfor his valuable and encouraging discussion.\nThe present work was supported by Grant-in-Aid for Scientific\nResearch from Ministry of Education, Science, Sports and\nCulture of Japan.\nM. N. was also supported by the Research Fellowships of the Japan \nSociety for the Promotion of Science for Young Scientists.\n\n\\appendix\n\n\\section{The correlation length of bond-alternating chain ($S=1/2$)}\n\\label{app}\n\nWe determined the correlation length of HAF bond-alternating chain\n($S=1/2$) in the ground state as a function of the ratio $J_1/J_2$. \nWe calculated the correlation function for various ratios ($J_1/J_2$) and\nestimated the correlation length using the relation \n\\begin{equation}\n{\\rm ln} \\langle S_i^z S_j^z \\rangle = {\\rm constant} - \\frac{r_{ij}}{\\xi},\n\\end{equation} \nwhere $r_{ij}$ denotes the distance between site $i$ and site $j$, and\n$\\xi$ denotes the correlation length of the system.\nWe treated periodic chains with 60 sites and \nsimulations were performed at $T=0.01$ in the $M_z=0$ space. \nWe show the correlation lengths for various ratios ($J_1/J_2$) in Table\n\\ref{corr_length}. Here we adopt the length of a pair of bonds $J_1$\nand $J_2$ to be a unit length, \ni.e., this unit is double of the bond length.\n\n\\begin{references}\n\n\\bibitem{Haldane}\nF. D. M. Haldane, Phys. Lett. {\\bf 93A}, 464 (1983); Phys. Rev. Lett. {\\bf 50}, 1153 (1983).\n\n\\bibitem{MG}\nC. K. Majumdar and D. K. Ghosh, J. Math Phys. {\\bf 10}, 1388 (1969).\n\n\\bibitem{AKLT}\nI. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. {\\bf 59},\n799 (1987); Commun. Math. Phys. {\\bf 115}, 477 (1988).\n\n\\bibitem{Anderson}\nP. W. Anderson, Mater. Res. Bull. {\\bf 8}, 153 (1973).\n\n\\bibitem{Fazekas}\nP. Fazekas and P. W. Anderson, Philos. Mag. {\\bf 30}, 423 (1974).\n\n\\bibitem{Sorensen}\nE. S. Sorensen and I. Affleck, Phys. Rev. B {\\bf 51}, 16115 (1995).\n\n\\bibitem{Pascal}\nP. Roos and S. Miyashita, Phys. Rev. B {\\bf 59}, 13782 (1999).\n\n\\bibitem{Kennedy}\nT. Kennedy, J. Phys. Condens. Matter {\\bf 2}, 5737 (1990).\n\n\\bibitem{Miyashita}\nS. Miyashita and S. Yamamoto, Phys. Rev. B {\\bf 48}, 913 (1993).\n\n%theory of alternating chain\n\\bibitem{Duffy_theory}\nW. Duffy and K. Barr, Phys. Rev. {\\bf 165}, 647 (1968).\n\n%\\bibitem{Bonner}\n%J. C. Bonner, H. W. Bl$\\ddot{o}$te, J. W. Bray, and I. S. Jacobs, J. Appl.\nPhys. {\\bf 50}, \n%1810 (1979).\n\n%experiments of alternating chain, aromatic free radical\n\\bibitem{Duffy_exp}\nW. Duffy and D. L. Dtrandburg, J. Chem. Phys. {\\bf 46}, 456 (1967) and\nreference therein.\n\n%Cu(NO3)2.5H20\n\\bibitem{Diedrix}\nK. M. Diedrix, H. W. J. Bl$\\ddot{o}$te, J. P. Groen, T. O. Klassen, and N.\nJ. Poulis, Phys. Rev. B \n{\\bf 19}, 420 (1979).\n\n%(VO)2P2O7\n\\bibitem{Garrett}\nA. W. Garrett, S. E. Nagler, D. A. Tennant, B. C. Sales, and T. Barnes,\nPhys. Rev. Lett. {\\bf 79}, 745 (1997).\n\n%Spin-Pierce\n\\bibitem{Bray}\nJ. W. Bray, H. R. Hart, Jr., L. V. Interrante, I. S. Jacobs,\nJ. S. Kasper, G. D. Watkins and S. H. Wee:\nPhys. Rev. Lett. {\\bf 35}, 744 (1975).\n\n\\bibitem{Uchinokura}\nM. Hase, I. Terasaki, Y. Sasago and K. Uchinokura:\nPhys. Rev. Lett. {\\bf 71}, 4059 (1993).\n\n\\bibitem{Nishi}\nM. Nishi, O. Fujita and J. Akimitsu:\nPhys. Rev. B{\\bf 50}, 6508 (1994).\n\n\\bibitem{Nishino}\nM. Nishino, H. Onishi, P. Roos, K. Yamaguchi, and S. Miyashita, Phys. Rev.\nB {\\bf 61}, 4033 (2000).\n\n\\bibitem{Friedman}\nJ. R. Friedman, M. P. Sarachik, T. Tejada, and R. Ziolo, Phys. Rev. Lett.\n{\\bf 76}, 3830 (1996).\n\n\\bibitem{Thomas}\nL. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara,\nNature ( London) {\\bf 383}, 145 (1996).\n\n\\bibitem{Hernandez}\nJ. M. Hernandez, X. X. Zhang, F. Luis, T. Tejada, J. R. Friedman, M. P.\nSarachik, and R. Ziolo, Phys. Rev. B {\\bf 55}, 5858 (1997).\n\n\\bibitem{Chiorescu}\nI. Chiorescu, W. Wernsdorfer, A. M\\\"{u}ller, H. B\\\"{o}gge, and B. Barbara,\nPhys. Rev. Lett. {\\bf 84} 3454 (2000).\n\n\\bibitem{Miya1}\nS. Miyashita, J. Phys. Soc. Jpn. {\\bf 64}, 3207 (1995).\n\n\\bibitem{DeRaedt}\nH. De Raedt, S. Miyashita, K. Saito, D. Garc\\'{\\i}a-Pablos and \nN. Garc\\'{\\i}a, Phys. Rev. B {\\bf 56}, 11761 (1997).\n\n\\bibitem{Miya2}\nS. Miyashita, K. Saito, and H. De Raedt, Phys. Rev. Lett. {\\bf 80}, 1525 (1998).\n\n\\bibitem{TFE}\nK. Saito and S. Miyashita, cond-mat/0004027.\n\n\\bibitem{Landau}\nL. Landau, Phys. Z. Sowjetunion {\\bf 2}, 46 (1932).\n\n\\bibitem{Zener}\nC. Zener, Proc. R. Soc. London, Ser. A {\\bf 137}, 696 (1932).\n\n\\bibitem{Stuckel}\nE. C. G. St$\\rm{\\ddot{u}}$ckelberg, Helv. Phys. Acta {\\bf 5}, 369 (1932).\n\n\\bibitem{Kaya1}\nY. Kayamuma, Phys. Rev. B {\\bf 47}, 9940 (1998).\n\n\\bibitem{Kaya2}\nY. Kayanuma and H. Nakayama, Phys. Rev. B {\\bf 57}, 13099 (1993); \nJ. Phys. Soc. Jpn. {\\bf 54}, 2037 (1985).\n\n\\bibitem{loop} \nN. Kawashima and J. E. Gubernatis,\nJ. Stat. Phys. {\\bf 90} 169 (1995) and reference therein;\\\\\nH. G. Evertz, cond-mat/9707221 and reference therein.\n\n\\bibitem{cont}\nB. B. Beard and U. -J. Wiese: Phys. Rev. Lett. {\\bf 77} 5130 (1996).\n\n%\\bibitem{onishi}\n%H. Onishi, M. Nishino, N. Kawashima, and S. Miyashita, J. Phys. Soc. Jpn.\n%{\\bf 68} 2547 (1999). \n\n\\bibitem{Suzuki}\nM. Suzuki, Phys. Lett. A {\\bf 146}, 319 (1990).\n\n\\bibitem{Liep}\nE. Lieb and D. Mattis, J. Math. Phys. {\\bf 3}, 749 (1962).\n\n%\\bibitem{Haldane}\n%F. D. M. Haldane, Phys. Lett. {\\bf 93A}, 464 (1983); Phys. Rev. Lett. {\\bf\n%50}, 1153 \n%(1983).\n\n%\\bibitem{Hagiwara}\n%M. Hagiwara, K. Katsumata I. Affleck, B. I. Halperin and J. P. Renard, \n%Phy. Rev. Lett. {\\bf 65}, 3181 (1990).\n\n\\end{references}\n\n\\begin{figure}\n\\caption{\nMagnetization profiles $\\{m_i\\}$ of the models (a)-(d) at $T=0.01$ in the\n$M_{z}=1/2$ space.\n(a) and (c) contain 63 sites ($L=63$) and (b) and (d) contain 65 sites ($L=65$) .\nThe diamonds denote the strength of bonds $\\{J_{i,i+1}\\}$, those at high\npositions \ndenote $J_1=1.3$ and those at the low positions denote $J_2=0.7$.\nDetails are shown in text.\n}\n\\label{fig_bondalt_mag}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\nSummation of magnetization per site from the\nleft edge site for the model (b).\n}\n\\label{fig_bondalt_sum}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Summation of magnetization per site from the left edge site for a bond-alternating chain of the type (b) in Fig. \\ref{fig_bondalt_mag} with $L=21$.\nThe strong and weak bonds are $J_{1}=2$ and $J_{2}=0.5$, respectively.}\n\\label{fig_bondalt_diagmag}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Distribution of $M_{z}$ for the model (b) in Fig.\n\\ref{fig_bondalt_mag}.\n}\n\\label{fig_bondalt_prob}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Magnetization profile $\\{m_i\\}$ for a model which has five\npositions of induced \nmoments in the $M_z=1/2$ space at $T=0.01$ with $L=101$.\nThe diamonds denote the strength of bonds $\\{J_{i,i+1}\\}$, those at high\npositions \ndenote $J_1=1.3$ and those at the low positions denote $J_2=0.7$.}\n\\label{five_moment}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\nGround state energy as a function\nof (a) $\\Delta_a$ and (b) $\\Delta_b$ obtained by an exact diagonalization ($L=24$, PBC).\n}\n\\label{fig_force_alt_diag}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Avoided level crossing.\n}\n\\label{level_cross}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\nTime evolution of magnetization.\nThin dotted line denotes the magnetization without noise.\nThick line denotes the magnetization of a local magnetic structure \n(effective $S=1/2$ spin) of $L=5$ with a random noise (averaged over 500 samples \n$\\{h_i\\}$).\nThick dotted line denotes the magnetization of a free $S=1/2$ \nspin with a random noise (averaged over 500 samples \n$\\{h_i\\}$). \n(a) $A=0.01, \\gamma=0.1$, (b) $A=0.02, \\gamma=0.1$.\n}\n\\label{comp_mag}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Energy structure of the two level system with a random \nnoise for $A=0.02, \\gamma=0.1$.\n}\n\\label{level_noise}\n\\end{figure}\n\n\n\\newpage\n\\begin{table}\n\\caption{Average energy gaps $\\langle \\Delta E \\rangle$ and their widths\n$\\delta E$ for several values of $A/\\sqrt{2\\gamma}$.\n(1), (5), and (9) mean a single free spin, the minimum model consisting of\n5 spins, and the next minimum model consisting of 9 spins of the type (d)\nin Fig. \\ref{fig_bondalt_mag}, \nrespectively.\nThe subscript p denotes that the quantities are obtained by the\nperturbation method and the subscript d denotes that the quantities are\nobtained by the exact diagonalization method.\n}\n\\label{Delta_E}\n\\end{table}\n\n\\begin{table}\n\\caption{Correlation length $\\xi$ as a function of the ratio of the \n$J_1/J_2$ in the bond-alternating chain ($S=1/2$)}\n\\label{corr_length}\n\\end{table}\n\n\\end{document}\n\n"
},
{
"name": "table.tex",
"string": "\\documentstyle{article}\n\\begin{document}\n\\pagestyle{empty}\nTable I \n\n\\vspace{1cm}\n\n\n\\def\\arraystretch{2}\n\\begin{center}\n\\hspace{-3.9cm}\n\\begin{tabular}{ccccccccc}\n\\hline\n$A/\\sqrt{2\\gamma}$ & $\\Delta E_{\\rm d} ({\\rm 1})$ & $\\delta E_{\\rm d} ({\\rm 1})$ & \n$\\Delta E_{\\rm p} ({\\rm 5})$ & \n$\\Delta E_{\\rm d} ({\\rm 5})$ & $\\delta E_{\\rm d} ({\\rm 5})$ & \n$\\Delta E_{\\rm p} ({\\rm 9})$ & \n$\\Delta E_{\\rm d} ({\\rm 9})$ & $\\delta E_{\\rm d} ({\\rm 9})$ \\\\\n\\hline\n0.01 &\t0.0412 & 0.0016 & \n0.0413 &\n0.0412 & 0.0016\t& \n0.0413 &\n0.0413\t& 0.0017 \\\\\n0.02 &\t0.0445 &\t0.0056 & \n0.0450 &\n0.0443\t& 0.0054\t&\n0.0450 &\n0.0446 & 0.0058 \\\\\n0.03 &\t0.0486 &\t0.0105 & \n0.0513 &\n0.0492\t& 0.0111\t&\n0.0514 &\n0.0492\t& 0.0112\\\\\n0.04 &\t0.0542 &\t0.0157 & \n0.0601 &\n0.0544\t& 0.0169\t&\n0.0602 &\n0.0546\t& 0.0171\\\\\n0.05 &\t0.0609 &\t0.0222 & \n0.0714 &\n0.0591 & 0.0213 & \n0.0716 &\n0.0605 & 0.0233 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{1cm}\n\nTable II\n\n\\vspace{1cm}\n\n\\begin{center}\n\\begin{tabular}{c|ccccccccc}\n\\hline\n$J_1/J_2$ & 1.2 & 1.3 &1.4 &1.5 &1.6 &1.7 &1.8 &1.9 &2.0 \\\\\n\\hline\n$\\xi$ & 2.02 & 1.51 & 1.31 & 1.18 & 1.01 & 0.93 & 0.84 & 0.80 & 0.77 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\n\n\n\n\n\n\n\n\n\\end{document}"
}
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[
{
"name": "cond-mat0002082.extracted_bib",
"string": "\\bibitem{Haldane}\nF. D. M. Haldane, Phys. Lett. {\\bf 93A}, 464 (1983); Phys. Rev. Lett. {\\bf 50}, 1153 (1983).\n\n\n\\bibitem{MG}\nC. K. Majumdar and D. K. Ghosh, J. Math Phys. {\\bf 10}, 1388 (1969).\n\n\n\\bibitem{AKLT}\nI. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. {\\bf 59},\n799 (1987); Commun. Math. Phys. {\\bf 115}, 477 (1988).\n\n\n\\bibitem{Anderson}\nP. W. Anderson, Mater. Res. Bull. {\\bf 8}, 153 (1973).\n\n\n\\bibitem{Fazekas}\nP. Fazekas and P. W. Anderson, Philos. Mag. {\\bf 30}, 423 (1974).\n\n\n\\bibitem{Sorensen}\nE. S. Sorensen and I. Affleck, Phys. Rev. B {\\bf 51}, 16115 (1995).\n\n\n\\bibitem{Pascal}\nP. Roos and S. Miyashita, Phys. Rev. B {\\bf 59}, 13782 (1999).\n\n\n\\bibitem{Kennedy}\nT. Kennedy, J. Phys. Condens. Matter {\\bf 2}, 5737 (1990).\n\n\n\\bibitem{Miyashita}\nS. Miyashita and S. Yamamoto, Phys. Rev. B {\\bf 48}, 913 (1993).\n\n%theory of alternating chain\n\n\\bibitem{Duffy_theory}\nW. Duffy and K. Barr, Phys. Rev. {\\bf 165}, 647 (1968).\n\n%\n\\bibitem{Bonner}\n%J. C. Bonner, H. W. Bl$\\ddot{o}$te, J. W. Bray, and I. S. Jacobs, J. Appl.\nPhys. {\\bf 50}, \n%1810 (1979).\n\n%experiments of alternating chain, aromatic free radical\n\n\\bibitem{Duffy_exp}\nW. Duffy and D. L. Dtrandburg, J. Chem. Phys. {\\bf 46}, 456 (1967) and\nreference therein.\n\n%Cu(NO3)2.5H20\n\n\\bibitem{Diedrix}\nK. M. Diedrix, H. W. J. Bl$\\ddot{o}$te, J. P. Groen, T. O. Klassen, and N.\nJ. Poulis, Phys. Rev. B \n{\\bf 19}, 420 (1979).\n\n%(VO)2P2O7\n\n\\bibitem{Garrett}\nA. W. Garrett, S. E. Nagler, D. A. Tennant, B. C. Sales, and T. Barnes,\nPhys. Rev. Lett. {\\bf 79}, 745 (1997).\n\n%Spin-Pierce\n\n\\bibitem{Bray}\nJ. W. Bray, H. R. Hart, Jr., L. V. Interrante, I. S. Jacobs,\nJ. S. Kasper, G. D. Watkins and S. H. Wee:\nPhys. Rev. Lett. {\\bf 35}, 744 (1975).\n\n\n\\bibitem{Uchinokura}\nM. Hase, I. Terasaki, Y. Sasago and K. Uchinokura:\nPhys. Rev. Lett. {\\bf 71}, 4059 (1993).\n\n\n\\bibitem{Nishi}\nM. Nishi, O. Fujita and J. Akimitsu:\nPhys. Rev. B{\\bf 50}, 6508 (1994).\n\n\n\\bibitem{Nishino}\nM. Nishino, H. Onishi, P. Roos, K. Yamaguchi, and S. Miyashita, Phys. Rev.\nB {\\bf 61}, 4033 (2000).\n\n\n\\bibitem{Friedman}\nJ. R. Friedman, M. P. Sarachik, T. Tejada, and R. Ziolo, Phys. Rev. Lett.\n{\\bf 76}, 3830 (1996).\n\n\n\\bibitem{Thomas}\nL. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara,\nNature ( London) {\\bf 383}, 145 (1996).\n\n\n\\bibitem{Hernandez}\nJ. M. Hernandez, X. X. Zhang, F. Luis, T. Tejada, J. R. Friedman, M. P.\nSarachik, and R. Ziolo, Phys. Rev. B {\\bf 55}, 5858 (1997).\n\n\n\\bibitem{Chiorescu}\nI. Chiorescu, W. Wernsdorfer, A. M\\\"{u}ller, H. B\\\"{o}gge, and B. Barbara,\nPhys. Rev. Lett. {\\bf 84} 3454 (2000).\n\n\n\\bibitem{Miya1}\nS. Miyashita, J. Phys. Soc. Jpn. {\\bf 64}, 3207 (1995).\n\n\n\\bibitem{DeRaedt}\nH. De Raedt, S. Miyashita, K. Saito, D. Garc\\'{\\i}a-Pablos and \nN. Garc\\'{\\i}a, Phys. Rev. B {\\bf 56}, 11761 (1997).\n\n\n\\bibitem{Miya2}\nS. Miyashita, K. Saito, and H. De Raedt, Phys. Rev. Lett. {\\bf 80}, 1525 (1998).\n\n\n\\bibitem{TFE}\nK. Saito and S. Miyashita, cond-mat/0004027.\n\n\n\\bibitem{Landau}\nL. Landau, Phys. Z. Sowjetunion {\\bf 2}, 46 (1932).\n\n\n\\bibitem{Zener}\nC. Zener, Proc. R. Soc. London, Ser. A {\\bf 137}, 696 (1932).\n\n\n\\bibitem{Stuckel}\nE. C. G. St$\\rm{\\ddot{u}}$ckelberg, Helv. Phys. Acta {\\bf 5}, 369 (1932).\n\n\n\\bibitem{Kaya1}\nY. Kayamuma, Phys. Rev. B {\\bf 47}, 9940 (1998).\n\n\n\\bibitem{Kaya2}\nY. Kayanuma and H. Nakayama, Phys. Rev. B {\\bf 57}, 13099 (1993); \nJ. Phys. Soc. Jpn. {\\bf 54}, 2037 (1985).\n\n\n\\bibitem{loop} \nN. Kawashima and J. E. Gubernatis,\nJ. Stat. Phys. {\\bf 90} 169 (1995) and reference therein;\\\\\nH. G. Evertz, cond-mat/9707221 and reference therein.\n\n\n\\bibitem{cont}\nB. B. Beard and U. -J. Wiese: Phys. Rev. Lett. {\\bf 77} 5130 (1996).\n\n%\n\\bibitem{onishi}\n%H. Onishi, M. Nishino, N. Kawashima, and S. Miyashita, J. Phys. Soc. Jpn.\n%{\\bf 68} 2547 (1999). \n\n\n\\bibitem{Suzuki}\nM. Suzuki, Phys. Lett. A {\\bf 146}, 319 (1990).\n\n\n\\bibitem{Liep}\nE. Lieb and D. Mattis, J. Math. Phys. {\\bf 3}, 749 (1962).\n\n%\n\\bibitem{Haldane}\n%F. D. M. Haldane, Phys. Lett. {\\bf 93A}, 464 (1983); Phys. Rev. Lett. {\\bf\n%50}, 1153 \n%(1983).\n\n%\n\\bibitem{Hagiwara}\n%M. Hagiwara, K. Katsumata I. Affleck, B. I. Halperin and J. P. Renard, \n%Phy. Rev. Lett. {\\bf 65}, 3181 (1990).\n\n"
}
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cond-mat0002083
|
Diffusion limited aggregation as a Markovian process. \\ Part I: bond-sticking conditions
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[
{
"author": "Boaz Kol and Amnon Aharony"
}
] |
Cylindrical lattice Diffusion Limited Aggregation (DLA), with a narrow width $N$, is solved using a Markovian matrix method. This matrix contains the probabilities that the front moves from one configuration to another at each growth step, calculated exactly by solving the Laplace equation and using the proper normalization. The method is applied for a series of approximations, which include only a finite number of rows near the front. The matrix is then used to find the weights of the steady state growing configurations and the rate of approaching this steady state stage. The former are then used to find the average upward growth probability, the average steady-state density and the fractal dimensionality of the aggregate, which is extrapolated to a value near 1.64.
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[
{
"name": "Kol2000a.tex",
"string": "%&latex209\n%\\documentstyle[preprint,eqsecnum,aps,pre,epsf]{revtex}\n\\documentstyle[multicol,eqsecnum,aps,prb,epsf]{revtex}\n%\\tighten\n\\begin{document}\n%\\draft\n%\\twocolumn\n%\\preprint{draft}\n\\title{ Diffusion limited aggregation as a Markovian process. \\\\\n\tPart I: bond-sticking conditions}\n\\author{Boaz Kol and Amnon Aharony}\n\\address{Raymond and Beverly Sackler Faculty of Exact Sciences, \n\tSchool of Physics and Astronomy,\\\\ Tel Aviv University, \n\t69978 Ramat Aviv, Israel}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nCylindrical lattice \nDiffusion Limited Aggregation (DLA), with a narrow width $N$, \nis solved using a Markovian matrix method.\nThis matrix contains the probabilities that the front moves from one\nconfiguration to another at each growth step, calculated\nexactly by solving the Laplace equation and using the proper normalization.\nThe method is applied for a series of approximations, which include\nonly a finite number of rows near the front.\nThe matrix is then\nused to find the weights of the steady state growing configurations\nand the rate of approaching this steady state stage.\nThe former are then used to find the average upward growth probability, the\naverage steady-state density and the fractal\ndimensionality of the aggregate, which is extrapolated to a value near\n1.64.\n\\end{abstract}\n\\pacs{PACS numbers: 61.43.Hv, 02.50.Ga, 05.20.-y, 02.50.-r}\n\\widetext\n\\begin{multicols}{2}\n\\section{Introduction}\nDiffusion limited aggregation (DLA) \\cite{Witten83} has been the subject of \nextensive study since it was first introduced. This model exhibits a growth\nprocess that produces highly ramified self similar patterns, which are \nbelieved to be fractals \\cite{Mandelbrot82}. It seems that DLA captures the \nessential mechanism in many natural growth processes, such as viscous \nfingering \\cite{Feder}, dielectric breakdown \\cite{DBM84a}, etc.\nIt is now understood that the Laplace equation, which is common to\nall of these processes and to DLA, has a major role in the resemblance \nbetween them. One of the interesting features of DLA is that there are no\nparameters to fine-tune in order to obtain a fractal. It thus differs from\nordinary critical phenomena, and belongs to the class of self organized\ncriticality (SOC) \\cite{Bak87}. In spite of\nthe apparent simplicity of the model, an analytic solution is still \nunavailable. Particularly, the exact value of the fractal dimension\nis not known.\n\nIn DLA there is a seed cluster of particles fixed somewhere. \nA particle is released at a distance from the cluster, and performs a random \nwalk until it attempts to penetrate the fixed cluster, in which case it \nsticks. Then the next particle is released and so on. There are two common \ntypes of sticking conditions. The sticking condition described above\nis called ``bond-DLA'', because it occurs when a particle goes into a \nperimeter bond. In ``site-DLA'', sticking occurs as soon as the particle \narrives in a perimeter site. This paper deals with bond-DLA, whereas part II\ndeals with site-DLA. The large scale\nstructure of DLA is not sensitive to the type of sticking conditions used \n\\cite{Cafiero93,Erzan95}.\n\nIt has been shown that bond-DLA is equivalent to the dielectric breakdown\nmodel (DBM) with $\\eta=1$ \\cite{DBM84a,DBM84b}. \nDBM is a cellular automaton that is defined on a lattice. It consists of the\nfollowing steps: one starts with a seed cluster of connected sites\nand with a boundary surface far away from it. \nA field\n$\\Phi$, which corresponds to the electrostatic potential, is found \nby solving the discrete Laplace equation on a lattice,\n\\begin{equation}\n\\nabla^{2} \\Phi = 0,\n\\label{Laplace}\n\\end{equation}\nwith the following boundary conditions:\nthe aggregate is considered to have a constant potential that is \nusually set to $0$, and the potential gradient on the distant boundary \nis set to $1$ in some arbitrary units (some use a constant potential on the \ndistant boundary instead). In this paper we set the distant boundary\nat infinity, and ignore the exponentially small finite size corrections.\nAfter solving the discrete Laplace equation (\\ref{Laplace}), the field $\\Phi$\ndetermines the growth probabilities per perimeter bond. More specifically,\nthe growth probabilities are proportional \nto the electric field to some power $\\eta$. The electric field is simply equal\nto the potential difference across each bond. Because the potential\nis set to $0$ on the aggregate, the electric-field is equal to \nthe potential value at the sites lying across the perimeter \nbonds. Thus,\n\\begin{equation}\nP_{b}={|\\Phi_{b}|^{\\eta} \\over \\sum_{b}|\\Phi_{b}|^{\\eta}}.\n\\label{growth probabilities}\n\\end{equation}\nHere, $b$ is the bond index.\n\nDLA and DBM can be grown in various geometries.\nBy geometry we refer to the dimensionality of the lattice,\nto the shapes of the boundaries and to the details of the seed for growth \n(usually a point or a line for two dimensional growth). \nFor instance, the case in which the distant boundary is \na sphere is called radial boundary conditions, and\nthe case in which the boundary is a distant plane at the top, while the \nseed cluster is a parallel plane at the bottom, with periodic boundary \nconditions on the sides, is called cylindrical boundary conditions.\nIn this paper we only consider the cylindrical case, with\na relatively short period length (width), from $N=2$ to about $N=7$, \nalthough the method described here could also be used for wider cases.\n\nRecently we published an exact\nsolution to DLA in cylindrical geometry of width\n$N=2$ \\cite{Kol98}. The present paper generalizes and extends that solution. \nOur approach follows the dynamics of\nthe interface. The interface alone determines the growth probabilities at\neach time step, and whatever lies behind it is irrelevant. This is because\nthe solution to the Laplace equation is unique, provided that the\nboundary conditions are well defined. We now give a brief summary of Ref. \n\\cite{Kol98}. The characterization of the interface\nfor $N=2$ is simple; The interface is {\\it fully} characterized by a single \nparameter (usually denoted by $i$ or $j$), which corresponds to the height \ndifference between the two columns.\nThis height difference, referred to as the step size, can be infinitely large; \nsee Fig. \\ref{stepfig}.\nIf the interface is flat ($j=0$), one can assume that the next particle will\nalways stick on the right side, without limiting the generality of this \ndiscussion. This means that the step size can always be considered as \nnonnegative. The Markovian dynamics is then presented using \nthe Master equation,\n\\begin{equation}\nP_i(t+1)=\\sum_{j=0}^{\\infty}E_{i,j}P_j(t),\n\\label{1.3}\n\\end{equation}\nwhere $P_j(t)$ is the probability that the step size is $j$ at time $t$,\nand $E_{i,j}$ is the time independent conditional probability that an initial \nstep size $j$ will become $i$ after the next growth process. An example with \nseveral possible transitions is shown in Fig. \\ref{transitionsfig}. \n${\\bf P}(t)$\nis called the state vector and ${\\bf E}$ is called the evolution matrix.\nIn principle, a similar Master equation can be written for more complex\ngrowth situations, provided the various configurations can be indexed\nwith a single index $j$.\nBeing made out of conditional probabilities, the elements of the evolution \nmatrix obey that, \n\\begin{eqnarray}\n&0&\\leq E_{i,j}\\leq 1, ~~ i,j=0,\\dots,\\infty, \\nonumber \\\\ \n&\\sum&_{i=0}^{\\infty}E_{i,j}=1, ~~j=0,\\dots,\\infty.\n\\end{eqnarray}\nAfter many iterations of Eq. (\\ref{1.3}) the system converges to a fixed point\n${\\bf P}^*$, also called the steady state, which represents the asymptotic time\ndistribution of the step sizes. From the steady state and the evolution matrix \nwe are able to extract the average upward growth probability \n$\\langle p_{\\rm up}\\rangle^*$, the average density $\\rho$ and the fractal \ndimension $D$.\n\nIn order to obtain an analytic expression for the elements of the evolution \nmatrix, one must first solve the Laplace equation. Having found the \nsolutions $\\Phi(m,n)$, the growth probabilities are found from Eq. \n(\\ref{growth probabilities}). The denominator there, which comes \nfrom the normalization, is particularly simple for the\nspecial case\nof $\\eta=1$, where the discrete version of the divergence theorem\nimplies that \\cite{Kol98}\n\\begin{equation}\n\\sum_b\\Phi_b=N.\n\\label{sum_bPhi_b=N=2}\n\\end{equation}\nThe actual growth probability into a site is then found\nfrom\n\\begin{equation}\np_{\\rm site}=\\sum_{\\rm bonds~into~site}p_b.\n\\end{equation}\n\nThe solution of the Laplace equation is now divided into two parts.\nIn the\nfirst part, we solve the Laplace equation for the 'upper' part of space,\nwhich starts just above the highest particle of the aggregate and continues\nupwards to infinity. In the example of Fig. 1, this part contains all the\nrows with $m>0$.\nAs we explain below, this solution is completely determined by the boundary\nconditions and by the values of the potential at the row with $m=0$,\ni.e. $\\{\\Phi(0,n)\\}$.\nWe then solve the Laplace equation for the 'lower' part ($m \\le 0$ in\nFig. 1), and find the values of $\\{\\Phi(0,n)\\}$ from matching the two regimes.\nThe solution in the 'upper' part is given as a combination of solutions\nof the form\n \\cite{Kol98}\n\\begin{equation}\n\\Phi(m,n)=e^{\\kappa m +ikn},\n\\end{equation}\nwith the dispersion relation\n\\begin{equation}\n\\sinh({\\kappa \\over 2})=\\pm \\sin({k \\over 2})\n\\label{dispersion}\n\\end{equation}\nand with the discrete set of allowed values $k_l={2\\pi \\over N}l$, which\nfollow from the periodic lateral boundary conditions, which require that \n$e^{ikN}=1$.\nThe boundary conditions at infinity have a uniform gradient, i.e.,\n\\begin{equation}\n\\lim_{m \\rightarrow \\infty} (\\Phi(m+1,n)-\\Phi(m,n))=1, ~~n=0,\\dots,N-1.\n\\end{equation}\nGiven the arbitrarily set of values $\\Phi(0,n)$,\nthe solution for the row $m=1$ is\n\\begin{equation}\n\\Phi(1,n)=1+\\sum_{n'=0}^{N-1}\\Phi(0,n')g_N(|n-n'|),\n\\label{g}\n\\end{equation}\nwhere\n\\begin{equation}\ng_N(n)\\equiv {1\\over N}\\sum_{l=0}^{N-1}e^{-\\kappa_l}\\cos(k_ln),~~n=0,\\dots,N-1,\n\\label{g_N(n)}\n\\end{equation}\nis the boundary Green's function, and $\\kappa_l$ corresponds to $k_l$ via the\ndispersion relation (\\ref{dispersion}). The solution is given only for $m=1$,\nbecause we are only interested in the potential at sites near the interface.\nNote that the \nGreen's function has the general property \n\\begin{equation}\n\\sum_{n=0}^{N-1}g_N(n)=1\n\\label{norm}\n\\end{equation}\n\\cite{Kol98}. It is therefore good practice to check this normalization\nfor each of the calculations presented below. Indeed, all our results\nobey this rule.\n\n\nIn general, the solution in the 'lower' regime is complicated by the variety of\nconfigurations. However, this solution is very simple for $N=2$, when\n$\\Phi(m,0)$ is a linear combination of $e^{\\kappa_f m}$ and $e^{-\\kappa_f m}$.\nSince $\\Phi(-j,0)=0$, one is left with one unknown $\\Phi(0,0)$, to be\ndetermined by the matching at row $0$.\n\nFor the special case $N=2$,\nthe above procedure has led to the exact solution \\cite{Kol98}\n\\begin{eqnarray} \nE_{i,j}&=&\\left\\{\n\\begin{array}{cll}\ny(\\infty)e^{-\\kappa_fi}{1-e^{-2\\kappa_f(j-i)}\\over 1+\\beta e^{-2\\kappa_fj}}\n&,&~0 \\leq i \\leq j-2 \\\\\n{3\\over 2}y(\\infty)e^{-\\kappa_f(j-1)}{1-e^{-2\\kappa_f} \\over\n1+ \\beta e^{-2\\kappa_fj}} &,& ~i=j-1 \\\\\nE_{\\infty+1,\\infty}\\left(1-\\alpha {e^{-2\\kappa_fj} \\over 1+\\beta \ne^{-2\\kappa_fj}} \\right) &,& ~i=j+1 \\\\\n0 & & {\\rm otherwise}\n\\end{array}\n\\right. , \\nonumber \\\\\n&&~~j \\geq 1,\n\\label{AnalyticE(i,j)}\n\\end{eqnarray} \nwhere \n\\begin{equation}\nE_{\\infty+1,\\infty}=\\lim_{j\\rightarrow \\infty}E_{j+1,j}=\n{1+g_2(1)y(\\infty)\\over 2}=0.5658\\dots, \n\\end{equation}\n$y(\\infty)=\\sqrt{3}-\\sqrt{2}=0.3178\\dots$, \n$e^{-\\kappa_f}=2-\\sqrt{3}=0.2679\\dots$, \n$\\alpha=(1+\\beta)g_2(1)y(\\infty)/(2E_{\\infty+1,\\infty})=0.1281\\dots$ and\n$\\beta=5-\\sqrt{24}=0.1010\\dots$. \nFor $j=0$, the interface will\ntransform into a step of size $j=1$ with probability $1$, hence $E_{1,0}=1$\nand $E_{i,0}=0$ for $i \\neq 1$.\nThe values of $E_{i,j}$ for $0 \\leq i,j \\leq 4$, up to \nthe fourth decimal digit, are \n\\begin{equation}\n{\\bf E}=\\left[\n\\begin{array}{cccccc}\n0 & 0.4393 & 0.3160 & 0.3177 & 0.3178 & \\cdots \\\\\n1 & 0 & 0.1185 & 0.0847 & 0.0851 & \\\\\n0 & 0.5607 & 0 & 0.0318 & 0.0227 & \\\\\n0 & 0 & 0.5655 & 0 & 0.0085 & \\\\\n0 & 0 & 0 & 0.5658 & 0 & \\\\\n\\vdots & & & & & \\ddots\n\\end{array}\n\\right].\n\\label{ExplicitE(i,j)}\n\\end{equation}\nThe first diagonal below the main, \nwhich represents the probabilities for the step to grow larger by one, \n$E_{j+1,j}$, approaches its asymptotic value of \n$E_{\\infty+1,\\infty}=0.5658...$ exponentially, as the third row of \nEq. (\\ref{AnalyticE(i,j)}) indicates.\nThe diagonal above the main represents the probabilities for growths at \nthe bottom of the fjord, $E_{j-1,j}$, and corresponds to the second row in\nEq. (\\ref{AnalyticE(i,j)}). \nThese probabilities decay exponentially as the step size $j$ grows. \nAccording to the first row in Eq. (\\ref{AnalyticE(i,j)}), \nthe elements $E_{i,j}$ converge exponentially for\nlarge $j$'s to a simple exponential function:\n\\begin{equation}\nE_{i,\\infty}=\\lim_{j\\rightarrow \\infty}E_{i,j}=y(\\infty)e^{-\\kappa_fi}.\n\\label{E_(j,infty)}\n\\end{equation}\nThese probabilities relate to the transition from a very large step to a \nstep of size $i$. \nNext, the steady state vector ${\\bf P}^*$ is computed and used \nto evaluate the average upward growth probability \n$\\langle p_{\\rm up} \\rangle^*$,\nwhich in turn, determines the average density $\\rho$ and the fractal dimension\n$D$. These computations are explained later in Sec. \\ref{frustrated}.\n\nOur previous paper does not specify details concerning the manner in which the\nsystem converges to the steady state in time. \nBesides addressing this issue, \nour present paper also treats DLA grown in wider geometrical periods \n(still in cylindrical geometry). \nThe basic approach is the same, i.e., we try to characterize the\npossible configurations of the interface for wider periods, and then write\nthe evolution matrix, which is composed of the growth probabilities, which are\ncomputed from the Laplace potential, after proper normalization. \nThe first difficulty encountered is in the characterization. For example,\nalready for a width of $N=3$ one cannot \ncharacterize the interface using a single\nparameter as in the case $N=2$, nor is it easy doing so using $2$ \nparameters, or more. Instead, we make a manual list of possible \nconfigurations of the interface, which we then\norder according to the difference\nin height between the highest and lowest points on the interface. \nThis difference is denoted by $\\Delta m$.\nOur \norder-$O$ approximation includes only the configurations with $\\Delta m \\leq O$.\nIn our approximation,\nsome of these configurations represent\nmany other (excluded) configurations, in the sense that they \nhave very similar growth probabilities, especially upward. \nThis is because of the screening quality of the Laplace equation, which \ncauses the potential to decay exponentially inside fjords. \nThus, the deeper parts of the interface have a very small effect on the upward\ngrowth probability. \nThe finite list of configurations is indexed arbitrarily, with an index \nusually denoted by $i$ or $j$. \nOur experience shows that accurate results are obtained,\nonly when the order of approximation $O$ is comparable to the width of the\ncylinder $N$. Thus, for wide periods, a high order calculation is called for.\nThis causes the method to be ineffective for very wide periods, because the \nnumber of configurations grows exponentially with the order of approximation.\nWe conducted calculation up to $N=7$. \n\nAfter selecting the finite list of configurations and obtaining the finite \nevolution matrix,\nwe compute the steady state vector, which is the fixed point of the matrix\n(the normalized eigenvector with an eigenvalue of $1$). For each configuration,\nwe identify the upward growth processes (when the newly attached particle is \nhigher than the rest). We then calculate the average upward growth probability\n$\\langle p_{\\rm up} \\rangle^*$ as a weighted average over the configurations.\n>From $\\langle p_{\\rm up} \\rangle^*$ we calculate the average density $\\rho$ and\nthe fractal dimension $D$. The computed values of \n$\\langle p_{\\rm up} \\rangle^*$, from different orders of the approximation,\nare compared with numerical\nsimulations in Table \\ref{2dresults}.\n\nIn Sec. \\ref{frustrated} we introduce a simple Markov process, called the\n``frustrated climber'', which we solve exactly. A slight modification of the\nmodel is equivalent to site-DLA with a period of $N=2$, which is presented\nin part II of this paper \\cite{Kol99}.\nWe then show a way of successively generalizing\nthe model to approximate bond-DLA with a period of $N=2$ and\nwith increasing orders $O$.\nWe are able to\ncheck the approximations by comparing with the exact results of Ref. \n\\cite{Kol98}. \nThis model also enables us to investigate the\nrate of convergence to the steady state. In this context we describe the \nconvergence in terms of other eigenvectors, with eigenvalues whose absolute\nvalues are smaller than $1$, and in terms of the infinite shift down operator.\nWe show that the average upward growth probability converges exponentially in \ntime to its steady state value, with a characteristic time constant on\nthe order of unity. In Sec. \\ref{N>2} we generalize our method \nto cylindrical DLA with $N>2$. We present in detail the\ncalculations for $N=3$ with $O=1$ and $O=2$,\nand for \n$N=4$ with $O=1$.\nNext we \nreport on numerical results for wider periods and higher orders.\nIn the final section we review the results and summarize.\n\n\\section{The frustrated climber model}\n\\label{frustrated}\nConsider someone trying to climb up a slippery infinite ladder. At each time \nstep the climber climbs up one step with probability $0\\leq p\\leq 1$, or falls \nall the way \ndown with probability $q\\equiv 1-p$. We call the climber ``frustrated'',\nbecause the probability to get very high is exponentially small.\nWe wish to compute the probability $P_i(t)$ for the climber to be at height \n$i$ after $t$ time steps, for $i=0,\\dots,\\infty$. \nThe Master equation for this problem is ${\\bf P}(t+1)={\\bf EP}(t)$, where the\nmatrix element $E_{i,j}$ is the conditional probability that\nthe climber moves from height $j$ to $i$ in a single\ntime step. The rules of the model imply that\n\\begin{equation}\nE_{i,j}=\\left\\{\n\\begin{array}{ccl}\np&,&~i=j+1 \\\\\nq&,&~i=0 \\\\\n0&,&~\\mbox{otherwise}\n\\end{array}\n\\right\\},~~ j \\geq 0,\n\\end{equation}\nso the matrix looks like this:\n\\begin{equation}\n{\\bf E}=\\left[\n\\begin{array}{ccccc}\nq&q&q&q&\\cdots \\\\ \np&0&0&0& \\\\\n0&p&0&0& \\\\\n0&0&p&0& \\\\\n\\vdots&&&&\\ddots\n\\end{array}\n\\right].\n\\label{Eclimb}\n\\end{equation}\nThis presentation helps us see the resemblance to the dynamics of DLA with \n$N=2$ in Eqs. (\\ref{AnalyticE(i,j)}, \\ref{ExplicitE(i,j)}): Eq.\n(\\ref{Eclimb}) would approximate these equations if we were to replace\n$E_{j+1,j}$ by $p \\approx E_{\\infty+1,\\infty}$ \nand $E_{0,j}$ by $q$ for all $j$, \nand neglect all other\ngrowth probabilities, which are indeed smaller. We shall discuss this\nand better approximations for DLA in the next subsections.\nBecause the \nMarkovian matrices for the two cases are similar for large $j$'s, we\nexpect that some of the dynamical features are similar as well. \nWe therefore present here an exact solution for the frustrated climber\nmodel, and then try to draw conclusions for generalized models which \nrepresent successive approximations for DLA.\nThe advantage is\nthat in the simple model of the frustrated climber it is possible to derive a\nsimple analytic expression for the steady state \nand a complete description of the temporal convergence.\n\nThe steady state equations for the frustrated climber model are\n\\begin{eqnarray}\n&P^*&_{i+1}=\\sum_{j=0}^{\\infty}E_{i+1,j}P^*_j=pP^*_i,~~i \\geq 0, \\\\\n\\Rightarrow &P&^*_j=qp^j,~~j \\geq 0.\n\\end{eqnarray}\nOne can easily check that this steady state is normalized,\n\\begin{equation}\n\\sum_{j=0}^{\\infty}P^*_j=\\sum_{j=0}^{\\infty}qp^j={q\\over 1-p}=1.\n\\end{equation}\nThe average upward growth probability in the steady state is\n\\begin{equation}\n\\langle p_{\\rm up} \\rangle^* =\\sum_{j=0}^{\\infty}P^*_jp_{\\rm up}(j)\n=\\sum_{j=0}^{\\infty}P_jp=p,\n\\end{equation}\nwhere $p_{\\rm up}(j)$ stands for the probability to move upwards when the \nheight of the climber is $j$. In this simple model $p_{\\rm up}(j)=p$ for\nall $j$'s.\n\nWe now investigate the temporal convergence to the steady state. We\ndefine the vector ${\\bf v}(t)$ by\n\\begin{equation}\n{\\bf P}(t)={\\bf P}^*+{\\bf v}(t).\n\\end{equation}\nBecause ${\\bf P}^*$ and ${\\bf P}(t)$ are probability vectors,\n$\\sum_{j=0}^{\\infty}P^*_j=\\sum_{j=0}^{\\infty}P_j(t)=1$, for any $t$, hence \n\\begin{equation}\n\\sum_{j=0}^{\\infty}v_j(t)=0.\n\\label{sum_v=0}\n\\end{equation}\nWe substitute ${\\bf v}$ into the dynamical equation and obtain\n\\begin{eqnarray}\n&{\\bf P}&(t+1)={\\bf EP}(t)={\\bf P}^*+{\\bf Ev}(t), \\\\\n\\Rightarrow &{\\bf v}&(t+1)={\\bf Ev}(t).\n\\end{eqnarray}\nNext, we look for\nthe rest of the eigenvectors of the evolution matrix\n(any eigenvector ${\\bf v}$ with an eigenvalue \n$\\lambda \\neq 1$, has to obey Eq. (\\ref{sum_v=0})). Surprisingly, \nthere are no eigenvectors besides the steady state in this case.\nThe eigenvector equations are\n\\begin{eqnarray}\n&&\\lambda v_0=q\\sum_{j=0}^{\\infty}v_j=0, \\nonumber \\\\\n&&\\lambda v_{i+1}=pv_i(t),~~ i \\geq 0.\n\\label{other_eigenvalues}\n\\end{eqnarray}\nThe first equation implies that either $\\lambda=0$ or $v_0=0$. In both \ncases, the last equation implies that {\\bf v}=0.\n\nWe next introduce the infinite shift-down operator:\n\\begin{equation}\n{\\bf S} \\equiv \\left[\n\\begin{array}{ccccc}\n0&0&0&0&\\cdots \\\\ \n1&0&0&0& \\\\\n0&1&0&0& \\\\\n0&0&1&0& \\\\\n\\vdots&&&&\\ddots\n\\end{array}\n\\right].\n\\end{equation}\nThis operator causes a vector to ``slide down'' and inserts a\nzero at the evacuated component at the top. {\\bf S} has no eigenvectors at all,\nnot even a fixed point (in spite of the fact that $\\sum_{i=0}^{\\infty}S_{i,j}\n=1$ for $j=0,\\dots,\\infty$).\nIn fact, ${\\bf Ev}=p{\\bf Sv}$ for all vectors ${\\bf v}$ with\n$\\sum_{j=0}^{\\infty}v_j=0$.\n\nNevertheless, the convergence of ${\\bf P}(t)$ to ${\\bf P}^*$ is simple. \nStarting from any initial state vector ${\\bf P}(t=0)$, the first application\nof ${\\bf E}$ causes the first component to be set to its steady state \nvalue $P_0(t=1)=q$. \nAt each subsequent iteration another components is set permanently: \n$P_1(t=2)=qp$, $P_2(t=3)=qp^2$, etc. $P_j$ becomes equal to $P^*_j$ \nafter no more than $j+1$ time steps. \nThe context we are interested in is wider. We wish to\ncompute the convergence of ``observables'', i.e., the average\nof an arbitrary function $a(j)$ over configurations. \nWe compute the average at time $t$\n\\begin{equation}\n\\langle a \\rangle(t) \\equiv \\sum_{j=0}^{\\infty}a(j)P_j(t)\n=\\langle a \\rangle ^* +\\sum_{j=0}^{\\infty}a(j)v_j(t),\n\\end{equation}\nwhere $\\langle a \\rangle^* \\equiv \\sum_{j=0}^{\\infty}a(j)P_j^*$ is the steady\nstate average. \nStarting from an initial deviation from the steady state ${\\bf v}(0)$, each\niteration causes a down shift and a multiplication by $p$, hence\n\\begin{equation}\n\\langle a \\rangle(t)=\\langle a \\rangle ^*+p^t\\sum_{j=0}^{\\infty}a(j+t)v_j(0).\n\\label{converge}\n\\end{equation}\nEquation (\\ref{converge}) is the analogue of the standard eigenvector \ndescription. We can also identify here the exponential decay of the factor \n$p^t$.\nFor example, the function $a(j)=\\delta_{j,j_0}$ ``measures'' the probability\nof the climber to be at height $j_0$ (at any time). At time $t$ the observed\naverage probability is\n\\begin{equation}\n\\langle a \\rangle (t)=P^*_{j_0}+p^tv_{j_0-t}(0),\n\\end{equation}\nfor $t \\leq j_0$, and $\\langle a \\rangle (t)=P^*_{j_0}$ for $t>j_0$ \n\\cite{remark}.\n\\subsection{First-order approximation for $N=2$}\n\\label{site stick}\nWe now return to Eq. (\\ref{AnalyticE(i,j)}), and try to approximate it\nby a sequence of models which are related to the frustrated climber model.\nThe simplest approximation would follow if we do not let\nthe particle penetrate into the fjord at all. This is equivalent to setting\n$\\kappa_f = \\infty$ in Eq. (\\ref{AnalyticE(i,j)}). According to these simplified\nrules, the particle can either stick at $(0,0)$\nand create a flat step of $i=0$, or it can stick at $(1,1)$ and increase the\nstep height by $1$. \nLet us denote the probability for the\nformer event by $q$ and the latter by $p$.\nIn the first-order approximation we take $p$ and $q$ to be\nindependent of the initial step size $j$, unless $j=0$, in which case\nthe step size increases with probability $1$. \nThe Markovian matrix {\\bf E} for\nthis case is almost identical to the case of the frustrated climber,\n\\begin{equation}\n{\\bf E}=\\left[\n\\begin{array}{ccccc}\nq_0&q&q&q&\\cdots \\\\ \np_0&0&0&0& \\\\\n0&p&0&0& \\\\\n0&0&p&0& \\\\\n\\vdots&&&&\\ddots\n\\end{array}\n\\right],\n\\label{q_0matrix}\n\\end{equation}\nthe only difference being in the first column, where we denote $q_0=0$ and\n$p_0=1$. In part II of this paper we show that this model is exact for \nthe case of site-sticking DLA for $N=2$\n\\cite{Kol99}.\n\nThe solution to this problem is very similar to that of the frustrated climber,\nwith small modifications. The steady state is\n\\begin{equation}\nP^*_j=P^*_0p_0p^{j-1},~~j \\geq 1,\n\\end{equation}\nwhere $P^*_0$ can be determined using the normalization condition\n\\begin{eqnarray}\n&&1=\\sum_{j=0}^{\\infty}P^*_j=P_0^*(1+p_0\\sum_{j=0}^{\\infty}p^j), \\nonumber \\\\\n&&\\Rightarrow P^*_0={1-p \\over 1-p+p_0}.\n\\end{eqnarray}\nThe average upward growth probability is evaluated by\n\\begin{equation}\n\\langle p_{\\rm up}^{(1)}\\rangle^*=P^*_0p_0+(1-P^*_0)p={p_0 \n\\over 1-p+p_0}.\n\\label{pup(1)}\n\\end{equation}\nThe superscript $(1)$ appears because it is the first-order approximation.\nWe now need to choose $p$. One possible choice would be\nto take $p=E_{\\infty+1,\\infty}=0.5658$, because this is the asymptotic\nupward growth probability, and then set $q=1-p$.\nThis would give $\\langle p_{\\rm up}^{(1)}\\rangle^*=0.6973$,\nto be compared with the exact value $0.6812$ \\cite{Kol98}. \nAn alternative approximation would return to Eq. (1.13), but \nreplace $y(\\infty)$ by $q$, and then find $q$ by solving\n$1=p+q=[1+g_2(1)q]/2+q$. This yields $p=1-q=2-\\sqrt{2}=0.5858$, and therefore\n$\\langle p_{\\rm up}^{(1)}\\rangle^*=\\sqrt{2}/2=0.7071$.\n\nWe next calculate the average density and the fractal dimensionality.\nSimilar to the argument used by Turkevich and Scher \\cite{Turkevich89}, we\nconsider a large number of growth processes $n$ in the steady state. \nDuring this growth the aggregate would grow higher by $h=\\langle p_{\\rm up}\n\\rangle^* n$. \nThe total volume covered by the new growth is $hN^{d-1}$, where $d=2$ is the \nEuclidean dimension. Thus, for $N=2$ and for our first approximation\nthe density is\n\\begin{equation}\n\\rho={n \\over hN^{d-1}}={n \\over \\langle p_{\\rm up} \\rangle^* nN^{d-1}}\n={1 \\over \\langle p_{\\rm up}\\rangle^* N^{d-1}}=0.7171,\n\\label{rho}\n\\end{equation}\nto be compared with the exact value $\\rho=0.7340$.\nAlthough our model does not really produce fractal structures (due to the \nnarrow width of our space), we can make an estimate of the fractal dimension\nin the same way Pietronero $et~al.$ estimated it in \n\\cite{Erzan95,Pietronero88b}. For a self similar fractal\nstructure, one expects that a change of scale by a factor $N$ will change the\naverage mass (number of occupied sites) of a $N \\times N$ cut by a factor \n$N^D$, where $D$ is the fractal dimension. Assuming that the above procedure\nrepresents a coarse graining of the sites into $N \\times N$ cells, we conclude\nthat asymptotically \n\\begin{equation}\n\\rho =N^{D-d},\n\\label{rho=N^(D-d)} \n\\end{equation}\nand this means that\n\\begin{equation}\nD=d+{\\ln(\\rho) \\over \\ln(N)}=1-{\\ln(\\langle p_{\\rm up} \\rangle^*) \\over\n\\ln(N)}=1.5202. \n\\label{estimate dimension}\n\\end{equation}\nIn Sec. IV we suggest a modified estimate of the fractal dimension, allowing\nfor corrections to the asymptotic form (\\ref{rho=N^(D-d)}). \n\nThe study of the convergence to the steady state is again limited to the \nsubspace of vectors ${\\bf v}$ with $\\sum_{j=0}^{\\infty}v_j=0$. \nThe dynamic equation for $i=0$ is,\n\\begin{eqnarray}\n&v&_0(t+1)=q_0v_0(t)+\\sum_{j=1}^{\\infty}qv_j(t)=(q_0-q)v_0(t), \\nonumber \\\\\n\\Rightarrow &v&_0(t)=(q_0-q)^tv_0(0).\n\\end{eqnarray}\nSince $q_0=0$, the exponentiated prefactor is negative, and therefore $v_0(t)$\nis oscillating during its decay.\nAfter the first iteration $v_1(1)=p_0v_0(0)$, regardless of its initial value.\nAfterwards it continues to follow $v_0$, i.e., $v_1(t)=p_0(q_0-q)^{t-1}v_0(0)$.\nAfter the second iteration $v_2(2)=p_0pv_0(0)$, and it also starts to decay\nexponentially with the factor $(q_0-q)$. This happens for any $j>1$; \nAfter more than $j$ time steps ($t>j$) one has,\n\\begin{equation}\nv_j(t)=p_0p^{j-1}(q_0-q)^{t-j}v_0(0).\n\\end{equation}\nFor short times and large indices $t<j$, the dynamics is governed by the shift\ndown operator:\n%, in fact we can use the following expression:\n\\begin{equation}\n{\\bf v}(t)=v_0(0)(q_0-q)^t{\\bf h}+p^t\\sum_{j=1}^{\\infty}c_j\n{\\bf e}^{(j+t)},\n\\label{v(t)(1)}\n\\end{equation}\nwhere ${\\bf e}^{(j)}$ are the standard basis vectors, the components\nof the vector ${\\bf h}$ are, \n\\begin{eqnarray}\nh_0&\\equiv& 1, \\nonumber \\\\\nh_j&\\equiv& {p_0 \\over p} \\left( {p \\over q_0-q}\\right)^j,~~j \\geq 1, \n\\end{eqnarray}\nand the constants $c_j$ are determined by the initial conditions,\n\\begin{equation}\nc_j=v_j(0)-v_0(0)h_j, ~~j=1,2,\\dots\n\\end{equation}\nFor $p>0.5$ the components of ${\\bf h}$ explode \nexponentially. However, $\\sum_{j=0}^{\\infty}v_j(0)=0$ and therefore \n$\\lim_{j\\to \\infty}v_j(0)=0$. Thus, in order to cancel the divergence of the\n$h_j$'s, the $c_j$'s must also explode exponentially and have an opposite\nsign. We note that because of this divergence ${\\bf h}$\ndoes not have a finite $L_1$ norm and thus\ndoes not belong to the domain of ${\\bf E}$. Therefore it is not\nan eigenvector.\n\n\\subsection{Higher-order approximations for $N=2$}\n\\label{approxN=2}\nAs mentioned earlier, the frustrated climber model resembles the bond-DLA \nevolution matrix (\\ref{AnalyticE(i,j)}, \\ref{ExplicitE(i,j)}).\nIn this section we approximate the full\ndynamics using increasingly more complex matrices.\nBy doing so we do not improve on the accuracy of our previously published\nresults \\cite{Kol98}, but rather learn about the rate of convergence to the\nsteady state. The method used in this section is generalized and applied\nto cylindrical DLA with wider periods in the next section. \nThe case $N=2$ is the simplest demonstration of this approach.\n\nThe second-order approximation is to allow also transitions of the kind\n$j \\rightarrow 1$ for $j \\geq 1$. \nWe also allow having arbitrary values in\nthe top left $2 \\times 2$ corner of the matrix, which we copy from\nthe original matrix of Eq. (\\ref{ExplicitE(i,j)}), i.e.,\n\\begin{equation}\n{\\bf E}=\\left[\n\\begin{array}{cccccc}\nq_0&q_1&q&q&q&\\cdots \\\\ \nr_0&r_1&r&r&r& \\\\\n0&p_1&0&0&0& \\\\\n0&0&p&0&0& \\\\\n0&0&0&p&0& \\\\\n\\vdots&&&&&\\ddots\n\\end{array}\n\\right],\n\\label{q_1matrix}\n\\end{equation}\nWe still require that the sum of the elements in each column be \nequal to $1$, i.e.,\n\\begin{eqnarray}\n&&q_0+r_0=1, \\nonumber \\\\\n&&q_1+r_1+p_1=1, \\nonumber \\\\\n&&q+r+p=1.\n\\end{eqnarray}\nIn terms of standard DLA this means that\nwe allow the particle to penetrate two sites into the fjord, but no more.\nIndeed it is exponentially improbable to penetrate deep into the fjord. This \nfact suggests a controlled approximation for DLA. In each order of the \napproximation we allow the depth of penetration into the fjord to grow by $1$.\nThis is done by copying the $(O+1)\\times O$ upper left block of the original \nmatrix (\\ref{AnalyticE(i,j)}, \\ref{ExplicitE(i,j)}), where $O$ is the order of \napproximation. Asymptotic values are used outside this block, i.e.,\n\\begin{eqnarray}\n&&E_{j+1,j}=E_{\\infty+1,\\infty},~~ j \\geq O, \\nonumber \\\\\n&&E_{i,j}=y(\\infty)e^{-\\kappa_fi},~~ j \\geq O,~ i \\leq O-2, \\nonumber \\\\\n&&E_{n-1,j}=1-\\sum_{i=0}^{n-2}y(\\infty)e^{-\\kappa_fi}-E_{\\infty+1,\\infty}\n\\nonumber \\\\\n&&~~=y(\\infty){e^{-\\kappa_f(n-1)} \\over 1-e^{-\\kappa_f}}, ~~ j \\geq O,\n\\end{eqnarray} \nand the rest of the matrix elements are null.\nFor example, in our case, $O=2$, the constants in the matrix (\\ref{q_1matrix})\nare\n\\begin{eqnarray}\n&&q_0=0, \\nonumber \\\\\n&&r_0=1, \\nonumber \\\\\n&&q_1={6-3\\sqrt{2} \\over 4}=0.4393, \\nonumber \\\\\n&&r_1=0, \\nonumber \\\\\n&&p_1={3\\sqrt{2}-2 \\over 4}=0.5607, \\nonumber \\\\\n&&q=y(\\infty)=\\sqrt{3}-\\sqrt{2}=0.3178, \\nonumber \\\\\n&&p=E_{\\infty+1,\\infty}=0.5658, \\nonumber \\\\\n&&r=y(\\infty){e^{-\\kappa_f} \\over 1-e^{-\\kappa_f}}=0.1163.\n\\label{q_1constants}\n\\end{eqnarray}\n\nFirst, the steady state is found by solving ${\\bf P}^*={\\bf EP}^*$, i.e.,\n\\begin{eqnarray}\n&&P^*_0=q_0P^*_0+q_1P^*_1+q\\sum_{j=2}^{\\infty}P^*_j, \\nonumber \\\\\n&&P^*_1=q_0P^*_0+q_1P^*_1+r\\sum_{j=2}^{\\infty}P^*_j, \\nonumber \\\\\n&&P^*_2=p_1P^*_1, \\nonumber \\\\\n&&P^*_{j+1}=pP^*_j,~~ j \\geq 2.\n\\label{q_1steady}\n\\end{eqnarray}\nThe solution to the last equation is\n\\begin{equation}\nP^*_j=P^*_2p^{j-2}, ~~ j \\geq 2.\n\\end{equation}\nKeeping this in mind it is possible to exchange the two last equations of the\nset (\\ref{q_1steady}) with\n\\begin{equation}\n\\sum_{j=2}^{\\infty}P^*_j=p_1P^*_1+p\\sum_{j=2}^{\\infty}P^*_j.\n\\end{equation}\nThus we obtain an autonomous and finite set of $3$ equations for $3$ unknowns,\nnamely, $P^*_0$, $P^*_1$ and $\\tilde P^*_2 \\equiv \\sum_{j=2}^{\\infty}P^*_j$. \nThe third parameter, $\\tilde P^*_2$, represents the total probability\nfor the infinitely many\n configurations with\n$j\\geq 2$. This reduction of the problem to three parameters\nbecame possible because all of\nthe configurations with $j\\geq 2$\nhave exactly the same transition probabilities to\nthe configurations $j=0$ and $j=1$, and because they have exactly the same\nupward growth probability. \nThus we obtain a fixed point equation for a $3\\times 3$ matrix,\n%which looks similar to the matrix (\\ref{q_1matrix}),\n\\begin{equation}\n\\left[\n\\begin{array}{c}\nP^*_0 \\\\\nP^*_1 \\\\\n\\tilde P^*_2 \\\\\n\\end{array}\n\\right]=\\left[\n\\begin{array}{ccc}\nq_0 & q_1 & q \\\\\nr_0 & r_1 & r \\\\\n0 & p_1 & p \\\\\n\\end{array}\n\\right] \\left[\n\\begin{array}{c}\nP^*_0 \\\\\nP^*_1 \\\\\n\\tilde P^*_2 \\\\\n\\end{array}\n\\right].\n\\label{q_1finite}\n\\end{equation}\nIt is guaranteed that a nontrivial solution exists, because the sum of \nthe terms in each \ncolumn of the finite matrix equals $1$. Using the constants from\nEqs. (\\ref{q_1constants}), the normalized solution obtained is,\n\\begin{eqnarray}\n&&P_0^{*(2)}=0.2705,~~(0.2696), \\nonumber \\\\\n&&P_1^{*(2)}=0.3184,~~(0.3113), \\nonumber \\\\\n&&\\tilde P_2^{*(2)}=0.4111,~~ (0.4191), \n\\end{eqnarray}\nwhere the superscript denotes the order of approximation and a comparison is\ndrawn to the exact values in parentheses. By ``exact'' we refer to very high\norder calculations, or to values from simulations (which are the same up to \nthe presented accuracy of $10^{-4}$) \\cite{Kol98}. The elements \n$P^*_j$ for $j \\geq 2$ are evaluated using\n\\begin{equation}\nP_j^{*(2)}=(1-p)\\tilde P_2^{*(2)}p^{j-2}, ~~j \\geq 2.\n\\end{equation}\nIt is now possible to evaluate the average upward growth probability\n\\begin{equation}\n\\langle p_{\\rm up}^{(2)} \\rangle^*=P^*_0r_0+P^*_1p_1+\\tilde P^*_2p=0.6816,\n\\end{equation}\nwhere the exact value is $0.6812$. The fractal dimension is evaluated \nas in Eq. (\\ref{estimate dimension}),\n\\begin{equation}\nD^{(2)}=1.5530,\n\\end{equation}\ncompared to the exact value $1.5538$. \n\nThe temporal convergence to the steady state in the second-order approximation\ncan be analyzed using both the shift down operator and eigenvectors.\nThe first eigenvector of the matrix in \nEq. (\\ref{q_1finite}) is the fixed point solution, \nwhich we denote by $\\tilde {\\bf P}^*$.\nLet us denote the other two (three-components) eigenvectors \nby $\\tilde {\\bf h}$ and $\\tilde {\\bf g}$, \nand their corresponding eigenvalues by $|\\lambda_0|\\geq|\\lambda_1|$.\nAfter $t$ iterations of the evolution matrix we have\n\\begin{equation}\n\\tilde {\\bf P}(t)=\\tilde {\\bf P}^*+c_0\\lambda_0^t\\tilde {\\bf h}\n+c_1\\lambda_1^t\\tilde {\\bf g},\n\\label{finite eigenvectors}\n\\end{equation}\nwhere $c_0$ and $c_1$ are constants determined by the initial conditions.\nThe configurational average of some function $a(j)$ with $a(j)=a(2)$\nfor $j>2$, can be expressed in terms of these eigenvalues only,\n\\begin{equation}\n\\langle a \\rangle(t)=\\langle a \\rangle ^*+k_0\\lambda_0^t +k_1\\lambda_1^t,\n\\end{equation}\nwhere $k_0$ and $k_1$ are some other constants. \nA special function of this type is the upward growth probability, \n$p_{\\rm up}(j)=(r_0, p_1, p, p, p,\\dots)$. \nThe eigenvalue with the largest absolute\nvalue other than $1$, $\\lambda_0$, makes the largest contribution to the\ndeviation from the steady state values, and thus controls the temporal \nconvergence. The characteristic time constant for the exponential convergence\nis,\n\\begin{equation}\n\\tau=-{1 \\over \\ln(|\\lambda_0|)}.\n\\label{tau=}\n\\end{equation}\nThe eigenvalues obtained are $\\lambda_0^{(2)}=-0.5599$ and \n$\\lambda_1^{(2)}=0.1257$, using the constants of Eqs. \n(\\ref{q_1constants}). Hence, $\\tau^{(2)}=1.7$.\nIn order to describe the convergence of $P_j(t)$ for $j \\geq 2$ we use \nthe vector ${\\bf v}(t)=\n{\\bf P}(t)-{\\bf P}^*$, once more, and we perform a decomposition similar to \nEq. (\\ref{v(t)(1)}):\n\\begin{equation}\n{\\bf v}(t)=c_0\\lambda_0^t{\\bf h}+c_1\\lambda_1^t{\\bf g}+p^t\\sum_{j=2}^{\\infty}\nc_j{\\bf e}^{(j+t)}, \n\\label{v(t)(2)}\n\\end{equation}\nwhere $c_0$ and $c_1$ are the same as in Eq. (\\ref{finite eigenvectors}) and\nthe constants $c_j$ for $j \\geq 2$ are determined by the initial condition\n${\\bf v}(0)$. The vectors ${\\bf h}$ and ${\\bf g}$ are infinite \ngeneralizations of the finite vectors $\\tilde {\\bf h}$ and $\\tilde {\\bf g}$,\naccording to\n\\begin{equation}\n\\begin{array}{lll}\nh_j=\\tilde h_j,&g_j=\\tilde g_j, & j=0,1, \\\\\nh_2=p_1\\tilde h_1,&g_2=p_1\\tilde g_1, &j=2,\\\\\nh_j=h_2\\left({p\\over \\lambda_0} \\right)^{j-2},\n&g_j=g_2\\left({p\\over \\lambda_1} \\right)^{j-2},& j\\geq 2.\n\\end{array}\n\\end{equation}\nBecause $p=E_{\\infty+1,\\infty}>|\\lambda_0|,|\\lambda_1|$, it is apparent that\nthe components $h_j$ and $g_j$ diverge exponentially for large $j$'s. This\nmeans that these vectors do not have a finite $L_1$ norm, and that they do\nnot belong to the domain of ${\\bf E}$. Therefore, they are not eigenvectors,\nand $\\lambda_0$ and $\\lambda_1$ are not eigenvalues of ${\\bf E}$. Nevertheless,\nEq. (\\ref{v(t)(2)}) is still true. The effect of the shift down operator\nis manifested in the sum $p^t\\sum_{j=2}^{\\infty}c_j{\\bf e}^{(j+t)}$. \n\nUsing the same method it is possible to make higher order calculations. The \nsteady state quantities resulting from the third order approximation are\npresented in Table \\ref{steady3}, in comparison with exact results.\nThe eigenvalue with the largest absolute value is $\\lambda_0^{(3)}=-0.5687$,\nwhich has a greater absolute value than $E_{\\infty+1,\\infty}=0.5658$. This\nmeans that a legitimate eigenvector exists for the infinite matrix.\nIn the fourth and fifth order approximation we get \n$\\lambda_0^{(4,5)}\\approx -0.5688$.\nThis suggests that the higher the order the more accurate is the evaluation of\n$\\lambda_0$ and that the accuracy obtained is better than $10^{-4}$. The \ntypical time needed to settle in the steady state from any initial condition \nis therefore as short as \n\\begin{equation}\n\\tau=1.8.\n\\end{equation}\n\n\\section{DLA with $N>2$}\n\\label{N>2}\nThe generalization of the exact methods from Ref.\n\\cite{Kol98} to $N>2$ is\nnot straightforward. Trying to proceed along a similar line, one would try to \nparameterize the interface with a parameter $i=1,2,\\dots,\\infty$, and write\nthe Master equation $P_i(t+1)=\\sum_{j=1}^{\\infty}E_{i,j}P_j(t)$. Unlike the \ncase $N=2$, the parameterization for $N>2$ is very complicated. \nFor instance, for the case $N=3$ it is reasonable to try\nusing two parameters, which indicate the height of two columns relative to\nthe highest (or lowest) third column. However,\nthis is insufficient because complex\nfjords (involving overhangs) might occur, as shown in Fig. \\ref{N3examplefig}.\nInstead of achieving\na perfect parameterization, we adopt the approximate approach of \nSec. \\ref{approxN=2},\ni.e., we take into account only a finite number of interface configurations. \nThese configurations are classified \naccording to the maximum height difference between the highest and lowest \nparticles on the interface $\\Delta m$. In the $O$th-order \napproximation all the configurations with $\\Delta m \\leq O$ are \nincluded. The excluded configurations with $\\Delta m> O$ are \ntransformed into a configuration with $\\Delta m=O$, by filling in\nthe $(O+1)$th row below the highest particle; see Fig. \\ref{truncatefig}. \nThis transformation does not\nchange the growth probabilities considerably. Especially, the upward growth \nprobability would hardly change for large $O$. \nThe variable $P_i(t)$, where $i$ \ncorresponds to a configuration with $\\Delta m=O$, actually represents\nthe sum of probabilities of all the configurations with \n$\\Delta m \\geq O$, that have the same $O$ uppermost rows, \nrather than represent the probability of the configuration $i$ alone.\nThis is analogous to $\\tilde P_2^*$ in the example above, see \nSec. \\ref{approxN=2}. \nAfter the finite set of configurations\nis chosen, the configurations are indexed with arbitrary \nconsecutive numbers. Then, the growth probabilities for each configuration are\ncomputed by solving the Laplace equation and by taking\ninto account the bond multiplicity. Each growth process results in a different\nfinal configuration, which must be identified with one of the configurations\nin the finite set. Special attention is required for the upward growth \nprocesses, which might result in configurations with $\\Delta m>O$,\nwhich do not belong to the finite set. This is rectified by truncating the\nbottom row of the interface (considering it as fully occupied). \nThe total upward probability for each \nconfiguration is added up and stored in a function $p_{\\rm up}(i)$, later\nto be averaged over the steady state distribution of configurations. The\ngrowth probabilities are arranged in the evolution matrix, ${\\bf E}$, whose\nfixed point corresponds to the steady state distribution of configurations, \nwhich is required for evaluating $\\langle p_{\\rm up} \\rangle^*$, $\\rho$ \nand $D$. Because the matrix\nis finite, the existence of at least one fixed point is guaranteed. The other \neigenvectors describe the rate of convergence to the steady state. \n\nThe best way to demonstrate this approach is by showing a few\nsample calculations. The easiest ones are the first and second order \napproximation for $N=3$ and the first order approximation for $N=4$.\nAfter that we explain the general algorithm for higher orders and widths,\nand report the results obtained numerically.\n\\subsection{First order approximation for $N=3$}\nIn the first order approximation we only look at the top row of the\naggregate. For $N=3$ there are only $3$ possible configurations (up to\nsymmetry), with the top row occupied by $1$, $2$ or $3$ particles. \nEach configuration is indexed and for each configuration we identify the \ngrowth processes and the final configurations resulting from them; see\nFig. \\ref{N3o1indexfig}. In part II of this paper we show that the calculation\npresented in this section can be used to solve exactly (no approximations)\nthe case of site-DLA with $N=3$.\n\nThe first configuration $(j=1)$ grows upward with probability $1$,\nthus $p_{\\rm up}(1)=1$. The resulting configuration is $i=2$, thus $E_{2,1}=1$\nand $E_{i,1}=0$ for $i \\neq 2$. This concludes the construction of the\nfirst column of the evolution matrix. \n\nIn order to obtain the other growth\nprobabilities we have to solve the relevant Laplace problems, for which we need\nthe Green's function according to Eq. (\\ref{g_N(n)}). For $N=3$ we have \n$k_l={2\\pi \\over 3}l$ for $l=0,1,2$. We recall that $e^{-\\kappa (k)}=\nq-\\sqrt{q^2-1}$, where $q \\equiv 2-\\cos(k)$ \\cite{Kol98} and find that\n\\begin{eqnarray}\n&&e^{-\\kappa_0}=1, \\nonumber \\\\\n&&e^{-\\kappa_1}=e^{-\\kappa_2}={5-\\sqrt{21}\\over 2},\n\\end{eqnarray}\nand thus\n\\begin{eqnarray}\ng_3(0)={1 \\over 3}\\left(1+2{5-\\sqrt{21}\\over 2}\\right)={6-\\sqrt{21}\\over 3},\n\\nonumber \\\\\ng_3(1)=g_3(2)={1-g_3(0) \\over 2}={\\sqrt{21}-3\\over 6}.\n\\end{eqnarray}\nThese values obey the normalization condition (\\ref{norm}).\n\nBecause of the symmetry of the configuration $j=2$, the potential can be \nexpressed in terms of one variable $x\\equiv \\Phi(0,0)=\\Phi(0,2)$, as shown\nin Fig. \\ref{N3o1j=2}. This kind of figure demonstrates the distribution of \nthe potential $\\Phi(m,n)$ over the lattice, and thus we call it a ``potential\ndiagram''. The potentials $\\Phi(1,0)=\\Phi(1,2)=1+(1-g_3(1))x$ do\nnot correspond to a growth process, but are important for solving for $x$. The\npotential $\\Phi(1,1)=1+2xg_3(1)$ corresponds to the upward growth process.\nThe Laplace equation for $x$ is\n\\begin{eqnarray}\n4x&=&x+(1-g_3(1))x+1, \\nonumber \\\\\n\\Rightarrow x&=&{9-\\sqrt{21}\\over 10}=0.4417.\n\\end{eqnarray}\nGrowth in both sites $(0,0)$ and $(0,2)$ results in configuration $i=3$,\nhence \n\\begin{equation}\nE_{3,2}={4 \\over 3}x={18-2\\sqrt{21} \\over 15}=0.5890,\n\\end{equation}\nwhere the numerator, $4$, is inserted because there are $2$ \nbonds for each of the $2$ growth sites, and the denominator is the \nnormalization factor $N=3$. A growth process in site $(1,1)$ results in an\ninterface that does not belong to our finite set. In this approximation we\nonly take into account the top most row of the interface, and therefore this\ninterface is identified with configuration $i=2$, i.e., \n\\begin{equation}\nE_{2,2}={2xg_3(1)+1\\over 3}={2\\sqrt{21}-3\\over 15}=0.4110.\n\\end{equation} \nThe transition to $i=1$ is impossible, hence, $E_{1,2}=0$. It is easy to check\nthat\nthe second column of the matrix is normalized, i.e., $\\sum_{i=1}^3E_{i,2}=1$. \nThe total upward growth probability for this configuration is \n\\begin{equation}\np_{\\rm up}(2)=E_{2,2}=0.4110.\n\\end{equation}\n\nThe potentials of configuration $j=3$ are described in terms of $x=\\Phi(0,1)$,\nas in Fig. \\ref{N3o1j=3}. The Laplace equation is \n\\begin{eqnarray}\n4x&=&g_3(0)x+1, \\nonumber \\\\\n\\Rightarrow x&=&{6-\\sqrt{21}\\over 5}=0.2835.\n\\end{eqnarray}\nThere are $3$ bonds leading to growth in site $(1,0)$, which results in the\nconfiguration $i=1$, hence \n\\begin{equation}\nE_{1,3}={3 \\over 3}x=0.2835.\n\\end{equation}\nThe upward growth process results in $i=2$ after truncation, and has \nprobability\n\\begin{equation}\np_{\\rm up}(3)=E_{2,3}={2 \\over 3}(1+g_3(1)x)=0.7165.\n\\end{equation}\nThe third element in the column is $E_{3,3}=0$, which concludes the calculation\nof the elements of the evolution matrix,\n\\begin{equation}\n{\\bf E}^{(3,1)}=\\left[\n\\begin{array}{ccc}\n0&0&0.2835 \\\\\n1&0.4110&0.7165 \\\\\n0&0.5890&0\n\\end{array}\n\\right],\n\\end{equation}\nwhere the superscript indicates that it is the first-order approximation\nfor $N=3$.\nThe upward growth probabilities series is \n\\begin{equation}\np_{\\rm up}=(1,0.4110,0.7165),\n\\end{equation}\nwhich happens to be equal to the second row of the matrix.\n\nThe normalized fixed point of the matrix is $P^*_1=0.0951$, $P^*_2=0.5695$\nand $P^*_3=0.3354$. The average upward growth probability is\n\\begin{equation}\n\\langle p_{\\rm up} \\rangle^*=\\sum_{i=1}^3P^*_ip_{\\rm up}(i)=0.5695.\n\\end{equation}\nWe have performed some DLA simulations in the cylindrical geometry for \nseveral values of $N$ and measured $\\langle p_{\\rm up} \\rangle^*$ \n\\cite{Kol2000}.\nThe value obtained from simulations for $N=3$ is $0.5462$. The typical\naccuracy is on the order of $10^{-4}$. \nThe steady state average density and fractal dimension are evaluated using\nEqs. (\\ref{rho}) and (\\ref{estimate dimension}), \n\\begin{eqnarray}\n&&\\rho={1\\over 3\\langle p_{\\rm up} \\rangle^*}=0.5853,~(0.6103), \\nonumber \\\\\n&&D=1-{\\ln(\\langle p_{\\rm up} \\rangle^*)\\over \\ln(3)}=1.5125,~(1.5506).\n\\end{eqnarray}\nThe values in parentheses are obtained from the same formulae, using the \nsimulation value of $\\langle p_{\\rm up} \\rangle^*$. \nThe two other eigenvalues are complex,\n$\\lambda_{0,1}=-0.29\\pm0.28i$, so according to Eq. (\\ref{tau=}) $\\tau=1.10$.\n\n\\subsection{Higher-order approximations for $N=3$}\nThe possible configurations of the interface in the second-order approximation\nare listed and\nindexed in Fig. \\ref{N3o2index}. The growth probabilities for the first $3$ \nconfigurations were already computed in the previous section, but a \nrearrangement of the upward growths is required in the evolution matrix.\nNow, the upward growth from configuration $j=2$ no longer stays at $i=2$, \nbut rather makes a transition to $i=4$, and the upward growth from $j=3$ \nresults in $i=5$ instead of $i=2$. Thus, we copy the previous\nevolution matrix ${\\bf E}^{(3,1)}$ into the upper left corner of the new\nmatrix ${\\bf E}^{(3,2)}$ with the replacements: $E^{(3,2)}_{2,2}=0$,\n$E^{(3,2)}_{4,2}=E^{(3,1)}_{2,2}$, $E^{(3,2)}_{2,3}=0$, and $E^{(3,2)}_{5,3}=\nE^{(3,1)}_{2,3}$. The unspecified elements in the first three columns are all\nequal to zero.\n\nThe next step is to go over each of the remaining configurations $i=4,\\dots,7$,\nand compute their probabilities, which are inserted into the \nevolution matrix according to the final configuration in which the relevant \ngrowth process results. Configuration $4$ is shown in Fig. \\ref{N3o2j=4}. \nThe Laplace equation is\n\\begin{eqnarray}\n4y&=&y+x,\\nonumber \\\\\n4x&=&x+y+1+(1-g_3(1))x, \\nonumber \\\\\n\\Rightarrow x&=&{3 \\over 14}(7-\\sqrt{21})=0.5180, \\nonumber \\\\\ny&=&x/3=0.1727.\n\\end{eqnarray}\nThe growth probabilities are\n\\begin{eqnarray}\nE_{6,4}&=&{2 \\over 3}x=0.3453, \\nonumber \\\\\nE_{5,4}&=&{4 \\over 3}y={4 \\over 9}x=0.2302, \\nonumber \\\\\nE_{4,4}&=&{1+2xg_3(1)\\over 3}=0.4244.\n\\end{eqnarray}\nThe upward growth probability is $p_{\\rm up}(4)=E_{4,4}=0.4244$.\n\nConfiguration $5$ is shown in Fig. \\ref{N3o2j=5}. The Laplace equations are\n\\begin{eqnarray}\n&4y=y/4+x+xg_3(1)+yg_3(0)+1,& \\nonumber \\\\\n&4x=y+g_3(0)x+g_3(1)y+1,& \\nonumber \\\\\n&\\Downarrow& \\nonumber \\\\\n&y=0.4808,& \\nonumber \\\\\n&x=0.4557.&\n\\end{eqnarray}\nThe growth probabilities are\n\\begin{eqnarray}\nE_{7,5}&=&{2 \\over 3}x=0.3038, \\nonumber \\\\\nE_{3,5}&=&{y \\over 3}=0.1603, \\nonumber \\\\\nE_{2,5}&=&{3 \\over 3}y/4=0.1202, \\nonumber \\\\\nE_{4,5}&=&{1+g_3(1)(x+y) \\over 3}=0.4157. \n\\end{eqnarray}\nThe upward growth probability is $p_{\\rm up}(5)=E_{4,5}=0.4157$.\n\nConfiguration $6$ is shown in Fig. \\ref{N3o2j=6}. The Laplace equations are\n\\begin{eqnarray}\n&4y=y/4+x,& \\nonumber \\\\\n&4x=y+g_3(0)x+1,& \\nonumber \\\\\n&\\Downarrow& \\nonumber \\\\\n&x={15 \\over 151}(26-5\\sqrt{21})=0.3067,& \\nonumber \\\\\n&y={4 \\over 15}x=0.0818.&\n\\end{eqnarray}\nThe growth probabilities are\n\\begin{eqnarray}\nE_{1,6}&=&{2 \\over 3}x=0.2044, \\nonumber \\\\\nE_{3,6}&=&{2 \\over 3}y={8 \\over 45}x=0.0545, \\nonumber \\\\\nE_{7,6}&=&{3 \\over 3}y/4=x/15=0.0204, \\nonumber \\\\\nE_{5,6}&=&{2 \\over 3}(1+g_3(1)x)=0.7206.\n\\end{eqnarray}\nThe upward growth probability is $p_{\\rm up}(6)=E_{5,6}=0.7206$.\n\nConfiguration $7$ is shown in Fig. \\ref{N3o2j=7}. The Laplace equations are\n\\begin{eqnarray}\n&4x=x/4+g_3(0)x+1,& \\nonumber \\\\\n&\\Downarrow& \\nonumber \\\\\n&x={12 \\over 105}(21-4\\sqrt{21})=0.3051.&\n\\end{eqnarray}\nThe growth probabilities are\n\\begin{eqnarray}\nE_{1,7}&=&{2 \\over 3}x=0.2304, \\nonumber \\\\\nE_{3,7}&=&{3 \\over 3}x/4=0.0763, \\nonumber \\\\\nE_{5,7}&=&{2 \\over 3}(1+g_3(1)x)=0.7203.\n\\end{eqnarray}\nThe upward growth probability is $p_{\\rm up}(7)=E_{5,7}=0.7203$.\n\nIn summary,\n\\begin{eqnarray}\n&&{\\bf E}^{(3,2)}= \\nonumber \\\\\n&&\\left[\n\\begin{array}{ccccccc}\n0&0 &0.2835 &0 &0 &0.2044 &0.2034 \\\\\n1&0 &0 &0 &0.1202 &0 &0 \\\\\n0&0.5890 &0 &0 &0.1603 &0.0545 &0.0763 \\\\\n0&0.4110 &0 &0.4244 &0.4157 &0 &0 \\\\\n0&0 &0.7165 &0.2302 &0 &0.7206 &0.7203 \\\\\n0&0 &0 &0.3453 &0 &0 &0 \\\\\n0&0 &0 &0 &0.3038 &0.0204 &0\n\\end{array}\n\\right], \\nonumber \\\\\n\\end{eqnarray}\n\\begin{eqnarray}\n&&p_{\\rm up}= \\nonumber \\\\\n&&\\left(\n\\begin{array}{ccccccc}\n1,&0.4110,&0.7165,&0.4244,&0.4157,&0.7206,&0.7203\n\\end{array}\n\\right). \\nonumber \\\\\n\\end{eqnarray} \nOne can check that elements in each column of the matrix sum up to $1$. \nNote that the majority of the elements are null.\nThe normalized fixed point is,\n\\begin{eqnarray}\n{\\bf P}^*=&\\left( \n\\begin{array}{cccc}\n0.0685,&0.1011,&0.1145,&0.2680,\n\\end{array} \\right.& \\nonumber \\\\\n&\\left.\n\\begin{array}{ccc}\n0.2711,&0.0925,&0.0843\n\\end{array}\n\\right)&\n\\end{eqnarray}\nwith which we compute some steady state quantities,\n\\begin{eqnarray}\n&&\\langle p_{\\rm up} \\rangle^*=\\sum_{j=1}^7P^*_jp_{\\rm up}(j)=0.5459,~(0.5462),\n\\nonumber \\\\\n&&\\rho={1 \\over 3\\langle p_{\\rm up} \\rangle^*}=0.6106,~(0.6103), \\nonumber \\\\\n&&D=1-{\\ln(\\langle p_{\\rm up} \\rangle^*)\\over \\ln(3)}=1.5510,~(1.5506),\n\\end{eqnarray}\nwhere once again, the values from simulation are shown in parentheses. \nIt is apparent that\nthe addition of configurations increases the accuracy of the results.\nThe eigenvalues with the largest absolute values (except for $1$) are\n$\\lambda_{0,1}=-0.34\\pm0.40i$, hence $\\tau=1.6$.\n \nThe third-order approximation yields $17$ configurations. The final\nresults are\n\\begin{eqnarray}\n&&\\langle p_{\\rm up} \\rangle^*=\\sum_{j=1}^{17}P^*_jp_{\\rm up}(j)=0.5460,\n~(0.5462), \\nonumber \\\\\n&&\\rho={1 \\over 3\\langle p_{\\rm up} \\rangle^*}=0.6104,~(0.6103), \\nonumber \\\\\n&&D=1-{\\ln(\\langle p_{\\rm up} \\rangle^*)\\over \\ln(3)}=1.5507,~(1.5506).\n\\end{eqnarray}\nThe eigenvalues with the largest absolute values (except for $1$) are\n$\\lambda_{0,1}=-0.34\\pm0.40i$, hence $\\tau=1.6$. \n\nIt is interesting to inspect the histogram of\nthe distribution of $p_{\\rm up}(j)$, illustrated in Fig. \\ref{pupdistribfig}. \nOne immediately observes that the upward growth \nprobabilities are clustered in three groups: the top one at $1$, the second \njust above $0.7$ and the third, just above $0.4$. It is easy to check that the\ntop one corresponds to the configuration $i=1$, the middle group corresponds to\nconfigurations that have two particles at the top row, and the bottom group\ncorresponds to configurations with one particle at the top row. This suggests,\nthat perhaps $17$ different configurations are excessive, and the real number\nof effective configurations is around $3$. An interesting question is whether \nit is possible to further reduce the number of configurations in higher-order \napproximations by including only ``effective'' ones.\n\n\\subsection{First-order approximation for $N=4$}\nOur last example is the case $N=4$, for which we present the first-order\ncalculation. First,\nwe calculate the Green's function $g_4(n)$ according to Eq. (\\ref{g_N(n)}).\nFor $N=4$, there are four possible values for $k$ and $\\kappa$, namely,\n$k_l={2\\pi \\over N}l=0,{\\pi \\over 2},\\pi,{3 \\over 2}\\pi$, $e^{-\\kappa_0}=1$,\n$e^{-\\kappa_1}=e^{-\\kappa_3}=2-\\sqrt{3}$, and $e^{-\\kappa_2}=3-\\sqrt{8}$.\nHence,\n\\begin{eqnarray}\ng_4(0)&=&{1+2(2-\\sqrt{3})+3-\\sqrt{8} \\over 4} \\nonumber \\\\\n&=&2-{\\sqrt{3}+\\sqrt{2} \\over 2}=0.4269, \\nonumber \\\\\ng_4(1)&=&g_4(3)={1-3+\\sqrt{8} \\over 4}={\\sqrt{2}-1 \\over 2}=0.2071, \n\\nonumber \\\\\ng_4(2)&=&{1-2(2-\\sqrt{3})+3-\\sqrt{8}\\over 4}\n={\\sqrt{3}-\\sqrt{2} \\over 2}=0.1589. \\nonumber \\\\\n\\end{eqnarray}\nOnce again, Eq. (\\ref{norm}) is obeyed.\n\nFigure \\ref{N4o1index} displays the relevant configurations. \nConfiguration \n$j=1$ grows into configuration $i=2$ with probability $1$, thus $E_{2,1}=1$ and\n$E_{i,1}=0$ for $i\\neq 2$. Also, $p_{\\rm up}(1)=1$. \n\nConfiguration $j=2$ is shown in Fig. \\ref{N4o1j=2}. The Laplace equations are\n\\begin{eqnarray}\n&4x=y+g_4(1)y+\\left(g_4\\left(0\\right)+g_4\\left(2\\right)\\right)x+1,& \n\\nonumber \\\\\n&4y=2x+g_4(0)y+2g_4(1)x+1,& \\nonumber \\\\\n&\\Downarrow & \\nonumber \\\\\n&x=0.5148,& \\nonumber \\\\\n&y=0.6277.&\n\\end{eqnarray}\nThe nonzero growth probabilities in the second column are\n$E_{3,2}={4 \\over 3}x=0.5148$, $E_{4,2}={1\\over 4}y=0.1569$, and\n$E_{2,2}={1 \\over 4}\\left(1+2g_4\\left(1\\right)x+g_4\\left(2\\right)y\\right)\n=0.3283=p_{\\rm up}(2)$.\n\nConfiguration $j=3$ is presented in Fig. \\ref{N4o1j=3}. The Laplace equation is\n\\begin{eqnarray}\n4x&=&x+\\left(g_4\\left(0\\right)+g_4\\left(1\\right)\\right)x+1, \\nonumber \\\\\n\\Rightarrow x&=&0.4226.\n\\end{eqnarray}\nThe nonzero growth probabilities in the third column are $E_{5,3}={4\\over 4}x\n=0.4226$ and $E_{2,3}={2\\over4}\\left[1+\\left(g_4\\left(1\\right)+g_4\\left(2\n\\right)\\right)x\\right]=0.5774=p_{\\rm up}(3)$.\n\nConfiguration $j=4$ is shown in Fig. \\ref{N4o1j=4}. The Laplace equation is\n\\begin{eqnarray}\n4x&=&\\left(g_4\\left(0\\right)+g_4\\left(2\\right)\\right)x+1, \\nonumber \\\\\n\\Rightarrow x&=&0.2929.\n\\end{eqnarray}\nThe nonzero growth probabilities in the fourth column are $E_{5,4}={6\\over 4}\nx=0.4393$ and $E_{2,4}={2\\over 4}\\left(1+2g_4\\left(1\\right)x\\right)=0.5607\n=p_{\\rm up}(4)$. Note that this configuration already appeared for $N=2$.\n\nThe last configuration is shown in Fig. \\ref{N4o1j=5}. The Laplace equation is\n\\begin{eqnarray}\n4x&=&1=g_4(0)x, \\nonumber \\\\\n\\Rightarrow x&=&0.2799.\n\\end{eqnarray}\nThe nonzero growth probabilities in the fifth column are $E_{1,5}={3\\over 4}x\n=0.2099$ and $E_{2,5}={1\\over 4}\\left[3+\\left(g_4\\left(2\\right)+2g_4\\left(1\n\\right)\\right)x\\right]=0.7901=p_{\\rm up}(5)$. This concludes the calculation\nof the $5\\times 5$ evolution matrix ${\\bf E}^{(4,1)}$.\n\nThe steady-state vector is \n\\begin{equation}\n{\\bf P}^*=\\left(\n\\begin{array}{ccccc}\n0.0298,&0.4954,&0.2551,&0.0777,&0.1420\n\\end{array}\n\\right).\n\\end{equation}\nIt enables to calculate the following steady-state quantities:\n\\begin{eqnarray}\n&&\\langle p_{\\rm up}\\rangle^*=P_2^*=0.4954,~(0.4657), \\nonumber \\\\\n&&\\rho={1\\over 4\\langle p_{\\rm up}\\rangle^*}=0.5046,~(0.5368) \\nonumber \\\\\n&&D=1-{\\ln(\\langle p_{\\rm up}\\rangle^*)\\over \\ln(4)}=1.5066,~(1.5512),\n\\end{eqnarray}\nwhere again, the values in parentheses are from simulation.\nThe eigenvalues with the largest absolute value after $1$ are $\\lambda_{0,1}=\n-0.16\\pm0.38i$, hence $\\tau=1.1$.\n\nIt is also possible to conduct these calculations using \ndifferent boundary conditions at the bottom; rather than assuming that there \nis a filled row of occupied sites below the configuration, it\nis possible to assume that each unoccupied site at the lowest row of the\nconfiguration is above an infinite fjord that extends all the way below.\nThe two possibilities are explained in Fig. \\ref{bottombcfig}. Performing the \ncalculations with infinite fjords is a bit simpler, because there are less\nconfigurations, e.g., the configuration $i=4$ would not appear in the \nfirst-order approximation for $N=4$ \\cite{Kol2000}.\n\n\\subsection{Higher order computations}\nAs one increases $N$ and the order of approximation $O$, the number of \nconfigurations increases exponentially, and it becomes harder to go over all of\nthem manually. However, it is possible to construct a computer algorithm\nto perform the procedure described here. The main challenges are the\nautomatic configuration recognition and automatic computation of the exact \ngrowth probabilities per configuration. \nIn this section we explain the algorithm and report some of the important \nresults.\n\nThe algorithm follows the method outlined in the examples of the \nprevious sections, i.e., it goes over all the possible configurations\nof the interface. In the sample calculations we have initially\nmade a list of all the possible configurations, called the index. \nInstead of doing this, the program \nstarts with only one configuration, namely the flat one\n(all the sites of the top row of the aggregate are occupied), which is indexed\nby $j=1$. This configuration grows with probability $1$ to a new configuration\nthat has one particle at the top row, while the row below it is fully \noccupied. This new configuration is inserted into the list of \nconfigurations with an index $j=2$. Therefore, the program sets \n$E_{2,1}=1$ and $p_{\\rm up}(1)=1$. Then the program continues by handling the\nnext configuration in the list, namely $j=2$. For each configuration, it \nsolves the Laplace equations and calculates the growth probabilities. Each\ngrowth process may create a new configuration. The resulting configuration is\nfirst checked for consistency with the desired order $O$; configurations\nwhich have $\\Delta m>O$ are truncated, as in Fig. \\ref{truncatefig}.\nOne then compares each\n'new' configuration with\nthe existing list of configurations. If it does not exist in that list it \nis added at the end of the list, and indexed consecutively. If the index of the\nconfiguration that results from the growth process is $i$ and the index of\nthe initial configuration is $j$ then the growth probability is inserted into\nthe matrix element $E_{i,j}$. The total sum of all the upward growth \nprobabilities of the initial configuration $j$ is stored in $p_{\\rm up}(j)$.\nThe main loop stops when the program finishes to process the last configuration\nin the index list. At this stage the Markovian evolution matrix ${\\bf E}$ is \nirreducible and closed, i.e., $\\sum_iE_{i,j}=1$ for every $j$. Then\nthe fixed point ${\\bf P}^*$ is calculated, by taking an\ninitial vector and iterating ${\\bf E}$ on it many times until it converges \n(for very large matrices this is much faster than using any of the \nMATLAB library functions). \nThe average upward growth probability is calculated using\n\\begin{equation}\n\\langle p_{\\rm up} \\rangle^*=\\sum_jP^*_jp_{\\rm up}(j),\n\\end {equation}\nthe average density and the fractal dimension are then computed using the left\nhand side of Eq. (\\ref{rho}) and Eq. (\\ref{estimate dimension}).\n\nOne of the challenges of the computer algorithm is the recognition of \nconfigurations. This recognition is important so that each growth process will\nbe inserted into the evolution matrix $E_{i,j}$ with the correct index $i$\n($j$ is the index of the configuration before growth). The recognition\nmaybe difficult because configurations that seem different may actually be\nequivalent. By equivalent we mean that they have the exact same set of \ntransition (growth) probabilities. The solution to the Laplace equations \nis determined uniquely by the shape of the interface, therefore all of the\nconfigurations with the same external interface are equivalent. \nThe description of the interface is not a trivial task though. We find that an\nefficient way to characterize an interface is by the set of empty sites that\nare connected to infinity. Of course, it is sufficient to specify only empty\nsites that are not higher than the highest particle in the aggregate, because\nall of the empty sites above it are connected to infinity. Figure\n\\ref{N3example2fig} shows an example of two configurations that are not \nidentical, but they have the same exterior contour. Both of them have a single\nempty site that is connected to infinity.\n\nIn order to reduce greatly the number of configurations it is\nadvisable to take symmetry into account, i.e., all the configurations which can\nbe obtained from one another using a rotation around the axis of the cylinder \nhave the same growth probabilities and the same steady state weights. The same\nis true for mirror images. Instead of taking all of them into account, we \nchoose one as a canonical representative of the whole set of symmetric\nconfigurations.\n\nThe results are summarized in Table \\ref{2dresults}. \nBy comparing the approximations to accurate results from simulations, \nit seems that in order\nto obtain a relative accuracy of about $10^{-3}$ one has to use at least \nan order of approximation of $O=N-2$ \n(except for $N=3$, where one still has to use the second-order approximation). \nThis becomes very difficult already for $N=6$,\nwhere in the fourth-order calculation there are $49678$ different \nconfigurations up to symmetry.\n\n\\section{Discussion}\n\\label{Discussionsec}\nThis paper treats DLA as a Markov process. The Markov states are the possible\nshapes of the interface, and the Markovian evolution matrix ${\\bf E}$\nis calculated analytically using exact solutions of the \nLaplace equations, with proper normalizations. We propose\na truncation scheme that takes into account only a finite number of \nstates. The states are ordered according to\nthe maximal difference in height between the highest and lowest points on\nthe interface, $\\Delta m$, and in each order of truncation $O$, \nonly the states with $\\Delta m\\leq O$ are included. \nWe justify this approach by the fact that the\npotential $\\Phi$ decays exponentially in deep fjords, and thus the shape of\nthe interface in its deeper parts has very little effect on the growth \nprobabilities. We perform this calculation for $N=2$, and verify that indeed \nit converges to the known analytic solution. We adopt the same approach for \nhigher values of the width $N$, between $3$ and $7$, and calculate the average\ndensity $\\rho$ in good agreement with simulations. The fact that the\nnumber of configurations grows exponentially with $N$ and with $O$, makes\nthe computation less effective than simulation for large $N$. \n\nWe observe that the method converges as a function of $O$, \nalso for higher values of $N$. \nLet us denote the calculated average steady-state density of an\naggregate of width $N$ in the $O$'th-order approximation by $\\rho_c(N,O)$.\nWe observe that $\\rho_c(N,O)$ converges to a finite limit very rapidly as\na function of $O$. In fact, a relative accuracy of $10^{-3}$ is achieved for \n$O=N-2$ (except for $N=3$). \nThis enables us to obtain accurate results for $3 \\leq N \\leq 6$.\nThe drawback of this method is that the number of configurations diverges \nexponentially with $O$ and $N$, and therefore it is possible to perform\nthe calculations only for relatively low $N$'s and $O$'s.\nOur computer was strong enough to perform the calculation only in the \nthird-order approximation for $N=7$, and therefore the result for $N=7$ \nis not very accurate.\nOne would hope that it may be possible to perform low-order approximations for\nlarge $N$'s and then extrapolate, in order to estimate the results for large \n$O$'s. Indeed, it is reasonable to conjecture \nthe scaling law $\\rho_c(N,O)=\\rho(N)f(N/O)$, where $\\rho(N)$ is the exact \n($O\\to \\infty$) density, as a function of $N$, and $f(N/O)$ is a universal\nscaling function that obeys $\\lim_{x\\to 0}f(x)=1$. Our investigation shows\nthat in spite of the fact that the conjecture is not very accurate for $O=1$\nand $O=2$, it is quite good for higher values of $O$, and presumably also for\nhigher values of $N$. This scaling relation may help to perform the \nextrapolation $O \\to \\infty$ for higher values of $N$. Paradoxically, it is \nvery hard to obtain data points for large $N$'s and $O$'s, and thus to extract\nthe scaling function accurately. Thus we are unable to make the extrapolation\neven for $N=7$, and we estimate $\\rho(N)$ by the highest-order approximation\navailable. \nHowever, we suggest an alternative way to obtain \n$\\rho_c(O,N)$, namely by simulation: it is possible to perform a regular\nDLA simulation in cylindrical geometry, only that one has to keep the $O$'th \nrow below the highest particle in the aggregate constantly filled. Measuring\nthe average density of the aggregate in such a simulation would approximate\n$\\rho_c(N,O)$. This simulation would be faster than a regular simulation, \nbecause particles would stick faster, due to the fact that they have less free\nspace. This study\nwould perhaps yield the scaling function $f(N/O)$, and enable extrapolation\nof lower order approximations for higher $N$'s, should anyone venture to \nperform them on more powerful computers. In light of this discussion we \nsuggest a more efficient way to perform DLA simulations in\ncylindrical geometry. We argue that one can obtain a relative accuracy\nof $10^{-3}$ if one follows just the $N-2$ top most rows of the aggregate.\nThis should save some time, because the diffusing particle would stick faster,\nand it would also require less memory. This is not to say that it is sufficient\nto grow the aggregate until it reaches a height of $N-2$, but rather, to \nperform many more growth processes, and each time the aggregate reaches\na height of $N-1$, truncate the bottom row.\n\nWe also discuss the temporal rate of convergence of the system to its\nsteady state. In this context we find that there is an\nexponential convergence to the steady state, and we calculate the\ncharacteristic time constant $\\tau$. This is demonstrated using the simple\nmodel of the frustrated climber.\nThe convergence is described in terms\nof the eigenvalues of the Markovian matrix, and in terms of the infinite \nshift-down operator. \n\nConsidering the fractal dimension, Pietronero {\\em et al.} suggested \nthat $\\rho(N)=N^{D-d}$, as mentioned in Eq. (\\ref{rho=N^(D-d)}). \nIn principle, one should always include an amplitude\nand finite size corrections of the form\n\\begin{equation}\n\\rho(N)=AN^{-\\alpha}\\left(1+B/N+\\dots\\right),\n\\label{rho=AN^}\n\\end{equation}\nwhere $\\alpha=d-D$, and $A$ and $B$ are constants.\nThe second term appearing in Eq. (\\ref{rho=AN^}) is a correction to scaling\nterm. Generally, there is an infinite sum of such terms with higher \nnegative powers of $N$. \nBecause we have data only for small values of $N$, these correction terms\nmay be large, but since we have only a few accurate data points \n($\\rho(N)$ for $N=2,3,\\dots,6$), we try to extract the parameters \n$\\alpha$, $A$ and $B$ only, and not higher order terms. \nUsing the three results for\n$N=4,5,6$, we determine the three unkown parameters to be\n$A=0.82$, $B=0.35$\nand $\\alpha=0.362$, hence $D=1.64$.\nThe deviation from the well know value of $D=1.66$ can\nbe attributed to systematic error due to the omission of higher order \nfinite size correction terms. We fit simulation data \\cite{Kol2000} for \n$N=3$, $4$, $5$, $6$, $7$, $32$, $48$, $64$, $96$, $128$, to a higher-order\napproximation $\\rho(N)=AN^{-\\alpha}\\left(1+B/N+C/N^2\\right)$, and find that\n$C=-0.205$, $B=0.561$, $A=0.761$ and $\\alpha=0.339$, which means that \n$D=1.661$. The maximum relative error of the fit is $1.2\\times 10^{-3}$, and\nthe average relative error is $1.0 \\times 10^{-3}$, which is in good\nagreement with estimated accuracy of the simulations.\n\n\n\\acknowledgments\nWe wish to thank Barak Kol and A. Vespignani for helpful\ndiscussions. We also wish to thank Yiftah Navot for helping with the\ncomputer program, by suggesting more efficient data structures and algorithms.\nWe thank Nadav Schnerb for offering the frustrated climber\nmetaphor.\nThis work was supported by a grant from the German-Israeli Foundation (GIF).\n\n\\begin{references}\n\\bibitem{Witten83}T. A. Witten and L. M. Sander, Phys. Rev. B {\\bf 27}, 5686 (1983).\n\\bibitem{Mandelbrot82}B. Mandelbrot, {\\it The Fractal Geometry of Nature} (Freeman, New York, 1982).\n\\bibitem{Feder}J. Feder, {\\it Fractals} (Plenum Press, New York, 1988).\n\\bibitem{DBM84a}L. Niemeyer, L. Pietronero, and H. J. Wiesmann, Phys. Rev. Lett. {\\bf 52}, 1033 (1984).\n\\bibitem{Bak87}P. Bak, C. Tang, and K. Weisenfeld, Phys. Rev. Lett. {\\bf 59}, 381 (1987); Phys. Rev. A {\\bf 38}, 364 (1988).\n\\bibitem{Cafiero93}R. Cafiero, L. Pietronero, and A. Vespignani, Phys. Rev. Lett. {\\bf 70}, 3939 (1993).\n\\bibitem{Erzan95}A. Erzan, L. Pietronero, and A. Vespignani, Rev. Mod. Phys. {\\bf 67}, 545 (1995). \n\\bibitem{DBM84b}L. Pietronero and H. J. Wiesmann, J. Stat. Phys. {\\bf 36}, 909 (1984).\n\\bibitem{Kol98}B. Kol and A. Aharony, Phys. Rev. E {\\bf 58}, 4716 (1998), cond-mat/9802048.\n\\bibitem{Kol99}B. Kol And A. Aharony, to be published.\n\\bibitem{remark}In \\cite{Kol98} there are functions of the form\n$a(j)=a(\\infty)+x_0e^{-\\alpha j}(1+x_1e^{-\\beta j}+x_2e^{-2\\beta j}+\\dots)$,\nwhere $a(\\infty),x_0,\\alpha>0,\\beta>0,x_1,x_2,$ etc. are arbitrary constants.\nFor the present model, the average converges exponentially\n$\\langle a \\rangle (t)=\\langle a \\rangle^*+p^t\\sum_{j=0}^{\\infty}a(j+t)v_j(0)\n= \\langle a \\rangle^*+x_0p^te^{-\\alpha t} \\left[f_0+f_1e^{-\\beta t}\n+f_2e^{-2\\beta t}+ \\dots \\right ]$,\nwhere $\\langle a \\rangle^*\\equiv \\sum_{j=0}^{\\infty}qp^ja(j)$, \nand\n$f_i=x_i\\sum_{j=0}^{\\infty}e^{-(\\alpha+i\\beta)j}v_j(0)$.\nNote that the\nconstant $a(\\infty)$ does not affect the rate of convergence.\n\\bibitem{Turkevich89}L. A. Turkevich and H. Scher, Phys. Rev. Lett. {\\bf 55}, 1026 (1985).\n\\bibitem{Pietronero88b}L. Pietronero, A. Erzan, and C. Evertsz, Physica A {\\bf 151}, 207 (1988).\n\\bibitem{Kol2000}B. Kol and A. Aharony, unpublished.\n\\end{references}\n\n\n%\\end{document}\n\\end{multicols}\n\\widetext\n\n\\begin{figure}\n\\epsfysize 9cm\n\\epsfbox{stepfig.eps}\n\\vspace{5mm}\n\\caption{The coordinates $(m,n)$ describe the location on a lattice\nthat is two sites wide. The gray sites belong to the interface of the \naggregate, which is shaped as a step of size $j$.}\n\\label{stepfig}\n\\end{figure}\n\n\\begin{figure}\n\\epsfysize 9cm\n\\epsfbox{transitionsfig.eps}\n\\vspace{5mm}\n\\caption{Possible growth processes that change the interface from an initial\nstep size $j=3$ to a final size $i=4,0,1,2$. \nThe growth probability is determined by the potential and the number \nof bonds associated with the site where growth is to occur. $E_{i,j}$ is\nthe conditional probability to grow from an initial step size $j$ to a final\nstep size $i$. The normalization follows from Eq. (1.10).}\n\\label{transitionsfig}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3examplefig.eps}\n\\vspace{5mm}\n\\caption{An example of an interface configuration for $N=3$ that cannot be\ncharacterized using the height differences of two columns relative to the \nthird.}\n\\label{N3examplefig}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{truncatefig.eps}\n\\vspace{5mm}\n\\caption{Configuration (a), with $\\Delta m=2$, is truncated by \ntaking only the top row, and turns into configuration (b), with \n$\\Delta m=1$, in the first-order approximation ($O=1$).}\n\\label{truncatefig}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3o1index.eps}\n\\vspace{5mm}\n\\caption{The three possible configurations in the first-order \napproximation for $N=3$, up to translation symmetry. \nThe arrows indicate the possible \ntransitions due to growth processes. The transition probability from\nconfiguration $j$ to $i$ is denoted by $E_{i,j}$.}\n\\label{N3o1indexfig}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3o1j=2.eps}\n\\vspace{5mm}\n\\caption{A ``potential diagram'': the potentials $\\Phi(m,n)$ \nof the configuration $j=2$, expressed in terms of the variable $x$.}\n\\label{N3o1j=2}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3o1j=3.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=3$.}\n\\label{N3o1j=3}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3o2index.eps}\n\\vspace{5mm}\n\\caption{Possible configurations in the second-order\napproximation for $N=3$. Note that the first three configurations, $j=1,2,3$, \nare the same as in the first-order approximation.}\n\\label{N3o2index}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3o2j=4.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=4$.}\n\\label{N3o2j=4}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3o2j=5.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=5$.}\n\\label{N3o2j=5}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3o2j=6.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=6$.}\n\\label{N3o2j=6}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3o2j=7.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=7$.}\n\\label{N3o2j=7}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{pupdistribfig.eps}\n\\vspace{5mm}\n\\caption{The distribution of $p_{\\rm up}$ over configurations for the \nthird-order approximation for $N=3$.}\n\\label{pupdistribfig}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N4o1index.eps}\n\\vspace{5mm}\n\\caption{Possible configurations in the first-order\napproximation for $N=4$.}\n\\label{N4o1index}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N4o1j=2.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=2$.}\n\\label{N4o1j=2}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N4o1j=3.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=3$.}\n\\label{N4o1j=3}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N4o1j=4.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=4$.}\n\\label{N4o1j=4}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N4o1j=5.eps}\n\\vspace{5mm}\n\\caption{The ``potential diagram'' for configuration $j=5$.}\n\\label{N4o1j=5}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{bottombcfig.eps}\n\\vspace{5mm}\n\\caption{The two top rows of a configuration are shown in (a). Two possible\nextensions for the rest of the configuration below are (b), with\na filled\nrow right below the configuration (this boundary condition is used in the\ncalculations presented in this paper), or (c), with the bottom row of the \nconfiguration repeating itself ad infinitum, creating an infinite fjord.}\n\\label{bottombcfig}\n\\end{figure}\n\n\\begin{figure}\n\\epsfbox{N3example2fig.eps}\n\\vspace{5mm}\n\\caption{Even though configuration (a) and (b) are not identical,\nthey are equivalent because they have the same growth probabilities.\nBoth configurations have the same external interface contour, which is\ncharacterized by the set of sites that are connected to infinity. In this \nexample there is only one such site, which is not higher that the aggregate,\nand it is marked by a circle}\n\\label{N3example2fig}\n\\end{figure} \n\n\\begin{table}\n\\caption{The two-dimensional approximate \nresults for various channel widths $N$ and for\ndifferent orders of approximation $O$. The quantities presented in each table\ncell are the average upward growth probability \n$\\langle p_{\\rm up} \\rangle ^*$ and the number of configurations $N_c$. \nThe approximate results are compared with simulations.}\n\\begin{tabular}{c|c|c|c|c|c|c|c}\n$N/O$&simulation&1&2&3&4&5&6\\\\\n\\tableline\n3&0.5462&0.569489&0.545911&0.546046&0.546126&0.546132&0.546132 \\\\\n&&3&7&17&45&127&371 \\\\\n\\tableline\n4&0.4657&0.495435&0.464571&0.465395&0.465730&0.465765&0.465768 \\\\\n&&5&20&98&575&3640&23676 \\\\\n\\tableline\n5&0.4106&0.444088&0.407582&0.409497&0.410414&0.410547& \\\\\n&&7&47&457&5539&69791& \\\\\n\\tableline\n6&0.3696&0.405619&0.364352&0.367295&0.369172&& \\\\\n&&12&131&2217&49678&& \\\\\n\\tableline\n7&0.3377&0.375448&0.330112&0.333622&&& \\\\\n&&17&337&10403&&& \\\\\n\\end{tabular}\n\\label{2dresults}\n\\end{table}\n\n\\begin{table}\n\\caption{Some steady state results of the third order approximation}\n\\begin{tabular}{c|ccccccc}\n &$\\langle p_{\\rm up} \\rangle^*$&$P^*_0$&$P^*_1$&$P^*_2$&$P^*_3$&$P^*_4$\n&$P^*_5$ \\\\\n\\tableline\n$3$rd order&0.6812&0.2696&0.3114&0.1820&0.1029&0.0582&0.0329 \\\\\n\\tableline\nAccurate&0.6812&0.2696&0.3113&0.1809&0.1032&0.0586&0.0332 \\\\\n\\end{tabular}\n\\label{steady3}\n\\end{table}\n\n\\end{document}\n\n\n\n\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002083.extracted_bib",
"string": "\\bibitem{Witten83}T. A. Witten and L. M. Sander, Phys. Rev. B {\\bf 27}, 5686 (1983).\n\n\\bibitem{Mandelbrot82}B. Mandelbrot, {\\it The Fractal Geometry of Nature} (Freeman, New York, 1982).\n\n\\bibitem{Feder}J. Feder, {\\it Fractals} (Plenum Press, New York, 1988).\n\n\\bibitem{DBM84a}L. Niemeyer, L. Pietronero, and H. J. Wiesmann, Phys. Rev. Lett. {\\bf 52}, 1033 (1984).\n\n\\bibitem{Bak87}P. Bak, C. Tang, and K. Weisenfeld, Phys. Rev. Lett. {\\bf 59}, 381 (1987); Phys. Rev. A {\\bf 38}, 364 (1988).\n\n\\bibitem{Cafiero93}R. Cafiero, L. Pietronero, and A. Vespignani, Phys. Rev. Lett. {\\bf 70}, 3939 (1993).\n\n\\bibitem{Erzan95}A. Erzan, L. Pietronero, and A. Vespignani, Rev. Mod. Phys. {\\bf 67}, 545 (1995). \n\n\\bibitem{DBM84b}L. Pietronero and H. J. Wiesmann, J. Stat. Phys. {\\bf 36}, 909 (1984).\n\n\\bibitem{Kol98}B. Kol and A. Aharony, Phys. Rev. E {\\bf 58}, 4716 (1998), cond-mat/9802048.\n\n\\bibitem{Kol99}B. Kol And A. Aharony, to be published.\n\n\\bibitem{remark}In \\cite{Kol98} there are functions of the form\n$a(j)=a(\\infty)+x_0e^{-\\alpha j}(1+x_1e^{-\\beta j}+x_2e^{-2\\beta j}+\\dots)$,\nwhere $a(\\infty),x_0,\\alpha>0,\\beta>0,x_1,x_2,$ etc. are arbitrary constants.\nFor the present model, the average converges exponentially\n$\\langle a \\rangle (t)=\\langle a \\rangle^*+p^t\\sum_{j=0}^{\\infty}a(j+t)v_j(0)\n= \\langle a \\rangle^*+x_0p^te^{-\\alpha t} \\left[f_0+f_1e^{-\\beta t}\n+f_2e^{-2\\beta t}+ \\dots \\right ]$,\nwhere $\\langle a \\rangle^*\\equiv \\sum_{j=0}^{\\infty}qp^ja(j)$, \nand\n$f_i=x_i\\sum_{j=0}^{\\infty}e^{-(\\alpha+i\\beta)j}v_j(0)$.\nNote that the\nconstant $a(\\infty)$ does not affect the rate of convergence.\n\n\\bibitem{Turkevich89}L. A. Turkevich and H. Scher, Phys. Rev. Lett. {\\bf 55}, 1026 (1985).\n\n\\bibitem{Pietronero88b}L. Pietronero, A. Erzan, and C. Evertsz, Physica A {\\bf 151}, 207 (1988).\n\n\\bibitem{Kol2000}B. Kol and A. Aharony, unpublished.\n"
}
] |
cond-mat0002084
|
One-Dimensional Stochastic L\'evy--Lorentz Gas.
|
[
{
"author": "E. Barkai$^a$"
},
{
"author": "V. Fleurov$^a$ and J. Klafter$^b$"
},
{
"author": "$^a$ School of Physics and Astronomy"
},
{
"author": "$^b$ School of Chemistry"
},
{
"author": "Beverly and Raymond Sackler Faculty of Exact Sciences"
},
{
"author": "Tel-Aviv 69978"
},
{
"author": "Israel"
}
] |
% We introduce a L\'evy--Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers $\{ \xi_i \}$ are independent random variables identically distributed according to the probability density function $\mu\left( \xi \right)\sim \xi^{-\left( 1 + \gamma\right)}$. We show that under certain conditions the mean square displacement of the particle obeys $\langle x^2 \left( t\right) \rangle \ge C t^{3 - \gamma}$ for $1 < \gamma < 2$. This behavior is compatible with a renewal L\'evy walk scheme. We discuss the importance of rare events in the proper characterization of the diffusion process. $$ $$
|
[
{
"name": "1LevyLorPRE.tex",
"string": "% IMPORTANT to create postscript file\n% 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{ \nOne-Dimensional Stochastic L\\'evy--Lorentz Gas.}\n\\author{E. Barkai$^a$, V. Fleurov$^a$ and J. Klafter$^b$ \\\\\n$^a$ School of Physics and Astronomy \\\\\n$^b$ School of Chemistry \\\\\nBeverly and Raymond Sackler Faculty of Exact Sciences\\\\\nTel-Aviv University\\\\\nTel-Aviv 69978, Israel}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\n%\n\n\n We introduce a L\\'evy--Lorentz gas \nin which a light particle is scattered by static point\nscatterers arranged on a line. We investigate the case where\nthe intervals between scatterers $\\{ \\xi_i \\}$ are independent\nrandom variables\nidentically\ndistributed \naccording to the probability\ndensity function $\\mu\\left( \\xi \\right)\\sim \\xi^{-\\left( 1 + \\gamma\\right)}$.\nWe show that under certain conditions \nthe mean square displacement of the particle obeys\n$\\langle x^2 \\left( t\\right) \\rangle \\ge C t^{3 - \\gamma}$ for $1 < \\gamma < 2$.\nThis behavior is compatible with \na renewal L\\'evy walk scheme. \nWe discuss the importance of rare events in the proper characterization\nof the diffusion process.\n\n$$ $$\n\n\\end{abstract} \n\nPACS numbers: 02.50.-r, 05.40.+j, 05.60.+w \\\\ \n\n%05.40.+j Fluctuation phenomena, random processes, and Brownian motion\n\n%05.60.+w Transport processes: theory\n\n%02.50.-r Probability theory, stochastic processes, and statistics \n\n\\section{Introduction}\n\n In recent years \nthere has been\na growing interest in anomalous diffusion\ndefined by \n%\n\\begin{equation}\n\\langle x^2 \\rangle = D_{\\delta} t ^ \\delta\n\\label{eq000}\n\\end{equation}\n%\nand $\\delta > 1$ \n\\cite{Bouch,Levy,Klafter1,Benkadda}. \nSuch a behavior was found\nin chaotic diffusion in low dimensional \nsystems \\cite{Geisel,Klafter5}, tracer diffusion in a rotating\nflow \\cite{Swin}, $N$ body Hamiltonian dynamics \\cite{Antony}, Lorentz\ngas with infinite horizon \\cite{Bouch,Mats} and\ndiffusion in egg crate potentials \\cite{Geisel2}.\nIn all these examples one observes long ballistic\nflights in which the diffusing particle\nmoves at a constant velocity. \nThe transport is characterized by a distribution\nof free flight times which follows a power law\ndecay. These processes have been usually analyzed using \nthe L\\'evy walk framework (see more details below)\n\\cite{Levy,Klafter1,Benkadda,Klafter5}.\n\n It has been recently suggested by Levitz \\cite{Levitz} that three-dimensional\nmolecular Knudsen diffusion, at very low pressures,\ninside porous media\ncan be described by L\\'evy walks.\nIt has been also shown \n\\cite{Levitz1} that pore chord distributions,\nin certain\nthree-dimensional porous media \ndecay as a power law,\nat least for several length scales.\nHence one can anticipate that a light test particle \ninjected into such a medium may exhibit \na L\\'evy walk.\nThis has motivated the investigation of a fractal\nLorentz gas.\nLevitz \\cite{Levitz} has simulated trajectories\nof a light particle\nreflected from a three-dimensional intersection of a \nfour-dimensional Weierstrass--Mandelbrot hyper surface,\nand found an enhanced L\\'evy type diffusion. \n\n \nHere we investigate a one dimensional stochastic\nLorentz gas\nwhich we call L\\'evy--Lorentz gas.\nIn this model a light particle is scattered by a fixed\narray of identical scatterers arranged randomly on a line.\nUpon each collision event the light particle can be\ntransmitted (or reflected) with probability $T$ (or $R=1-T$). \nWe investigate the case when the intervals between the scatterers\nare independent identically distributed random variables\nwith a diverging variance.\n\n We find: ({\\bf a}) a lower bound for the mean square displacement\nwhich is compatible with the L\\'evy walk model, and ({\\bf b})\nthat the generalized diffusion coefficient $D_\\delta$ is\nvery sensitive to the way the system has been prepared at\ntime $t=0$. \nAs expected, we show that the transport\nis not Gaussian. \nIn systems that exhibit normal diffusion,\nthe contribution\nfrom ballistic motion, $x^2 = v^2 t^2$, is\nimportant only for short times; here we show that\nthe ballistic motion cannot be neglected even at $t \\to \\infty$.\nThe ballistic paths contribute to the generalized diffusion\ncoefficient $D_\\delta$ exhibiting a behavior\ndifferent than normal.\n\n\\section{Model and Numerical Procedure}\n\n Assume a light particle which moves with a constant speed\n($v=\\pm 1$) among identical point scatterers arranged randomly \non a line. \nUpon each collision, \nthe probability that the light particle\nis transmitted (reflected) is $T$ $(R=1-T)$.\n%No renewal assumption is made.\nThe intervals between scattering points,\n$\\xi_i> 0$ with $\\left( i=\\cdots,-n,\\cdots,-1,0,1,\\cdots\\right)$,\nare independent identically distributed random variables\ndescribed by a probability density function $\\mu\\left( \\xi \\right)$.\nAn important random variable\nis $x_f$\ndefined to be the distance between\nthe initial location of the light particle $(x=0)$\nand the first scatterer in the sequence \nlocated at $x>0$.\nThe random variable $x_f$ is described by \nthe probability density function $h(x_f)$.\nA set of scatterers (black dots) is given schematically by:\n%\n $$ $$\n$$ \\cdots \\ \\ \\overbrace{ \\bullet \\ \\ \\ \\ \\ \\ \\bullet }^{\\xi_{\n-2}} \\overbrace{ \\ \\ \\ \\ \\ \\ \\bullet }^{\\xi_{-1}} \\overbrace\n{ \\ \\ \\ \\ \\ \\underbrace{\\circ \\ \\ \\ \\ \\bullet}_{x_f} }^{\\xi_0} \\overbrace{ \\ \\ \\ \\ \\ \\ \\ \\bullet }^{\\xi_1}\n\\overbrace{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\bullet}^{\\xi_2} \\ \\ \\ \\cdots $$\n$$ $$\n%\nwhere the open circle represents the light test particle at time $t=0$.\n%\nWe consider the case when\nfor large $\\xi$,\n$\\mu\\left(\\xi\\right) \\sim \\xi^{-\\left( 1 + \\gamma\\right)}$,\nwith $0< \\gamma < 2$. Thus the variance of the length intervals\n$\\{ \\xi_i \\}$ diverges.\nA realization of the scatterers is shown in Fig. \\ref{fitLevy0},\nfor the case $\\gamma=3/2$.\nWe observe large gaps which are of the order\nof the length of the system.\n\n\n The case for which the variance converges\nhas been investigated thoroughly\nin \\cite{Henk,Gras,Spohn,Ernst,barkaiJSP}, resulting in:\n$(i)$\na normal Gaussian diffusion \nas expected from the central limit theorem,\nand $(ii)$\na $3/2$ power law decay in $t$ of\nthe velocity autocorrelation function. \n \n Along this work we present numerical results for the\ncase $\\gamma=3/2$ and $T=1/2$.\nWe use the following numerical procedure.\nFirst we generate a set of scatterers \non a one dimensional lattice with a lattice\nspacing equal unity.\nUsing a discrete time\nand space iteration scheme we find an exact expression\nfor \nthe probability of finding the particle on $x$\nat time $t$, \n$p(x,t|x=0,t=0)$, given that at $t=0$\nthe particle is on $x=0$.\nThe initial location of the particle is determined\nusing equilibrium initial conditions (see details below).\nThe initial velocity is $v=1$ or $v=-1$ with equal \nprobabilities.\n$p(x,t|x=0,t=0)$ depends on the realization of disorder\nwe have generated in the first step. \nThis procedure is repeated many times.\n\n\nIn appendix A we explain how we generated\nthe random intervals $\\{\\xi_i\\}$. \nWhen $\\gamma=3/2$ the\nmean $\\langle \\xi \\rangle \\equiv \\int_0^{\\infty} \\mu(\\xi) d \\xi$\nis finite while the second moment\n$\\langle \\xi^2 \\rangle=\\infty$. \nSince\n$|v| = 1$\nthe characteristic microscopic time scale is $\\langle \\xi \\rangle$\nwhich is referred to as the mean collision time,\nand our simulations are for times $t \\sim 1000 \\langle \\xi \\rangle$.\nFor our\nchoice of parameters $\\langle \\xi \\rangle \\sim 4$ (see more details\nin Appendix A). \n\n\\section{Results}\n\n Let us analyze our one-dimensional L\\'evy--Lorentz model using\nthe L\\'evy walk approach\n\\cite{Levy,Klafter1,Benkadda,Klafter5,previous4}.\nL\\'evy walks describe random walks which \nexhibit enhanced diffusion and are based on the generalized\ncentral limit theorem and L\\'evy stable distributions \\cite{Feller}.\nBriefly, a particle moves with a constant velocity\n$v=+1$ or $v=-1$ and then at a random time $\\tau_1$ its velocity is changed.\nThen the process is renewed. Each collision is independent of\nthe previous collisions. The \ntimes between collision events \n$\\{ \\tau_i \\}$\nare assumed to be independent identically distributed random\nvariables,\ngiven in terms of a probability density function $q\\left( \\tau \\right)$.\nOne might expect that the dynamics of L\\'evy--Lorentz gas\ncan be analyzed using the L\\'evy walk renewal approach \nwith\n$q\\left( \\tau \\right) \\sim \\tau^{ -\\left( 1 + \\gamma\\right)}$,\nfor large $\\tau$ and $0< \\gamma < 2$,\nwhich leads to \n%\n\\begin{equation}\n\\langle x^2 \\rangle \\sim \\left\\{ \n\\begin{array}{cc}\nt^{3 - \\gamma}, & \\ \\ \\ 1< \\gamma < 2 \\\\\n%\n \\ & \\ \\ \\ \\\\\n%\nt^{2}, & \\ \\ \\ 0< \\gamma < 1. \n\\end{array}\n\\right.\n\\label{eqLL02}\n\\end{equation}\n%\nFor $\\gamma>2$ one finds normal diffusion.\nIt is clear that the renewal L\\'evy walk approach and the\nL\\'evy--Lorentz gas are very different. Within the L\\'evy--Lorentz gas\ncollisions are not independent and correlations are important.\nHence it is interesting to check whether the renewal L\\'evy\nwalk model is suitable for the description of the L\\'evy--Lorentz\ngas.\nIn this context it is interesting to recall that Sokolov \net al \\cite{Sokolov}\nhave shown that correlations between jumps\nin a L\\'evy flight in a chemical space \ndestroy the L\\'evy statistics of the walk.\n\n\n% This is the way figure are entered. You put them in the correct\n% place. The number 13 is the size of the figure (play with it).\n% The path is the path to the postscript file of the figure\n%\n% see sscg Levy/Static/REAL/eli/(real,real1,real2) fort.299 and use NLL1.f\n%\n%\n\\begin{figure}[htb]\n\\epsfxsize=21\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{real.ps} }}\n\\caption {%\\protect\\footnotesize\nA realization of a set of scatterers with $\\gamma=3/2$ exhibiting\ngaps on many scales.\nThe horizontal axis is the $x$\ncoordinate. All along this work\nwe consider dimensionless units. \n}\n\\label{fitLevy0}\n\\end{figure}\n\n\nWe consider a continuum model to derive\nour analytical results; the generalization to\nthe lattice case is straightforward.\n Let $\\langle p\\left( x , t | x=0, t = 0 \\right) \\rangle dx$\nbe the probability, averaged over disorder, of finding\nthe test particle at time $t$, in the interval $(x,x+dx)$.\nInitially, at time $t=0$, the particle is\nat $x=0$, and there is an equal probability\nof the particle having a velocity $v=+1$ or $v=-1$.\nFigs. \\ref{FIG.pxens1} and \\ref{fitLevy1} present numerical simulations\nwhich show\n$\\langle p\\left( x , t | x=0, t = 0 \\right) \\rangle$.\nOne can see that in addition\nto the central peak on $x=0$, two other peaks\nappear at locations\n$x=\\pm t$. These peaks, known as ballistic\npeaks, were observed in a similar context\nin other systems exhibiting enhanced diffusion \n\\cite{Klafter5,Levitz,West}. The peaks are stable on the\ntime scale of the numerical simulation.\n The height of these peaks decays with time, \nand according to our finite time numerics\nthe central peak and the ballistic peaks\ndecay according to the same power law when\n$\\gamma=3/2$.\n\n\nLet us analyze analytically the time dependence of\nthe ballistic peaks and calculate\ntheir contribution to the mean square displacement.\n Since in our model $|v|=1$ it is clear that \n%\n\\begin{equation}\n \\langle p\\left( x , t | x=0, t = 0 \\right) \\rangle = 0, \\ \\ \\ \\mbox{for} \\ \\ |x|> t. \n\\label{eqLL03}\n\\end{equation}\n%\nWe decompose the ensemble averaged probability density into \ntwo terms\n%\n $$ \\langle p\\left( x , t | x=0, t = 0 \\right) \\rangle = $$\n%\n\\begin{equation}\n \\langle \\tilde{p}\\left( x , t | x=0, t = 0 \\right) \\rangle + \n{1 \\over 2} Q_b\\left( t \\right) \\left[ \\delta\\left(x+t \\right) + \\delta\\left(x-t\\right) \\right].\n\\label{eqLL04}\n\\end{equation}\n%\nThe first term on the RHS,\n $\\langle \\tilde{p}\\left( x , t | x=0, t = 0 \\right) \\rangle$,\n is the probability density of finding\nthe light particle at $|x|< t$. $Q_b\\left( t \\right)$ is the \nprobability of finding the light particle at $x=t$\n($x=-t$) if initially \nat $x=0$ and its velocity\nis\n$+1$ ($-1$). \nThe left-right symmetry in Eq. (\\ref{eqLL04})\nmeans that we have used the symmetric\ninitial condition (i.e., $v=+1$ or $v=-1$\nwith equal probabilities) and the assumption that the\nsystem of scatterers is isotropic in an averaged sense.\nUsing a similar notation we write\n%\n\\begin{equation}\n\\langle x^2 \\rangle = \\langle \\tilde{x}^2 \\rangle + \n\\langle x^2 \\rangle_b,\n\\label{eqLL05}\n\\end{equation}\n%\nwhere $\\langle x^2 \\rangle_b$ is the ballistic contribution\nto the mean square displacement.\nFrom Eq. (\\ref{eqLL04}) we have\n%\n\\begin{equation}\nQ_b\\left( t \\right) t^2 \\le \\langle x^2\\left( t \\right) \\rangle \\le t^2.\n\\label{eqLL06}\n\\end{equation}\n%\nThe upper bound is an obvious consequence \nof the fact that $|v|=1$. The lower bound,\nfound using $\\langle x^2(t)\\rangle_b \\le \\langle x^2(t) \\rangle$,\n is of no\nuse when all moments of $\\mu\\left( \\xi \\right)$ converge,\nsince then $Q_b \\left( t \\right)$ usually \ndecays exponentially for long times. \nEq. \n(\\ref{eqLL06}) is useful when the \nmoments of $\\mu(\\xi)$ diverge, a case we consider here.\n\n%\n\\begin{figure}[htb]\n\\epsfxsize=21\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{pxens1new.ps} }}\n\\caption {%\\protect\\footnotesize\nA histogram presenting the \n$\\langle p(x,t|x=0,t=0) \\rangle$ versus $x$\nfor time $t=1024$, $\\gamma=3/2$\nand $T=1/2$.\nNotice the ballistic peaks of the propagator at $x=\\pm t$.\nThe average is over $1.8*10^5$ realizations of disorder.\nThe bin length is unity.\n}\n\\label{FIG.pxens1}\n\\end{figure}\n\n% This is the way figure are entered. You put them in the correct\n% place. The number 20 is the size of the figure (play with it).\n% The path is the path to the postscript file of the figure\n%\n% see ccsg Levy/logeli/pxens2\n% see file Px*out\n%\n%\n\\begin{figure}[htb]\n\\epsfxsize=20\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{pxens2.ps} }}\n\\caption {%\\protect\\footnotesize\nThe same as Fig. \\protect\\ref{FIG.pxens1}\nfor time $t=4096$. The probability \nof finding a ballistic path, \n$\\langle\np(x=\\pm t, t |x=0,t=0)\\rangle \\simeq 0.004$,\nis small but yet of statistical\nsignificance when $\\langle x^2 (t) \\rangle$ is calculated.\n}\n\\label{fitLevy1}\n\\end{figure}\n\n\n To find $Q_b \\left( t \\right)$ consider a test particle\nwhich is initially of velocity $+1$\nand located at $x=0$. The probability it\nreaches $x=t$, at time $t$,\nis $T^r$ where $r$ is the number of scatterers\nin the interval of length $(0,t)$.\nHence,\n%\n\\begin{equation}\nQ_b \\left( t \\right) = \\sum_{r = 0}^{\\infty} T^r G_r \\left( t \\right),\n\\label{eqLL07}\n\\end{equation}\n%\nand $G_r \\left( t \\right)$ is the probability\nof finding $r$ scatterers in $(0,t)$.\n$G_r \\left( t \\right)$ \ncan be calculated in terms of $\\mu\\left( \\xi \\right)$ and of \n$h\\left(x_f\\right)$.\nIn appendix B we use renewal theory to calculate\nthe Laplace $t \\to u$ transform \nof $Q_b\\left( t \\right)$\n%\n\\begin{equation}\n\\hat{Q}_b\\left( u \\right) =\n { 1 \\over u} + { \\left ( T - 1 \\right) \\hat{h}\\left( u \\right) \\over\n\\left[ 1 - T \\hat{\\mu}\\left( u \\right) \\right] u}.\n\\label{eqLL08}\n\\end{equation}\n%\nWhen $T=1$, $\\hat{Q}_b\\left( u \\right) = 1/u$, as expected from a\ntransmitting set of scatterers.\nIn deriving Eq. (\\ref{eqLL08}) we have used the model assumptions\nthat the intervals $\\{ \\xi_i \\}$ are statistically\nindependent and identically distributed.\n\n\n The function $h(x_f)$ depends on the way the system\nof scatterers and light particle are initially\nprepared. Consider the following preparation process.\nA scatterer is assigned at the location $x=-L$\n(eventually $L \\to \\infty$), then random independent length intervals\nare generated using the probability density $\\mu(\\xi)$. These\nlength intervals determine the location of scatterers on the line.\nWhen the sum of the length intervals exceeds $2L$ the\nprocess is \nstopped. As mentioned, at time $t=0$ the light particle is\nassigned to the point $x=0$. \nWhen the mean distance between scatterers\n$\\langle \\xi \\rangle = \\int_0^{\\infty} \\xi \\mu(\\xi) d\\xi$\nconverges (i.e., $1< \\gamma$), and $L \\to \\infty$ then \naccording to \\cite{Feller,COX}\n%\n\\begin{equation}\nh\\left( x_f \\right) = { 1 - \\int_0^{x_f} \\mu(\\xi) d\\xi \\over \\langle \\xi \\rangle},\n\\label{eqLL09}\n\\end{equation}\n%\nwhich is standard in the context of \nthe Lorentz gas \nwhen the moments of $\\mu(\\xi)$\nconverge \n\\cite{Henk,Spohn}. \nThis type of initial condition\nis called equilibrium initial condition. \n When $1< \\gamma < 2$, Eq. (\\ref{eqLL09})\nimplies that\n$h(x_f) \\sim \\left( x_f\\right)^{ - \\gamma}$ and hence \n$\\langle x_f \\rangle = \\int_0^{\\infty} x_f h(x_f) dx_f \\to \\infty$.\nAt first sight\nthis divergence\nmight seem to be paradoxical, since \nthe mean distance between scatterers, \n$\\langle \\xi \\rangle$,\nconverges.\nWe notice however that the point $x=0$ \nhas a higher probability to be situated\nin a large gap.\nHence, statistically the interval $\\xi_0$ is much\nlarger than the\nothers and in our case\n$\\langle x_f \\rangle= \\infty$.\nEq. (\\ref{eqLL09}) implies that on average\none has to wait an infinite time for the \nfirst collision event. \n\n\n% This is the way figure are entered. You put them in the correct\n% place. The number 20 is the size of the figure (play with it).\n% The path is the path to the postscript file of the figure\n%\n%\n\\begin{figure}[htb]\n\\epsfxsize=20\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{hxnew.ps} }}\n\\caption {%\\protect\\footnotesize\nThe probability to find the first scatterer\nat a distance $x_f$ from the origin. Here the average is over $3*10^5$\nrealizations, and half the length of the system is $L=10^5$.\nWe use a bin of length $32$ (dimensionless units).\nThe solid curve is the theoretical prediction,\nEq. (\\protect\\ref{eqLL09}),\nwith no fitting parameters.\nFor large $x_f$, $h(x_f)\\sim x_f^{ -\\gamma}$ \nand $\\gamma=3/2$,\nwhich implies that $\\langle x_f \\rangle$ diverges.}\n\\label{fitLevy2}\n\\end{figure}\n\n\n In numerical simulations the system's length $L$\nis finite, so that Eq. (\\ref{eqLL09}) is only an \napproximation which we expect to be valid for \n$x_f<<L$.\nHowever, if we observe a system for time $t <<L$\nthe boundary condition is not expected\nto influence the anomalous dynamics.\nWe have generated numerically many random systems,\nusing $\\mu(\\xi) \\sim \\xi^{ - \\left( 1 + \\gamma \\right)}$ and $\\gamma=3/2$.\nAs shown in \nFig. \\ref{fitLevy2},\n$h(x_f) \\sim \\left( x_f\\right)^{ - \\gamma}$ as predicted\nin Eq. (\\ref{eqLL09}). \n\n\n We consider the small $u$ expansion, \nof the Laplace transform of $\\mu(\\xi)$,\n%\n\\begin{equation}\n\\hat{\\mu}\\left( u \\right) = 1 - \\langle \\xi \\rangle u + a \\left( \\langle \\xi \\rangle u \\right)^{\\gamma} \\cdots, \n\\label{eqLL10}\n\\end{equation}\n%\nwhere $1 < \\gamma < 2$ and $a$ is a constant.\nUsing a Tauberian theorem and \nEqs. \n(\\ref{eqLL08}), \n(\\ref{eqLL09})\nit can be shown that for long times $t$\n%\n$$ Q_b \\left( t \\right)=$$\n\\begin{equation}\n {a \\over \\Gamma\\left( 2 - \\gamma \\right)} \\left({ t \\over \\langle \\xi \\rangle}\\right)^{1 - \\gamma} + { 2 a \\left( \\gamma - 1 \\right) \\over \\Gamma\\left( 2 - \\gamma \\right) }{ T \\over 1 - T} \\left( { t \\over \\langle \\xi \\rangle} \\right)^{ - \\gamma} + \\cdots .\n\\label{eqLL11}\n\\end{equation}\n%\nInserting Eq. \n(\\ref{eqLL11})\nin Eq. (\\ref{eqLL06}) we find \n%\n\\begin{equation}\n{a \\langle \\xi \\rangle^2 \\over \\Gamma\\left( 2 - \\gamma \\right) }\\left( { t \\over \\langle \\xi \\rangle} \\right)^{3 - \\gamma} \\le \\langle x^2 \\left( t \\right) \\rangle \\le t^2\n\\label{eqLL12}\n\\end{equation}\n%\nThis bound demonstrates that the diffusion is enhanced, namely the mean square\ndisplacement increases faster than linearly with time.\n\nIn Fig. \n\\ref{fitLevy3}, we present the mean square displacement\nof the light particle obtained by numerical simulations\nfor the case $\\gamma = 3/2$. We see that for the chosen \nvalues of parameters the asymptotic $t^{3 - \\gamma}$ \nbehavior can be observed for times which are accessible \non our computer. Fig. \\ref{fitLevy3} clearly shows\n that the ballistic contribution\n$\\langle x^2 \\rangle_b$ to the mean square displacement\n$\\langle x^2 \\rangle$ is significant.\nNotice that our numerical results are\npresented for times which are much larger than \nthe mean collision time\n$\\langle \\xi \\rangle \\sim 4$.\n\nIn Fig. \\ref{fitLevy4} we show the probability of\nfinding a ballistic path, namely,\nthe probability of finding the light particle at time $t$ at $x=+t$,\nor at $x=-t$.\nBy definition these probabilities are\nequal to $Q_b(t)/2$.\nWe observe the $t^{1- \\gamma}$ behavior of $Q_b(t)$, Eq. \n(\\ref{eqLL11}), with which our lower bound was found.\n The fact that \nthe probability of finding the particle at $x=t$ is\nequal to the probability of finding the particle\nat $x=-t$ means that our system\nis isotropic in an averaged sense.\nThis is achieved by choosing large values of $L$. \n\n% This is the way figure are entered. You put them in the correct\n% place. The number 13 is the size of the figure (play with it).\n% The path is the path to the postscript file of the figure\n%\n% see ccsg Levy/logeli/logx2 x2.out and for*155\n%\n%\n\\begin{figure}[htb]\n\\epsfxsize=21\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{logx2.ps} }}\n\\caption {%\\protect\\footnotesize\n$\\log_{10}\\left[ \\langle x^2 \\rangle \\right]$ versus $\\log_{10}(t)$.\nThe points are numerical results. The straight curve is the\nasymptotic behavior of the lower bound,\nEq. (\\protect\\ref{eqLL12}) (i.e., $\\langle x^2\\rangle_b$).\nWe use $\\gamma=3/2$ and so $\\langle x^2 \\rangle \\ge C t^{3/2}$.\n}\n\\label{fitLevy3}\n\\end{figure}\n\n The lower bound in Eq. (\\ref{eqLL12}) does not depend on the\ntransmission coefficient $T$. Thus, even when all\nthe scatterers are perfect reflectors, with $R=1$, \nthe diffusion is enhanced. Large gaps which are of the order of \nthe length $t$ are responsible for this behavior. \n The transmission coefficient has an important role\nin determining what is the asymptotic\ntime of the problem. The condition that the first term\nin Eq. (\\ref{eqLL11}) dominates over the second reads:\n%\n\\begin{equation}\n{ t \\over \\langle \\xi \\rangle }\\gg 2 \\left( \\gamma - 1 \\right) { T \\over 1 - T}.\n\\label{eqLL13}\n\\end{equation}\n%\nOnly under this condition the behavior in Eq. (\\ref{eqLL12})\nis expected to be valid. \n\nThe lower bound in Eq. (\\ref{eqLL12}) is compatible with the renewal L\\'evy\nwalk approach Eq. (\\ref{eqLL02}). Other stochastic models\n\\cite{Bouch,Fog}\nfor enhanced diffusion based on L\\'evy scaling arguments\npredict\n%\n\\begin{equation}\n\\langle x^2 \\rangle \\sim t^{ 2 / \\gamma} \\ \\ \\mbox{for} \\ \\ \\ 1< \\gamma < 2\n\\label{eqLL14}\n\\end{equation}\n%\nwhich is different from Eq. (\\ref{eqLL02}).\nThis approach is based upon a fractional\nFokker--Planck equation (FFPE)\n%\n\\begin{equation}\n{\\partial p\\left( x, t \\right) \\over \\partial t} = D_{\\gamma}\n \\nabla^{\\gamma} p \\left(x, t\\right)\n\\label{eqLL15}\n\\end{equation}\n%\nused in \\cite{Fog}\nto predict an enhanced diffusion.\nThe non-local fractional operator in Eq. (\\ref{eqLL15})\nis defined in Fourier $k$ space according\nto the transformation $\\nabla^{\\gamma} \\to -|k|^{\\gamma}$.\nOur findings here show that Eq. (\\ref{eqLL14})\ndoes not describe the dynamics of the\nL\\'evy--Lorentz gas, since $3-\\gamma \\ge 2/\\gamma$\nfor $ 1\\le \\gamma \\le 2$.\n\n% This is the way figure are entered. You put them in the correct\n% place. The number 13 is the size of the figure (play with it).\n% The path is the path to the postscript file of the figure\n%\n% see ccsg Levy/logeli/pxl\n% see file pxl.out fort.15 as*f\n%\n%\n\\begin{figure}[htb]\n\\epsfxsize=21\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{wing1.ps} }}\n\\caption {%\\protect\\footnotesize\nThe probability to find the light particle\nat time $t$ at $x = + t$ (stars) and at $x=-t$\n(dots) versus time.\nThe solid curve is the theoretical prediction,\nEq. (\\protect\\ref{eqLL11})\n(no fitting parameters) which gives\n$Q_b(t)/2 \\sim t^{(1-\\gamma)}$,\nwith $\\gamma=3/2$. \n}\n\\label{fitLevy4}\n\\end{figure}\n\n\n Consider now the case when \nthe light particle is initially located at a scattering point.\nSuch an initial condition is called non equilibrium initial \ncondition.\nUnder this condition $h(x_f)=\\mu(x_f)$ instead of \nEq. (\\ref{eqLL09}).\nThis means that the particle has to wait an average\ntime $\\langle \\xi \\rangle$ before\nthe first collision event instead of the infinite time\nwhen the equilibrium initial conditions were used.\nUsing Eqs.\n(\\ref{eqLL05}), (\\ref{eqLL08})\nand \n(\\ref{eqLL10}),\n and \n$\\hat{\\mu}( u ) = 1 - \\left( A u \\right)^{\\gamma} + \\cdots$\nfor $0< \\gamma < 1$ and small $u$, we find\n%\n\\begin{equation}\n\\langle x^2 \\rangle \\ge \\left\\{\n\\begin{array}{cc}\n{1 \\over 1 - T} a { \\left( \\gamma - 1 \\right) \\over \\Gamma\\left( 2 - \\gamma \\right)} \\langle \\xi \\rangle^2 \\left( { t \\over \\langle \\xi \\rangle }\\right)^{2 - \\gamma}, & \\ \\ \\ \\ 1< \\gamma < 2 \\\\\n%\n & \\\\\n{ 1 \\over 1 - T } { \\left( 1 - \\gamma\\right) \\over \\Gamma\\left( 2 - \\gamma \\right) } A^2 \\left( { t \\over A } \\right)^{2 - \\gamma}, & \\ \\ \\ \\ 0< \\gamma < 1. \n\\end{array}\n\\right.\n\\label{eqLL16}\n\\end{equation}\n%\nFor $1< \\gamma < 2$ the bound differs\nfrom the $t^{3 - \\gamma}$ found \nin Eq. (\\ref{eqLL12}),\nwhere we chose $h(x_f)$ \naccording to Eq. (\\ref{eqLL09}).\nThus the ballistic contribution $\\langle x^2 \\rangle_b$,\ndefined in Eq. (\\ref{eqLL05}), behaves differently\nfor the two ensembles even when $t \\to \\infty$.\nThis is very different from regular Lorentz gases,\nwhich in the limit $t\\to \\infty$ are not sensitive to the\nchoice of $h(x_f)$.\n\n Finally, Fig. \\ref{px0} shows the behavior of the correlation\nfunction $\\langle p(x=0,t | x=0,t = 0) \\rangle$ \nobtained from the numerical\nsimulation with equilibrium initial conditions.\nWe observe a\n$t^{-1/2}$ decay of the correlation function. This behavior is compatible with\nstandard Gaussian diffusion which gives the well known \n$t^{-d/2}$ \nresult\nin $d$ dimensions. We find this behavior\nfor time scales which are much larger than the mean\ncollision time $\\langle \\xi \\rangle$, however we have no proof that this\nbehavior is asymptotic. On the other hand the L\\'evy walk model\npredicts\n$\\langle p(x=0,t | x=0,t = 0) \\rangle\\sim t^{-1/\\gamma}$\n\\cite{Klafter5}.\n\n\n% We have no analytical explanation\n%for the observed time dependence of the\n%correlation function. One may speculate that\n%a fraction of samples, look on the\n%time scale of the simulation, as if they\n%are 'normally' distributed. \n% These 'normal' samples contribute\n%a $t^{-1/2}$ dependence to the correlation\n%function while other samples which are\n%much more dilute are not contributing \n%since for such dilute systems $p(x=0,t|x=0,t=0) \\sim 0$.\n% We would also like to mention\n%that sub diffusive random walks on fractals \\cite{Aharony,Havlin}\n%are usually characterized by two independent\n%exponents the first is the mean square displacement exponent\n% $\\delta$ and the second\n%is the correlation function\n%exponent. Hence we might expect that the\n%L\\'evy--Lorentz gas which exhibits enhanced diffusion\n%is also described by\n%two exponents which cannot be easily related.\n%\n\n% This is the way figure are entered. You put them in the correct\n% place. The number 13 is the size of the figure (play with it).\n% The path is the path to the postscript file of the figure\n%\n% see ccsg Levy/logeli/logx2 x2.out and for*155\n%\n%\n\\begin{figure}[htb]\n\\epsfxsize=21\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{px0.ps} }}\n\\caption {%\\protect\\footnotesize\n$\\log_{10}\\left[ \\langle p(x=0,t|x=0,t=0) \\rangle \\right]$ \nversus $\\log_{10}(t)$.\nThe points are numerical results. The straight curve is \na fit exhibiting the $t^{-1/2}$ behavior.\n}\n\\label{px0}\n\\end{figure}\n\n\n\\section{Summary and Discussion}\n\n\n In this work we have considered a one dimensional\nL\\'evy--Lorentz gas. \nWe have shown that:\\\\ \n{\\bf (a)} the mean square displacement in the L\\'evy--Lorentz \ngas \nis compatible with the L\\'evy walk framework \nand not with the FFPE. \\\\\n{\\bf (b)} Ballistic contributions to the\nmean square displacement are important\neven for large times. \\\\\n({\\bf c}) The ballistic peaks at $x=+t$ and\n$x=-t$ can be analyzed analytically. They decay\nas power laws. \\\\\n({\\bf d}) The way in which the system\nis prepared at $t=0$ (i.e., equilibrium versus\nnon equilibrium initial conditions) determines the behavior\nof the ballistic peaks. Since these peaks contribute\nto the mean square displacement even at large times\n, we conclude that the diffusion coefficient $D_{\\delta}$\nis sensitive to the way the system is prepared.\n\n% In different L\\'evy processes the mean square displacement\n%is many times used to characterize the stochastic motion. The mean\n%square displacement, and higher moments,\n%depend strongly on the wings of \n%the appropriate distribution function.\n%For Gaussian Lorentz gases these wings decay\n%exponentially and may be neglected while\n%for the L\\'evy--Lorentz gas the wings are decaying\n%slowly and cannot be neglected at large times.\n%Hence the mean square displacement cannot give\n%statistical information which can characterize\n%the majority of trajectories and samples.\n%We have shown this here by explicitly calculating\n%the dependence of the mean square displacement on the rare \n%ballistic paths and showing that they are non negligible\n%even at long times.\n\n In our work we considered an initial condition\n$v=1$ or $v=-1$ with equal probabilities.\nIt is clear that if we\nassign a velocity $v=+1$ to the light particle,\nat $t=0$, $\\langle p(x,t|x=0,t=0) \\rangle$\nwill never become symmetric, even approximately. \nInstead of the three peaks in Fig. \\ref{fitLevy1} one will\nobserve only two peaks one at $x=0$ and the other at $x=+t$.\n\n The reason for these behaviors in the L\\'evy--Lorentz gas\nstems from the statistical importance of ballistic paths.\nThis is different from the systems in which diffusion is normal in which\nthese paths are of no significance at long times.\nThus, similarly to Newtonian dynamics, the system exhibits\na strong sensitivity to initial conditions.\n \n\n Experiments measuring diffusion phenomena \nusually sample data \nonly in a scaling regime (e.g., $-\\sqrt{D_1t} < x < \\sqrt{D_1t}$\n). Rare events where the diffusing particle is\nfound outside this regime are many times assumed to be of\nno statistical importance. Here we showed that for the \nL\\'evy--Lorentz gas rare events found in the outer most part\nof $\\langle p(x,t|x=0,t=0) \\rangle$ are of statistical importance.\n%The L\\'evy--Lorentz gas is an example of a system where had\n%we assumed a standard sampling regime in the first\n%place our conclusions on the\n%diffusion process would be wrong.\n\n$$ $$\n{\\em Note added in proof.} Recently related theoretical\nwork on enhanced diffusion was published \\cite{latora}\n\n\n{\\bf Acknowledgment} We thank A. Aharony, P. Levitz, I. Sokolov and R. Metzler\nfor helpful discussions.\n\n\\subsection{Appendix A}\n\n\nAs mentioned we use a lattice model for\nthe simulation so that $\\xi$ is an integer.\nWe use the transformation\n%\n\\begin{equation}\n\\xi=\\mbox{INT}\\left\\{ \\left[ \\tan\\left( { u \\pi \\over 2} \\right) \\right]\n^{1 / \\gamma} \\right\\}+1.\n\\label{eqAP1}\n\\end{equation}\n%\nHere $\\mbox{I}=\\mbox{INT}\\{ z \\}$ is the integer\nclosest to $z$ satisfying $I \\le z$.\nIn Eq. (\\ref{eqAP1})\n$u$ is a random variable distributed uniformly\naccording to\n%\n\\begin{equation}\n 0\\le u_{min} \\le u \\le u_{max} \\le 1,\n\\label{eqAP1a}\n\\end{equation}\n%\nwhere $u_{min}$ and $u_{max}$ are cutoffs. It is easy\nto generate the random variable $u$ on a computer.\nThe probability to find an interval of length $\\xi$ is,\n%\n\\begin{equation}\n\\mu\\left( \\xi \\right)= \\int_{\\xi-1}^{\\xi} \\mu_c(y) dy\n\\label{eqAP2}\n\\end{equation}\n%\nwith\n%\n\\begin{equation}\n\\mu_c(y) = \\left\\{\n\\begin{array}{ccc}\n0 & \\ \\ y< y_{min} \\\\\n \\ & \\ \\ \\\\\n { 2 \\gamma \\over \\pi \\Delta } { y^{\\gamma - 1} \\over 1 + y^{2 \\gamma}} & \\ \\\ny_{min}< y < y_{max} \\\\\n \\ & \\ \\ \\\\\n 0 & \\ \\ y > y_{max}. \\\\\n\\end{array}\n\\right.\n\\label{eqAP3}\n\\end{equation}\n%\nHere\n\\begin{equation}\ny_{min} = \\left[ \\tan\\left( {u_{min} \\pi \\over 2} \\right)\\right]^{1/ \\gamma},\n\\ \\ \\ \\\ny_{max} = \\left[ \\tan\\left( {u_{max} \\pi \\over 2} \\right)\\right]^{1/ \\gamma}\n\\label{eqAP4}\n\\end{equation}\n%\nare the cutoffs of $\\mu_c(y)$. When $u_{min}=0$ and $u_{max}=1$ we\nhave $y_{min}=0$ and $y_{max}= \\infty$.\nIn Eq. (\\ref{eqAP3}) $\\Delta=u_{max} - u_{min}$ determines the normalization\ncondition $\\int_{0}^{\\infty} \\mu_c(y) dy = 1$.\n\n To derive Eqs. (\\ref{eqAP2})-(\\ref{eqAP4}) we use the transformation\n$ y = \\left[ \\tan\\left( u \\pi / 2 \\right) \\right]^{1/ \\gamma}$,\nand then $\\mu_c\\left( y \\right) = g\\left( u \\right) | du/dy|$,\nwhere $g\\left( u \\right)$ is the uniform probability density\n of $u$.\n\n For large $\\xi$ we find\n%\n\\begin{equation}\n\\mu\\left(\\xi \\right) \\sim \\left\\{\n\\begin{array}{ccc}\n{ 2 \\gamma \\over \\pi \\Delta}\\xi ^{ - 1 - \\gamma} & \\ \\ \\xi< \\xi_{max} \\\\\n \\ & \\ \\ \\\\\n0 & \\ \\ \\xi > \\xi_{max},\n\\end{array}\n\\right.\n\\label{eqP5}\n\\end{equation}\n%\nwith\n$\\xi_{max} = \\mbox{INT} \\left\\{ \\left[ \\tan\\left( u_{max} \\pi/2 \\right) \\right]^\n{1 / \\gamma} \\right\\} + 1$.\nWhen $u_{max}=1$ the\nsecond moment of $\\mu(\\xi)$ diverges.\n\n In our numerical simulations we consider\n$u_{min}=1/2$, $u_{max}=1$ and $\\gamma=3/2$. Then\n$ \\langle \\xi \\rangle \\simeq 4.031$ and for large $\\xi$,\nwe find \n%\n\\begin{equation}\n\\mu(\\xi)\\sim (6/ \\pi) \\xi^{- 5/2} \\ \\ \\ \\ \\ \\ \\ \\xi>>1.\n\\label{eqEX01}\n\\end{equation}\n%\n\n\\subsection{Appendix B}\n\n The calculation of $\\hat{Q}_b(u)$ can be found in\n\\cite{Feller,COX}. The probability that \nthe interval $(0,t)$ is empty is\n%\n\\begin{equation}\nG_0(t)=1-\\int_0^t h(\\tau) d \\tau \n\\label{eqAPB1}\n\\end{equation}\n%\nand in Laplace space $\\hat{G}_0(u)=[1 - \\hat{h}(u) ] / u$.\nThe Laplace transform of $G_r(t)$ for $r\\ge 1$\nis found using convolution\n%\n\\begin{equation}\n\\hat{G}_r(u) = \\hat{h}(u) \\hat{\\mu}^{r - 1}(u) \\hat{W}(u).\n\\label{eqAPB2}\n\\end{equation}\n%\n$W(t)=1- \\int_0^{t} \\mu(\\xi) d \\xi$ is the probability \nthat an interval of length $(0,t)$ is empty, given that\na scatterer occupies $0^-$. In Laplace space $\\hat{W}(u)=[1- \\hat{\\mu}(u)]/u$.\nUsing Eqs. (\\ref{eqAPB1}), (\\ref{eqAPB2})\nand (\\ref{eqLL07})\nwe find (\\ref{eqLL08}).\n\n\n%\n\\begin{references}\n\n\n\\bibitem[]{} \nAuthor e-mail: barkai@mit.edu\n\n\\bibitem[]{} \n\n\\bibitem[1]{Bouch} J. P. Bouchaud and A. Georges, {\\em Phys. Rep.}\n {\\bf 195} 127 (1990).\n\n\\bibitem[2]{Levy} \n M.F. Shlesinger, G. M. Zaslavsky and U. Frisch ed. \n {\\em L\\'evy Flights and Related Topics in Physics}\n (Springer-Verlag, Berlin, 1994).\n\n\\bibitem[3]{Klafter1} \nJ. Klafter, M. F. Shlesinger and G. Zumofen, \n {\\em Physics Today} {\\bf 49} (2) 33 (1996).\n\n\\bibitem[4]{Benkadda} E. Barkai and J. Klafter,\nin {\\em Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas}\n G. M. Zaslavsky and S. Benkadda ed.\n(Springer-Verlag, Berlin 1998).\n\n%\n\\bibitem[5]{Geisel} T. Geisel, J. Nierwetberg and A. Zacherl, {\\em Phys. Rev. Lett.\n} {\\bf 54}, 616 (1985).\n\n\\bibitem[6]{Klafter5} G. Zumofen and J. Klafter, {\\em Phys.Rev. E} {\\bf 47},\n 851 (1993).\n\n%\n\\bibitem[7]{Swin} T. H. Solomon, E. R. Weeks and H. L. Swinney,\n {\\em Phys. Rev. Lett.} {\\bf 71}, 23 (1995).\n\n%\n\\bibitem[8]{Antony} A. Torcini and M. Antony,\n {\\em Phys. Rev. E} {\\bf 57}, R6233 (1998).\n%\n\\bibitem[9]{Mats} Matsuoka and R. F. Martin,\n {\\em J. Stat. Phys.} {\\bf 88} 81 (1997).\n\n%\n\\bibitem[10]{Geisel2} T. Geisel, A. Zacharel and G. Radons {\\em Z. Phys. B} {\\bf 71} 117 (1988). \n\n\\bibitem[11]{Levitz} P. Levitz, {\\em Europhys. Lett.} {\\bf 39} 6 593(1997).\n\n\\bibitem[12]{Levitz1} P. Levitz and D. Tchoubar, {\\em J. Phys. I} {\\bf 2} 771 (1992).\n\n%\\bibitem[13]{Hovi} J. P. Hovi, A. Aharony, D. Stauffer and B. B. Mandelbrot, {\\em Phys. Rev. Lett.} {\\bf 77} 877 (1996).\n\n\n\\bibitem[13]{Henk} H. van Beijeren, {\\em Rev. Mod. Phys.} {\\bf 54} 195 (1982).\n%\n\n\\bibitem[14]{Gras} P. Grassberger, {\\em Physica A} {\\bf 103} 558 (1980).\n\n\\bibitem[15]{Spohn} H. van Beijeren and H. Spohn, {\\em J. Stat. Phys.} {\\bf 31} 231 (1983).\n\n\\bibitem[16]{Ernst} M. H. Ernst, J. R. Dorfman, R. Nix and D. Jacobs {\\em Phys. Rev\n. Lett.}\n {\\bf 74} 4416 (1995).\n\n\n\\bibitem[17]{barkaiJSP} E. Barkai and V. Fleurov, {\\em J. Stat. Phys.} {\\bf 96}, 325 (1999).\n\n\\bibitem[18]{previous4} E. Barkai and V. Fleurov, {\\em Phys. Rev. E}\n {\\bf 56} 6355 (1997).\n\n\\bibitem[19]{Feller} W. Feller,{\\em An introduction to probability Theory and\nIts Applications} Vol. 2 (John Wiley and Sons 1970).\n\n\\bibitem[20]{Sokolov} I. M. Sokolov, J. Mai and A. Blumen, {\\em Phys. Rev. Lett.} {\\bf 79} 857 (1997). \n\n\\bibitem[21]{West} M. Stefancich, P. Allegrini, L. Bonci, P. Grigolini and B. J. west {\\em Phys. Rev. E} {\\bf 57} 6625 (1998) \n\n%\n\\bibitem[22]{COX} D. R. Cox, {\\em Renewal Theory},\n (Methuen \\& Co. London, 1970).\n\n\n\\bibitem[23]{Fog} H. C. Fogedby, {\\em Phys. Rev. Lett.} {\\bf 73} 2517 (1994),\n{\\em Phys. Rev. E} {\\bf 58} 1690 (1998).\n\n\\bibitem[24]{latora}\nV. Latora, P. Rapisarda and S. Ruffo\n{\\em Phys. Rev. Lett.} {\\bf 83} 2104 (1999),\nB.A. Carreras, V.E. Lynch, D.E. Newman and G. M. Zaslavsky\n{\\em Phys. Rev. E} {\\bf 69} 4770 (1999),\nP. Castiglione, A. Mazzino, P. Muratore-Ginanneschi and A. Vulpiani\n{\\em Physica D} {\\bf 134} 75 (1999)\n\n\n%\\bibitem[24]{Gefen1} Y. Gefen, A. Aharony and S. Alexander,\n% {\\em Phys. Rev. Lett} {\\bf 50} 77 (1983).\n%\n%bibitem[24]{Aharony} D. Stauffer and A. Aharony, {\\em Percolation Theory}\n% (Taylor and Francis, London 1994).\n\n%\n%bibitem[25]{Havlin} S. Havlin and D. Ben Avraham,\n% {\\em Adv. Phys.}, {\\bf 36}, 695 (1987).\n\n\n\\end{references}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002084.extracted_bib",
"string": "\\bibitem[]{} \nAuthor e-mail: barkai@mit.edu\n\n\n\\bibitem[]{} \n\n\n\\bibitem[1]{Bouch} J. P. Bouchaud and A. Georges, {\\em Phys. Rep.}\n {\\bf 195} 127 (1990).\n\n\n\\bibitem[2]{Levy} \n M.F. Shlesinger, G. M. Zaslavsky and U. Frisch ed. \n {\\em L\\'evy Flights and Related Topics in Physics}\n (Springer-Verlag, Berlin, 1994).\n\n\n\\bibitem[3]{Klafter1} \nJ. Klafter, M. F. Shlesinger and G. Zumofen, \n {\\em Physics Today} {\\bf 49} (2) 33 (1996).\n\n\n\\bibitem[4]{Benkadda} E. Barkai and J. Klafter,\nin {\\em Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas}\n G. M. Zaslavsky and S. Benkadda ed.\n(Springer-Verlag, Berlin 1998).\n\n%\n\n\\bibitem[5]{Geisel} T. Geisel, J. Nierwetberg and A. Zacherl, {\\em Phys. Rev. Lett.\n} {\\bf 54}, 616 (1985).\n\n\n\\bibitem[6]{Klafter5} G. Zumofen and J. Klafter, {\\em Phys.Rev. E} {\\bf 47},\n 851 (1993).\n\n%\n\n\\bibitem[7]{Swin} T. H. Solomon, E. R. Weeks and H. L. Swinney,\n {\\em Phys. Rev. Lett.} {\\bf 71}, 23 (1995).\n\n%\n\n\\bibitem[8]{Antony} A. Torcini and M. Antony,\n {\\em Phys. Rev. E} {\\bf 57}, R6233 (1998).\n%\n\n\\bibitem[9]{Mats} Matsuoka and R. F. Martin,\n {\\em J. Stat. Phys.} {\\bf 88} 81 (1997).\n\n%\n\n\\bibitem[10]{Geisel2} T. Geisel, A. Zacharel and G. Radons {\\em Z. Phys. B} {\\bf 71} 117 (1988). \n\n\n\\bibitem[11]{Levitz} P. Levitz, {\\em Europhys. Lett.} {\\bf 39} 6 593(1997).\n\n\n\\bibitem[12]{Levitz1} P. Levitz and D. Tchoubar, {\\em J. Phys. I} {\\bf 2} 771 (1992).\n\n%\n\\bibitem[13]{Hovi} J. P. Hovi, A. Aharony, D. Stauffer and B. B. Mandelbrot, {\\em Phys. Rev. Lett.} {\\bf 77} 877 (1996).\n\n\n\n\\bibitem[13]{Henk} H. van Beijeren, {\\em Rev. Mod. Phys.} {\\bf 54} 195 (1982).\n%\n\n\n\\bibitem[14]{Gras} P. Grassberger, {\\em Physica A} {\\bf 103} 558 (1980).\n\n\n\\bibitem[15]{Spohn} H. van Beijeren and H. Spohn, {\\em J. Stat. Phys.} {\\bf 31} 231 (1983).\n\n\n\\bibitem[16]{Ernst} M. H. Ernst, J. R. Dorfman, R. Nix and D. Jacobs {\\em Phys. Rev\n. Lett.}\n {\\bf 74} 4416 (1995).\n\n\n\n\\bibitem[17]{barkaiJSP} E. Barkai and V. Fleurov, {\\em J. Stat. Phys.} {\\bf 96}, 325 (1999).\n\n\n\\bibitem[18]{previous4} E. Barkai and V. Fleurov, {\\em Phys. Rev. E}\n {\\bf 56} 6355 (1997).\n\n\n\\bibitem[19]{Feller} W. Feller,{\\em An introduction to probability Theory and\nIts Applications} Vol. 2 (John Wiley and Sons 1970).\n\n\n\\bibitem[20]{Sokolov} I. M. Sokolov, J. Mai and A. Blumen, {\\em Phys. Rev. Lett.} {\\bf 79} 857 (1997). \n\n\n\\bibitem[21]{West} M. Stefancich, P. Allegrini, L. Bonci, P. Grigolini and B. J. west {\\em Phys. Rev. E} {\\bf 57} 6625 (1998) \n\n%\n\n\\bibitem[22]{COX} D. R. Cox, {\\em Renewal Theory},\n (Methuen \\& Co. London, 1970).\n\n\n\n\\bibitem[23]{Fog} H. C. Fogedby, {\\em Phys. Rev. Lett.} {\\bf 73} 2517 (1994),\n{\\em Phys. Rev. E} {\\bf 58} 1690 (1998).\n\n\n\\bibitem[24]{latora}\nV. Latora, P. Rapisarda and S. Ruffo\n{\\em Phys. Rev. Lett.} {\\bf 83} 2104 (1999),\nB.A. Carreras, V.E. Lynch, D.E. Newman and G. M. Zaslavsky\n{\\em Phys. Rev. E} {\\bf 69} 4770 (1999),\nP. Castiglione, A. Mazzino, P. Muratore-Ginanneschi and A. Vulpiani\n{\\em Physica D} {\\bf 134} 75 (1999)\n\n\n%\n\\bibitem[24]{Gefen1} Y. Gefen, A. Aharony and S. Alexander,\n% {\\em Phys. Rev. Lett} {\\bf 50} 77 (1983).\n%\n%bibitem[24]{Aharony} D. Stauffer and A. Aharony, {\\em Percolation Theory}\n% (Taylor and Francis, London 1994).\n\n%\n%bibitem[25]{Havlin} S. Havlin and D. Ben Avraham,\n% {\\em Adv. Phys.}, {\\bf 36}, 695 (1987).\n\n\n"
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cond-mat0002085
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Nucleation theory and the phase diagram of the magnetization-reversal transition
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The phase diagram of the dynamic magnetization-reversal transition in pure Ising systems under a pulsed field competing with the existing order can be explained satisfactorily using the classical nucleation theory. Indications of single-domain and multi-domain nucleation and of the corresponding changes in the nucleation rates are clearly observed. The nature of the second time scale of relaxation, apart from the field driven nucleation time, and the origin of its unusual large values at the phase boundary are explained from the disappearing tendency of kinks on the domain wall surfaces after the withdrawal of the pulse. The possibility of scaling behaviour in the multi-domain regime is identified and compared with the earlier observations.\newpage
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"name": "paper.tex",
"string": "%% This LaTeX-file was created by <arko> Mon Feb 7 11:13:33 2000\n%% LyX 1.0 (C) 1995-1999 by Matthias Ettrich and the LyX Team\n\n%% Do not edit this file unless you know what you are doing.\n\\documentclass[12pt]{article}\n\\usepackage{a4wide}\n\\usepackage{graphics}\n\n\\makeatletter\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.\n\\newcommand{\\LyX}{L\\kern-.1667em\\lower.25em\\hbox{Y}\\kern-.125emX\\@}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.\n\\topmargin -1cm\n\n\\makeatother\n\n\\begin{document}\n\n\n\\title{Nucleation theory and the phase diagram of the magnetization-reversal transition}\n\n\\maketitle\n{\\par\\centering Arkajyoti Misra\\footnote{\ne-mail : arko@cmp.saha.ernet.in\n} and Bikas K. Chakrabarti\\footnote{\ne-mail : bikas@cmp.saha.ernet.in\n} \\par}\n\n{\\par\\centering \\emph{Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta\n700 064, India.}\\par}\n\nPACS. 64.60 Ht - Dynamic Critical Phenomena.\n\nPACS. 64.60 Kw - Mulicritical Points.\n\nPACS. 64.60.Qb - Nucleation.\n\n\\begin{abstract}\nThe phase diagram of the dynamic magnetization-reversal transition in pure Ising\nsystems under a pulsed field competing with the existing order can be explained\nsatisfactorily using the classical nucleation theory. Indications of single-domain\nand multi-domain nucleation and of the corresponding changes in the nucleation\nrates are clearly observed. The nature of the second time scale of relaxation,\napart from the field driven nucleation time, and the origin of its unusual large\nvalues at the phase boundary are explained from the disappearing tendency of\nkinks on the domain wall surfaces after the withdrawal of the pulse. The possibility\nof scaling behaviour in the multi-domain regime is identified and compared with\nthe earlier observations.\\newpage\n\n\\end{abstract}\nThe study of dynamical phase transitions in pure Ising systems, with ferromagnetic\nshort range interactions, under the influence of time dependent external magnetic\nfield has recently become one of the significant areas of interest in statistical\nphysics \\cite{acrmp}\\cite{srnpre}\\cite{mcpre}. The effect of external magnetic\nfields which are periodic in time was first dealt with mean field theory \\cite{to}.\nSubsequently, through extensive Monte Carlo studies, the existence of a dynamic\nphase transition was established and properly characterized \\cite{srnpre}\\cite{acprb}\\cite{srnprl}.\nA relevant investigation in this context was the study of the effect of external\npulsed field which is uniform in space but applied for a finite duration. All\nthese pulsed field studies are concerned with the system below its static critical\ntemperature \\( T_{c}^{0} \\), where the system has a long range order characterized\nby a non-zero magnetization \\( m_{0} \\). Under the influence of a `positive'\nfield or a field applied along the direction of prevalent order, the system\ndoes not show any new phase transition \\cite{abc}. However, a `negative' pulsed\nfield competes with the existing order and the system may show a transition\nfrom an initial equilibrium state with magnetization \\( m_{0} \\) to a final\nequilibrium state with magnetization \\( -m_{0} \\) \\cite{mcpa}. Such a transition\ncan be brought about by tuning the duration (\\( \\Delta t \\)) and strength (\\( h_{p} \\))\nof the pulse, and the `phase diagram' in the \\( h_{p}-\\Delta t \\) plane gives\nthe minimal combination of these two parameters to bring about the transition\nat any temperature \\( T \\) below \\( T_{c}^{0} \\). The transition is dynamic\nin nature and both length and time scales are seen to diverge across this transition\n\\cite{mcpre}\\cite{smc}. \n\nAccording to classical theory of nucleation, there could be two mechanisms through\nwhich the droplet of a particular spin grows in the sea of opposite spins. When\nthe external magnetic field is relatively weak, a single droplet grow to cover\nthe whole system size and this regime is called the single-droplet (SD) \\cite{srnprl}\nor nucleation regime \\cite{as}. On the other hand, under stronger magnetic\nfields, many small droplets grow simultaneously and eventually they coalesce\nto span the entire system. This is called the multi-droplet (MD) or coalescence\nregime\\cite{srnprl}\\cite{as}. The crossover from the SD to the MD regime takes\nplace at the dynamic spinodal field or \\( H_{DSP} \\), which is a function of\nthe system size \\( L \\) and of the temperature \\( T \\). There are two time\nscales in the problem : (i) the nucleation time \\( \\tau _{N} \\) is the time\ntaken by the system to leave the metastable state under the influence of the\nexternal field and (ii) relaxation time \\( \\tau _{R} \\) which is defined as\nthe time taken by the system to reach the final equilibrium state after the\nexternal field is withdrawn. In this letter, we show that the phase diagram\nof the magnetization-reversal transition can be explained satisfactorily by\nemploying the classical nucleation theory \\cite{gd}. The nature of the transition\nchanges form a discontinuous to a continuous one, giving rise to a `tricritical'\npoint as the system goes from the SD regime to the MD regime, depending on the\ntemperature and the strength of the applied pulse. The dimension dependent factor\nby which the nucleation time differs in the two regimes is confirmed by Monte\nCarlo simulation of \\( 2d \\) kinetic Ising model evolving under Glauber dynamics\nbelow \\( T^{0}_{c} \\). Our observations here also support the validity of the\nfinite size scaling of the of the order parameter used earlier to obtain the\ndynamic critical exponents of the transition \\cite{mcpre}.\n\nMonte Carlo simulation using single spin-flip Glauber dynamics have been used\nto study pure Ising system in 2\\( d \\) below \\( T_{c}^{0} \\) (\\( \\simeq 2.27,\\textrm{ } \\)\nin units of the strength of the cooperative interaction). The system is subjected\nto a magnetic field \\( h(t) \\) of finite strength \\( h_{p} \\) for a finite\nduration \\( \\Delta t \\) : \n\n\n\\begin{equation}\n\\label{eqn1}\n\\begin{array}{ccrlc}\nh(t) & = & -h_{p} & \\textrm{for }t_{0}\\leq t\\leq t_{0}+\\Delta t & \\\\\n & = & 0 & \\textrm{otherwise}.\n\\end{array}\n\\end{equation}\nIn order to prepare the system in an ordered state corresponding to a temperature\n\\( T<T_{c}^{0} \\), \\( t_{0} \\) is taken much larger than the equilibrium relaxation\ntime of the system at that temperature. The negative sign signifies that the\napplied field competes with the prevalent order characterized by equilibrium\nmagnetization \\( m_{0} \\), which is a function of \\( T \\) only. Depending\non the values of \\( h_{p} \\) and \\( \\Delta t \\), the system can either go\nback to the original state (with magnetization \\( m_{0} \\)) or to the other\nequivalent ordered state, characterized by equilibrium magnetization \\( -m_{0} \\),\neventually after the withdrawal of the field. This gives rise to the magnetization-reversal\ntransition which brings the system from one equilibrium ordered state to the\nother, below \\( T_{c}^{0} \\). \n\nIt appears that the average magnetization at the time of withdrawal of the pulse\n\\( m_{w}\\equiv m(t_{0}+\\Delta t) \\) is a very relevant quantity : depending\non the sign of \\( m_{w} \\), the system chooses the final equilibrium state.\nOn an average, the transition to the \\( -m_{0} \\) state occurs for negative\nvalues of \\( m_{w} \\), whereas for positive \\( m_{w} \\) the system goes back\nto the original \\( +m_{0} \\) state. There occur of course fluctuations, where\nthe transition to the opposite order takes place even for small positive \\( m_{w} \\)\nvalues or vice versa. The phase boundary is therefore identified as the \\( m_{w}=0 \\)\nline. Fig. 1 shows the Monte Carlo phase boundaries at a few different temperatures\nbelow \\( T_{c}^{0} \\) for a 100\\( \\times \\)100 lattice with periodic boundary\ncondition. At very low temperatures (typically for \\( T<0.5 \\)), the transition\nis discontinuous in nature all along the phase boundary. Crossover to a continuous\ntransition region appears along the phase boundary for higher temperatures,\nthereby giving rise to a tricritical point. Along a typical phase boundary where\nboth kinds of transition are observed, the continuous transition region appears\nfor smaller values of \\( \\Delta t \\) (higher values of \\( h_{p} \\)), whereas\nthe discontinuous region appears for higher values of \\( \\Delta t \\) (smaller\nvalues of \\( h_{p} \\)). \n\nIt is instructive to look at the droplet picture of the classical nucleation\ntheory to describe the nature of the phase diagram for the magnetization-reversal\ntransition. The typical configuration of a ferromagnet below \\( T_{c}^{0} \\)\nconsists of clusters or droplets of down spins in a background of up spins or\nvice versa. According to the classical nucleation theory \\cite{gd}, the equilibrium\nnumber of droplets of size \\( l \\) is then given by \\( n_{l}=N\\exp (-\\beta \\epsilon _{l}) \\),\nwhere \\( \\beta =1/k_{B}T \\) and \\( \\epsilon _{l} \\) is the free energy of\nformation of a droplet of size \\( l \\). Assuming spherical shape of the droplets,\none can write \\( \\epsilon _{l}=2hl+\\sigma (T)l^{(d-1)/d} \\), where \\( \\sigma \\)\nis the temperature dependent surface tension and \\( h \\) is the external magnetic\nfield. Droplets of size greater than a critical value \\( l_{c} \\) are then\nfavoured to grow. One obtains \\( l_{c} \\) by maximizing \\( \\epsilon _{l} \\)\n: \\( l_{c}=\\left[ \\sigma (d-1)/2d\\left| h\\right| \\right] ^{d} \\). The number\nof supercritical droplets is then given by \\( n_{l}^{*}=N\\exp \\left( -\\beta K_{d}\\sigma ^{d}/\\left| h\\right| ^{d-1}\\right) \\),\nwhere \\( K_{d} \\) is a constant depending on dimension only. In the nucleation\nregime or the SD regime, there is only one supercritical droplet that grows\nand engulfs the entire system. The nucleation time \\( \\tau ^{SD}_{N} \\) is\ninversely proportional to the nucleation rate \\( I \\). According to the Becker-D\\( \\ddot{\\textrm{o}} \\)ring\ntheory \\cite{gd}, \\( I \\) in turn is proportional to \\( n^{*}_{l} \\), and\nthus\n\n\\[\n\\tau ^{SD}_{N}\\propto I^{-1}\\propto \\exp \\left( \\frac{\\beta K_{d}\\sigma ^{d}}{\\left| h\\right| ^{d-1}}\\right) .\\]\n In the coalescence or MD regime, where due to stronger magnetic field many\ndroplets grow simultaneously and eventually coalesce to form a system spanning\ndroplet, the nucleation time is given by \\cite{as} \n\\[\n\\tau ^{MD}_{N}\\propto I^{-1/(d+1)}\\propto \\exp \\left( \\frac{\\beta K_{d}\\sigma ^{d}}{(d+1)\\left| h\\right| ^{d-1}}\\right) .\\]\nDuring the time when the field is `on', the only relevant time scale of the\nproblem is the nucleation time. The phase boundary of the magnetization-reversal\ntransition corresponds to the threshold value of the field pulse (\\( h_{p}^{c} \\)),\nwhich can bring the system from an equilibrium state with magnetization \\( m_{0} \\)\nto a state with magnetization \\( m_{w}=0_{-} \\) in time \\( \\Delta t \\), so\nthat the system eventually evolves to a state with magnetization \\( -m_{0} \\).\nEquating therefore \\( \\Delta t \\) with the nucleation time \\( \\tau _{N} \\),\none gets for the magnetization-reversal phase diagram \n\\begin{equation}\n\\label{eqn2}\n\\begin{array}{cccccl}\n\\ln (\\Delta t) & = & c_{1} & + & \\frac{C}{h_{p}^{d-1}} & \\textrm{in SD regime}\\\\\n & = & c_{2} & + & \\frac{C}{(d+1)h_{p}^{d-1}} & \\textrm{in MD regime},\n\\end{array}\n\\end{equation}\nwhere \\( C=\\beta K_{d}\\sigma ^{d} \\) and \\( c_{1} \\), \\( c_{2} \\) are constants.\nTherefore a plot of \\( \\ln (\\Delta t) \\) against \\( \\left( h_{p}^{c}\\right) ^{d-1} \\)\nwould show different slopes in the two regimes. This is indeed the case as shown\nin Fig. 2, where two distinct slopes are seen at higher temperatures (typically\nfor \\( T>0.5 \\)). The ratio of the slopes \\( R \\) corresponding to the two\nregimes has a value close to \\( 3 \\) as predicted for \\( 2d \\) by the classical\nnucleation theory. The point of intersection of the two straight lines gives\nthe value of \\( H_{DSP} \\) at that temperature and system size. At lower temperatures,\nhowever, the MD region is absent which is marked by a single slope (Fig. 2(d)).\n\nAccording to the classical nucleation theory \\( l_{c}\\propto \\left| h_{p}\\right| ^{-d} \\)\nand at any fixed temperature, therefore, one expects stronger (weaker) fields\nto bring the system to the MD (SD) regime. Fig. 3 shows snapshots of the spin\nconfiguration at a particular temperature, where the dots correspond to down\nspins. It is clear from the figure that many droplets of down spins are formed\nfor a smaller value of \\( \\Delta t \\) or equivalently for large \\( h_{p} \\)\nwhereas only a single down spin droplet is formed for a larger \\( \\Delta t \\)\nor weaker \\( h_{p} \\). The snapshots of the system are taken at the time of\nwithdrawal of the pulse, beyond which the system is left to itself to relax\nback to either of the equilibrium states. The time taken by the system to reach\nthe final equilibrium after the withdrawal of the pulse is defined as the relaxation\ntime (\\( \\tau _{R} \\)). It is observed \\cite{mcpre} that \\( \\tau _{R}\\sim \\kappa (T,L)\\exp \\left[ -\\lambda (T)\\left| m_{w}\\right| \\right] \\),\nwhere \\( \\kappa (T,L) \\) is a constant depending on temperature and system\nsize (\\( \\kappa \\rightarrow \\infty \\textrm{ as }L\\rightarrow \\infty \\)) and\n\\( \\lambda (T) \\) is a constant depending on temperature. This is in distinct\ncontrast with the normal relaxation of a ferromagnet to the equilibrium state\nwith magnetization \\( m_{0} \\), starting from a random initial state where\nthe average magnetization is close to zero. The effect of the field is to initiate\nthe nucleation process and by the time the pulse is withdrawn, the droplet(s)\nhas (have) very few kinks along the surface(s). Once a droplet or a domain forms\na flat boundary, it becomes a rather stable configuration and the domain wall\nmovement practically stops; thereby restricting further nucleation. When the\nsystem is trapped in such a metastable state, it is left for large fluctuations\nto initiate further movement of the domain walls and resume the process of nucleation.\nThe closer to zero is \\( m_{w} \\), more is the chance that the system will\nbe trapped in a metastable state owing to more number of flat domain walls.\nThus the effect of the pulse is to create a domain structure which does not\nfavour fast nucleation and even for \\( T\\ll T_{c}^{0} \\), \\( \\tau _{R} \\)\ntherefore diverges at the phase boundary.\n\nTo determine the order of the phase transition one can look at the probability\ndistribution \\( P(m_{w}) \\) of \\( m_{w} \\), as one approaches a phase boundary.\nFig. 4 shows the variation of \\( P(m_{w}) \\) across the phase boundary corresponding\nto a particular \\( T \\) at two different values of \\( \\Delta t \\). The existence\nof a single peak of \\( P(m_{w}) \\) in (a), which shifts its position continuously\nfrom \\( +1 \\) to \\( -1 \\), indicates the continuous nature of the transition;\nwhereas in (b), one obtains two peaks simultaneously, close to \\( \\pm m_{0}(T) \\),\nindicating that the system can simultaneously reside in two different phases\nwith finite probabilities. This is a sure indication of discontinuous phase\ntransition. The crossover from the discontinuous transition to a continuous\none along the phase boundary is not very sharp; instead one gets a `tricritical\nregion' where the determination of the nature of the transition is not very\nconclusive. As is evident from Fig. 2, the data points in this particular region\ndo not fit to either of the two straight lines corresponding to the two regimes.\nThus for \\( h_{p}^{c}\\ll H_{DSP} \\) the system belongs to the MD regime and\nthe nature of the transition is continuous, whereas for \\( h_{p}^{c}\\gg H_{DSP} \\)\nthe system is brought to the SD regime where the transition is discontinuous\nin nature.\n\nIn our earlier work \\cite{mcpre}, we clearly noticed that finite size scaling\nof the order parameter was possible only at higher temperatures for moderately\nlow \\( \\Delta t \\). The reason behind that becomes clear now, as at higher\ntemperatures the tricritical point shifts towards higher values of \\( \\Delta t \\)\nand hence in the lower \\( \\Delta t \\) region the system belongs to the MD regime\nand the transition is continuous in nature. This supports the scaling. Since\nat lower temperatures the continuous region gradually shrinks before disappearing\naltogether, all attempts for a finite size scaling fit failed. This observation\nalso compares with that of Sides et al. \\cite{srnprl} for the scaling behaviour\nin the case of dynamic transition under periodic fields.\n\nIn this letter we have shown that the phase diagram of the magnetization-reversal\ntransition in the Ising model can be explained satisfactorily from the classical\nnucleation theory. Across \\( H_{DSP} \\), the system goes from MD or coalescence\nregime where the transition is continuous in nature to SD or nucleation regime\nwhere the transition is discontinuous. A tricritical point separates these two\nregimes which moves towards smaller \\( \\Delta t \\) (larger \\( h_{p} \\)) values\nas one decreases \\( T \\) until it disappears altogether at low temperatures.\nThe dimensional factor in the nucleation time of the two regimes is found to\nbe quite accurately reproduced in the Monte Carlo simulations. There are two\ntime scales in the problem, viz. \\( \\tau _{N} \\) and \\( \\tau _{R} \\). \\( \\tau _{N} \\),\nwhich is determined by the pulse duration \\( \\Delta t \\), brings the system\nfrom one regime to the other depending on the temperature; while \\( \\tau _{R} \\)\ndetermines the relaxation of the field-released system and it diverges at the\nphase boundary even for \\( T\\ll T_{c}^{0} \\). Finite size scaling of the order\nparameter for the transition is also justified in the MD regime where the transition\nis continuous in nature.\n\n{\\par\\centering {*}{*}{*}\\par}\n\nA. M. would like to thank Muktish Acharyya and Burkhard Duenweg for useful discussions. \n\\newpage\n\n\\begin{thebibliography}{}\n\\bibitem{acrmp}Chakrabarti B. K. and Acharyya M., \\emph{Rev. Mod. Phys.} \\textbf{71} (1999)\n847.\n\\bibitem{srnpre}Sides W., Rikvold P. A. and Novotny M. A., \\emph{Phys. Rev. E} \\textbf{57} (1998)\n6512.\n\\bibitem{mcpre}Misra A. and Chakrabarti B. K., \\emph{Phys. Rev. E} \\textbf{58} (1998) 4277.\n\\bibitem{to}Tome T. and de Oliveira M. J., \\emph{Phys. Rev. A} \\textbf{41} (1990) 4251.\n\\bibitem{acprb}Acharyya M. and Chakrabarti B. K., \\emph{Phys. Rev. B} \\textbf{52} (1995) 6550.\n\\bibitem{srnprl}Sides S. W., Rikvold P. A. and Novotny M. A., \\emph{Phys. Rev. Lett.} \\textbf{81}\n(1998) 834, \\emph{Phys. Rev. E} \\textbf{59} (1999) 2710.\n\\bibitem{abc}Acharyya M., Bhattacharjee J. K. and Chakrabarti B. K., \\emph{Phys. Rev. E}\n\\textbf{55} (1997) 2392.\n\\bibitem{mcpa}Misra A. and Chakrabarti B. K. , \\emph{Physica A} \\textbf{247} (1997) 510.\n\\bibitem{smc}Stinchcombe R. B., Misra A. and Chakrabarti B. K., \\emph{Phys. Rev. E} \\textbf{59}\n(1999) R4717.\n\\bibitem{as}Acharyya M. and Stauffer D., \\emph{Euro. Phys. J. B} \\textbf{5} (1998) 571.\n\\bibitem{gd}See e.g., Gunton J. D. and Droz M., \\emph{Introduction to the Theory of Metastable\nand Unstable States}, Lecture Notes in Physics, vol. 183, Springer, Heidelberg,\n1983.\\newpage\n\n\\end{thebibliography}\n\\textbf{\\large Figure Captions}{\\large \\par}\n\n\\noindent Fig. 1. Monte Carlo phase diagram for magnetization-reversal transition\nin a \\( 100\\times 100 \\) lattice: (a) \\( T=0.2 \\), (b) \\( T=0.6 \\), (c) \\( T=1.0 \\),\n(d) \\( T=1.4 \\), (e) \\( T=2.0 \\).\n\n\\noindent Fig. 2. Plot of \\( \\ln (\\Delta t) \\) against \\( \\left[ h_{p}^{c}(\\Delta t,T)\\right] ^{-1} \\),\nobtained from the phase diagram in Fig. 1, at different temperatures. (a) \\( T=1.4 \\),\n(b) \\( T=1.0 \\), (c) \\( T=0.7 \\), (d) \\( T=0.2 \\). The observed values of\nthe slope ratio \\( R \\) are close to \\( 3.38 \\), \\( 3.09 \\) and \\( 3.24 \\)\nin (a), (b) and (c) respectively.\n\n\\noindent Fig. 3. Snapshots of the spin configuration of the system at \\( t=t_{0}+\\Delta t \\)\nfor \\( T=1.0 \\); (a) \\( \\Delta t=2 \\), \\( h_{p}=1.91 \\) (MD regime) (b) \\( \\Delta t=100 \\),\n\\( h_{p}=0.65 \\) (SD regime).\n\n\\noindent Fig. 4. Probability distribution \\( P(m_{w}) \\) of \\( m_{w} \\) at\n\\( T=0.6 \\). In (a) where \\( \\Delta t=1 \\) (SD regime) , the position of the\npeak changes sharply from \\( +1 \\) to \\( -1 \\) indicating a discontinuous\ntransition: (i) \\( h_{p}=2.05 \\), (ii) \\( h_{p}=2.15 \\), (iii) \\( h_{p}=2.25 \\).\nIn (b) where \\( \\Delta t=500 \\) (MD regime) , the position of the peak changes\ncontinuously from \\( +1 \\) to \\( -1 \\) indicating a continuous transition:\n(i) \\( h_{p}=0.92 \\), (ii) \\( h_{p}=1.02 \\), (iii) \\( h_{p}=1.12 \\).\n\\newpage\n\n\\vspace{0.3cm}\n{\\par\\centering \\includegraphics{fig1.eps} \\par}\n\\vspace{0.3cm}\\newpage\n\n\\vspace{0.3cm}\n{\\par\\centering \\includegraphics{fig3.eps} \\par}\n\\vspace{0.3cm}\\newpage\n\n\\vspace{0.3cm}\n{\\par\\centering \\includegraphics{fig2.eps} \\par}\n\\vspace{0.3cm}\\newpage\n\n\\vspace{0.3cm}\n{\\par\\centering \\includegraphics{fig4a.eps} \\par}\n\\vspace{0.3cm}\\newpage\n\n\\vspace{0.3cm}\n{\\par\\centering \\includegraphics{fig4b.eps} \\par}\n\\vspace{0.3cm}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002085.extracted_bib",
"string": "\\begin{thebibliography}{}\n\\bibitem{acrmp}Chakrabarti B. K. and Acharyya M., \\emph{Rev. Mod. Phys.} \\textbf{71} (1999)\n847.\n\\bibitem{srnpre}Sides W., Rikvold P. A. and Novotny M. A., \\emph{Phys. Rev. E} \\textbf{57} (1998)\n6512.\n\\bibitem{mcpre}Misra A. and Chakrabarti B. K., \\emph{Phys. Rev. E} \\textbf{58} (1998) 4277.\n\\bibitem{to}Tome T. and de Oliveira M. J., \\emph{Phys. Rev. A} \\textbf{41} (1990) 4251.\n\\bibitem{acprb}Acharyya M. and Chakrabarti B. K., \\emph{Phys. Rev. B} \\textbf{52} (1995) 6550.\n\\bibitem{srnprl}Sides S. W., Rikvold P. A. and Novotny M. A., \\emph{Phys. Rev. Lett.} \\textbf{81}\n(1998) 834, \\emph{Phys. Rev. E} \\textbf{59} (1999) 2710.\n\\bibitem{abc}Acharyya M., Bhattacharjee J. K. and Chakrabarti B. K., \\emph{Phys. Rev. E}\n\\textbf{55} (1997) 2392.\n\\bibitem{mcpa}Misra A. and Chakrabarti B. K. , \\emph{Physica A} \\textbf{247} (1997) 510.\n\\bibitem{smc}Stinchcombe R. B., Misra A. and Chakrabarti B. K., \\emph{Phys. Rev. E} \\textbf{59}\n(1999) R4717.\n\\bibitem{as}Acharyya M. and Stauffer D., \\emph{Euro. Phys. J. B} \\textbf{5} (1998) 571.\n\\bibitem{gd}See e.g., Gunton J. D. and Droz M., \\emph{Introduction to the Theory of Metastable\nand Unstable States}, Lecture Notes in Physics, vol. 183, Springer, Heidelberg,\n1983.\\newpage\n\n\\end{thebibliography}"
}
] |
cond-mat0002086
|
Applications of Chern-Simons Ward Identities to $ \log(T\tau) $ Conductivity Calculations of the $\nu=1/2$ System
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[
{
"author": "J\\\"urgen Dietel"
},
{
"author": "Institut f\\\"ur Theoretische Physik"
},
{
"author": "Universit\\\"at Leipzig"
},
{
"author": "Germany"
}
] |
We reconsider the theory of the half-filled Landau level with impurities using the Chern-Simons formulation and study Ward identities for the Chern-Simons theory. From these we get conductivity diagrams with impurities which obey the continuity equation. We calculate the conductivity of these diagrams for which we obtain $ \log(T\tau) $ divergent conductivity diagrams for low temperature $ T $. We compare our result with the experimental values of the low temperature conductivities. Finally we calculate the conductivity for small deviations of the magnetic field $ B $ from the value $\nu=1/2 $. In this case we get a singularity in the conductivity.
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[
{
"name": "prep.tex",
"string": "\n\\documentclass[12pt,twoside]{article} % Specifies the document class\n \\usepackage{amsmath, amsfonts,pst-feyn,pst-key}\n\n\\newfont{\\banner}{cmssdc10 scaled 1820}\n\\newfont{\\figuretitle}{cmssdc10 scaled 1440}\n\\textwidth15cm\n\\oddsidemargin0cm\n\\evensidemargin1cm\n\\title{Applications of Chern-Simons Ward Identities to \n$ \\log(T\\tau) $ Conductivity \nCalculations of the $\\nu=1/2$ System } \n % Declares the document's title.\n\\author{J\\\"urgen Dietel \\\\ \nInstitut f\\\"ur Theoretische Physik, \\\\ Universit\\\"at Leipzig, \nGermany } % Declares the author's name.\n\n\\date{} % Deleting this command produces today's date.\n\n\\begin{document} % End of preamble and beginning of text.\n\\psset{ArrowInside=->, ArrowAdjust=true}\n\n\\maketitle \n\\begin{abstract}\nWe reconsider the theory of the half-filled Landau level with impurities \nusing the Chern-Simons formulation and study Ward identities for \nthe Chern-Simons\ntheory. From these we get conductivity diagrams with impurities \nwhich obey the continuity equation. We calculate the conductivity of these \ndiagrams for which we obtain $ \\log(T\\tau) $ divergent \nconductivity diagrams for \nlow temperature $ T $. We compare our result with the experimental values\nof the low temperature conductivities. Finally we calculate the conductivity \nfor small deviations of the magnetic field $ B $ from the value $\\nu=1/2 $.\nIn this case we get a singularity in the conductivity. \n\\end{abstract}\n\n\\section{Introduction}\nIn this paper we consider a system of electrons in a strong magnetic \nfield in two dimensions. This system is characterized by \nthe filling factor $ \\nu $, defined as the electron density divided by the \ndensity of a completely \nfilled Landau level. In the case of $ \\nu \\approx 1/2 $ the \nbehavior of the system resembles that of a Fermi liquid in the \nabsence of a magnet field or at small magnetic fields. Over past years an intriguing picture has emerged: \nAt $ \\nu=1/2 $ each electron combines with two flux quanta of the magnetic field to form a composite fermion (CF); these composite fermions \nthen move in an effective\nmagnetic field which is zero on the average. The interpretation of \nmany experiments support this picture. We mention some transport experiments\nfor illustration \\cite{wi1}.\\\\\nThe theoretical framework for the understanding of the $ \\nu=1/2 $-system \nwas \ndeveloped by Halperin, Lee and Read \\cite{hlr}. In their theory one has \nto pay a \nprice to get a mean-field free system for the CFs. The CFs \ninteract via long-ranged gauge interactions. Further in real $ GaAs/Al_xGa_{1-x}As$ \nheterojunctions one also has a large number of Coulomb impurities which \nhas to be integrated in the theory, too. For the case of the Coulomb problem \nthis was done about 20 years ago \\cite{al1, al2}. \\\\\nOur aim is to calculate the low temperature conductivity of CFs \nwith impurities. \nIn the first section of this work we care about which Feynman graphs should be \ncalculated to get a good conductivity for the $ \\nu=1/2 $ system. This \nquestion will be treated with the help of Ward identities \n\\cite{ab1} of Chern-Simons systems. In the second section we will \napply the results of section 1 to get some Feynman-graphs which should\ngive a good approximation of the conductivity for CFs. At least \n we will calculate the conductivity of these Feynman graphs \nat low temperature $ T $ and make comparison with experimental results. \n \n\\section{The Impact of Ward Identities of CFs on the \nperturbational continuity equation}\n\n\\begin{figure}[t]\n {\n \\psset{unit=1cm}\n\\begin{pspicture}(10,3)\n\\psset{linewidth=1.5pt,arrowinset=0}\n\\psline[linewidth=1.3pt,ArrowInside=->](0.5,1.7)(2.0,1.7)\n\\psline[linewidth=1.3pt,ArrowInside=-](0.,1.5)(0.5,1.7)\n\\psline[linewidth=1.3pt,ArrowInside=-](2.0,1.7)(2.5,1.5)\n\\WavyLine[n=25,beta=180](0.5,1.7)(0.5,1.2)\n\\WavyLine[n=25,beta=180](2.0,1.7)(2.0,1.2)\n\\rput(3.25,1.3){$\\rightarrow$}\n\n% \n\\psline[linewidth=1.3pt,ArrowInside=->](4.5,1.7)(5.25,1.8)\n\\psline[linewidth=1.3pt,ArrowInside=->](5.25,1.8)(6.,1.7)\n\\rput(5.35,2.3) {$(1,\\frac{2k_i+q_i}{2m})$ }\n\\psline[linewidth=1.3pt,ArrowInside=-](4.,1.5)(4.5,1.7)\n\\psline[linewidth=1.3pt,ArrowInside=-](6.,1.7)(6.5,1.5)\n\\WavyLine[n=25,beta=180](4.5,1.7)(4.5,1.2)\n\\WavyLine[n=25,beta=180](6.,1.7)(6.,1.2)\n\\psdots[dotstyle=*,dotscale=0.6](5.25,1.8)\n\\rput(3.25,0.2) { (a) } \n\n%\n\\psline[linewidth=1.3pt,ArrowInside=->](8.5,1.7)(9.5,1.7)\n\\psline[linewidth=1.3pt,ArrowInside=->](9.5,1.7)(10.5,1.7)\n\\WavyLine[n=25,beta=180](9.5,1.7)(9.5,0.7)\n\\psline[linewidth=1.3pt,ArrowInside=->](8.5,0.7)(9.5,0.7)\n\\psline[linewidth=1.3pt,ArrowInside=->](9.5,0.7)(10.5,0.7)\n\\psdots[dotstyle=pentagon,dotscale=1.1](9.5,1.7)\n\\rput(11.3,1.4){$\\rightarrow$}\n\\rput(9.5,2.3) {$\\frac{\\left(2\\vec{k}+\\vec{q}\\right)\\cdot\\vec{f}(q)}{2m}$ }\n%\n\\psline[linewidth=1.3pt,ArrowInside=->](12,1.7)(13,1.7)\n\\psline[linewidth=1.3pt,ArrowInside=->](13,1.7)(14,1.7)\n\\WavyLine[n=25,beta=180](13,1.7)(13,0.7)\n\\psline[linewidth=1.3pt,ArrowInside=->](12,0.7)(13,0.7)\n\\psline[linewidth=1.3pt,ArrowInside=->](13,0.7)(14,0.7)\n\\psdots[dotstyle=pentagon,dotscale=1.1](13,1.7)\n\\rput(11.3,0.2) { (b) } \n\\rput(13,2.3) {$\\frac{\\vec{e}_i\\cdot\\vec{f}(q)}{2m}$ }\n\\end{pspicture}\n \\caption{Vertexoperations for constructing $ \\Lambda^b $ from a \n self energy graph $ \\Sigma^b $. For $ \\Lambda^b_0 $ one has to do \n vertexoperation (a) with a \n density-vertex between the two Green's functions. \n For $ \\Lambda^b_i $ one has to do vertex operation (a) with a \n current-vertex between the two Green's functions as well as the \n vertex operation (b).}\n}\n\\end{figure}\n\nIn this section we try to answer the question which subset of \nFeynman diagrams should be calculated, in order to get a good estimate for \nthe conductivity of the $ \\nu=1/2 $-\nsystem. \nTo answer this question we consider at first some \nself energy diagrams $ \\Sigma^p(k) $ of \nfrequency $ k_0 $ and impulse $(k_1,k_2)$ (Index $ p $ for perturbational). \nIn the following we will construct $ T $-product response Feynman-graphs \nfrom $ \\Sigma^p $. \n$ \\Lambda_\\mu $ ( $ \\mu=0..2 $ ) is defined as \n\\begin{equation} \\label{eq1}\n\\left\\langle\\left|T\\left[j_\\mu(q),\\Psi^*(k)\\Psi(k+q)\\right]\\right|\\right\\rangle\n= G(k) \\Lambda_\\mu(k+q,q) G(k+q)\\;, \n\\end{equation}\nwhere $ G(k) $ is the Chern-Simons Green's function.\n $ \\vec{j} $, $ \\Psi $ are the current operator and field operator \nof the Chern-Simons theory. \nIn the following one has to notice that the Chern-Simons vertices \ncontain current vertices \n$ \\frac{(2\\vec{k}+\\vec{q})\\vec{f}(q)}{2m} $ where \n$ f_i(q)=i 2 \\pi \\tilde{\\phi}\\epsilon_{ij} \\frac{q_j}{q^2}$. \nWe now apply some transformations on $ \\Sigma^p $ to get \n$ \\Lambda_\\mu $ graphs. \nWe use the notation $ \\Lambda^p_0 $ for \n$ \\Lambda_0 $ diagrams which originate from \n$ \\Sigma^p $ through the operation (a) in figure 1 for every Green's \nfunction in $ \\Sigma^p $ .\n This means that one \none has to 'divide' every Green's function in $ \\Sigma^p $ making two \nGreen's function. Similarly $ \\Lambda^p_i$ ($ i=1,2 $) denotes the \ndiagrams which are originating through operation (a), (b) in figure 1.\nThis has to be done for every \nGreen's function and current vertex in $ \\Sigma^p$.\nIn contrast to $ \\Lambda^p_0 $ operation (a) puts for $ \\Lambda^p_i$ \na current operator between the two Green's functions. \nThe vertices in diagram (b) are current CS-vertices. In (b) \nevery current vertex of $ \\Sigma^b $ is transformed into a density-density \nCS current coupling. \nThen one can derive the following Ward identities \n\\begin{eqnarray} \n-q_0 \\left(\\Lambda^p_0(k,k+q_0e_0)-1\\right) & = & \\Sigma^p(k+q_0e_0)\n-\\Sigma^p(k) \\label{eq2}\\; , \\\\ \nq_i \\left(\\Lambda^p_{i}(k,k+q_ie_i)-q_i\\frac{2 k_{i}+q_i}{2 m}\\right) & = & \n\\Sigma^p(k+q_ie_i)-\\Sigma^p(k)\\;. \\label{eq3}\n\\end{eqnarray} \nTo prove these relations we classify the Green's functions $ G^0_i $\n(the free Green's functions) \nof the self energy graph $ \\Sigma^p $. $ G^0_{i_1} \\sim G^0_{i_2} \\in W_j \\in {\\cal W} $\nif there is a path in the graph $ \\Sigma^p $ which do not pass a vertex, \nconnecting $ G^0_{i_1} $ with $ G^0_{i_2} $. \n$ G^0_a $, $G^0_e \\in W_{ae} $ denote the Green's functions which are at \nthe continuation of the outer truncated Green's functions\nof the graph $\\Sigma^p $. In the following we show at first \nrelation (\\ref{eq2}).\n$ \\Sigma(k+q_0e_0) $ is the graph in which one makes the substitution \n$ G^0_i (k') \\rightarrow G^0_i(k'+q_0 e_0) $ for every Green's function in \n$\\Sigma^p$.\nFurthermore we denote as $ \\Gamma^p_i(k+q,k;k',k'+q) $ \nthe graph in which \na Green's function $ G^0_i $ is eleminated from $ \\Sigma^p $. \nWith the help of $ \\{G^0_{1},G^0_{2}.. \\}=W_j \\not= W_{ae}$ one gets \n\\begin{equation} \\label{eq4}\n\\sum\\limits_{G^0_i \\in W_j} \\sum\\limits_{k'} \n\\Gamma_{i}(k,k+q_0e_0;k'+qe_0,k')\\left(\nG^0_{i}(k'+q_0e_0)-G^0_{i}(k')\\right)\\quad = \\quad 0 \\;.\n\\end{equation}\nThis is valid because the Green's functions in $ W_j \\not= W_{ae}$\nare forming a circle.\nFurthermore one gets for $ \\{G^0_{1},G^0_{2}.. \\}= W_{ae}$\n\\begin{eqnarray} \n\\lefteqn{\\sum\\limits_{G^0_i \\in W_{ae}} \\sum\\limits_{k'} \n\\Gamma^p_{i}(k,k+q_0e_0;k'+q_0e_0,k')\n\\left(G^0_{i}(k'+q_0e_0)-G^0_{i}(k')\\right)}\n\\label{eq5}\\\\ \n& = & \\sum\\limits_{k'}\n \\Gamma^p_{e}(k,k+q_0e_0;k'+q_0e_0,k')G^0_{e}(k'+q_0e_0) \n-\\Gamma^p_{a}(k,k+q_0e_0;k'+q_0e_0,k')G^0_{a}(k') \\nonumber \\\\\n & = & \\Sigma^p(k+q_0 e_0)-\\Sigma^p(k) \\;. \\nonumber \n\\end{eqnarray}\nWith the help of (\\ref{eq4}), (\\ref{eq5}), one gets (\\ref{eq2}). \\\\\nWe now prove the current Ward identity (\\ref{eq3}). \nIf one has no current vertex in $ \\Sigma^p $ one gets immediately \n(\\ref{eq3}) in the same manner as (\\ref{eq2}). Otherwise \nwe denote $ C $ as the number of current vertices.\nFurthermore we denote $ B(G^0_i) $ ($ E(G^0(i)$) as the beginning- (end-)\nvertex of the directed Green's function $ G^0_i $. \nIf $ B(G^0_i) \\in C $ ($E(G^0_i)\\in C$), we denote \n$ \\vec{j}(B(G^0_i)) $ ($\\vec{j}(E(G^0_i)$) as the current of the vertex \n$ B(G^0_i)$ ($ E(G^0_i) $). \nRelations (\\ref{eq4}), (\\ref{eq5}) keep their validity if one makes \nthe following substitutions: Every Green's function $ G^0_i(k'+q_0 e_0) $\n($ G^0_i(k'+q_0 e_0) $)\nin the sums of (\\ref{eq4}), (\\ref{eq5})\nshould be replaced by $ G^0_i(k'+q_i e_i)$ ($ G^0_i(k')$) if \n$ E(G^0_i) \\notin S $ ($B(G^0_i)\\notin S$), \notherwise by \n$ G^0_i(k'+q_i e_i)\\left( 1+ \n\\left[\\vec{j}(E(G^0_i)) \\rightarrow \\frac{q_i}{2m} e_i \\right]\\right) $ \n$ \\left(G^0_i(k')\\left( 1- \n\\left[\\vec{j}(B(G^0_i)) \\rightarrow \\frac{q_i}{2m} e_i \\right]\\right) \\right)$.\nThe brackets have the meaning of \nreplacing the \ncurrent $ \\vec{j} $ of the vertex $ B(G_i) $ ($E(G_i)$) in \n$ \\Gamma^p_i $ by $ \\frac{q_i}{2m} e_i $. With the help of \nthese substitutions in the expressions (\\ref{eq4}), (\\ref{eq5}) one immediately gets the \ncurrent Ward identity (\\ref{eq3}).\\\\\n\\vspace{0.5cm}\nSimilar to (\\ref{eq2}), (\\ref{eq3}) one can show with $ q=(q_0,q_1,q_2) $ \nthe following combined Ward identity \n\\begin{equation} \\label{eq8}\n-q_0\\Lambda^p_0(k,k+q)+\\sum\\limits_{i=1}^2 q_i\\Lambda^p_{i}(k,k+qe_i)\n+q_0-\\sum\\limits_{i=1}^2 q_i\\frac{2 k_i+q_i}{2 m}= \\Sigma^p(k+q)-\\Sigma^p(k)\n\\;. \n\\end{equation} \nWe now consider our primary problem. It is clear that the \nresponse (density or current) to an external \n$ \\vec{A} $-field should obey \nthe continuity equation. We now define the approximation of the \ndensity-current and \ncurrent-current $ T $-product as \n\\begin{eqnarray} \\label{eq9}\n\\lefteqn{\\hspace{-2cm}\n\\left\\langle\\left|j_\\mu(q)j_\\nu(-q)\\right|\\right\\rangle \\approx \n\\Pi^p_{\\mu i}(q)\n= \\int \\sum\\limits_{k'} G^{\\Sigma}(k'+q) \\Lambda ^p_\\mu(k'+q,k')\nG^{\\Sigma}(k')} \\\\\n& & \\hspace{3cm}\n\\times \\left(\\delta_{\\mu,0}+(1-\\delta_{\\mu,0})\\frac{2\n k_\\mu'+q_\\mu}{2m}\\right) \\;,\\nonumber\n\\end{eqnarray}\nwhere $ G^\\Sigma(k)=(k_0-\\xi(\\vec{k})-\\Sigma^p(k))^{-1} $ \\\\\nWithin the help of the combined Ward identities (\\ref{eq8}), \nthe Kubo-formula and \nthe relation of $ T $-products and retarded correlation functions \n\\cite{ab1},\none sees that the continuity equation is valid \nif $ \\Pi^p_{\\mu \\nu} $ fulfills the equation\n\\begin{equation} \\label{eq10}\n e^2 q_0 \\Pi^p_{0 i}(q)-e^2 \\sum\\limits_{j=1}^2 q_j \\Pi^p_{j i}(q)=-i q_i \n\\frac{e^2 n_e}{m}\\;,\n\\end{equation}\nwhere $ n_e $ is the electron density. \\\\\nThis equation can easily checked with the help of (\\ref{eq8}).\nNow we have the problem that the graphs $\\Pi^p_{\\mu \\nu}(q) $ are not \nsymmetric in the coupling of the currents. \nOn one side of $\\Pi^p_{\\mu \\nu}(q) $ one\nhas density-density Chern-Simons current-couplings. \nThese current couplings are missing \non the other side of the graphs $\\Pi^p_{\\mu \\nu}(q) $. To get rid of \nthis problem we name $ \\Lambda^{b,4}_{i} $ as the members of\n$ \\Lambda^{b}_i $ which are originating through operation (b) in figure 1 from \n$ \\Sigma^b $. We now close the open ends of $\\Lambda^{b,4}_{i} $ with a \nGreen's function $ G^0 $and call these graphs ${\\Lambda}^{b,4}_{i} $. \nThese graphs consists of closed Green's function loops.\nTo these graphs we apply the operations (a) or (b) of figure 1, respectively, \nand call these graphs $ \\Xi_{0 i}(q)$ or $ \\Xi_{j i}(q) $, respectively .\nThan one sees similar to the proof of (\\ref{eq2}), (\\ref{eq3}) \nthat $ \\Xi_{0 i}(q) $, $ \\Xi_{j i}(q) $ fulfills the equation \n$ e^2 q_0 \\Xi^p_{0 i}(q)-e^2 \\sum\\limits_{j=1}^2 q_j \\Xi^p_{j i}(q)=0 $\n($ \\Xi_{0 i}(q) $, $ \\Xi_{j i}(q) $ connsists only of closed Green's function \ncircles). \nWith the definition $ {\\Pi}^{b,2}_{\\mu \\nu}=\\Pi^{b}_{\\mu \\nu}+\\Xi^p_{\\mu \\nu} $one sees that $ {\\Pi}^{b,2}$ consists of current symmetric graphs. \nFurthermore $ {\\Pi}^{b,2}$ fulfills equation (\\ref{eq10}). \\\\\nIt is also clear that the continity equation is fulfilled \nif one only considers the graphs in $ {\\Pi}^{b,2}$ which have the same number \nof vertices. \n\n\\section{The Calculation of the Conductivity} \n\nNext we calculate the conductivity of a $ \\nu=1/2 $ \nChern-Simons gas with impurities. The calculable quantity is the conductivity \nof CFs which is related to the physical conductivity through \n$ \\sigma^{CS}=\\sigma^2_{xx}+\\sigma_{xy}/\\sigma_{xx} $ \\cite{hlr}.\nFor the mean-field Green's function of CFs in an impurity \nbackground one has \n$ G_{\\pm}(\\omega,p)=\n\\left(\\omega-\\xi(p) \\pm i/(2\\tau)\\right)^{-1} $. \nIn the following calculation \nwe need momentum integrals $ \\int\\limits_{0}^{\\infty} dp $ of products of \nsuch Green's functions. One can approximate\nthese momentum integrals (also for $ k_F l\\approx 1 $) by extending the range \nof such integrals to infinity, $ \\int\\limits_{-\\infty}^{\\infty} dp $. \nWe use this approximation in the following. \nIn doing that we hope to get better results for smaller $ k_F l $\nin contrast to the standard $ k_F l \\gg 1 $ approximation. \nTo calculate physical \nquantities also for smaller densities of the $ \\nu=1/2 $-system is suggestive \nbecause one could reduce the effective $ k_F l $ of a $\\nu=1/2$-system \nin increasing \nthe magnetic field to get a $ \\nu=(5/2,9/2,13/2 ..)$-system. Systems with these fillings behave similar to $ \\nu=1/2 $-systems with a reduced density \n$ n_e=(n_e/5, n_e/9,n_e/13..) $.\\\\\nIn the following we limit our calculation to the case of \nimpuritis with a $ \\delta $-corrolation \\cite{hlr}. \nIt could be shown \\cite{fa1,kh1} that one gets the same \nresults in the case of RMF-scattering for $ k_Fl\\gg 1 $.\nFurther we will only discuss the particle-hole channel conductivity graphs \nbecause we don't expect any weak localization correction due to the magnetic \nbroken time reversal symmetry \\cite{hlr}. \\\\ \nIn the Coulomb gauge ($ div \\vec{A}=0 $), the gauge interaction between \nCFs at $ \\omega \\tau \\ll 1 $ and $ql \\ll 1 $ is \ndescribed by \n\\begin{equation} \\label{eq11}\n\\begin{array}{c c c}\nD^{-1}_{\\mu \\nu} & = & \\left(\n\\begin{array} {c c}\n\\frac{m}{2\\pi} \\frac{Dq^2}{Dq^2-i \\omega} & -i \\frac{q}{4 \\pi} \\\\\ni \\frac{q}{4 \\pi} & -i \\gamma_q |\\omega|+\\chi_q q^2\n\\end{array}\n\\right) \\;, \n\\end{array}\n\\end{equation}\nwhere $ D=\\frac{1}{2} v_F^2 \\tau $ is a diffusion coefficient, \n$ \\chi_q=\\frac{1}{24 \\pi m} +\\frac{V_q}{(4 \\pi)^2} $ is an effective \ndiamagnetic susceptibility given in terms of the \npairwise electron potential $ V_q $, \n$ \\gamma_q=\\frac{mD}{2\\pi} $ is proportional\n to the CF mean free path $ l= v_F \\tau $ and $m$ stands for the \neffective mass.\nComparing the gauge interaction functions (\\ref{eq11}) with the gauge \ninteraction functions in \\cite{hlr,kh1} one sees that we get a difference \nby a factor $ 1/2 $ in $ \\chi_q $. \nIn \\cite{di1} we made an exakt calculation of the polarisator \n$ \\Pi_{11} $ without impurities. In this polarisator one also gets\nhalf the value of the magnetic susceptibility in \\cite{hlr}. \nSo we believe that our calculation is correct.\nAs in \\cite{kh1} we will discuss in the following the two \nregimes $ V_q \\approx V_0=2 \\pi \\frac{e^2}{\\kappa} $ and \n$ V_q = 2\\pi \\frac{e^2}{q} $. \\\\\nWe now generate the conductivity diagrams from \nthe left hand side self energy diagrams in figure 2. These diagrams are also \nthe starting point in the case of the Coulomb gas \\cite{al1,al2}. Then one \ngets\namong others the diagrams in the right hand side of figure 2. \nIn \\cite{al2} was \nshown that the conductivity of the diagrams not listed \nin figure 2 adds to zero. This can also be shown for our \n$ k_F l\\approx 1 $ approximation. This is also significant for CFs. \nWe now invert the matrix (\\ref{eq11}). So we get \n\\begin{eqnarray} \nD_{00} & = & \\frac{-i|\\omega|+Dq^2}{\\frac{m}{2\\pi}D q^2-\\frac{q^2}{16\\pi^2} \n \\frac{-i\\, |\\omega|+Dq^2}{-i \\gamma_q |\\omega|+\\chi_q q^2}}\\;, \\label{eq12}\\\\\n D_{11} & = & \\frac{1}{-i\\gamma_q |\\omega|+\\chi_q q^2-\\frac{q^2}{16\\pi^2} \n \\frac{-i\\,|\\omega|+Dq^2}{\\frac{m}{2\\pi}D q^2}} \\approx\n \\frac{1}{-i \\gamma_q |\\omega|+\\chi'_q q^2} \\,\\theta(Dq^2-|\\omega|) \\;,\n\\label{eq13}\n\\end{eqnarray}\nwhere $ \\chi'_q=\\chi_q-\\frac{1}{8\\pi m} $.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%% Die Strom-Strom Korrelationsgraphen\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\\begin{figure}[t]\n {\n \\psset{unit=0.9cm}\n% \\begin{pspicture}[2](12,10)\n\\begin{pspicture}[-0.2](10,2)\n\\psset{ArrowInside=->, ArrowAdjust=true}\n\\psset{linewidth=1.5pt,arrowinset=0}\n\\WavyArc[linewidth=1.5pt,n=20,beta=180](1.3,0.7){0.8}{0}{180}\n\\psline[ArrowInside=->](0.5,0.7)(2.1,0.7)\n\\psline[ArrowInside=-](0.,0.7)(0.5,0.7)\n\\psline[ArrowInside=-](2.1,0.7)(2.6,0.7)\n\\psset{linewidth=0.5pt,arrowinset=0}\n\\psarcn[linewidth=0.5pt,linestyle=dashed]{-}(0.5,0.7){0.1}{180}{0}\n\\psarcn[linewidth=0.5pt,linestyle=dashed]{-}(0.5,0.7){0.25}{180}{0}\n\\psarcn[linewidth=0.5pt,linestyle=dashed]{-}(2.1,0.7){0.1}{180}{0}\n\\psarcn[linewidth=0.5pt,linestyle=dashed]{-}(2.1,0.7){0.25}{180}{0}\n\\psarcn[linewidth=0.5pt,linestyle=dashed]{-}(1.3,0.7){1.2}{180}{0}\n\\rput(0.3,0.9){\\footnotesize x}\n\\rput(2.3,0.9){\\footnotesize x}\n\\rput(1.3,1.9){\\footnotesize x}\n\\rput(3.2,0.9){$\\rightarrow$}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\\psset{linewidth=1pt,arrowinset=0}\n\\psellipse(5,1)(1.2,0.7)\n\\WavyArc[linewidth=1.5pt,n=20,beta=180](5,1.6){0.6}{0}{180}\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](5,0.3)(5,1.7)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](4.8,0.3)(4.8,1.7)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](5.2,0.3)(5.2,1.7)\n\\rput(5.05,1){\\footnotesize x }\n\\rput(4.85,1){\\footnotesize x }\n\\rput(5.25,1){\\footnotesize x }\n\\psdots[dotstyle=*,dotscale=0.4](3.8,1)\n\\psdots[dotstyle=*,dotscale=0.4](6.2,1)\n\\rput(5,-0.3) { (a) } \n\\rput(3.7,1.6) { $ \\scriptstyle \\epsilon+\\Omega $ }\n\\rput(6.2,1.6) { $ \\scriptstyle \\epsilon+\\Omega $ }\n\\rput(4.2,0.3) { $ \\scriptstyle \\epsilon$ }\n\\rput(5.8,0.3) { $ \\scriptstyle \\epsilon$ }\n\\rput(4.9,1.45) { $ \\scriptstyle \\epsilon+\\omega+\\Omega $ }\n\\psline[linewidth=0.9pt] {->}(3.88,1.2)(3.98,1.35)\n%\n\\rput(7.5,0.7){\\footnotesize x}\n\\rput(7.95,0.7){\\footnotesize x}\n\\rput(7.75,0.7){\\footnotesize x}\n\\psellipse(8,1)(1.2,0.7)\n\\WavyLine[linewidth=1.5pt,n=20,beta=180](7.5,1.6)(8.5,0.4)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](7.5,0.4)(8.5,1.6)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](7.3,0.43)(8.3,1.63)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](7.7,0.37)(8.7,1.57)\n\\psdots[dotstyle=*,dotscale=0.4](6.8,1)\n\\psdots[dotstyle=*,dotscale=0.4](9.2,1)\n\\rput(8,-0.3) { (b) } \n\\rput(7.0,1.8) { $ \\scriptstyle \\epsilon+\\Omega $ }\n\\rput(8.8,1.9) { $ \\scriptstyle \\epsilon+\\omega+\\Omega $ }\n\\rput(7.2,0.3) { $ \\scriptstyle \\epsilon$ }\n\\rput(9.1,0.4) { $ \\scriptstyle \\epsilon+\\omega$ }\n\\psline[linewidth=0.9pt] {->}(6.88,1.2)(6.98,1.35)\n% \n\\psellipse(11,1)(1.2,0.7)\n\\rput(10.5,0.7){\\footnotesize x}\n\\rput(10.95,0.7){\\footnotesize x}\n\\rput(10.75,0.7){\\footnotesize x}\n\\WavyLine[linewidth=1.5pt,n=20,beta=30](10.1,1.4)(12.2,1)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](10.5,0.4)(11.5,1.6)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](10.3,0.43)(11.3,1.63)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](10.7,0.37)(11.7,1.57)\n\\psdots[dotstyle=*,dotscale=0.4](9.8,1)\n\\psdots[dotstyle=*,dotscale=0.4](12.2,1)\n\\rput(11,-0.3) { (c) } \n% \\rput(10.2,0.3) { $ \\scriptstyle \\epsilon$ }\n% \\rput(9.6,1.3) { $ \\scriptstyle \\epsilon+\\Omega $ }\n% \\rput(12.0,1.8) { $ \\scriptstyle \\epsilon+\\omega+\\Omega $ }\n%\n\n\\psellipse(14,1)(1.2,0.7)\n\\WavyLine[linewidth=1.5pt,n=20,beta=180](12.8,1)(15.2,1)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](14,0.3)(14,1.7)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](13.8,0.3)(13.8,1.7)\n\\psline[linewidth=0.5pt,linestyle=dashed,ArrowInside=-](14.2,0.3)(14.2,1.7)\n\\rput(14.05,0.65){\\footnotesize x }\n\\rput(13.85,0.65){\\footnotesize x }\n\\rput(14.265,0.65){\\footnotesize x }\n\\psdots[dotstyle=*,dotscale=0.4](12.8,1)\n\\psdots[dotstyle=*,dotscale=0.4](15.2,1)\n\\rput(14,-0.3) { (d) } \n% \\rput(14,7.1) { $ \\scriptstyle \\epsilon$ }\n% \\rput(14.0,8.9) { $ \\scriptstyle \\epsilon+\\Omega$ }\n\\end{pspicture}\n \\caption{The relevant conductivity diagrams discussed in the text. \nOn the left hand side one sees the self energy diagram from which the \nconductivity diagrams are extracted through the formalism of section 2.}\n}\n\\end{figure} \nOne can further see from (\\ref{eq11}) that the vertex $ D_{01} $ isn't \nas much divergent as $ D_{00} $, $D_{11} $. \nBecause of this one can show, that this vertex gives no \n$ \\log(T \\tau) $-term in the conductivity. \nWe now calculate the graphs (a), (b) of figure 2.\nAt first we will calculate the graphs (a), (b) with the $ D_{00} $-vertex. \nTo get the divergent part of the graphs one has to put the vertex correction \n$ \\Gamma(\\epsilon,\\omega,q)=(m \\tau)^{-1}(D q^2-i |\\omega|)^{-1} $ \nfor $ \\epsilon (\\epsilon+\\omega) <0 $ \n at the endpoints of the vertex $ D_{00} $. One can then \nuse the analysis of the Coulomb problem \\cite{al2}.\nAfter integration one gets for $ D \\gamma_q \\gg \\chi_q $ \n\\begin{equation} \\label{eq14}\n\\delta \\sigma^{D_{00}}_{ii}\\approx\\frac{1}{\\pi h} \\left(\\frac{(k_F l)^2}\n{(\\frac{5}{3}+(k_F l)^2)}+ \\frac{2}{3} \\frac{(k_F l)^2}{(\\frac{5}{3}+(k_F l)^2)^2}\n\\log\\left(\\frac{3}{2}(k_F l)^2\\right)\\right)\\log(T \\tau) \\;,\n \\end{equation}\nwhere $ l=(h k_F \\tau)/(2\\pi m) $.\\\\ \nFrom (\\ref{eq14}) one sees that in the case $ (k_F l) \\gg 1 $, \n$ \\sigma_{ii}^{D_0} $ goes to the conductivity of the Coulomb problem \\cite{al1}.\nWe now calculate the conductivity of the $ D_{11} $-vertex. \nTo calculate the diagrams (a), (b) of figure 2 one needs the quantities \n$ J^\\pm_i(\\vec{q})=\n\\sum_{\\vec{p}} G_{\\pm}^2(p) G_{\\mp}(p)\\frac{\\vec{p}_i}{m}(\n\\frac{\\vec{p}}{m} \n\\times \\frac{\\vec{q}}{q})=(\\mp i \\epsilon_F \\tau^2-\\tau/2) \n(\\vec{e}_i \\times \\frac{\\vec{q}}{q}) $. \nSimilar for the graphs (c), (d) one needs the quantity \n$ J_i(\\vec{q})=\n\\sum_{\\vec{p}} G_{\\pm}(p) G_{\\mp}(p)(\n\\vec{e}_i \n\\times \\frac{\\vec{q}}{q})=\\tau (\\vec{e}_i \\times \\frac{\\vec{q}}{q}) $. \nOne now has to discuss the different frequency combinations of (a), (b).\nWe make in the following the convention $ \\Omega >0 $.\nBecause of the frequency constraints of the pole of the particle-hole channel\n$ (1/\\tau) \\Gamma(\\epsilon,\\omega,q) $ one has \nsome frequency constraint on the graphs to get this pole. \nDu to the different prefactor of the first terms \n$ J^{\\pm}_1 $ of $ J^\\pm $ one immediately gets for the nonvanishing \n$ (J^{\\pm}_1)^2 $-\nterms in $ \\sigma^{D_{11},a}+ \\sigma^{D_{11},b}$ \nthe two frequency combinations $\\epsilon < 0, \\epsilon+\\Omega>0,\n\\epsilon+\\omega +\\Omega>0, \\epsilon+\\omega<0 $ and \n $\\epsilon < 0, \\epsilon+\\Omega<0,\n\\epsilon+\\omega +\\Omega>0, \\epsilon+\\omega>0 $. In each one of these \ntwo frequency combinations one gets \n$\\sigma^{D_{11},a}+ \\sigma^{D_{11},b}=2\\sigma^{D_{11},a} $.\nFor the diagrams (a) (b) with reversed Green's function direction one gets \ntwo similar frequency combinations of additive $ \\sigma $ from (a), (b). \nThen one can carry out the $ \\Omega $-integration \\cite{al1} and \ngets for the conductivity of the diagrams (a), (b) of \ncurrent-combination $ (J_1^{\\pm})^2 $ \n\\begin{equation} \\label{eq15}\n \\delta \\sigma_{ii}^{D_{11}}\\approx-\\frac{i 2 e^2}{2 \\pi}\n\\int\\limits_\\Omega^{\\frac{1}{\\tau}}\n \\frac{d \\omega}{2\\pi}\\int \\frac{d \\vec{q}}{(2 \\pi)^2}\n\\frac{\\left(4J_{1,i}^+(q)J_{1,i}^+(q)\\right)\n\\;\\theta(Dq^2-|\\omega|) }\n{m \\tau^2 \\left[ D q^2-i\\left|\\omega+\\Omega\\right|\\right]\n \\left(-i |\\omega| \\gamma_q+\\chi'_q q^2\\right)} \\;. \n\\end{equation}\nAt finite temperature \n$ T \\gg \\Omega $ we can calculate the frequency-integral in \n$ \\omega $ by using imaginary frequencies with an $ \\omega $ integral \ncut-off at low frequencies $ \\frac{1}{T} $. For $ V_q=(2\\pi)\n\\frac{e^2}{\\kappa} $ \none gets \n\\begin{equation} \\label{eq16}\n\\delta \\sigma_{ii}^{D_{11}}\\approx\\frac{2}{\\pi h} \n\\left(\\log(k_F l)+i \\frac{3}{8} \\pi\\right) \\log(T \\tau) \\;.\n\\end{equation} \nFor $ V_q=(2\\pi)\\frac{e^2}{q} $ we make a partial fraction decomposition \nof the denominator of (\\ref{eq15}) and get three additive terms\n($ \\Omega \\to 0 $)\n$ 1/2\\, ( \\pm 8 \\pi i |\\omega| \\gamma_q +\\sqrt{i|\\omega|/D})\\,\n(i|\\omega|/D)^{-\\frac{3}{2}}\\, (q\\mp \\sqrt{i|\\omega|/D})^{-1} $ and \n$ i D/|\\omega|\\, (- 8 \\pi i |\\omega|\\gamma_q +q)^{-1} $. Then one sees \nthat the integral (\\ref{eq15}) results in a finite value for $ T \\to 0 $. \nNext we calculate $\\sigma^{D_{11},a}+ \\sigma^{D_{11},b}$ for the \ncurrent combinations $ (J^{\\pm}_2)^2 $ and $ J^{\\pm}_1 J^{\\pm}_2 $. \nFor example for the \n$ (J^{\\pm}_2)^2 $ combination one gets for the conductivity to be nonvanishing\nthe two regimes \n$ \\epsilon >0 $, $ \\epsilon+\\Omega<0 $ and $ \\epsilon <0 $, \n$ \\epsilon+\\Omega>0 $. Integrating out the $ \\Omega $-integral one \ngets two terms $ \\Omega \\int d\\omega $, $ \\int d\\omega \\,\\omega $. The \n$ \\log(T\\tau) $-singularity is cancelled for the \n$ \\Omega \\int d\\omega $-terms of \nthe two frequency ranges and \nthe remaining two $ \\int d\\omega \\,\\omega $-integrals are infinite. These \ninfinities are cancelled by similar infinities of the graphs $ (c), (d)$. \nFurthermore one can see with the help of \n$ J^{\\pm} $, $ J $ that the $ \\Omega \\int d\\omega $ integrals are cancel \nfor every single graph $ (c) $, $ (d) $. \nIn summary one can see, \nthat the sum of the conductivities of the graphs (a), (b) for $ (J^{\\pm}_2)^2$, $ J^{\\pm}_1 J^{\\pm}_2 $ \nand of the graphs (c), (d) is zero for \n$ T \\to 0$. \nOne sees from the above analysis that the typical $\\rho \\rho $ \npart of the CS-current is important to get a finite result for lower \n$ k_Fl $. \\\\ \nWe now compare our results with the experimental results of Rokhinson et al \n\\cite{ro1}. They get a $ \\log (T) $ dependence for $ \\sigma^{CS}_{xx} $ \nfor temperature $ T < 500$mK . \nSo we use $ V_q=2 \\pi e^2/\\kappa $ to compare our results with their \nmeasurements. \nWith $ \\sigma^{CS}_{xx}=\n\\sigma_0^{CS}+\\lambda \\frac{e^2}{h} \\log(T \\tau) $ they get \n$ \\lambda_{ex}=(0.1,0.4,1.6) $ for Drude conductivities \n$ \\sigma_0^{CS}=e^2/(2h) (k_F l)=(5.4 e^2/h, 16.0 e^2/h, 39.0 e^2/h) $.\nFrom (\\ref{eq14}), (\\ref{eq16}) we get for the theoretical value \n$ \\lambda_{th} $, $ \\lambda_{th}=(1.81,2.53,3.1) $.\nWe see from these results that $ \\lambda_{th} $ is getting better for \n larger $ k_F l $. \\\\\nAn improvement of the above calculation would be the calculation of \nthe conductivity of diagrams which are\noriginating through the operations of section 2 from the \ncorresponding Hartree self energy diagrams of figure 2. \nFor the $ D_{00} $-vertex this was done by Castellani et al. \\cite{fi1}. \nThey calculated higher order $ D_{00} $-vertex graphs \nand got for $ k_Fl \\gg 1 $, \n$ \\sigma_{xx}^{CS} = e^2/(\\pi h) (2-2\\log(2)) \\log(T \\tau) $. This is \nonly a small correction to (\\ref{eq14}) and (\\ref{eq16}) for $ k_F l \\gg 1 $. \nFurther one sees from $ D_{11}(0,q) \\sim 1/q^2 $ \nthat the $ D_{11} $-Vertex can not give any contribution to \n$ \\sigma^{CS}_{xx} $ in Hartree-diagrams \\cite{al1}. \\\\\nAt least one could calculate the CS-conductivity from the diagrams of \nfigure 2 for small deviations of the magnetic field $ B_{1/2} $. \nThis means that the CFs are subject to an effective magnetic field \n $ B^d=B-B_{1/2} $. For $ \\omega^d_c\\tau \\ll 1 $, $ k_F l \\gg1 $\none gets with the help of the formalism of Houghton et al. \\cite{ho1},\n$ J^\\pm_{1,i}(\\vec{q},B^d)=i \\epsilon_F \\tau^2(\\mp\n1/(1+(\\omega_c^d \\tau)^2)(\\vec{e}_i \\times \\frac{\\vec{q}}{q})\n-(\\omega^d_c \\tau) \n(\\vec{e_i}\\cdot \\frac{\\vec{q}}{q})) $. \nAlso one gets for \nthe magnetic correction to the \ndiffusion constant $ D $, \n$ D(B^d)=D/(1+(\\omega_c^d \\tau)^2) $ (this should also be considered \nin $\\gamma_q$). \nIn this $ k_F l \\gg 1 $ limit, the only relevant graphs in figure (2) \nare (a), (b).\nBecause the second term in $ J^\\pm_{1,i}(\\vec{q},B^d) $ has the same \nbehaviour as $ J_2^{\\pm} $ for $ B^d=0 $ one sees from the \nabove analysis that the conductivity \n$ \\sigma^{a}_{xx}(B^d)+\\sigma^{b}_{xx}(B^d) $\nis infinite for \n$ |\\omega^d_c \\tau|>0 $. Since the graphs (c),(d) of figure 2 \ngive only lower order terms in $ k_F l $ this singularity can not be \ncancelled. So the next stage of improvement should be \nto consider Feynman-graphs which make $ \\sigma_{xx}(B^d)$ finite for \n$ |\\omega^d_c \\tau|>0 $. This should also give better results for \n$ \\sigma_{xx}(0)$. Further we have to remark that we don't\nuse a density of states correction at the fermi-level due to $ B_d \\not=0 $\nin the above formulas because the general statement of an infinite\nconductivity is still correct.\\\\ \nAt least we have to remark that Khveshchenko \\cite{kh1, kh2} calculated the \nconductivity of the graphs (a), (b) of figure 2 for $ k_F l \\gg 1 $. He comes to \nother results than (\\ref{eq16}) and also for the conductivity \nof $ V_q=2 \\pi e^2/q $ in the large $ k_F l $ limit.\nFor example the difference to (\\ref{eq16}) is a factor $ 1/4 $. \nThis is exakt the result one gets if one assumes \nfrequency constraints through vertex corrections $ \\Gamma $\nat the endpoints of the $ D_{11} $ vertex as in the case of the \n$ D_{00}$ vertex \\cite{al2}. These Feynman graphs \nhave a vanishing conductivity \n because of the current couplings at the endpoints of the $ D_{11} $ vertices. \nIf one calculates the graphs (a), (b) without vertex corrections, \none gets another result, as we have shown in this paper.\n \n\\begin{thebibliography}{99}\n\\bibitem{wi1} R.L. Willett et al. , Phys. Rev. Lett. {\\bf 65},112 (1990);\nW. Kang et al., Phys. Rev. Lett. {\\bf 71},3850 (1993); V.J. Goldman et al., Phys. Rev. Lett. {\\bf 72},2065 (1994) \n \\bibitem{hlr} B.I. Halperin, P.A. Lee, and N. Read, Phys. Rev. B {\\bf 47},7312 (1993)\n\\bibitem{ab1} A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of quantum field theory in statistical physics, Dover publication, New York (1975); \nJ.R. Schrieffer, Theory of Superconductivity (Benjaming, Reading, Mass., 1964)\n; P. Nozieres, Theory of interacting Fermi systems, W.A. Benjamin, New York (1964) \n\\bibitem{di1} J. Dietel, Th. Koschny, W. Apel and W. Weller, \nEur. Phys. J. B {\\bf 5},439 (1998)\n\\bibitem{fa1} \nA.G. Aronov, A.D. Mirlin; and P. W\\\"olfle, Phys. Rev. B {\\bf 49},16609 (1994);\nV. Fal'ko, Phys. Rev. B {\\bf 50},17406 (1994) \n\\bibitem{al1} B.L. Altshuler, A.G. Aronov and P.A. Lee, Phys. Rev. Lett. \n{\\bf 44},1288 (1980)\n\\bibitem{al2} B.L. Altshuler, D. Khmel'nitzkii, A.I. Larkin and P.A. Lee, Phys. Rev. B {\\bf 22},5142 (1980)\n\\bibitem{ro1} L.P. Rokhinson, B. Su and V.J. Goldman,\n Phys. Rev. B {\\bf 52},11588 (1995)\n\\bibitem{ho1} A. Houghton, J. R. Senna, and S.C. Ying, \n Phys. Rev. B {\\bf 25},2196 (1982) \n\\bibitem{kh1} D.V. Khveshchenko,\n Phys. Rev. Lett. {\\bf 77},1817 (1996) \n\\bibitem{kh2} D.V. Khveshchenko,\n preprint cond-mat/9605051 \n\\bibitem{fi1} A.M. Finkelstein, Sov. Phys. Jetp {\\bf 57},97 (1983); \n C. Castellani, C. Di Castro, P.A. Lee and M. Ma, \n Phys. Rev. B {\\bf 30},527 (1984) \n\\end{thebibliography} \n\\end{document}\n\n"
},
{
"name": "pst-feyn.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*- Mode: Latex -*- %%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% pst-feyn.tex --- Generation of feynman diagrams with PSTricks\n%%\n%% Author : Thomas SIEGEL <siegel@aix550.informatik.uni-leipzig.de>\n%% Created the : Mon Aug 25 16:02:02 MET DST 1997\n%% Last mod. by : Thomas SIEGEL <siegel@aix550.informatik.uni-leipzig.de>\n%% Last mod. the : Thu Feb 19 17:11:38 MET 1998\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\def\\fileversion{0.94}\n\\def\\filedate{1998/02/19}\n\n\\message{`PST-Feynman' v\\fileversion , \\filedate\\space (Thomas Siegel)}\n\n\\csname PSTFeynmanLoaded\\endcsname\n\\let\\PSTFeynmanLoaded\\endinput\n\n% Require PSTricks and pst-plot packages\n\\ifx\\PSTricksLoaded\\endinput\\else\\input pstricks.tex\\fi\n\\ifx\\PSTplotLoaded\\endinput\\else\\input pst-plot.tex\\fi\n\\ifx\\PSTnodeLoaded\\endinput\\else\\input pst-node.tex\\fi\n\n% DPC interface to the KeyVal package (until keyval based version of PSTricks)\n\\input pst-key.tex\n\n\\edef\\PstAtCode{\\the\\catcode`\\@}\n\\catcode`\\@=11\\relax\n\n% Some macro definitions:\n% -----------------------\n\n% Denis Girou addition begin - Dec. 18, 1997\n\n% Adapted from \\psset@arrows\n\\def\\psset@ArrowInside#1{%\n\\begingroup\n\\pst@activearrows\n\\xdef\\pst@tempg{<#1}%\n\\endgroup\n\\expandafter\\psset@@ArrowInside\\pst@tempg\\@empty-\\@empty\\@nil\n\\if@pst\\else\n\\@pstrickserr{Bad intermediate arrow specification: #1}\\@ehpa\n\\fi}\n\n% Adapted from \\psset@@arrows\n\\def\\psset@@ArrowInside#1-#2\\@empty#3\\@nil{%\n\\@psttrue\n\\def\\next##1,#1-##2,##3\\@nil{\\def\\pst@tempg{##2}}%\n\\expandafter\\next\\pst@arrowtable,#1-#1,\\@nil\n\\@ifundefined{psas@#2}%\n{\\@pstfalse\\def\\psk@ArrowInside{}}%\n{\\def\\psk@ArrowInside{#2}}}\n\\def\\psk@ArrowInside{}\n\n\n% Modified version of \\pst@addarrowdef\n\\def\\pst@addarrowdef{%\n\\addto@pscode{%\n/ArrowA {\n\\ifx\\psk@arrowA\\@empty\n\\pst@oplineto\n\\else\n\\pst@arrowdef{A}\nmoveto\n\\fi\n} def\n/ArrowB {\n\\ifx\\psk@arrowB\\@empty \\else \\pst@arrowdef{B} \\fi\n} def\n% DG addition\n/ArrowInside {\\ifx\\psk@ArrowInside\\@empty \\else \\pst@arrowdefA{Inside} \\fi} def\n}}\n\n% Adapted from \\pst@arrowdef\n\\def\\pst@arrowdefA#1{%\n\\ifnum\\pst@repeatarrowsflag>\\z@\n/Arrow#1c [ 6 2 roll ] cvx def Arrow#1c\n\\fi\n\\tx@BeginArrow\n\\psk@arrowscale\n\\@nameuse{psas@\\@nameuse{psk@Arrow#1}}\n\\tx@EndArrow}\n\n% ArrowInsidePos parameter\n\\def\\psset@ArrowInsidePos#1{\\pst@checknum{#1}\\psk@ArrowInsidePos}%\n\\psset@ArrowInsidePos{0.5}\n\n% ArrowAdjust parameter\n% T.S. addition begin \n\\newif\\ifPst@ArrowAdjust\n\\define@key{psset}{ArrowAdjust}[false]{%\n\\@nameuse{Pst@ArrowAdjust#1}}\n% T.S. addition end\n\n% T.S. addition begin\n\\def\\pst@ArrowScaleX{1 }%\n\\def\\pst@ArrowScaleY{1 }%\n% T.S. addition end\n\n% Modified version of \\pst@getscale\n\\def\\pst@getscale#1#2{%\n\\pst@expandafter\\pst@getnumii{#1 #1} {} {} {}\\@nil\n\\@psttrue\n\\ifdim\\pst@tempg\\p@=\\z@\n\\@pstrickserr{Bad scaling argument #1'}\\@ehpa\n\\def\\pst@tempg{1 }%\n\\@pstfalse\n\\fi\n\\ifdim\\pst@temph\\p@=\\z@\n\\if@pst\\@pstrickserr{Bad scaling argument #1'}\\@ehpa\\fi\n\\def\\pst@temph{1 }%\n\\fi\n% T.S. addition begin\n\\edef\\pst@ArrowScaleX{\\pst@tempg}%\n\\edef\\pst@ArrowScaleY{\\pst@temph}%\n% T.S. addition end\n\\edef#2{\\pst@tempg\\space \\pst@temph\\space scale }%\n\\ifdim\\pst@tempg\\p@=\\p@ \\ifdim\\pst@temph\\p@=\\p@\n\\def#2{}%\n\\fi\\fi}\n\n\n\n% Redefinition of the PostScript /Line macro to print the intermediate arrow\n% on each segment of the line\n\\pst@def{Line}<{%\nNArray\nn 0 eq not\n { n 1 eq { 0 0 /n 2 def } if\n 2 copy\n /y1 ED\n /x1 ED\n ArrowA\n x1 y1\n /n n 1 sub def\n n { 4 copy\n /y1 ED\n /x1 ED\n /y2 ED\n /x2 ED\n% T.S. addition begin\n /dx x2 x1 sub def\n /dy y2 y1 sub def\n \\ifPst@ArrowAdjust\n /@arrowwidth\n \\psk@arrowsize\n \\pst@number\\pslinewidth mul\n add\n def\n /@arrowlength \n \\psk@arrowlength \n @arrowwidth mul \\pst@ArrowScaleY\\space mul\n def\n \\ifx\\pslinestyle\\@none\n /@adjustment 0 def\n \\else\n /@adjustment\n \\pst@number\\pslinewidth \n @arrowlength mul \n @arrowwidth \\pst@ArrowScaleX\\space mul div\n def\n \\fi\n /@arrowlength \n @arrowlength \n @adjustment add \n def\n /t % ��red�mybox�$t=�frac�@arrowlength��2�sqrt�dx^2 + dy^2��$��\n @arrowlength\n dx dx mul dy dy mul add \n sqrt 2 mul \n div \n def\n \\else\n /t 0 def\n \\fi\n /aposx dx \\psk@ArrowInsidePos t add mul x1 add def\n /aposy dy \\psk@ArrowInsidePos t add mul y1 add def\n x1 y1 aposx aposy ArrowInside pop pop pop pop Lineto\n% T.S. addition end\n } repeat\n CP 4 2 roll ArrowB L pop pop } if}>\n\n% Denis Girou addition end\n\n\n%\n% common variables for \\WavyLine and \\WavyArc :\n%\n\n\\define@key{psset}{A}{% \n\\edef\\psk@A{#1}}\n\n\\define@key{psset}{n}{%\n\\edef\\psk@n{#1}}\n\n\\define@key{psset}{alpha}{%\n\\edef\\psk@alpha{#1}}\n\n\\define@key{psset}{beta}{%\n\\edef\\psk@beta{#1}}\n\n\\def\\WavyLine{\\pst@object{WavyLine}}%\n\n\\def\\WavyLine@i{%\n\\setkeys{psset}{plotpoints=200,\n alpha=0,beta=0,\n A=.1,n=10}%\n\\begin@OpenObj\n\\pst@getcoors[\\WavyLine@ii}\n\\def\\WavyLine@ii{%\n\\addto@pscode{\\WavyLine@iii}%\n\\showpointsfalse\n\\end@OpenObj}\n\n\\def\\WavyLine@iii{%\n/CoorField 4 array def\nCoorField astore pop\n/x1 { CoorField 2 get } def\n/y1 { CoorField 3 get } def\n/x2 { CoorField 0 get } def\n/y2 { CoorField 1 get } def\n/hypot { x2 x1 sub dup mul y2 y1 sub dup mul add sqrt } def\n/ep { \\psk@n\\space 2 mul hypot mul } def\n/bn { ep 360 div round 360 mul \\psk@beta\\space add } def\n/korr { bn \\psk@alpha\\space sub ep div } def\n/Shape {\n 2 hypot mul korr mul t mul \\psk@n\\space mul \n \\psk@alpha\\space add sin \\psk@A\\space mul \\pst@number\\psunit mul \n } \ndef\n/denome { Shape hypot div } def\n/xOrtho { y2 y1 sub denome mul } def\n/yOrtho { x2 x1 sub denome mul } def\n/step { 1 \\psk@plotpoints\\space div } def\n/i 0 def\n\\psk@plotpoints 1 add {\n /t i step mul def \n /i i 1 add def\n x2 x1 sub t mul\n x1 add\n xOrtho sub\n y2 y1 sub t mul\n y1 add\n yOrtho add\n t 0 eq { moveto } { lineto } ifelse\n} repeat }\n\n\n\\def\\WavyArc{\\pst@object{WavyArc}}\n\n\\def\\WavyArc@i{%\n\\setkeys{psset}{plotpoints=400,\n alpha=0,beta=0,\n A=.1,n=10}%\n\\@ifnextchar({\\WavyArc@ii}{\\WavyArc@ii(0,0)}}\n\n\\def\\WavyArc@ii(#1)#2#3#4{%\n\\begin@OpenObj\n\\pst@getangle{#3}\\pst@tempa\n\\pst@getangle{#4}\\pst@tempb\n\\pst@@getcoor{#1}%\n\\edef\\@Radius{#2 \\pst@number\\psunit mul \\space}%\n\\addto@pscode{\\WavyArc@iii}%\n\\showpointsfalse\n\\end@OpenObj}\n\n\\def\\WavyArc@iii{%\n\\pst@coor\n/CoorField 2 array def\nCoorField astore pop\n/x1 { CoorField 0 get } def\n/y1 { CoorField 1 get } def\n/r \\@Radius def\n/c 57.2957 r \\tx@Div def\n/angleA \\pst@tempa def\n/angleB \\pst@tempb def\nangleA angleB gt { /angleB angleB 360 add def } if\n/n { \\psk@n\\space \\pst@number\\psunit div } def\n/A { \\psk@A\\space \\pst@number\\psunit mul } def\n/dp { angleB angleA sub } def\n/a { angleA n mul r mul } def\n/b { dp n mul r mul a add } def\n/an {\n a 360 div round 360 mul \\psk@alpha\\space\n add n r mul div } \ndef\n/bn {\n b 360 div round 360 mul \\psk@beta\\space\n add n r mul div } \ndef\n/dpn { bn an sub } def\n/arg { dpn t mul an add n mul r mul } def\n/T { dp t mul angleA add } def\n/step { 1 \\psk@plotpoints\\space div } def\n/i 0 def\n\\psk@plotpoints 1 add {\n /t i step mul def\n /i i 1 add def\n T cos arg sin A mul r add mul x1 add\n T sin arg sin A mul r add mul y1 add\n t 0 eq { moveto } { lineto } ifelse\n} repeat }\n\n\n\\newif\\ifPst@turn\n\\define@key{psset}{turn}[false]{%\n\\@nameuse{Pst@turn#1}}\n\n\\def\\Photon{%\n\\@ifnextchar[\\@Photon{\\@Photon[]}}\n\n\\def\\@Photon[#1](#2)(#3)#4{%\n\\setkeys{psset}{turn=false}%\n\\setkeys{psset}{#1}%\n\\SpecialCoor%\n\\pnode(#2){PhotonBegin}%\n\\pnode(#3){PhotonEnd}%\n\\pnode(0,0){Origin}%\n\\psline(PhotonBegin)(PhotonEnd)%\n\\nc@object{Open}{PhotonBegin}{PhotonEnd}{.5}{%\ntx@Dict begin \n/@xA xA def\n/@yA yA def\n/@xB xB def\n/@yB yB def\n/@r #4 def\nend}\n\\nc@object{Open}{Origin}{PhotonBegin}{.5}{%\ntx@Dict begin \n/@ox xB xA sub def\n/@oy yB yA sub def\nend}\n\\ifPst@turn\n\\rput(!@xB @xA sub 2 div \\pst@number\\psxunit div @ox \\pst@number\\psxunit div add\n@yB @yA sub 2 div \\pst@number\\psyunit div @oy \\pst@number\\psyunit div add){%\n\\WavyArc[beta=180,#1]\n{@xB @xA sub dup mul @yB @yA sub dup mul add sqrt \n2 div @r mul \\pst@number\\psunit div}%\n{(PhotonBegin)}{(PhotonEnd)}}\n\\else\n\\rput(!@xB @xA sub 2 div \\pst@number\\psxunit div @ox \\pst@number\\psxunit div add\n@yB @yA sub 2 div \\pst@number\\psyunit div @oy \\pst@number\\psyunit div add){%\n\\WavyArc[beta=180,#1]\n{@xB @xA sub dup mul @yB @yA sub dup mul add sqrt \n2 div @r mul \\pst@number\\psunit div}%\n{(PhotonEnd)}{(PhotonBegin)}}\n\\fi\n}\n\n\n\n%\n% common variables for \\CycloidLine, \\EpiCycloid \n% and HypoCycloid :\n%\n\n\\newif\\ifadaption\n\\def\\psset@adaption#1{\\@nameuse{adaption#1}}\n\\psset@adaption{true}\n\n\\define@key{psset}{r}{%\n\\edef\\psk@r{#1}}\n\n\\define@key{psset}{a}{%\n\\edef\\psk@a{#1}}\n\n\\def\\CycloidLine{\\pst@object{CycloidLine}}%\n\n\\def\\CycloidLine@i{%\n\\setkeys{psset}{plotpoints=200,\n r=0.05,a=0.15}%\n\\begin@OpenObj\n\\pst@getcoors[\\CycloidLine@ii}\n\\def\\CycloidLine@ii{%\n\\addto@pscode{\\CycloidLine@iii}%\n\\ifadaption\n\\addto@pscode{\\CycloidLine@iv}%\n\\else\n\\addto@pscode{\\CycloidLine@v}%\n\\fi\n\\addto@pscode{\\CycloidLine@vi}%\n\\showpointsfalse\n\\end@OpenObj}\n\n\\def\\CycloidLine@iii{%\n/CoorField 4 array def\nCoorField astore pop\n/x1 { CoorField 2 get } def\n/y1 { CoorField 3 get } def\n/x2 { CoorField 0 get } def\n/y2 { CoorField 1 get } def\n/pipi 6.283185307179586477 def\n/Sin { 360 mul pipi div sin } def\n/Cos { 360 mul pipi div cos } def\n/hypot { x2 x1 sub dup mul y2 y1 sub dup mul add sqrt } def\n/Radius { \\psk@r\\space \\pst@number\\psunit mul } def\n/Amplitude { \\psk@a\\space \\pst@number\\psunit mul } def\n/fkt_x { dup Radius mul exch Sin Amplitude mul sub } def\n/fkt_y { Cos Amplitude mul Radius exch sub } def\n}\n\n\\def\\CycloidLine@iv{%\n /newton { % define newton iteration to solve a diophantic equation:\n dup dup dup\n Sin Amplitude mul exch Radius mul sub exch\n Cos Amplitude mul Radius sub div sub\n } def\n /start\n 1 1 15 { % 15 should be enough ...\n 1 eq {\n 1 Radius Amplitude div sub 6 mul sqrt\n } if\n newton\n } for\n def\n /offset start fkt_y def\n /n hypot pipi Radius mul div round def\n /f hypot n pipi mul Radius mul div 1 mul def\n}\n\n\\def\\CycloidLine@v{%\n /start 0 def\n /offset 0 def\n /n hypot pipi Radius mul div def\n /f 1 def\n} \n\n\\def\\CycloidLine@vi{%\nx1 y1 translate % assume the centimeter mode\ny2 y1 sub x2 x1 sub atan rotate\n/ende n pipi mul start sub def \n/step ende start sub \\psk@plotpoints\\space div def\n/i 0 def\n \\psk@plotpoints 1 add {\n /t start i step mul add def\n /i i 1 add def\n t fkt_x f mul\n t fkt_y offset sub\n t start eq { moveto } { lineto } ifelse\n } repeat \n}\n\n\n\\def\\EpiCycloid{\\pst@object{EpiCycloid}}%\n\n\\def\\EpiCycloid@i{%\n\\setkeys{psset}{plotpoints=300,\n r=.05,a=.15}%\n\\@ifnextchar({\\EpiCycloid@ii}{\\EpiCycloid@ii(0,0)}}\n\n\\def\\EpiCycloid@ii(#1)#2#3#4{%\n\\begin@OpenObj\n\\pst@@getcoor{#1}%\n\\pst@getangle{#3}\\pst@tempa\n\\pst@getangle{#4}\\pst@tempb\n\\edef\\@Radius{#2 \\pst@number\\psunit mul \\space}%\n\\addto@pscode{\\EpiCycloid@iii}%\n\\ifadaption\n\\addto@pscode{\\EpiCycloid@iv}%\n\\else\n\\addto@pscode{\\EpiCycloid@v}%\n\\fi\n\\addto@pscode{\\EpiCycloid@vi}%\n\\ifadaption\n\\addto@pscode{\\EpiCycloid@vii}%\n\\else\n\\addto@pscode{\\EpiCycloid@viii}%\n\\fi\n\\addto@pscode{\\EpiCycloid@ix}%\n\\ifadaption\n\\addto@pscode{\\EpiCycloid@x}%\n\\else\n\\addto@pscode{\\EpiCycloid@xi}%\n\\fi\n\\addto@pscode{\\EpiCycloid@xii}%\n\\showpointsfalse\n\\end@OpenObj}\n\n\n\\def\\EpiCycloid@iii{%\n\\pst@coor\n/CoorField 2 array def\nCoorField astore pop\n/x1 { CoorField 0 get } def\n/y1 { CoorField 1 get } def\n/angleA \\pst@tempa def\n/angleB \\pst@tempb def\nangleA angleB gt { /angleB angleB 360 add def } if\n/r {\\psk@r\\space \\pst@number\\psunit mul} def\n/a {\\psk@a\\space \\pst@number\\psunit mul} def\n/pipi 6.283185307179586477 def % �red$2�pi$\n/end angleB angleA sub pipi mul 360 div def % �red$��scriptstyle�frac�2�pi(�beta-�alpha)��360��$\n/Sin { 360 mul pipi div sin } def % �red$�sin��scriptstyle�frac�360��t��2�pi��$\n/Cos { 360 mul pipi div cos } def % �red$�cos��scriptstyle�frac�360��t��2�pi��$\n}\n\n\\def\\EpiCycloid@iv{%\n/f\n \\@Radius end mul pipi r mul\n div round pipi mul r mul\n end \\@Radius mul div % �red�mybox�$f=�frac�2�pi��r��R�cdot�mbox�end���cdot�mbox�Round��left(�frac�R�cdot�mbox�end���2�pi��r��right)$�\ndef\n} \n\n\\def\\EpiCycloid@v{/f 1 def} % �red$f=1$\n\n\\def\\EpiCycloid@vi{\n/fkt_x { dup Cos \\@Radius r add mul\n exch \\@Radius r div f mul 1\n add mul Cos a mul sub } def\n/fkt_y { dup Sin \\@Radius r add mul\n exch \\@Radius r div f mul 1\n add mul Sin a mul sub } def\nx1 y1 translate\n}\n\n\\def\\EpiCycloid@vii{angleA}\n\n\\def\\EpiCycloid@viii{0}\n\n\\def\\EpiCycloid@ix{\nrotate\n/newton { % �red�textit�define newton iteration to solve a�\n dup dup dup % �red�textit�diophantic equation:�\n Sin r \\@Radius add mul exch f \\@Radius mul\n r div 1 add mul Sin a mul sub\n exch dup\n Cos r \\@Radius add mul exch f \\@Radius mul\n r div 1 add mul Cos a mul\n f \\@Radius mul r div 1 add mul sub div sub\n} def\n}\n\n\\def\\EpiCycloid@x{\n/start\n 1 1 15 { % �red�textit�15 should be enough ...�\n 1 eq {\n r \\@Radius add r mul f \\@Radius mul\n r add a mul sub\n r dup mul mul\n r \\@Radius add r dup dup mul mul mul\n f \\@Radius mul r add dup dup mul mul\n a mul sub div\n 6 mul sqrt\n } if\n newton\n } for\ndef\n}\n\n\\def\\EpiCycloid@xi{/start 0 def}\n\n\\def\\EpiCycloid@xii{\n/step { \n end start sub \n start sub \n \\psk@plotpoints\\space div \n} def\n/i 0 def\n\\psk@plotpoints 1 add {\n /t start i step mul add def\n /i i 1 add def\n t fkt_x\n t fkt_y\n t start eq { moveto } { lineto } ifelse\n} repeat }\n\n\n\\newif\\ifadaption\n\\def\\psset@adaption#1{\\@nameuse{adaption#1}}\n\\psset@adaption{true}\n\n\\define@key{psset}{r}{%\n\\edef\\psk@r{#1}}\n\n\\define@key{psset}{a}{%\n\\edef\\psk@a{#1}}\n\n\n\n\\def\\HypoCycloid{\\pst@object{HypoCycloid}}%\n\n\\def\\HypoCycloid@i{%\n\\setkeys{psset}{plotpoints=300,\n r=.05,a=.15}%\n\\@ifnextchar({\\HypoCycloid@ii}{\\HypoCycloid@ii(0,0)}}\n\n\\def\\HypoCycloid@ii(#1)#2#3#4{%\n\\begin@OpenObj\n\\pst@@getcoor{#1}%\n\\edef\\@Radius{#2 \\pst@number\\psunit mul \\space}%\n\\pst@getangle{#3}\\pst@tempa\n\\pst@getangle{#4}\\pst@tempb\n\\addto@pscode{\\HypoCycloid@iii}%\n\\ifadaption\n\\addto@pscode{\\HypoCycloid@iv}%\n\\else\n\\addto@pscode{\\HypoCycloid@v}%\n\\fi\n\\addto@pscode{\\HypoCycloid@vi}%\n\\ifadaption\n\\addto@pscode{\\HypoCycloid@vii}%\n\\else\n\\addto@pscode{\\HypoCycloid@viii}%\n\\fi\n\\addto@pscode{\\HypoCycloid@ix}%\n\\ifadaption\n\\addto@pscode{\\HypoCycloid@x}%\n\\else\n\\addto@pscode{\\HypoCycloid@xi}%\n\\fi\n\\addto@pscode{\\HypoCycloid@xii}%\n\\showpointsfalse\n\\end@OpenObj}\n\n\n\\def\\HypoCycloid@iii{%\n\\pst@coor\n/CoorField 2 array def\nCoorField astore pop\n/x1 { CoorField 0 get } def\n/y1 { CoorField 1 get } def\n/angleA \\pst@tempa def\n/angleB \\pst@tempb def\nangleA angleB gt { /angleB angleB 360 add def } if\n/r {\\psk@r\\space \\pst@number\\psunit mul} def\n/a {\\psk@a\\space \\pst@number\\psunit mul} def\n/pipi 6.283185307179586477 def\n/end angleB angleA sub pipi mul 360 div def\n/Sin { 360 mul pipi div sin } def\n/Cos { 360 mul pipi div cos } def\n}\n\n\\def\\HypoCycloid@iv{%\n/f\n \\@Radius end mul pipi r mul \n div round pipi mul r mul \n end \\@Radius mul div\ndef\n} \n\n\\def\\HypoCycloid@v{/f 1 def}\n\n\\def\\HypoCycloid@vi{\n/fkt_x { dup Cos \\@Radius r sub mul \n exch \\@Radius r div f mul 1\n sub mul Cos a mul add } def\n/fkt_y { dup Sin \\@Radius r sub mul \n exch \\@Radius r div f mul 1\n sub mul Sin a mul sub } def\nx1 y1 translate\n}\n\n\\def\\HypoCycloid@vii{angleA}\n\n\\def\\HypoCycloid@viii{0}\n\n\\def\\HypoCycloid@ix{\nrotate\n/newton { % define newton iteration to solve a diophantic equation:\n dup dup dup\n Sin \\@Radius r sub mul exch f \\@Radius mul \n r div 1 sub mul Sin a mul sub\n exch dup\n Cos \\@Radius r sub mul exch f \\@Radius mul \n r div 1 sub mul Cos a mul \n f \\@Radius mul r div 1 sub mul sub div sub\n} def\n}\n\n\\def\\HypoCycloid@x{\n/start\n 1 1 15 { % 15 should be enough ...\n 1 eq {\n r \\@Radius sub 1 f \\@Radius mul \n r div sub a mul sub\n r \\@Radius sub 6 div f \\@Radius mul \n r div 1 sub dup dup mul mul \n a mul 6 div add\n div sqrt\n } if\n newton\n } for\ndef\n}\n\n\\def\\HypoCycloid@xi{/start 0 def}\n\n\\def\\HypoCycloid@xii{\n/step { \n end start sub \n start sub \n \\psk@plotpoints\\space div \n} def\n/i 0 def\n\\psk@plotpoints 1 add {\n /t start i step mul add def\n /i i 1 add def\n t fkt_x\n t fkt_y\n t start eq { moveto } { lineto } ifelse\n} repeat }\n\n\\catcode`\\@=\\PstAtCode\\relax\n\\endinput\n%%\n%% END: pst-feyn.tex\n\n%%% Local Variables: \n%%% mode: latex\n%%% TeX-master: \"pst-key.tex \"\n%%% End: \n"
},
{
"name": "pst-key.tex",
"string": "%%\n%% This is file `pst-key.tex',\n%% generated with the docstrip utility.\n%%\n%% The original source files were:\n%%\n%% keyval.dtx (with options: `package,plain,pstricks')\n%% \n%% This file is based on keyval.dtx from the LaTeX tools distribution.\n%% It may be distributed and used with the conditions applying to the\n%% PSTricks distribution. See the comments in pstricks.tex for details.\n%% \n%% File: keyval.dtx Copyright (C) 1993 1994 1995 1997 David Carlisle\n\\def\\next[#1]{\\catcode`\\@=11\n \\expandafter\\let\\csname ver@keyval.sty\\endcsname\\empty\n \\wlog{keyval: #1}}\\next\n [1997/06/13 v1.10 key=value parser (DPC)]\n\\def\\setkeys{%\n \\@ifnextchar[%\n \\KV@list\n {\\let\\KV@undefined\\KV@error\n \\KV@setkeys}}\n\\def\\KV@psset{psset}\n\\def\\KV@setkeys#1#2{%\n \\def\\@tempa{#1}%\n \\edef\\KV@prefix{%\n \\ifx\\@tempa\\KV@psset\\else\n KV@%\n \\fi\n #1@}%\n \\KV@do#2,\\relax,}\n\\def\\psset#1{%\n \\def\\KV@prefix{psset@}%\n \\KV@do#1,\\relax,}\n\\def\\use@par{\\expandafter\\psset\\expandafter{\\pst@par}}\n\\def\\KV@list[#1]{%\n \\def\\KV@undefined{\\KV@add#1}%\n \\ifx#1\\@undefined\n \\let#1\\@empty\n \\fi\n \\KV@setkeys}\n\\def\\KV@do#1,{%\n \\ifx\\relax#1\\empty\\else\n \\KV@split#1==\\relax\n \\expandafter\\KV@do\\fi}\n\\def\\KV@split#1=#2=#3\\relax{%\n \\KV@@sp@def\\@tempa{#1}%\n \\ifx\\@tempa\\@empty\\else\n \\expandafter\\let\\expandafter\\@tempc\n \\csname\\KV@prefix\\@tempa\\endcsname\n \\ifx\\@tempc\\relax\n \\KV@undefined{#2}{#3}%\n \\else\n \\ifx\\@empty#3\\@empty\n \\KV@default\n \\else\n \\KV@@sp@def\\@tempb{#2}%\n \\expandafter\\@tempc\\expandafter{\\@tempb}\\relax\n \\fi\n \\fi\n \\fi}\n\\def\\KV@default{%\n \\expandafter\\let\\expandafter\\@tempb\n \\csname\\KV@prefix\\@tempa @default\\endcsname\n \\ifx\\@tempb\\relax\n \\KV@err{No value specified for \\@tempa}%\n \\else\n \\@tempb\\relax\n \\fi}\n\\def\\KV@add#1#2#3{%\n \\toks2\\expandafter{#1}%\n \\KV@@sp@def\\@tempb{#2}%\n \\toks4\\expandafter{\\@tempb}%\n \\edef#1{\\the\\toks2\n \\ifx#1\\@empty\\else,\\fi\n \\@tempa\n \\ifx\\KV@add#3\\KV@add\\else={\\the\\toks4}\\fi}}\n\\def\\KV@err#1{\\errmessage{keyval: #1}}\n\\def\\KV@error#1#2{\\KV@err{\\@tempa\\space undefined}}\n\\def\\@tempa#1{%\n\\def\\KV@@sp@def##1##2{\\KV@@sp@b##2\\@nil\\@nil#1\\@nil\\relax##1}}\n\\@tempa{ }\n\\def\\KV@@sp@b#1#2 \\@nil{\\KV@@sp@c#1#2}\n\\def\\KV@@sp@c#1\\@nil#2\\relax#3{\\def#3{#1}}\n\\def\\define@key#1#2{%\n \\def\\KV@prefix{#1}%\n \\edef\\KV@prefix{%\n \\ifx\\KV@prefix\\KV@psset\\else\n KV@%\n \\fi\n #1@#2}%\n \\@ifnextchar[\\KV@def{\\@namedef\\KV@prefix####1}}\n\\def\\KV@def[#1]{%\n \\@namedef{\\KV@prefix @default\\expandafter}\\expandafter\n {\\csname \\KV@prefix\\endcsname{#1}}%\n \\@namedef\\KV@prefix##1}\n\\endinput\n%%\n%% End of file `pst-key.tex'.\n"
}
] |
[
{
"name": "cond-mat0002086.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{wi1} R.L. Willett et al. , Phys. Rev. Lett. {\\bf 65},112 (1990);\nW. Kang et al., Phys. Rev. Lett. {\\bf 71},3850 (1993); V.J. Goldman et al., Phys. Rev. Lett. {\\bf 72},2065 (1994) \n \\bibitem{hlr} B.I. Halperin, P.A. Lee, and N. Read, Phys. Rev. B {\\bf 47},7312 (1993)\n\\bibitem{ab1} A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of quantum field theory in statistical physics, Dover publication, New York (1975); \nJ.R. Schrieffer, Theory of Superconductivity (Benjaming, Reading, Mass., 1964)\n; P. Nozieres, Theory of interacting Fermi systems, W.A. Benjamin, New York (1964) \n\\bibitem{di1} J. Dietel, Th. Koschny, W. Apel and W. Weller, \nEur. Phys. J. B {\\bf 5},439 (1998)\n\\bibitem{fa1} \nA.G. Aronov, A.D. Mirlin; and P. W\\\"olfle, Phys. Rev. B {\\bf 49},16609 (1994);\nV. Fal'ko, Phys. Rev. B {\\bf 50},17406 (1994) \n\\bibitem{al1} B.L. Altshuler, A.G. Aronov and P.A. Lee, Phys. Rev. Lett. \n{\\bf 44},1288 (1980)\n\\bibitem{al2} B.L. Altshuler, D. Khmel'nitzkii, A.I. Larkin and P.A. Lee, Phys. Rev. B {\\bf 22},5142 (1980)\n\\bibitem{ro1} L.P. Rokhinson, B. Su and V.J. Goldman,\n Phys. Rev. B {\\bf 52},11588 (1995)\n\\bibitem{ho1} A. Houghton, J. R. Senna, and S.C. Ying, \n Phys. Rev. B {\\bf 25},2196 (1982) \n\\bibitem{kh1} D.V. Khveshchenko,\n Phys. Rev. Lett. {\\bf 77},1817 (1996) \n\\bibitem{kh2} D.V. Khveshchenko,\n preprint cond-mat/9605051 \n\\bibitem{fi1} A.M. Finkelstein, Sov. Phys. Jetp {\\bf 57},97 (1983); \n C. Castellani, C. Di Castro, P.A. Lee and M. Ma, \n Phys. Rev. B {\\bf 30},527 (1984) \n\\end{thebibliography}"
}
] |
cond-mat0002087
|
Striped quantum Hall phases
|
[
{
"author": "Felix von Oppen"
},
{
"author": "$^{(1)}$ Bertrand I.\\ Halperin"
},
{
"author": "$^{(2)}$ and Ady Stern $^{(3)}$"
}
] |
Recent experiments seem to confirm predictions that interactions lead to charge density wave ground states in higher Landau levels. These new ``correlated'' ground states of the quantum Hall system manifest themselves for example in a strongly anisotropic resistivity tensor. We give a brief introduction and overview of this new and emerging field.
|
[
{
"name": "mbx-proc.tex",
"string": "\n\\documentclass{ws-p9-75x6-50}\n%\\documentstyle[prb,aps,preprint,psfig]{revtex}\n\\begin{document}\n\n\\title{Striped quantum Hall phases}\n\n\\author{Felix von Oppen,$^{(1)}$ Bertrand I.\\ Halperin,$^{(2)}$ and Ady Stern\n$^{(3)}$}\n\n\\address{$^{(1)}$ Institut f\\\"ur Theoretische Physik, Universit\\\"at \nzu K\\\"oln, Z\\\"ulpicher Str.\\ 77, 50937 K\\\"oln, Germany\\\\\n$^{(2)}$ Physics Department, Harvard University, Cambridge, Massachusetts \n02139,USA\\\\\n$^{(3)}$ Department of Condensed Matter Physics, The Weizmann Institute of \nScience, 76100 Rehovot, Israel} \n\n\n\\maketitle\n\n\\abstracts{Recent experiments seem to confirm predictions that\n interactions lead to charge density wave ground states in higher\n Landau levels. These new ``correlated'' ground states of the\n quantum Hall system manifest themselves for example in a strongly\n anisotropic resistivity tensor. We give a brief introduction and\n overview of this new and emerging field.}\n\n\n\\section{Introduction}\n\nWhile a large number of fractional quantized Hall states have been\ndiscovered in the lowest Landau level, such states become increasingly\nrare in higher Landau levels. Few quantized Hall states have been\nobserved in the first Landau level (Landau level filling factor\n$2\\le\\nu<4$) and no such states have so far been found for filling\nfactors $\\nu\\ge4$. Correspondingly, the role of electron-electron\ninteractions in this filling factor range has long remained almost\nuncharted territory.\n\nThis is now rapidly changing, following recent experiments by Lilly\n{\\it et al.}\\cite{Lilly} and by Du {\\it et al.}\\cite{Du} on extremely\nclean two-dimensional electron systems (2DES) for Landau level (LL)\nfilling factor $\\nu>4$. The central observation is that the\nresistivity becomes strongly anisotropic close to half filling of the\ntopmost Landau level in this filling factor range. This is believed\nto be a signature of a novel Coulomb-induced charge density wave\nground state whose existence had been predicted by Fogler {\\it et\n al.}\\cite{Fogler} and by Moessner and Chalker.\\cite{Moessner} Unlike\nthe Laughlin states,\\cite{Laughlin} these states can be partially\nunderstood within the Hartree-Fock (HF) approximation. In fact, it was\nalready suggested at the end of the '70s by Fukuyama {\\it et\n al.}\\cite{Fukuyama} that the HF ground state of the lowest Landau\nlevel is a charge density wave (CDW). While in the lowest Landau\nlevel, the CDW ultimately turned out to be preempted by the Laughlin\nstates, it seems to be more favorable in higher Landau levels.\n\nDifferent ground states have been predicted depending on filling\nfactor.\\cite{Fogler,Moessner} Near half filling of higher Landau\nlevels ($\\nu\\simeq N+1/2$), the HF ground state should be a\nunidirectional charge density wave (UCDW). In this state, the filling\nfactor alternates between stripes of filling factor $N$ and $N+1$.\nThe period of the density modulation is of the order of the cyclotron\nradius $R_c$. Further away from half filling one expects a so-called\nbubble phase, in which clusters of the minority filling factor ($N$ or\n$N+1$) order on a triangular lattice with characteristic length $R_c$.\nVery close to integer filling factors, a Wigner crystal phase should\nform.\n\nThe strongly anisotropic transport properties\\cite{Lilly,Du,Shayegan}\nnear half filling of higher Landau levels are believed to be associated\nwith the unidirectional charge density wave phase. In this paper we\nbriefly review some of the recent work on these states. The\nexperimental results are reviewed in Sec.\\ \\ref{sec-experiment}.\nSec.\\ \\ref{theory} focuses on theoretical developments. In Sec.\\ \n\\ref{sec-theory} we sketch early theoretical work which suggested\nthat there exists an instability towards CDW formation. It\nis well known that Hartree-Fock calculations tend to overestimate the\ndegree of ordering. Some insights into this can be gained from an\nanalogy with two-dimensional liquid crystal phases which we review in\nSec.\\ \\ref{sec-liquid}. A focus of recent theoretical work are the\ntransport properties of the UCDW states, as reviewed in Sec.\\ \n\\ref{sec-transport}. It has been shown under rather general\nassumptions that the conductivity tensor should satisfy non-trivial\nrelations, which are independent of microscopic\nparameters.\\cite{MacDonald,Oppen-cdw} Finally we summarize and\nconclude in Sec.\\ \\ref{sec-conclusions} by mentioning some open\nproblems.\n\n\\section{Experiment}\n\\label{sec-experiment}\n\nThe most prominent observation\n\\cite{Lilly,Du,Shayegan,Pan,Lilly2,Eisenstein} is the development of\nlarge anisotropies in the resistivity close to half filling of the\ntopmost Landau level. Observation of this effect requires extremely\nclean samples and very low temperatures ($T<150$mK). The anisotropy\ndevelops even though the sample behaves essentially isotropically at\nvery large and very low magnetic fields. The case for a new\n``correlated'' ground state in this filling factor range is further\nstrengthened by the observation that the width of the peak in the\nresistivity around half filling does not decrease with decreasing\ntemperature. This is very different from what one would expect for\nthe integer quantum Hall effect (QHE) plateau transition.\n\nTo date, the principal experimental features of these novel states\nare:\n\n(a) The anisotropic resistivity has so far been observed near half\nfilling of the topmost LL in the filling factor range $4\\le\\nu\\le12$.\nThe anisotropy is largest for $\\nu=9/2$ and decreases monotonically\nwith increasing LL index $N$.\\cite{Lilly,Du}\n\n(b) The principal axes of the anisotropic resistivity tensor seem to\nbe consistently oriented along certain crystallographic axes of the\nGaAs crystal. The high-resistance direction is always along the\n$1{\\bar1}0$ direction while the low-resistance direction is aligned\nwith the $110$ direction.\\cite{Lilly,Du}\n\n(c) The resistances in the two principal directions differ by up to\nseveral orders of magnitude in van-der-Pauw\nmeasurements.\\cite{Lilly,Du} However, van-der-Pauw measurements\nsignificantly overestimate the anisotropy of the resistivity tensor\n(assuming that a local resistivity tensor is an appropriate\ndescription).\\cite{Simon,Lilly-comment} This is associated with the\ndetailed current distribution in the device. Accordingly, Hall-bar\nmeasurements which should be free of this problem\\footnote{However,\n they currently measure the two diagonal resistivities on different\n samples.} show a much smaller but still very significant anisotropy\nof the order of five.\\cite{Lilly}\n\n(d) The anisotropy usually appears more stable against temperature for\nfilling factors corresponding to the lower spin component of each\nLL.\\cite{Lilly,Du}\n\n(e) A striking set of experiments\\cite{Pan,Lilly2} revealed that even\nweak in-plane magnetic fields $B_\\parallel$ can have a strong effect\non the anisotropic states (and on the even denominator fractional QHE\nstate at $\\nu=5/2$.) Applying the in-plane field in the\nlow-resistance direction leads to an interchange of easy and hard\ndirections for $B_\\parallel$ of the order of $0.5$T. In-plane fields\nin the high-resistance direction affect the measured resistivities in\nthe upper spin component of each LL only very weakly, but suppress the\nanisotropy in the lower spin component.\n\n(f) Transport in the high-resistance direction is strongly non-linear\nwith the differential resistance rising with increasing bias current.\nThe change in resistivity with applied current is smooth, which would\nseem inconsistent with a depinning transition of the UCDW. The\nnon-linearity is more pronounced in the lower spin component of each\nLL.\\cite{Lilly}\n\n(g) Intriguing reentrant {\\it integer} QHE states have been found near\nquarter and three quarter filling of the uppermost LL.\\cite{Cooper}\nThe Hall resistance in these states is quantized at the value of the\nclosest integer plateau. Current-voltage characteristics in this\nfilling factor regime exhibit discontinuous and hysteretic behavior.\nIt has been suggested that this may be related to depinning of a CDW\nstate.\\cite{Cooper}\n\n\\section{Theory}\n\\label{theory}\n\n\\subsection{Hartree-Fock calculations and numerical exact-diagonalization \nstudies}\n\\label{sec-theory}\n\nIt turns out that essential features of the CDW ground states can be\nunderstood within the Hartree-Fock approximation (HFA). Most\ncalculations assume that one can project to a single (partially\nfilled) Landau level. This is natural for the lowest LL. For higher\nLLs, this relies on an effective interaction derived by Aleiner and\nGlazman\\cite{Aleiner} which includes the effects of screening by the\nlower (filled) LLs. This effective interaction turns out to be\nsufficiently weak compared to the Landau level spacing so that\nprojection to a single LL should be a reasonable starting point.\n\nThe zero-temperature HF equations within the single-LL approximation\nhave been studied in detail by Fogler {\\it et al.}\\cite{Fogler} In\nparticular, these authors compared various possible CDW ground states.\nIn the vicinity of half filling of the topmost LL ($\\nu\\simeq N+1/2$)\nthey find that a unidirectional CDW state whose period is of the order\nof the cyclotron radius is most favorable. In this state,\none-dimensional stripes of filling factor $N$ alternate with stripes\nof filling factor $N+1$. The analysis of this state is simplified by\nthe fact that the exact HF eigenfunctions are still the usual Landau\ngauge wave functions for electrons in a magnetic field. All Landau\ngauge states with centers in the filling factor $N+1$ range are\noccupied, while those wavefunctions with centers in the filling factor\n$N$ region remain empty. Clearly, the relative modulation in the\nfilling factor is larger than that of the electron density due to the\nfinite width of the wavefunctions.\n\nWhile CDW formation leads to a cost in Hartree energy due to the\nassociated charge density modulation, this energy cost is more than\noffset by the gain in exchange energy due to the closer packing of the\nelectrons. It turns out that this mechanism is particularly effective\nin higher LLs where the (Hermite-polynomial) wavefunctions have\nzeroes. In this case, there are wavevectors of the charge density wave\nfor which the associated {\\it charge} density modulation is\nparticularly small. Essentially, this happen when a state centered in\nthe middle of a filling factor $N+1$ region has its first side maximum\nin the filling factor $N$ region. It is for this reason that the CDW\nstates are more favorable in higher LLs than in the lowest LL. \n\nThe bulk of this paper will be concerned with the unidirectional CDW.\nFurther away from half filling, Fogler {\\it et al.}\\cite{Fogler}\npredict a triangular CDW, termed bubble phase, consisting of clusters\nof minority filling factor ordering in a background of majority\nfilling factor on a triangular lattice. Again, cluster size and period\nof the lattice are given by the cyclotron radius $R_c$. Very close to\ninteger filling factors, the electrons (or holes) in the topmost\nLandau level are predicted to form a triangular Wigner crystal.\n\nMoessner and Chalker\\cite{Moessner} arrived at similar conclusions by\nderiving Landau theories for various CDW states and comparing their\nfree energy. Following Fukuyama {et al.},\\cite{Fukuyama} these authors\nexpand the free energy in powers of the appropriate order parameter.\nThis procedure is justified in the vicinity of the Hartree-Fock\ntransition temperature $T_c^{hf}$. In addition, they show by\ndiagrammatic arguments that the Hartree-Fock approximation becomes\nexact for the uniform phase in the limit of high Landau levels.\n\nThese conclusions have been partly supported by numerical exact\ndiagonalization studies.\\cite{Rezayi} In these calculations, systems\nof up to 12 electrons on a torus are diagonalized exactly and strong\npeaks indicating CDW ordering have been found in the wave vector \ndependence of the static density susceptibility and the equal-time\ndensity-density correlation function.\n\nMotivated by the dramatic effect of in-plane magnetic fields\n$B_\\parallel$ in experiment, two groups\\cite{Jungwirth,Stanescu}\nextended the Hartree-Fock calculations to include the finite thickness\nof the 2DES and the resulting orbital effects of $B_\\parallel$. While\nthese calculations suggest that the influence of $B_\\parallel$ is\nsensitive to sample details, the perhaps more realistic of the two\ncalculations\\cite{Jungwirth} shows that in-plane fields can rotate the\nstripe pattern in the appropriate manner. These calculations do not\nexplain the experimentally observed dependences on the spin of the LL.\n\n\\subsection{Analogy with liquid crystal systems}\n\\label{sec-liquid}\n\nHartree-Fock calculations tend to overestimate the degree of ordering\nof a system. The influence of quantum and thermal fluctuations in the\npresent system can be assessed to some degree by an analogy with\ntwo-dimensional (2d) liquid crystals.\\cite{Fradkin} The analogy is\nbased on the fact that the UCDW shares its symmetry with 2d smectic\nliquid crystals. The appropriate elastic variable of both systems is\nthe phase $u$ of the density oscillations, $\\delta\\rho({\\bf\n r})=\\rho_0\\cos(q_0 x-u({\\bf r}))$. In terms of $u$, the\nlong-wavelength elastic free energy in the absence of forces tending\nto align the stripes reads\\cite{Nelson}\n\\begin{equation}\n\\label{free-energy}\n F={A\\over 2}\\int d{\\bf r}\\left\\{\\left(\\partial u\\over\\partial\n x\\right)^2 +\\lambda^2\\left(\\partial^2u\\over\\partial\n y^2\\right)^2\\right\\},\n\\end{equation}\nwhere $A$ is an elastic constant and $\\lambda$ a length which is\npresumably of the order of the UCDW period. The absence of a term\ninvolving $(\\partial u/\\partial y)^2$ is a consequence of the global\nrotation symmetry of the system. (Note that $u=ay$ only leads to a\nglobal rotation of the CDW to linear order in $a$ and thus leaves the\nfree energy unchanged.)\n\nIt is an immediate consequence of this elastic free energy that\ndislocations will cost only a finite energy.\\cite{Nelson} Thus there\nwill be a finite density of dislocations at any non-zero temperature\nand the stripes should be broken down to stripe segments. This is\nexpected to destroy translational long-range order but preserve\nquasi-long-range orientational order of the remaining stripe segments.\nThus, the smectic order predicted by the HFA is preserved only at zero\ntemperature. At non-zero temperatures, the system is analogous in terms\nof symmetries to a 2d nematic liquid crystal.\n\nAs the temperature increases the system should undergo a\nKosterlitz-Thouless transition from the nematic phase to an isotropic\nphase in which the orientational order of the stripe segments\ndisappears. Nevertheless, short-range stripe ordering should still\npersist up to the presumably much higher Hartree-Fock transition\ntemperature $T_c^{hf}$.\n\nMacDonald and Fisher\\cite{MacDonald} have recently studied the\nstability of the UCDW against quantum fluctuations in the framework of\nthe elastic theory and found that quantum fluctuations are not strong\nenough to destroy the smectic order predicted by the HFA.\n\nWhile much can be learned about the influence of quantum and thermal\nfluctuations from this analogy, much less is known about the influence\nof disorder on the UCDW.\\cite{Radzihovsky} One may expect that\ntransport properties of the striped phases should be affected by even\nsmall amounts of disorder on the substrate, which will pin the stripe\npositions at low temperatures. Disorder should also lead to a finite\ndensity of dislocations, even at zero temperature. Moreover, since the\nforces aligning the stripes are believed to be very weak, steps or\nother large-scale features of the GaAs-AlGaAs interface may lead to\nlarge regions where the stripes are oriented differently from the\naverage preferred direction.\n\n\\subsection{Transport properties}\n\\label{sec-transport}\n\nAn important problem is a quantitative understanding of transport in\nthe striped phases. It had been previously noted in various\ncontexts\\cite{modulation} that an {\\it externally imposed} density\nmodulation can lead to strongly anisotropic transport in the presence\nof a magnetic field. While this effect is indeed related to the\ndevelopment of anisotropic transport in higher Landau levels, none of\nthe developed formalisms apply directly to the present system.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig1-mbx.ps,width=5.0cm,angle=270}}\n\\caption{(a) Ideal and (b) realistic stripe structures. \n Shaded and unshaded regions represent stripes with filling factors\n $N + 1$ and $N$, respectively. Arrows indicate the direction of\n electron flow on edge states. The dashed lines are scattering\n centers allowing electrons to tunnel between neighboring edge\n states. The realistic stripe structure contains a dislocation (A)\n and a large angle grain boundary (B-B$^\\prime$).}\n\\label{stripe-structure}\n\\end{figure}\n\nWe start with the idealized case of a {\\it pinned} unidirectional CDW\nnear $\\nu=N+1/2$, first studied by MacDonald and\nFisher\\cite{MacDonald} (cf.\\ Fig.\\ \\ref{stripe-structure}a). The\nstripes are taken parallel to the $y$ axis and the CDW has period $a$.\nCurrent can be carried by the {\\it internal} edge channels belonging\nto the $(N+1)$st LL which travel along each stripe edge, with velocity\n$v_F$ and density of states $N(0)=1/hv_F$. Impurities lead to\nscattering between these edge channels. The scattering rates should\ndecrease rapidly with increasing distance between edges. Thus, we\ninclude only scattering between neighboring edges and denote the\nscattering rate across electron stripes (stripes of filling factor\n$N+1$) as $1/\\tau_e$ and across hole stripes (stripes of filling\nfactor $N$) as $1/\\tau_h$. Each scattering event acts as\nbackscattering with regard to the $y$ direction and a single\nrandom-walk step in the $x$ direction. In addition, we assume that\nquantum interference does not play an important role. In this case,\nwe can obtain the conductivity tensor via the Einstein relation from\nthe classical diffusion constants.\n\nThe density of excitations on the up-moving and down-moving edge\nchannels of the $i$-th electron stripe, denoted by $P^+_t(y,i)$ and\n$P^-_t(y,i)$ respectively, satisfy the rate equations\n\\begin{eqnarray}\n (\\partial_t+v_F\\partial_y)P^+_t(i)&=&-\\left({1\\over\\tau_e}+{1\\over\\tau_h}\n \\right)P^+_t(i)\n +{1\\over\\tau_h}P_t^-(i-1)+{1\\over\\tau_e}P_t^-(i) \n \\label{rate1}\\\\\n (\\partial_t-v_F\\partial_y)P^-_t(i)&=&-\\left({1\\over\\tau_e}+{1\\over\\tau_h}\n \\right)P^-_t(i)\n +{1\\over\\tau_e}P_t^+(i)+{1\\over\\tau_h}P_t^+(i+1) .\n\\label{rate2}\n\\end{eqnarray}\nActing on Eq.\\ (\\ref{rate1}) with $\\partial_t-v_F\\partial_y\n+(1/\\tau_h+1/\\tau_e)$, inserting Eq.\\ (\\ref{rate2}), and dropping\nsecond-order time derivatives, one obtains the diffusion equation\n\\begin{eqnarray}\n \\partial_t P^+_t(i)={v_F^2\\over2}{\\tau_e\\tau_h\\over\\tau_e+\\tau_h}\n \\partial_y^2 P^+_t(i)\n +{1\\over 2(\\tau_e+\\tau_h)}[P_t^+(i+1)-2P_t^+(i)+P_t^+(i-1)].\n\\label{diffusion}\n\\end{eqnarray}\nThe same equation is satisfied by $P^-_t(i)$. One can now read off the \ndiffusion constants \n\\begin{eqnarray}\n D_{xx}={a^2\\over2}{1\\over\\tau_e+\\tau_h} \n \\qquad\\qquad\n D_{yy}={v_F^2\\over2}{\\tau_e\\tau_h\\over\\tau_e+\\tau_h}.\n\\end{eqnarray}\nAccording to the Einstein relation, the conductivity is related to the\ndiffusion constants by $\\sigma_{\\alpha\\alpha}=e^2(2/a)(1/hv_F)\nD_{\\alpha\\alpha}$ and one obtains\\cite{MacDonald}\n\\begin{eqnarray}\n\\label{conductivity-ideal}\n \\sigma_{xx}={e^2\\over h}{a\\over v_F(\\tau_e+\\tau_h)} \\qquad\\qquad\n \\sigma_{yy}={e^2\\over h}{v_F\\over a}{\\tau_e\\tau_h\\over\\tau_e+\\tau_h}.\n\\end{eqnarray}\n\nTo obtain the Hall conductivity, we note that the $N$ completely\nfilled LL's contribute $Ne^2/h$. To find the contribution of the\npartially filled topmost LL, we apply a chemical potential gradient in\nthe $x$ direction. Assuming a chemical potential drop of $ev$ between\nthe two edges of an electron stripe, we have a Hall current\n$j_y=(e^2/h)(v/a)$ in the $y$ direction and a diffusion current\n$j_x=(e/\\tau_e)(1/hv_F)ev$ in the $x$ direction. Comparing with\n$j_x=\\sigma_{xx}E_x$, we find $v=[\\tau_e/(\\tau_e+\\tau_h)] E_x a$ and\ntherefore a Hall conductivity of\\cite{MacDonald}\n\\begin{equation}\n\\label{hall-ideal}\n \\sigma_{xy}={e^2\\over h}\\left(N+{\\tau_e\\over\\tau_e+\\tau_h}\\right).\n\\end{equation}\n\nRemarkably, these results make a number of predictions which are\nindependent of the microscopic parameters.\\cite{MacDonald,Oppen-cdw}\nFollowing MacDonald and Fisher\\cite{MacDonald} we first focus on the\nsymmetric point $\\tau_e=\\tau_h\\equiv\\tau$. If one assumes\nparticle-hole symmetry in the partially filled LL, this would\ncorrespond to half filling of the topmost Landau level, $\\nu=N+1/2$.\nIn this case, we deduce from Eq.\\ (\\ref{conductivity-ideal}) that the\nproduct of the diagonal conductivities takes on a universal\nvalue\\cite{MacDonald}\n\\begin{equation}\n \\sigma_{xx}\\sigma_{yy}=(e^2/2h)^2,\n\\label{product-rule}\n\\end{equation}\nindependent of the period, Fermi velocity, or the scattering rate.\nLikewise, one finds that the Hall conductivity becomes independent of\nthe microscopic parameters, $\\sigma_{xy}={e^2\\over h}(N+1/2)$.\nHence, we can also rewrite the product rule in terms of the diagonal \nresistivities as\\cite{MacDonald}\n\\begin{equation}\n \\rho_{xx}\\rho_{yy}=(h/e^2)^2{1\\over [N^2+(N+1)^2]^2}.\n\\label{product-rule-rho}\n\\end{equation}\nFor the Hall resistivity at the symmetric point one finds \n$\\rho_{xy}=(h/e^2)(2N+1)/[N^2+(N+1)^2]$. \n\nWhile the product of the diagonal resistivities is universal, the\nanisotropy $\\rho_{xx}/\\rho_{yy}$ depends on the microscopic parameters,\n\\begin{equation}\n\\label{anisotropy}\n {\\rho_{xx}\\over\\rho_{yy}}=\\left(v_F\\tau\\over a\\right)^2,\n\\end{equation}\nand is given by the square of the ratio of the basic diffusion steps\nin the $y$ and $x$ directions.\n\nIt is also interesting to note that if we assume that the diagonal\nconductivity (or, equivalently, the scattering rates $1/\\tau_e$ and\n$1/\\tau_h$) do not depend significantly on the Landau level index\n$N$, say at the symmetric point, then the resistivities decrease with\nincreasing Landau level index roughly as $1/N^2$. This seems to be in\nrather good agreement with the experimental results for the peak\nheights in the high-resistance direction.\n\nInitial results indicate that the product rule Eq.\\ \n(\\ref{product-rule-rho}) is reasonably well satisfied in\nexperiment.\\cite{Eisenstein-priv} This may be surprising in view of\nthe expectation that the experimental samples should be quite far from\nthe perfect stripe ordering assumed here. Thus, experiment raises the\nquestion whether the product rule is not in fact valid much more\ngenerally than indicated by this derivation. An additional question is\nrelated to the neglect of quantum interference. Since the\nexperimental anisotropy in the resistivity is about\nfive,\\cite{Lilly,Simon} these results [cf.\\ Eq.\\ (\\ref{anisotropy})] would\nimply that the electrons hop between edges after traveling only a\ndistance of a few cyclotron radii along the edge. For such a\nsituation, quantum interference effects should be important,\nparticularly since the experiments are performed at extremely low\ntemperatures. While it is currently not known how quantum interference\naffects the validity of the product rule, it is certainly possible\nthat it leads to significant deviations.\n\nTransport in more realistic stripe structures has been studied by von\nOppen {\\it et al.}\\cite{Oppen-cdw} The starting point is that the\nconductivity tensor of the perfect stripe structure actually satisfies\nan even more general relation, namely {\\it the semicircle\n law}\\cite{Oppen-cdw}\n\\begin{equation}\n \\sigma_{xx}\\sigma_{yy}+(\\sigma_{xy}-\\sigma_h^0)^2=(e^2/2h)^2,\n\\label{semicircle}\n\\end{equation}\nwith $\\sigma_h^0={e^2\\over h}(N+1/2)$, which {\\it holds also away from\n the symmetric point.} This relation can be generalized to a wide\nclass of stripe structures. In particular, a semicircle law holds even\nin the presence of topological and orientational defects such as\ndislocations and grain boundaries if one assumes that the defects are\npinned by disorder. An example of such a more general stripe structure\nis shown in Fig.\\ \\ref{stripe-structure}(b).\n\nStrictly speaking, one finds that the macroscopic conductivity tensor\n$\\hat\\sigma^*$ (relating the spatially averaged currents and fields)\nsatisfies the semicircle law\\cite{Oppen-cdw}\n\\begin{equation}\n \\sigma^*_1\\sigma^*_2+(\\sigma^*_h-\\sigma_h^0)^2 = (e^2/2h)^2.\n\\label{semicircle*}\n\\end{equation}\nHere, the macroscopic conductivity tensor\n$\\hat\\sigma^*=\\hat\\sigma^*_d+\\sigma_h^*\\hat\\epsilon$ is written as the\nsum of its dissipative part $\\hat\\sigma_d^*$ and Hall component\n$\\sigma_h^*\\hat\\epsilon$ with $\\hat\\epsilon$ the totally antisymmetric\ntensor. $\\sigma_1^*$ and $\\sigma_2^*$ are the eigenvalues of the\n(real symmetric) dissipative part $\\hat\\sigma_d$.\n\nThe product rule Eq.\\ (\\ref{product-rule}) is a special case of the\nsemicircle law for the symmetric point $\\sigma_{xy}=(e^2/h)(N+1/2)$.\nThus, the more general validity of the semicircle law also implies the\nsame for the product rule, thereby explaining its agreement with\nexperiment. Moreover, this makes the experimental results consistent\nwith a picture where electrons hop between edges much more rarely,\nwhile the anisotropy is reduced by the presence of defects such as\ndislocations and grain boundaries. In such a picture, neglecting\nquantum interference may indeed be justified.\\cite{Oppen-cdw}\n\nThe generalized semicircle law Eq.\\ (\\ref{semicircle*}) can be\nsupported by two different arguments.\\cite{Oppen-cdw} The most general\nderivation uses results obtained by Shimshoni and\nAuerbach\\cite{Shimshoni} for a model of the ``quantized Hall\ninsulator.'' In this argument, one maps the problem to a network of\npuddles of filling factor $\\nu=1$ in vacuum. Neglecting quantum\ninterference, assuming that the network is planar, and that the stripe\nstructure is pinned, it was shown by Shimshoni and Auerbach that the\ncorresponding (properly defined) Hall resistance equals\n$R_{xy}=h/e^2$. Following the mapping from the stripe structure to the\npuddle network in reverse, one finds that this implies the semicircle\nlaw (\\ref{semicircle*}).\\cite{Oppen-cdw} The assumption that the\nnetwork is planar would for example be violated if one included\nnext-to-nearest neighbor hopping between edges.\n\nAlternatively, one can also give a continuum argument\\cite{Oppen-cdw}\nfor the semicircle law Eq.\\ (\\ref{semicircle*}) based on a duality\ntransformation first exploited by Dykhne and Ruzin.\\cite{Dykhne} In\nthis argument, one assumes that the defect density is sufficiently low\nso that one can define a local conductivity tensor which everywhere\nsatisfies the semicircle relation (\\ref{semicircle}). The defects lead\nto spatial variations of the scattering rates and of the principal\naxes of the dissipative part of the conductivity tensor. Using a\nduality transformation, involving new currents and fields which are\nlinear combinations of the original currents and fields, and choosing\nthe dual system to be the time reverse of the original one, it can\nthen be shown\\cite{Oppen-cdw} that the macroscopic conductivity tensor\nsatisfies the semicircle law Eq.\\ (\\ref{semicircle*}).\n\nSeveral authors observed that the internal edge modes should be\nLuttinger liquids.\\cite{Fradkin,MacDonald} It has been\nargued\\cite{MacDonald} that the stripe phase is unstable (possibly\ntowards an anisotropic Wigner crystal) due to backscattering, albeit\nonly below experimentally accessible temperatures. MacDonald and\nFisher\\cite{MacDonald} have also suggested that the Luttinger liquid\nbehavior may explain the experimentally observed nonlinearities in the\ndiagonal resistivity. Within the transport theory sketched above,\nLuttinger liquid correlations imply that the scattering rates\n$1/\\tau_e$ and $1/\\tau_h$ decrease with increasing voltage. Thus, the\nanisotropy increases with increasing voltage (or applied current), as\nwas observed in experiment. However, there are not yet detailed\npredictions for the full temperature and current dependences which\nmight be compared with experiment.\n\nFradkin {\\it et al.}\\cite{Fradkin2} have used the liquid-crystal\nanalogy to study the temperature dependence of the anisotropy for the\ntransition from the nematic to the isotropic phase. They argue on the\nbasis of symmetry that close to $T_c$ the anisotropy should be\nproportional to the order parameter of an xy model with director order\nparameter in the presence of a small background anisotropy. Using\nMonte-Carlo results for this model they attempt to fit the\nexperimental anisotropy, using two free parameters. The validity of\nthe analysis is hard to assess, however, because it is unknown for\nwhich range around $T_c$ the assumed proportionality holds.\n\n\\section{Conclusions and open questions}\n\\label{sec-conclusions}\n\nThe discovery of the anisotropic phases in higher Landau\nlevels\\cite{Lilly,Du}, combined with the earlier theoretical\npredictions of CDW states\\cite{Fogler,Moessner} in this filling factor\nrange, has opened a new chapter of quantum Hall physics. Initially,\nthe experimental evidence for charge density wave ordering has been\npurely qualitative. The product rule\\cite{MacDonald} and the\nsemicircle law\\cite{Oppen-cdw} discussed in Sec.\\ \\ref{sec-transport}\nare the first predictions for the striped CDW phases which can be\ntested {\\it quantitatively} in experiment. Initial results seem to\nindicate reasonable agreement.\\cite{Eisenstein-priv}\n\nIt seems clear that the investigation of the new charge density wave\nstates in higher Landau levels has only started and much interesting\nphysics remains to be studied. Obvious open questions include the\nfollowing: The mechanism which causes the stripes to line up\npreferentially with a particular axis of the GaAs substrate is not\nwell understood. The influence of disorder on the striped phases has\nnot been studied. There is currently no explanation for the\nobservation that various quantities have a prominent dependence on\nwhether the Fermi energy is in the lower or upper spin component of a\nLL. The influence of quantum interference on transport has not been\nstudied. Filling factors away from half filling of the topmost LL\nhave received relatively little attention.\n\n\\section*{Acknowledgments}\n\nWe benefitted from discussions with J. Eisenstein, B. Huckestein, M.\nShayegan, S. Simon, D. Shahar and H. Stormer. The work was supported\nin part by SFB 341, NSF grant DMR-94-16910, US-Israel BSF grant\n98-354, DIP-BMBF grant, a Minerva foundation grant, and the Israeli \nAcademy of Science.\n%Initial phases of the work were carried out at the ITP, Santa Barbara,\n%in August 1998, with support from NSF grant PHY94-07194.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Lilly} M.P.\\ Lilly, K.B.\\ Cooper, J.P.\\ Eisenstein, L.N.\\ Pfeiffer, \nand K.W.\\ West, Phys.\\ Rev.\\ Lett.\\ {\\bf 82}, 394 (1999).\n\n\\bibitem{Du} R.R.\\ Du, D.C.\\ Tsui, H.L.\\ Stormer, L.N.\\ Pfeiffer,\nK.W.\\ Baldwin, and K.W.\\ West, Solid State Comm.\\ {\\bf 109}, 389 (1999).\n\n\\bibitem{Fogler} M.M.\\ Fogler, A.A.\\ Koulakov, and B.I.\\ Shklovskii,\nPhys.\\ Rev.\\ B {\\bf 54}, 1853 (1996); A.A.\\ Koulakov, M.M.\\ Fogler,\nand B.I.\\ Shklovskii, Phys.\\ Rev.\\ Lett.\\ {\\bf 76}, 499 (1996). \n\n\\bibitem{Moessner} R.\\ Moessner, J.T.\\ Chalker, Phys.\\ Rev.\\ B\n{\\bf 54}, 5006 (1996). \n\n\\bibitem{Laughlin} R.B.\\ Laughlin, Phys.\\ Rev.\\ Lett.\\ {\\bf 50}, \n1395 (1983).\n\n\\bibitem{Fukuyama} H.\\ Fukuyama, P.M.\\ Platzman, and P.W.\\ Anderson, \nPhys.\\ Rev.\\ B {\\bf 19}, 5211 (1979).\n\n\\bibitem{Rezayi} E.H.\\ Rezayi, F.D.M.\\ Haldane, and K.\\ Yang, \nPhys.\\ Rev.\\ Lett.\\ {\\bf 83}, 1219 (1999).\n\n\\bibitem{Shayegan} M.\\ Shayegan and H.C.\\ Manoharan, cond-mat/9903405.\n\n\\bibitem{MacDonald} A.H.\\ MacDonald and M.P.A.\\ Fisher, cond-mat/9907278.\n\n\\bibitem{Oppen-cdw} F.\\ von Oppen, B.I.\\ Halperin, and A.\\ Stern, \ncond-mat/9910132.\n\n\\bibitem{Pan} W.\\ Pan, R.R.\\ Du, H.L.\\ Stormer, D.C.\\ Tsui, L.N.\\ Pfeiffer,\nK.W.\\ Baldwin, and K.W.\\ West, cond-mat/9903160.\n\n\\bibitem{Lilly2} M.P.\\ Lilly, K.B.\\ Cooper, J.P.\\ Eisenstein, L.N.\\ Pfeiffer, \nand K.W.\\ West, cond-mat/9903196.\n\n\\bibitem{Eisenstein} J.P.\\ Eisenstein, M.P.\\ Lilly, K.B.\\ Cooper,\nL.N.\\ Pfeiffer, and K.W.\\ West, cond-mat/9909238.\n\n\\bibitem{Simon} S.H.\\ Simon, cond-mat/9903086.\n\n\\bibitem{Lilly-comment} M.P.\\ Lilly, K.B.\\ Cooper, and J.P.\\ \nEisenstein, cond-mat/9903153.\n\n\\bibitem{Cooper} K.B.\\ Cooper, M.P.\\ Lilly, J.P.\\ \nEisenstein, L.N.\\ Pfeiffer, and K.W.\\ West, cond-mat/9907374.\n\n\\bibitem{Aleiner} I.L.\\ Aleiner and L.I.\\ Glazman, Phys.\\ Rev.\\ B {\\bf 52},\n11296 (1995).\n\n\\bibitem{Jungwirth} T.\\ Jungwirth, A.H.\\ MacDonald, L.\\ Smrcka, and \nS.M.\\ Girvin, cond-mat/9905353.\n\n\\bibitem{Stanescu} T.\\ Stanescu, I.\\ Martin, and P.\\ Philipps,\ncond-mat/9905116.\n\n\\bibitem{Fradkin} E.\\ Fradkin, S.A.\\ Kivelson, Phys.\\ Rev.\\ B\n{\\bf 59}, 8065 (1999).\n\n\\bibitem{Nelson} D.R.\\ Nelson, in {\\it Phase Transitions and Critical\nPhenomena}, ed.\\ by C.\\ Domb and J.L.\\ Lebowitz (Academic Press, London, \n1983).\n\n\\bibitem{Radzihovsky} See, however, L.\\ Radzihovsky and J.\\ Toner,\n Phys.\\ Rev.\\ B {\\bf 60}, 206 (1999) for a recent study of smectic\n liquid crystals in the presence of disorder.\n\n\\bibitem{modulation} See, e.g., G.R.\\ Aizin and V.A.\\ Volkov, Sov.\\ \n Phys.\\ JETP {\\bf 65}, 188 (1987); and F.\\ von Oppen, A.\\ Stern, and\n B.I.\\ Halperin, Phys.\\ Rev.\\ Lett.\\ {\\bf 80}, 4494 (1998) and\n references therein.\n\n\\bibitem{Eisenstein-priv} J.\\ Eisenstein, private communication \nand cond-mat/0003405 (to appear in Physica E).\n\n\\bibitem{Shimshoni} E.\\ Shimshoni and A.\\ Auerbach, Phys.\\ Rev.\\ B {\\bf 55},\n9817 (1997).\n\n\\bibitem{Dykhne} A.M.\\ Dykhne and I.M.\\ Ruzin, Phys.\\ Rev.\\ B {\\bf 50},\n2369 (1994).\n\n\\bibitem{Fradkin2} E.\\ Fradkin, S.A.\\ Kivelson, E.\\ Manousakis, and \nK.\\ Nho, cond-mat/9906064.\n\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n"
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[
{
"name": "cond-mat0002087.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Lilly} M.P.\\ Lilly, K.B.\\ Cooper, J.P.\\ Eisenstein, L.N.\\ Pfeiffer, \nand K.W.\\ West, Phys.\\ Rev.\\ Lett.\\ {\\bf 82}, 394 (1999).\n\n\\bibitem{Du} R.R.\\ Du, D.C.\\ Tsui, H.L.\\ Stormer, L.N.\\ Pfeiffer,\nK.W.\\ Baldwin, and K.W.\\ West, Solid State Comm.\\ {\\bf 109}, 389 (1999).\n\n\\bibitem{Fogler} M.M.\\ Fogler, A.A.\\ Koulakov, and B.I.\\ Shklovskii,\nPhys.\\ Rev.\\ B {\\bf 54}, 1853 (1996); A.A.\\ Koulakov, M.M.\\ Fogler,\nand B.I.\\ Shklovskii, Phys.\\ Rev.\\ Lett.\\ {\\bf 76}, 499 (1996). \n\n\\bibitem{Moessner} R.\\ Moessner, J.T.\\ Chalker, Phys.\\ Rev.\\ B\n{\\bf 54}, 5006 (1996). \n\n\\bibitem{Laughlin} R.B.\\ Laughlin, Phys.\\ Rev.\\ Lett.\\ {\\bf 50}, \n1395 (1983).\n\n\\bibitem{Fukuyama} H.\\ Fukuyama, P.M.\\ Platzman, and P.W.\\ Anderson, \nPhys.\\ Rev.\\ B {\\bf 19}, 5211 (1979).\n\n\\bibitem{Rezayi} E.H.\\ Rezayi, F.D.M.\\ Haldane, and K.\\ Yang, \nPhys.\\ Rev.\\ Lett.\\ {\\bf 83}, 1219 (1999).\n\n\\bibitem{Shayegan} M.\\ Shayegan and H.C.\\ Manoharan, cond-mat/9903405.\n\n\\bibitem{MacDonald} A.H.\\ MacDonald and M.P.A.\\ Fisher, cond-mat/9907278.\n\n\\bibitem{Oppen-cdw} F.\\ von Oppen, B.I.\\ Halperin, and A.\\ Stern, \ncond-mat/9910132.\n\n\\bibitem{Pan} W.\\ Pan, R.R.\\ Du, H.L.\\ Stormer, D.C.\\ Tsui, L.N.\\ Pfeiffer,\nK.W.\\ Baldwin, and K.W.\\ West, cond-mat/9903160.\n\n\\bibitem{Lilly2} M.P.\\ Lilly, K.B.\\ Cooper, J.P.\\ Eisenstein, L.N.\\ Pfeiffer, \nand K.W.\\ West, cond-mat/9903196.\n\n\\bibitem{Eisenstein} J.P.\\ Eisenstein, M.P.\\ Lilly, K.B.\\ Cooper,\nL.N.\\ Pfeiffer, and K.W.\\ West, cond-mat/9909238.\n\n\\bibitem{Simon} S.H.\\ Simon, cond-mat/9903086.\n\n\\bibitem{Lilly-comment} M.P.\\ Lilly, K.B.\\ Cooper, and J.P.\\ \nEisenstein, cond-mat/9903153.\n\n\\bibitem{Cooper} K.B.\\ Cooper, M.P.\\ Lilly, J.P.\\ \nEisenstein, L.N.\\ Pfeiffer, and K.W.\\ West, cond-mat/9907374.\n\n\\bibitem{Aleiner} I.L.\\ Aleiner and L.I.\\ Glazman, Phys.\\ Rev.\\ B {\\bf 52},\n11296 (1995).\n\n\\bibitem{Jungwirth} T.\\ Jungwirth, A.H.\\ MacDonald, L.\\ Smrcka, and \nS.M.\\ Girvin, cond-mat/9905353.\n\n\\bibitem{Stanescu} T.\\ Stanescu, I.\\ Martin, and P.\\ Philipps,\ncond-mat/9905116.\n\n\\bibitem{Fradkin} E.\\ Fradkin, S.A.\\ Kivelson, Phys.\\ Rev.\\ B\n{\\bf 59}, 8065 (1999).\n\n\\bibitem{Nelson} D.R.\\ Nelson, in {\\it Phase Transitions and Critical\nPhenomena}, ed.\\ by C.\\ Domb and J.L.\\ Lebowitz (Academic Press, London, \n1983).\n\n\\bibitem{Radzihovsky} See, however, L.\\ Radzihovsky and J.\\ Toner,\n Phys.\\ Rev.\\ B {\\bf 60}, 206 (1999) for a recent study of smectic\n liquid crystals in the presence of disorder.\n\n\\bibitem{modulation} See, e.g., G.R.\\ Aizin and V.A.\\ Volkov, Sov.\\ \n Phys.\\ JETP {\\bf 65}, 188 (1987); and F.\\ von Oppen, A.\\ Stern, and\n B.I.\\ Halperin, Phys.\\ Rev.\\ Lett.\\ {\\bf 80}, 4494 (1998) and\n references therein.\n\n\\bibitem{Eisenstein-priv} J.\\ Eisenstein, private communication \nand cond-mat/0003405 (to appear in Physica E).\n\n\\bibitem{Shimshoni} E.\\ Shimshoni and A.\\ Auerbach, Phys.\\ Rev.\\ B {\\bf 55},\n9817 (1997).\n\n\\bibitem{Dykhne} A.M.\\ Dykhne and I.M.\\ Ruzin, Phys.\\ Rev.\\ B {\\bf 50},\n2369 (1994).\n\n\\bibitem{Fradkin2} E.\\ Fradkin, S.A.\\ Kivelson, E.\\ Manousakis, and \nK.\\ Nho, cond-mat/9906064.\n\n\n\\end{thebibliography}"
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cond-mat0002088
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Density functional theory of phase coexistence in\\ weakly polydisperse fluids
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"author": "Hong Xu $^{\\dag}$ and Marc Baus $^{\\dag\\dag}$"
}
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The recently proposed universal relations between the moments of the polydispersity distributions of a phase-separated weakly polydisperse system are analyzed in detail using the numerical results obtained by solving a simple density functional theory of a polydisperse fluid. It is shown that universal properties are the exception rather than the rule.\par
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"name": "fracs.tex",
"string": "%\\documentstyle[prl,aps,epsfig,twocolumn]{revtex}\n\\documentclass[11pt]{article}\n%\\documentclass[twocolumn]{article}\n\n\\textwidth 14.5true cm\n\\textheight 8.5true in\n\\oddsidemargin 1.7true cm\n\\evensidemargin 1.7true cm\n\\topmargin -0.3true in\n\\headsep 0.4true in\n\n\\usepackage[draft]{graphics}\n%\\usepackage[final]{graphics}\n\\usepackage{latexsym}\n\n\\renewcommand{\\baselinestretch}{2}\n%\n\\baselineskip=30pt\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\n\\newcommand{\\ba}{\\begin{eqnarray}}\n\\newcommand{\\ea}{\\end{eqnarray}}\n\n\\newcommand{\\bat}{\\begin{tabular}{lr}}\n\\newcommand{\\eat}{\\end{tabular}}\n\n\\newcommand{\\bay}{\\[\\begin{array}{lr}}\n\\newcommand{\\eay}{\\end{array}\\]}\n\n\\newcommand{\\baa}{\\begin{eqnarray*}}\n\\newcommand{\\eaa}{\\end{eqnarray*}}\n\n\\def\\lg{{\\lambda}}\n\\def\\gam{{\\Gamma}}\n\\def\\rs{{r_s}}\n\\def\\chil{{\\chi_e^{(0)}}}\n\n\\def\\br{{\\mathbf{r}}}\n\\def\\ror{{\\rho(\\br)}}\n\\def\\sg{{\\sigma}}\n\\def\\ros{{\\rho(\\sg)}}\n\\def\\rror{{\\left[\\ror\\right]}}\n\\def\\brp{{\\mathbf{r}'}}\n\\def\\roa{{\\rho_1({\\br}) }}\n\\def\\rob{{\\rho_2({\\br}) }}\n\\def\\fibr{{\\phi_2(r) }}\n\\def\\rs{{r_s}}\n\\def\\ia{{_\\alpha}}\n\\def\\ag{{\\alpha}}\n\\def\\bg{{\\beta}}\n\n\\def\\rrp{{|\\br}-{\\br}'|}\n\\def\\hab{{h_{\\ag\\bg}}}\n\\def\\gab{{g_{\\ag\\bg}}}\n\\def\\rpr{{(\\br,\\brp)}}\n\\def\\dr{d\\br}\n\\def\\drp{d\\brp}\n\\def\\intt{{\\int d\\br\\int d\\brp}}\n\n\n\\def\\um{{\\frac{1}{2}}}\n\n\\def\\xid{\\xi^\\dagger}\n\\def\\xis{\\xi^*}\n\\def\\xip{\\xi '}\n\n\\def\\ranglev{\\rangle_{\\xid}}\n\\def\\ranglevm{\\rangle_{\\xid,M}}\n\\def\\ranglevpm{\\rangle_{\\xip,M}}\n\\def\\llangle{\\left\\langle}\n\\def\\rrangle{\\right\\rangle}\n\\def\\rranglev{\\right\\rangle_{\\tilde V}}\n\\def\\rranglevp{\\right\\rangle_{\\xip}}\n\\def\\rranglesp{\\right\\rangle_{\\sigma=0}}\n\\def\\rranglevm{\\right\\rangle_{\\xid,M}}\n\\def\\rranglevpm{\\right\\rangle_{\\xip,M}}\n\\def\\rranglespm{\\right\\rangle_{\\sigma=0,M}}\n\n\\begin{document}\n\\bibliographystyle{prbsty}\n \n\\title{Density functional theory of phase coexistence in\\\\\nweakly polydisperse fluids} \n \n\\author{Hong Xu $^{\\dag}$ and Marc Baus $^{\\dag\\dag}$}\n\\date{}\n\\maketitle\n\n\\noindent\n$^{\\dag}$ D\\'epartement de Physique des Mat\\'eriaux (UMR 5586 du CNRS),\\\\\nUniversit\\'e Claude Bernard-Lyon1, 69622 Villeurbanne Cedex, France\\\\\n$^{\\dag\\dag}$ Physique des Polym\\`eres, Universit\\'e Libre de Bruxelles,\\\\\nCampus Plaine, CP 223, B-1050 Brussels, Belgium\\\\\n\n\\vspace{2truecm}\n\\noindent\nPACS numbers: 05.70.-a, 64.75.+g, 82.60.Lf\n\\pagebreak\n \n\\begin{abstract}\nThe recently proposed universal relations between the moments\nof the polydispersity distributions of a phase-separated weakly\npolydisperse system are analyzed in detail using the numerical\nresults obtained by solving a simple density functional theory\nof a polydisperse fluid. It is shown that universal properties\nare the exception rather than the rule.\\par\n\\end{abstract} \n\n\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\vspace{1truecm}\nMany, natural or man-made, systems are mixtures of similar instead of\nidentical objects. For example, in a colloidal dispersion\\cite{rus}\nthe size and surface charge of the colloidal particles are usually\ndistributed in an almost continuous fashion around some mean value. When\nthis distribution is very narrow the system can often be assimilated\n\\cite{poon} to a one-component system of identical objects.\nSuch a system is usually called monodisperse whereas otherwise\nit is termed polydisperse. Since polydispersity is a direct consequence\nof the physico-chemical production process it is an intrinsic\nproperty of many industrial systems. Therefore, many author\\cite{laro}\nhave included polydispersity into the description of a given phase\nof such systems. More recently, a renewed interest can be witnessed\nfor the study of phase transitions occuring in weakly polydisperse\nsystems\\cite{evan}. The phase behavior of polydisperse systems is\nof course much richer than that of its monodisperse counterpart. It is\nalso more difficult to study theoretically, essentially because\none has to cope with an infinity of thermodynamic coexistence\nconditions\\cite{laro}. Therefore, several authors have proposed\napproximation schemes\\cite{sol} which try to bypass this difficulty.\nIn the present study we take to opposite point of view by solving\nnumerically the infinitely many thermodynamic coexistence conditions\nfor a simple model polydisperse system. On this basis we have studied\nthe radius of convergence of the weak polydispersity expansion\nused in ref.4 and found that their ``universal law of fractionation\" and\nsome of their conclusions have to be modified in several cases.\\par\n\nThe statistical mechanical description of a polydisperse equilibrium\nsystem is equivalent to a density functional theory\\cite{han} for a\nsystem whose number density, $\\rho(\\br,\\sg)$, depends besides the\nposition variable $\\br$ (assuming spherical particles) also on\nat least one polydispersity variable $\\sg$ (which we consider to be\ndimensionless). Such a theory is completely determined once the\nintrinsic Helmholtz free-energy per unit volume, $f[\\rho]$, has been\nspecified as a functional of $\\rho(\\br,\\sg)$ (for notational\nconvenience the dependence on the temperature $T$ will not be\nindicated explicitly). For the spatially uniform fluid phases\nconsidered here (and also implicitly in ref.4) we have,\n$\\rho(\\br,\\sg)\\rightarrow\\ros$, and the pressure can be written\nas, $p[\\rho]=\\int d\\sg\\ros\\mu(\\sg;[\\rho])-f[\\rho]$, where\n$\\mu(\\sg;[\\rho])=\\delta f[\\rho]/\\delta \\ros$, is the chemical potential of\n``species\" $\\sg$. When a parent phase of density $\\rho_0(\\sg)$\nphase separates into $n$ daughter phases of density $\\rho_i(\\sg)$\n($i=1,\\ldots,n$) the phase coexistence conditions imply that,\n$p[\\rho_1]=p[\\rho_2]=\\ldots=p[\\rho_n]$, and \n$\\mu(\\sg;[\\rho_1])=\\mu(\\sg;[\\rho_2])=\\ldots=\\mu(\\sg;[\\rho_n])$.\nFor simplicity we consider here only the case of two daughter\nphases ($n=2$) and rewrite moreover $\\rho_i(\\sg)=\\rho_i h_i(\\sg) (i=0,1,2)$\nin terms of the average density $\\rho_i$ and a polydispersity\ndistribution $h_i(\\sg)$ such that $\\int d\\sg h_i(\\sg)=1$.\nSince the ideal gas contribution to $f[\\rho]$ is exactly known\\cite{han} \none has, $\\mu(\\sg;[\\rho])=k_B T\\ln\\{\\Lambda^3(\\sg)\\ros\\}\n+\\mu_{ex}(\\sg;[\\rho])$, where $k_B$ is Boltzmann's constant,\n$\\Lambda(\\sg)$ is the thermal de Broglie wavelength of species\n$\\sg$ and $\\mu_{ex}$ the excess (ex) contribution to $\\mu$.\nThis allows us to rewrite the equality of the chemical potentials\nof the two daughter phases as, $h_1(\\sg)=h_2(\\sg)A(\\sg)$, where\n$A(\\sg)$ is a shorthand notation for:\n\\be\nA(\\sg)=\\frac{\\rho_2}{\\rho_1}\\exp\\beta\\left\\{\\mu_{ex}(\\sg;[\\rho_2])\n-\\mu_{ex}(\\sg;[\\rho_1])\\right\\}\n\\ee\nwith $\\beta=1/k_B T$. The polydispersity distributions are further\nconstrained by the relation, $x_1 h_1(\\sg)+x_2 h_2(\\sg)=h_0(\\sg)$,\nwhich expresses particle number conservation. The number\nconcentration of phase 1, $x_1=1-x_2$, is\ngiven by the lever rule: \n$x_1=\\frac{\\rho_1}{\\rho_1-\\rho_2}\\cdot\\frac{\\rho_0-\\rho_2}{\\rho_0}$.\nCombining these two relations one finds:\n\\be\nh_2(\\sg)-h_1(\\sg)=h_0(\\sg)\\cdot H(\\sg)\n\\ee\nwhere $H(\\sg)\\equiv (1-A(\\sg))/(x_2+x_1 A(\\sg))$. Eq.(2) is the starting\npoint to relate the difference between the moments \nof the daughter phases, $\\Delta_k=\\int d\\sg\\,\\sg^k(h_2(\\sg)-h_1(\\sg))$,\nto the moments, $\\xi_k=\\int d\\sg\\,\\sg^k h_0(\\sg)$ ($k=0,1,2,\\ldots$), of\nthe parent phase distribution $h_0(\\sg)$. Indeed, when $\\sg$ is\nchosen such that $h_0(\\sg)$ tends to the Dirac delta function $\\delta(\\sg)$\nin the monodisperse limit, $\\Delta_k$ can be obtained from (2) by\nexpanding $H(\\sg)$ around $\\sg=0$, $H(\\sg)=\\sum_{l=0}^{\\infty} a_l\\sg^l$,\nyielding for a weakly polydisperse system, \n$\\Delta_k=\\sum_{l=0}^{\\infty} a_l\\xi_{l+k}$. \nThe normalization of the $h_i(\\sg)$ ($i=0,1,2$) implies\n$\\Delta_0=0$, $\\xi_0=1$ or $a_0=-\\sum_{l=1}^{\\infty}a_l\\xi_l$,\nand eliminating $a_0$ from $\\Delta_k$ \nyields the general moment relation:\n\\be\n\\Delta_k=a_1\\xi_{k+1}+\\sum_{l=2}^{\\infty} a_l(\\xi_{k+l}-\\xi_l\\xi_k).\n\\ee\nwhere we took moreover into account that\n$\\sg$ can always be chosen such that $\\xi_1=0$.\nWhen only the first term in the r.h.s. of (3) is retained we recover\nthe universal law $\\Delta_k/\\Delta_l=\\xi_{k+1}/\\xi_{l+1}$, put\nforward in ref.4. The question left unanswered by the study of\nref.4 concerns the radius of convergence of the weak polydispersity\nexpansion (3). In order to study this problem in more detail\nwe now consider\na simple model system for which we can determine the $h_i(\\sg)(i=1,2)$\nnumerically and compare the results with (3). The free energy\ndensity functional chosen here corresponds to a simple van der Waals\n(vdW) model\\cite{gua} for the liquid-vapor transition in polydisperse\nsystems of spherical particles of variable size:\n\\ba\nf[\\rho]=&k_B\\,T\\int d\\sg\\ros\\left\\{\\ln(\n\\frac{\\Lambda^3(\\sg)\\ros}{E[\\rho]})-1\\right\\}\\nonumber\\\\\n&+\\frac{1}{2}\\int d\\sg\\int d\\sg'\\,V(\\sg,\\sg')\\ros\\rho(\\sg')\n\\ea\nwhere, $E[\\rho]=1-\\int d\\sg\\,v(\\sg)\\ros$, describes the average excluded volume correction for\nparticles of radius $R_\\sg$ and volume $v(\\sg)=\\frac{4\\pi}{3}R_\\sg^3$,\nwhile $V(\\sg,\\sg')=\\int d\\br\\,V(r;\\sg,\\sg')$ is the integrated attraction\nbetween two particles of species $\\sg$ and $\\sg'$, for which we took\nthe usual vdW form, $V(r;\\sg,\\sg')=-\\epsilon_0(R_\\sg+R_{\\sg'})^6/r^6$\nfor $r\\ge R_\\sg+R_{\\sg'}$ and zero otherwise, $\\epsilon_0$ being the\namplitude of the attraction at the contact of the two particles.\nThe size-polydispersity can be described\nin terms of the dimensionless variable, $\\sg=R_\\sg/R\\,-1$, with\n$R$ the mean value of $R_\\sg$ in the parent phase, hence\n$\\xi_1=\\int d\\sg\\,\\sg h_0(\\sg)=0$. The thermodynamics is given in terms\nof $h_0(\\sg)$, the dimensionless temperature $t=k_B\\,T/\\epsilon_0$ and the dimensionless density $\\eta=v_0\\rho$, with\n$v_0=\\frac{4\\pi}{3}R_0^3$ and $R_0$ the value of $R_\\sg$ \nin the monodisperse limit.\nThe coexistence conditions are integral equations which can be solved\nnumerically using, for instance, an iterative algorithm\\cite{mix}\nfor any $t$, $\\eta_0=v_0\\rho_0$ and $h_0(\\sg)$. For $h_0(\\sg)$ we took\na Schulz distribution\\cite{laro} with zero mean. The normalized \ndistribution is given, for $-1\\le\\sg<\\infty$, by \n$h_0(\\sg)=\\ag^\\ag(1+\\sg)^{\\ag-1}e^{-\\ag(1+\\sg)}/\\Gamma(\\ag)$,\nwith $\\Gamma(\\ag)$ the gamma function and $1/\\ag$ a width parameter which\nmeasures the distance to the monodisperse limit, $h_0(\\sg)\\rightarrow\\delta(\\sg)$\nwhen $\\ag\\rightarrow\\infty$. We then have: $\\xi_0=1$, $\\xi_1=0$, $\\xi_2=1/\\ag$,\n$\\xi_3=2/\\ag^2$, $\\xi_4=\\frac{3}{\\ag^2}+\\frac{6}{\\ag^3}$,\n$\\xi_5=\\frac{20}{\\ag^3}+\\frac{24}{\\ag^4}$, etc. For a weakly polydisperse\nsystem we retain only the dominant terms of (3) in a $1/\\ag$ expansion.\nFrom (3) we obtain then:\n$\\Delta_1=a_1(\\infty)\\xi_2+O(1/\\ag^2)$,\n$\\Delta_2=a_1(\\infty)\\xi_3+a_2(\\infty)(\\xi_4-\\xi_2^2)+O(1/\\ag^3)\n=\\{a_1(\\infty)+a_2(\\infty)\\}\\xi_3+O(1/\\ag^3)$,\n$\\Delta_3=a_1(\\infty)\\xi_4+O(1/\\ag^3)$, etc, where $a_l(\\infty)$ are\nthe values of $a_l$ for $\\ag\\rightarrow\\infty$. \nUsing the vdW expression (4) to evaluate (1) one finds, for ex. for\n$t=1.0$ and $\\eta_0=0.5$, $a_1(\\infty)=1.75$ and $a_2(\\infty)=-2.68$.\nUsing the corresponding numerical solutions found for $h_1(\\sg)$ and\n$h_2(\\sg)$ (see Fig.1) it can be seen from Fig.2 that \n$\\Delta_1/\\xi_2\\approx 1.75$, $\\Delta_2/\\xi_3\\approx -0.93$ and $\\Delta_3/\\xi_4\\approx 1.75$ are\nobeyed to within ten percent for $\\ag$ larger than, respectively,\n40, 80 and 150.\nWe can conclude thus that the weak\npolydispersity expansion (3) is valid (to dominant order) for Schulz\ndistributions $h_0(\\sg)$ with a dispersion $\\left((\\xi_2-\\xi_1^2)^{1/2}\\right)$\nsmaller than, say, 0.1 ($\\ag\\approx 100$). These values do of course depend on the thermodynamic\nstate but the case considered here ($t=1$, $\\eta_0=0.5$) is representative\nof other $t,\\eta_0$ values. Note also that we have verified \nnumerically that the radius of convergence of (3) with respect to\n$1/\\ag$ is fairly sensitive to the total amount of polydispersity\npresent. Allowing, for instance, the amplitude $\\epsilon_0$ of the pair\npotential $V(r;\\sg,\\sg')$ to depend on $\\sg$ and $\\sg'$ does reduce the\nradius of convergence of (3) considerably. From the above it follows that,\n$\\frac{\\Delta_3}{\\Delta_1}$ follows the universal law, \n$\\frac{\\Delta_3}{\\Delta_1}=\\frac{\\xi_4}{\\xi_2}$, put forward in\nref.4 whereas $\\frac{\\Delta_2}{\\Delta_1}$ follows the non-universal\nlaw, $\\frac{\\Delta_2}{\\Delta_1}\n=\\{1+\\frac{a_2(\\infty)}{a_1(\\infty)}\\}\\frac{\\xi_3}{\\xi_2}$.\nWe have verified that similar results can be obtained for different\n$h_0(\\sg)$ distributions. Taking, for instance, a Gaussian for\n$h_0(\\sg)$ similar results are found although $\\xi_3=0$\nfor this case. This invalidates the conclusion of ref.4 that a particular\nimportance should be attached to the skewness of $h_0(\\sg)$. In conclusion,\nthe general moment relation (3) can yield useful information about the\nphase behavior of weakly polydisperse systems but this information\nis in general not universal.\\par\n\\pagebreak\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\begin{references} % %\n\\begin{thebibliography}{999}\n%1 \n\\bibitem{rus} W.B. Russel, D.A. Saville and W.R. Schowalter,\n{\\sl Colloidal Dispersions},\n(Cambridge University Press, Cambridge 1998); A.P. Gast and\nW.B. Russel, {\\sl Physics Today}, {\\bf 51}($n^\\circ$ 12), 24(1998).\n%2\n\\bibitem{poon}W.C.K. Poon and P. Pusey, {\\sl Observation, Prediction\nand Simulation of Phase Transitions in Complex Fluids}, (Edited by\nM. Baus, L.F. Rull and J.P. Ryckaert) p.3 (Kluwer Academic Publishers,\nDordrecht, 1995).\n%3\n\\bibitem{laro} S. Leroch, G. Kahl and F. Lado, {\\sl Phys. Rev. E}\n{\\bf 59}, 6937(1999); S.E. Phan, W.B. Russel, J. Zhu and\nP.M. Chaikin, {\\sl J. Chem. Phys.}, {\\bf 108}, 9789(1998);\nP. Bartlett, {\\sl J. Chem. Phys.} {\\bf 107}, 188(1997);\nR. McRae and A.D.J. Haymet, {\\sl J.Chem.Phys.} {\\bf 88}, 1114(1988);\nJ.L. Barrat and J.P. Hansen, {\\sl J. Physique} {\\bf 47}, 1547(1986);\nJ.J. Salacuse and G. Stell. {\\sl J. Chem. Phys.} {\\bf 77}, 3714(1982);\nand references therein.\n%4\n\\bibitem{evan} R.M.L. Evans, D.J. Fairhurst and W.C.K. Poon,\n{\\sl Phys. Rev. Lett.} {\\bf 81},1326(1998); R.M.L. Evans,\n{\\sl Phys. Rev. E} {\\bf 59}, 3192(1999); G.H. Fredrickson,\n{\\sl Nature} {\\bf 395}, 323(1998).\n%5\n\\bibitem{sol} P. Sollich and M.E. Cates, {\\sl Phys. Rev. Lett.} {\\bf 80},\n1365(1998); P.B. Warren, {\\sl Phys. Rev. Lett.} {\\bf 80},\n1369(1998).\n%6\n\\bibitem{han} see e.g. J.P. Hansen and I.R. McDonald,\n{\\sl Theory of simple liquids}, 2nd ed. (Academic Press,\nLondon, 1986)\n%7\n\\bibitem{gua} J.A. Gualtieri, J.M. Kincaid and G. Morrison, {\\sl J. Chem. Phys.}\n{\\bf 77}, 521(1989); A. Daanoun, C.F. Tejero and M. Baus,\n{\\sl Phys. Rev.}{\\bf E50}, 2913(1994); R. Lovett and M. Baus, {\\sl J. Chem. Phys.}, to appear; and references therein. \n%8\n\\bibitem{mix} D.D. Johnson, {\\sl Phys. Rev.} {\\bf B38}, 12807(1988)\n%\\end{references} % %\n\\end{thebibliography} % %\n\\pagebreak\n%\\include{hydrogbib}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\noindent{\\bf Figure Captions}\\par\n\\vspace{1truecm}\n\\noindent\n{\\bf FIG. 1.} The polydispersity distributions $h_n(\\sg)$ of the\nparent phase ($n=0$: full curve) (a Schulz distribution with the width parameter\n$\\ag=50$), the low-density (n=1: dotted curve) and the high-density (n=2: circles) daughter\nphases, \nas obtained by numerically solving the coexistence conditions\nof the van der Waals model of eq.(4) for $t=1$, $\\eta_0=0.5$.\nThe corresponding dimensionless densities of the coexisting\ndaughter phases are $\\eta_1=0.106$, $\\eta_2=0.521$ \nwhereas for the monodisperse\nsystem one has, $\\eta_1=0.103$, $\\eta_2=0.608$. Also shown are $h_1(\\sg)-h_0(\\sg)$ \n(dashed curve) and $[h_2(\\sg)-h_0(\\sg)]\\cdot 50$ (triangles).\\par\n%\\includegraphics*[10mm,15mm][160mm,165mm]{hydrofig2.ps}\n\\vspace{0.3truecm}\n\\noindent\n{\\bf FIG. 2.} The ratio $\\Delta_k/\\xi_{k+1}$ ($k=1,2,3$) versus $1/\\ag$\nas obtained from the numerical solution of the van der Waals model\nof eq.(4) for $t=1$, $\\eta_0=0.5$ and a Schulz distribution\nfor $h_0(\\sg)$. The symbols are as follows: circles(k=1),\nsquares(k=2) and triangles(k=3). \nThe dotted lines indicate their asymptotic\n($\\ag\\rightarrow\\infty$) values. The arrows indicate\nfor each case the\nradius of convergence of the weak polydispersity expansion of eq.(3). \\par\n\n\\end{document}\n \n \n\n\n"
}
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[
{
"name": "cond-mat0002088.extracted_bib",
"string": "\\begin{thebibliography}{999}\n%1 \n\\bibitem{rus} W.B. Russel, D.A. Saville and W.R. Schowalter,\n{\\sl Colloidal Dispersions},\n(Cambridge University Press, Cambridge 1998); A.P. Gast and\nW.B. Russel, {\\sl Physics Today}, {\\bf 51}($n^\\circ$ 12), 24(1998).\n%2\n\\bibitem{poon}W.C.K. Poon and P. Pusey, {\\sl Observation, Prediction\nand Simulation of Phase Transitions in Complex Fluids}, (Edited by\nM. Baus, L.F. Rull and J.P. Ryckaert) p.3 (Kluwer Academic Publishers,\nDordrecht, 1995).\n%3\n\\bibitem{laro} S. Leroch, G. Kahl and F. Lado, {\\sl Phys. Rev. E}\n{\\bf 59}, 6937(1999); S.E. Phan, W.B. Russel, J. Zhu and\nP.M. Chaikin, {\\sl J. Chem. Phys.}, {\\bf 108}, 9789(1998);\nP. Bartlett, {\\sl J. Chem. Phys.} {\\bf 107}, 188(1997);\nR. McRae and A.D.J. Haymet, {\\sl J.Chem.Phys.} {\\bf 88}, 1114(1988);\nJ.L. Barrat and J.P. Hansen, {\\sl J. Physique} {\\bf 47}, 1547(1986);\nJ.J. Salacuse and G. Stell. {\\sl J. Chem. Phys.} {\\bf 77}, 3714(1982);\nand references therein.\n%4\n\\bibitem{evan} R.M.L. Evans, D.J. Fairhurst and W.C.K. Poon,\n{\\sl Phys. Rev. Lett.} {\\bf 81},1326(1998); R.M.L. Evans,\n{\\sl Phys. Rev. E} {\\bf 59}, 3192(1999); G.H. Fredrickson,\n{\\sl Nature} {\\bf 395}, 323(1998).\n%5\n\\bibitem{sol} P. Sollich and M.E. Cates, {\\sl Phys. Rev. Lett.} {\\bf 80},\n1365(1998); P.B. Warren, {\\sl Phys. Rev. Lett.} {\\bf 80},\n1369(1998).\n%6\n\\bibitem{han} see e.g. J.P. Hansen and I.R. McDonald,\n{\\sl Theory of simple liquids}, 2nd ed. (Academic Press,\nLondon, 1986)\n%7\n\\bibitem{gua} J.A. Gualtieri, J.M. Kincaid and G. Morrison, {\\sl J. Chem. Phys.}\n{\\bf 77}, 521(1989); A. Daanoun, C.F. Tejero and M. Baus,\n{\\sl Phys. Rev.}{\\bf E50}, 2913(1994); R. Lovett and M. Baus, {\\sl J. Chem. Phys.}, to appear; and references therein. \n%8\n\\bibitem{mix} D.D. Johnson, {\\sl Phys. Rev.} {\\bf B38}, 12807(1988)\n%\\end{references} % %\n\\end{thebibliography}"
}
] |
cond-mat0002089
|
Random Fixed Point of Three-Dimensional Random-Bond Ising Models
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[
{
"author": "Koji {\\sc Hukushima}"
}
] |
[
{
"name": "dwrg-3dsg.tex",
"string": "\\documentstyle[twocolumn,epsf]{jpsj}\n\\newcommand{\\figwidth}{8.5cm}\n\\newcommand{\\vup}{\\vspace*{-1pc}}\n%\\documentstyle[preprint,epsf]{jpsj}\n%\\newcommand{\\figwidth}{0.8\\textwidth}\n\n\n%%%%%%%%%%%%%%%%%\n\\def\\runtitle{Random Fixed Point of Three-Dimensional Random-Bond Ising\nModels }\n\\def\\runauthor{Koji {\\sc Hukushima}}\n%%%%%%%%%%%%%%%\n\n\n\\title{\nRandom Fixed Point of Three-Dimensional Random-Bond Ising Models\n}\n\\author{\nKoji {\\sc Hukushima}\n}\n\\inst{\nInstitute for Solid State Physics, Univ. of Tokyo, 7-22-1\n Roppongi, Minato-ku, Tokyo 106-8666\n}\n\\recdate{November 12, 1999}\n\n\\abst{\nThe fixed-point structure of three-dimensional bond-disordered Ising models\n is investigated using the numerical domain-wall renormalization-group\n method. \nIt is found that, in the $\\pm J$ Ising model, there exists a non-trivial\n fixed point along the phase boundary between the paramagnetic and\n ferromagnetic phases. The fixed-point Hamiltonian of the $\\pm J$ model\n numerically coincides with that of the unfrustrated random Ising\n models, strongly suggesting that both belong to the same universality class. \nAnother fixed point corresponding to the multicritical point is also\n found in the $\\pm J$ model. \nCritical properties associated with the fixed point are qualitatively\n consistent with theoretical predictions. \n}\n\\kword{domain wall renormalization group, fixed point, random system, \nspin glass, MC simulation}\n\n\\begin{document}\n\\sloppy\n\\maketitle\n\nThe influence of quenched disorder on model systems has attracted \nconsiderable interest in the field of statistical physics. \nThe first remarkable criterion was given by Harris\\cite{Harris}, who\nclaimed that if the specific-heat exponent $\\alpha_{\\rm pure}$ of the\npure system is positive, disorder becomes relevant, implying that a\nnew random fixed point governs critical phenomena of the random\nsystem. One of the simplest models belonging to such a class is the\nthree-dimensional ($3D$) Ising model. \nCritical exponents associated with the random fixed point have been\ninvestigated for dilution-type disorder by \nexperimental\\cite{Experiment1,Experiment2,Experiment3},\ntheoretical\\cite{GrinsteinLuther,Janssen} and numerical\napproaches\\cite{Heuer,BFMMRG}. \nRecently, extensive Monte Carlo (MC) studies\\cite{BFMMRG}\n have clarified that the critical exponents of the $3D$ site-diluted\n Ising model are independent of the concentration of the site dilution\n $p$, suggesting the existence of a random fixed point. \nThese results were obtained by carefully taking into account correction\nfor finite-size scaling, unless the exponents clearly depended \non $p$\\cite{BFMMRG}. \nExperimentally, the critical exponents of randomly diluted\nantiferromagnetic Ising compounds,\nFe$_x$Zn$_{1-x}$F$_2$\\cite{Experiment2,Experiment3} and \nMn$_x$Zn$_{1-x}$F$_2$\\cite{Experiment1}, are distinct from those of\nthe pure $3D$ Ising model. \n\nWhile the existence of the random fixed point has been established for\nthe $3D$ site-diluted Ising model, the idea of the universality class\nfor random systems, namely classification by fixed points, has not\nbeen explored yet as compared with various pure systems. \nIn particular, the question as to whether the random fixed point is\nuniversal irrespective of the type of disorder or not is a non-trivial\nproblem. \nIn the present work, we study critical phenomena associated with the\nferromagnetic phase transition in $3D$ site- and bond-diluted and $\\pm\nJ$ Ising models. \nThe main purpose is to determine the fixed-point structure of $3D$\nrandom-bond Ising models by making use of a numerical\nrenormalization-group (RG) analysis. \nOur strategy is based on the domain-wall RG (DWRG) method proposed by\nMcMillan\\cite{McMillan,McMillan2}. \nThis method has been applied to a $2D$ frustrated random-bond Ising\nmodel\\cite{McMillan,McMillan2} where there is no random fixed point, and\nrecently \nto a $2D \\pm J$ frustrated random-bond three-state Potts model\\cite{SGH}\nwhich displays a non-trivial random fixed point. \nIn this paper, we show systematic RG flow diagrams for $3D$ \nrandom Ising spin systems,\n which convinces us of the existence of the random fixed point.\n\nLet us first explain briefly the RG scheme and the numerical method used \nby us. \nIn the DWRG\\cite{McMillan,McMillan2}, \nthe following domain-wall free energy $\\Delta F_J$ of a spin system on a\ncube with the size $L$ is regarded as an effective coupling associated\nwith a length scale $L$ for a particular bond configuration, denoted by\n$J$; \n\\begin{equation}\n\\frac{\\Delta F_J(T)}{T} = \\ln \\frac{Z_{\\rm P}(T)}{Z_{\\rm AP}(T)},\n\\end{equation}\nwhere $Z_{\\rm P(AP)}(T)$ is the partition function of the cube at\ntemperature $T$ under (anti-) periodic boundary conditions for a given\ndirection, while the periodic boundary conditions are imposed for the\nremaining directions. \nIn disordered systems, the distribution $P(\\Delta F)$ of the\neffective couplings over the bond configurations is considered to be a\nrelevant quantity. \nTherefore, in the DWRG scheme, we are interested in how the distribution\nis renormalized as $L$ increases. \nIn the ferromagnetic phase, the expectation value of the distribution\napproaches infinity as $L$ increases, while it vanishes in the paramagnetic\nphase. \nFixed points are characterized by an invariant distribution of the\ncoupling under a DWRG transformation, namely, increasing $L$. \nFor example, the unstable fixed-point distribution corresponding to the\npure-ferromagnetic phase transition is a delta function with a non-zero\nmean. The random fixed point, if any, is expected to have a \nnon-trivial distribution with a finite width. \n\nIt is convenient to consider typical quantities characterizing the\ndistribution $P(\\Delta F)$, instead of the distribution itself. \nWe discuss here the mean $\\overline{\\Delta F}$ and the width\n$\\sigma(\\Delta F)$ of the distribution which are evaluated as\n\\begin{eqnarray}\n \\overline{\\Delta F} & = & \\int \\mbox{d} \\Delta F\\ \\Delta F\\ P(\\Delta F), \\\\\n\\sigma^2 (\\Delta F) & = & \\overline{\\Delta F^2} - \\overline{\\Delta F}^2,\n\\end{eqnarray}\nrespectively. \nUsing these quantities, we define two reduced parameters, \n$r=\\sigma(\\Delta F)/\\overline{\\Delta F}$ and $t=T/\\overline{\\Delta F}$, \nas in the previous works\\cite{McMillan,McMillan2,SGH}. \nThe renormalized parameters $(r(L),t(L))$ of the length scale $L$ are\nestimated numerically with bare parameters in a model Hamiltonian fixed. \nThen, the RG flow is represented by an arrow connected from the point\n$(r(L),t(L))$ of size $L$ to $(r(L'),t(L'))$ of a larger size $L'$. \nFixed points should be observed as points where the position is\ninvariant under the RG transformation. \nLinearizing the flow about a fixed point $(r^*,t^*)$, we obtain \n\\begin{equation}\n \\left(\n\\matrix{r(L')-r^* \\cr t(L')-t^* }\n\\right)\n=\\hat{T}\n\\left(\n\\matrix{r(L)-r^* \\cr t(L)-t^* }\n\\right),\n\\end{equation}\nwhere $\\hat{T}$ is a RG transformation matrix whose eigenvectors are\nscaling axes of the RG flow. \nThe matrix elements of $\\hat{T}$ as well as the fixed point $(r^*,t^*)$\ncan be determined by least-squares fitting from the data point\nnumerically obtained. \nThe critical exponents $y$ are obtained as $\\log\\lambda/\\log(L'/L)$ \nwith the eigenvalue $\\lambda$ of $\\hat{T}$. \n\nWe consider Ising models with quenched disorder which is defined\non a simple cubic lattice. The model Hamiltonian is \n\\begin{equation}\n \\label{eqn:model}\n {\\cal H}(J_{ij},S_i) = -\\sum_{\\langle ij\\rangle}J_{ij}S_iS_j, \n\\end{equation} \nwhere the sum runs over nearest-neighbor sites. \nIn order to obtain the domain-wall free energy, we use a recently\nproposed MC method\\cite{KH} which enables us to evaluate the\nfree-energy {\\it difference} directly with sufficient accuracy. \nIn the novel MC algorithm, \nwe introduce a dynamical variable specifying the boundary conditions,\nwhich is the sign of the interactions between the first and the last\nlayer of the cube for the given direction. \nThe algorithm is based on the exchange MC method\\cite{EMC},\nsometimes called the parallel tempering\\cite{Marinari}, which turns out\nto be reasonably efficient for randomly frustrated spin systems. \n\\begin{figure}\n\\epsfxsize=\\figwidth\n\\epsffile{fig-flow-3dbond.eps}\n\\vup\n\\caption{\nFlow diagram for the $3D$ unfrustrated bond-diluted Ising model. \nThe random fixed point is found numerically at $(0.63(2), 1.77(5))$. \nThe bold arrows indicate the eigenvectors, whose \nexponents are $y_1=1.47(4)$ and $y_2=-1.3(4)$. \n}\n\\vup\n\\label{fig:flow-bond}\n\\end{figure}\n\nFirst we study a $3D$ bond-diluted Ising model, which is given by\neq.~(\\ref{eqn:model}) with the bond distribution, $P(J_{ij}) = p\\delta\n(J_{ij}-J)+(1-p)\\delta (J_{ij})$. \nThe bond concentrations studied are $p=1.00$, $0.90$, $0.65$, $0.45$ and\n$0.35$ with the system sizes $L=8$ and $L=12$. \nSample averages are taken over $128-1984$ samples depending on the size\nand the concentration. \nErrors are estimated from statistical fluctuation over samples.\nThe number of temperature points in the exchange MC is\nfixed at $20$. \nWe distribute these temperatures to replicas for each $p$ such that the\nacceptance ratio for each exchange process becomes constant. \nWe show the RG flow diagram in Fig. \\ref{fig:flow-bond}.\nWhen disorder is absent, corresponding to the $x=0$ axis in\nFig.~\\ref{fig:flow-bond}, the pure fixed point is observed at \n$(0,1.63)$, denoted by $P$. \nNear the fixed point $P$, the arrows flow away from $P$ as disorder\nis introduced. This means that the pure fixed point of the $3D$ Ising\nmodel is unstable against the disorder, consistent with the Harris\ncriterion\\cite{Harris}. \nMeanwhile, this finding suggests that \na characteristic feature of the RG flow is reproduced within the system\nsizes studied. \nApart from the pure fixed point $P$, we find another fixed point,\ndenoted by $R$, along the phase boundary. \nThe RG flows approach $R$ from both sides, indicating that it is an\nattractive fixed point along the critical surface. \nThe fixed point $R$ governs critical phenomena of the disordered\nferromagnetic phase transition. Therefore, it should be called the\nrandom fixed point. \nThe position of $R$ in the parameter space is obtained numerically as\n$(0.63(2),1.77(5))$. At $R$, the exponents $y_{1}=1.47(4)$, $y_{2}=-1.3(4)$\nand the corresponding eigenvectors are indicated by the bold arrows in\nFig.~\\ref{fig:flow-bond}.\nThe critical exponent $\\nu$, the inverse of the larger eigenvalue $y_1$, \n is compatible with that of the site-diluted Ising model\\cite{BFMMRG},\n although our estimate is not so accurate. \nAll the arrows below $T_{\\rm c}$ go to the unique ferromagnetic fixed\npoint at $(0,0)$. \n\\begin{figure}\n\\epsfxsize=\\figwidth\n\\epsffile{fig-pd-3dI.eps}\n\\vup\n\\caption{\nPhase diagram of the $3D$ random-bond Ising models \nwith the bond (triangle) and the site (box) dilution. \nIn the inset, a schematic flow diagram for the diluted Ising models \nis shown as a function of temperature and disorder. \nThe fixed points and the RG flow are indicated.\n}\n\\vup\n\\label{fig:pd-dilute}\n\\end{figure}\n\nThe transition temperatures are estimated by a naive finite-size scaling\nassumption\\cite{McMillan,McMillan2}, \n$\\overline{\\Delta F}=f((T-T_{\\rm c})L^{1/\\nu})$ with the\ncorrelation-length exponent $\\nu$, \nnamely, the crossing point of $\\overline{\\Delta F}$ with $L=8$ and $12$\nas a function of temperature is located on $T_{\\rm c}$. \nThis scaling form is similar to that of the Binder parameter frequently\nused. \n%%%%%%%%%%%%%%%%%%%\nThe estimated $T_{\\rm c}$ is shown in Fig.~\\ref{fig:pd-dilute}. \n%%%%%%%%%%%%%%\nThe exponent $\\nu$ obtained by the scaling depends significantly on the\nconcentration $p$, similar to those observed as an \neffective exponent in the MC simulation of the site-disordered Ising\nmodel\\cite{BFMMRG}. \nIt is found from the RG flow diagram that the system has a subleading\nscaling parameter which gives rise to a systematic correction to the\nscaling. \n%%%%%%%%%%%%%%%%%%%%\nTherefore, we conclude that the continuously varying exponent observed\nin the naive scaling is due to the subleading parameter. \n\\begin{figure}\n\\epsfxsize=\\figwidth\n\\epsffile{fig-site-flow.eps}\n\\vup\n\\caption{\nFlow diagram for the $3D$ unfrustrated site-diluted Ising model. \nThe random fixed point is located on the point $(0.64(4), 1.71(9))$, whose\nexponents are $y_1=1.37(9)$ and $y_2=-0.9(5)$. \n}\n\\vup\n\\label{fig:flow-site}\n\\end{figure}\n\n\nWe also investigate a $3D$ site-diluted Ising model by the same\nprocedure as in the bond-diluted Ising model described above. The model\nis given by eq.~(\\ref{eqn:model}) but with the bond\n$J_{ij}=J\\epsilon_i\\epsilon_j$ \nand the distribution, $P(\\epsilon_{i}) = p\\delta (\\epsilon_{i}-1)+(1-p)\\delta\n(\\epsilon_{i})$. \nAs shown in Fig.~\\ref{fig:flow-site}, there exists a random fixed\npoint, which is consistent with the universality scenario observed in\nthis $3D$ site-diluted Ising model\\cite{BFMMRG}. \nIn the MC simulation\\cite{BFMMRG}, the correction to the scaling becomes \nsmaller around $p=0.80$. \nThis can be explained by the finding that the random fixed point we found is\nlocated on the position near the bare coupling with $p=0.80$. \nAn interesting point to note is that the position of the fixed point is\nclose to that observed in the $3D$ bond-diluted Ising model, though\nthe corresponding bare parameters such as the concentration $p$ and the \ntemperature differ between these two models, \n%%%%%%%%%%%%\nas seen in Fig.~\\ref{fig:pd-dilute}. \n%%%%%%%%%%%%\nThis finding implies that\nthe random fixed point is universal for a large class of the $3D$\nunfrustrated random Ising models. \n\n\nNext we consider a $3D \\pm J$ Ising model, where the\ninteractions $J_{ij}$ are randomly distributed according to the \nbimodal distribution, $P(J_{ij}) = p\\delta (J_{ij}-J)+(1-p)\\delta\n(J_{ij}+J)$. \nThe multi-spin coding technique can be easily implemented in this\nmodel. For that purpose, we consider $32$ temperature points in the exchange MC\nsimulation. \nIn this model, the ferromagnetically ordered state survives at\nlow temperatures up to a critical concentration, recently estimated at \n$p_{\\rm c}=0.7673(3)$\\cite{non-eq}, while below $p_{\\rm c}$ a spin glass \n(SG) phase appears.\nWe estimate the domain-wall free energy with $L=8$ and $12$ for a wide\nrange of concentration $p$ including the critical concentration. \nSample averages are taken over $240-6800$ samples depending on the size\nand the concentration. \n\\begin{figure}\n\\epsfxsize=\\figwidth\n\\epsffile{fig-flow-3dsg.eps}\n\\vup\n\\caption{\nRG flow diagram for the $3D \\pm J$ Ising SG model. \nThe pure critical fixed point is denoted by $P$, the random fixed\n point by $R$ and the multicritical fixed point by $N$. \nThe bold arrows represent the eigenvector of the RG transformation\n matrix $\\hat{T}$ at $R$ and $N$. \nThe broken line represents a critical surface as a guide for the eyes. \nThe random fixed point $R$ at $(0.66(2),1.74(1))$ is characterized by the\n eigenvalues $y_1=1.52(2)$ and $y_2=-0.42(13)$. \nThe exponents associated with the fixed point $N$ at $(1.66(6),1.17(2))$ are\n $y_1=1.2(2)$ and $y_2=0.62(6)$. \nData points on the Nishimori line are marked by gray symbols. \n}\n\\vup\n\\label{fig:flow-sg3d}\n\\end{figure}\n\nWe show in Fig.~\\ref{fig:flow-sg3d} the RG flow diagram for the $3D \\pm\nJ$ Ising SG model. The arrow connects results for $L=8$ and $L'=12$. \nHere, we again find the random fixed point $R$ as an attractor along the\nferromagnetic phase boundary. The position of $R$ is also close to that\nobserved in the diluted Ising models. \nThis finding indicates that both belong to the same universality class as\nthe ferromagnetic phase transition. \nIn other words, the renormalized coupling constants (or coarse-grained\nspin configurations) are independent of the details of the microscopic\nHamiltonian and of whether it is frustrated or not. \nOne of the inherent characteristics of the $\\pm J$ model is\nthe existence of a highly symmetric line, which we call the Nishimori\nline\\cite{Nishimori}. There were several theoretical works concerning \nthe Nishimori line\\cite{Nishimori,DH,DH2,HTE,HTE2}. \nIt was suggested by the $\\epsilon$-expansion method and a symmetry\nargument\\cite{DH,DH2} that the multicritical point must be located on\nthe line. \nThis was confirmed by MC simulations\\cite{non-eq,ON} and\nhigh-temperature series expansions\\cite{HTE,HTE2}. As shown in\nFig.~\\ref{fig:flow-sg3d}, we also observe a fixed point, denoted by $N$,\ncorresponding to the multicritical point, where both scaling axes have\na positive eigenvalue. \nOne of the scaling axes governed by the larger eigenvalue at $N$ almost\ncoincides with the Nishimori line, which is in agreement with the\npredictions by the $\\epsilon$-expansion\\cite{DH,DH2}. \n\nAs seen in Fig.~\\ref{fig:phase}, the estimated critical temperatures for\nthe concentrations simulated are consistent with those obtained by\nthe large-scale MC simulations up to size $L=101$\\cite{non-eq,non-eq2}. \nOur analysis can also be performed for the SG transition at smaller\nvalues of $p$. \nThe SG transition temperature $T_{\\rm SG}$ is determined from the\ncrossing of $\\sigma (\\Delta F)$. \nThe value of $T_{\\rm SG}$ at $p=0.50$ is in good agreement with \nthe recent estimation\\cite{KY}. \nAlong the ferromagnetic phase boundary, it is natural to expect that\nthere exists another fixed point $Z$ at zero temperature, which\nseparates the ferromagnetic phase from the SG one. \n%%%%%%%%\nBecause the eigenvalues of the RG matrix at $N$ are positive for both\nscaling axes, the thermal axis would be irrelevant in contrast to the\ndiluted Ising models, namely, an irrelevant flow into the\nzero-temperature fixed point $Z$ originates at $N$, as shown in the\ninset of Fig.~\\ref{fig:phase}. \nThe zero-temperature fixed point governs the critical behavior \nbetween the ferromagnetic and SG phase. \nIn the present study, however, we cannot reach temperatures that are \nsufficiently low to extract the zero-temperature fixed point. \nA modern optimization technique such as a genetic algorithm \nwould be useful in searching for the location of the fixed point.\n\\begin{figure}\n\\epsfxsize=\\figwidth\n\\epsffile{fig-sg3d-tc.eps}\n\\vup\n\\caption{\nPhase diagram of the $3D \\pm J$ Ising SG model. Open triangles are\n transition temperature estimated \nin the present work. \nThe SG transition temperatures marked by the solid triangles at\n $p=0.50$ and $0.76$ are determined from $\\sigma(\\Delta F)$. \nThe open circles represent estimations by the nonequilibrium MC relaxation\n method{\\protect\\cite{non-eq,non-eq2}}. \nThe dotted line represents the so-called Nishimori line. \nA schematic flow diagram for the $\\pm J$ model is drawn in the inset. \nThe fixed points $R$ and $N$, and the RG flow around them are supported\n in the present work.\n}\n\\vup\n\\label{fig:phase}\n\\end{figure}\n\n\nNow we comment on the correction to the finite-size scaling. \nIn order to obtain the critical exponent, one uses a range of the system\nsize frequently independent of the bare parameters in the model\nHamiltonian. \nWhen the system has a subleading scaling parameter, \nit causes systematic corrections to the scaling. \nHowever, it is not known how\nsuch corrections affect the leading scaling {\\it a priori}. \nThe RG flow diagram gives a good indication of necessity and\njustification for corrections to the leading scaling term. \nOnce the fixed-point structure of interest is clarified from the flow,\none of the best numerical approaches to obtain the critical exponents, within\nrestricted computer facilities, is to choose a model parameter close to\nthe fixed Hamiltonian. \n\nIn experiments, the $3D \\pm J$ Ising model is approximately realized in\na mixed compound with strong anisotropy such as\nFe$_{x}$Mn$_{1-x}$TiO$_3$. \nOur findings suggest that the critical phenomena of the compound\nFe$_{x}$Mn$_{1-x}$TiO$_3$ near $x\\sim 1$ belong to the same\nuniversality class as the $3D$ random Ising models. \nWe expect that various random Ising systems such as a mixed Ising spin\ncompound Fe$_{x}$Co$_{1-x}$F$_{2}$\\cite{Nash}, though beyond the scope\nof he present study, are also categorized into the same universality\nclass. \n\nIn conclusion, \nwe have studied the fixed-point structure of the $3D$ random-bond Ising\nmodels using the numerical DWRG method. \nWe have found in the $\\pm J$ Ising model, a random fixed point besides the\npure and multicritical ones along the ferromagnetic phase boundary. \nFurthermore, the observed random fixed point is found to be very close\nto that of the $3D$ non-frustrated dilute Ising models, while bare\nparameters in the model Hamiltonians differ entirely. \nThis fact strongly suggests that there exists a universal fixed point\ncharacterizing the $3D$ disordered ferromagnetic Ising model\nirrespective of the type of disorder. \nThe present work is, to our knowledge, the first of its kind performed\nfor determining the fixed-point structure of a $3D$ random spin system. \nWe consider that the present numerical RG approach is quite useful\nfor understanding the universality class of random spin systems,\nincluding spin glasses. \n\n\nThe author would like to thank H.~Takayama, K.~Nemoto and M.~Itakura for\nhelpful discussions. \nNumerical simulations have been performed mainly on Compaq and SGI\nworkstations at the Supercomputer Center, Institute of Solid State\nPhysics, University of Tokyo. \nPart of the simulations has been performed on Fujitsu VPP500 at the\nSupercomputer Center, ISSP, University of Tokyo and Hitachi SR2201 at\nthe Supercomputer Center, University of Tokyo. \nThe present work is supported by a Grant-in-Aid for the Encouragement of\nYoung Scientists from the Ministry of Education (No. 11740220),\nScience, Sports and Culture of Japan. \n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5\n\\begin{thebibliography}{99}\n\\bibitem{Harris}\nA.~B.~Harris: J.~Phys.~C {\\bf 7} (1974) 1671.\n \\bibitem{Experiment1}\nP.~W.~Mitchell, R.~A.~Cowley, H.~Yoshizawa, P.~B\\\"oni, Y.~J.~Uemura and\n\tR.~J.~Birgeneau : Phys.~Rev.~B {\\bf 34} (1986) 4719. \n \\bibitem{Experiment2}\nR.~J.~Birgeneau, R.~A.~Cowley, G.~Shirane, H.~Yoshizawa, D.~P.~Belanger, \n\tA.~R.~King and V.~Jaccarino: Phys.~Rev.~B {\\bf 27} (1983) 6747. \n \\bibitem{Experiment3}\nP.~H.~Barret: Phys.~Rev.~B {\\bf 34} (1986) 3513. \n \\bibitem{GrinsteinLuther}\nG.~Grinstein and A.~Luther: Phys.~Rev.~B {\\bf 13} (1976) 1329.\n \\bibitem{Janssen}\nH.~K.~Janssen, K.~Oerding and E.~Sengespeick: \nJ.~Phys.~A {\\bf 28} (1995) 6073. \n \\bibitem{Heuer}\nH.~-O.~Heuer: J.~Phys.~A {\\bf 26} (1993) L333. \n\\bibitem{BFMMRG}\nH.~G.~Ballesteros, L.~A.~Fern\\'andez, V.~Mart\\'in-Mayor, \nA.~Mu\\~noz Sudupe, J.~J.~Ruiz-Lorenzo and G.~Parisi: Phys.~Rev.~B {\\bf\n\t58} (1998) 2740. \n\\bibitem{McMillan}\nW.~L.~McMillan: Phys.~Rev.~B {\\bf 29} (1984) 4026\n\\bibitem{McMillan2}\nW.~L.~McMillan: Phys.~Rev.B {\\bf 30} (1984) 476.\n\\bibitem{SGH}\nE.~S.~S\\o rensen, M.~J.~P.~Gingras and D.~A.~Huse:\n\tEuro.~Phys.~Lett. {\\bf 44} (1998) 504. \n\\bibitem{KH}\nK.~Hukushima: Phys.~Rev.~E {\\bf 60} (1999) 3606. \n\\bibitem{EMC}\nK.~Hukushima and K.~Nemoto: J.~Phys.~Soc.~Jpn. {\\bf 65} (1996) 1604. \n\n \\bibitem{Marinari}\nE.~Marinari, G.~Parisi and J.~J.~Ruiz-Lorenzo: {\\it Numerical\n\tSimulations of Spin Glass Systems}, in ``{\\it Spin Glasses and\n\tRandom Fields}'', ed. A.~P.~Young (World Scientific,\n\tSingapore, 1997), p.~59.\n\\bibitem{non-eq}\nY.~Ozeki and N.~Ito: J.~Phys.~A {\\bf 31} (1998) 5451.\n \\bibitem{Nishimori}\nH.~Hishimori: Prog.~Theor.~Phys. {\\bf 66} (1981) 1169. \n\\bibitem{DH}\nP.~Le~Doussal and A.~B.~Harris: \nPhys.~Rev.~Lett. {\\bf 61} (1988) 625. \n\\bibitem{DH2}\nP.~Le~Doussal and A.~B.~Harris: Phys.~Rev.~B {\\bf 40} (1989) 9249. \n\\bibitem{HTE}\nR.~R.~P.~Singh and J.~Adler: Phys.~Rev.~B {\\bf 54} (1996) 364. \n\\bibitem{HTE2}\nR.~R.~P.~Singh: Phys.~Rev.~Lett. {\\bf 67} (1991) 899.\n\\bibitem{ON}\nY.~Ozeki and H.~Nishimori: J.~Phys.~Soc.~Jpn. {\\bf 56} (1987) 1568.\n \\bibitem{non-eq2}\nN.~Ito, Y.~Ozeki and H.~Kitatani: J. Phys. Soc. Jpn. \n\t{\\bf 68} (1998) 803.\n \\bibitem{KY}\nN.~Kawashima and A.~P.~Young: Phys.~Rev.~B {\\bf 53} (1996) R484. \n\n \\bibitem{Nash}\nA.~E.~Nash, C.~A.~Ramos and V.~Jaccarino: Phys.~Rev.~B {\\bf 47} (1993) 5805. \n\\end{thebibliography}\n\n\\clearpage\n\n\\end{document}\n\n"
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{
"name": "cond-mat0002089.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{Harris}\nA.~B.~Harris: J.~Phys.~C {\\bf 7} (1974) 1671.\n \\bibitem{Experiment1}\nP.~W.~Mitchell, R.~A.~Cowley, H.~Yoshizawa, P.~B\\\"oni, Y.~J.~Uemura and\n\tR.~J.~Birgeneau : Phys.~Rev.~B {\\bf 34} (1986) 4719. \n \\bibitem{Experiment2}\nR.~J.~Birgeneau, R.~A.~Cowley, G.~Shirane, H.~Yoshizawa, D.~P.~Belanger, \n\tA.~R.~King and V.~Jaccarino: Phys.~Rev.~B {\\bf 27} (1983) 6747. \n \\bibitem{Experiment3}\nP.~H.~Barret: Phys.~Rev.~B {\\bf 34} (1986) 3513. \n \\bibitem{GrinsteinLuther}\nG.~Grinstein and A.~Luther: Phys.~Rev.~B {\\bf 13} (1976) 1329.\n \\bibitem{Janssen}\nH.~K.~Janssen, K.~Oerding and E.~Sengespeick: \nJ.~Phys.~A {\\bf 28} (1995) 6073. \n \\bibitem{Heuer}\nH.~-O.~Heuer: J.~Phys.~A {\\bf 26} (1993) L333. \n\\bibitem{BFMMRG}\nH.~G.~Ballesteros, L.~A.~Fern\\'andez, V.~Mart\\'in-Mayor, \nA.~Mu\\~noz Sudupe, J.~J.~Ruiz-Lorenzo and G.~Parisi: Phys.~Rev.~B {\\bf\n\t58} (1998) 2740. \n\\bibitem{McMillan}\nW.~L.~McMillan: Phys.~Rev.~B {\\bf 29} (1984) 4026\n\\bibitem{McMillan2}\nW.~L.~McMillan: Phys.~Rev.B {\\bf 30} (1984) 476.\n\\bibitem{SGH}\nE.~S.~S\\o rensen, M.~J.~P.~Gingras and D.~A.~Huse:\n\tEuro.~Phys.~Lett. {\\bf 44} (1998) 504. \n\\bibitem{KH}\nK.~Hukushima: Phys.~Rev.~E {\\bf 60} (1999) 3606. \n\\bibitem{EMC}\nK.~Hukushima and K.~Nemoto: J.~Phys.~Soc.~Jpn. {\\bf 65} (1996) 1604. \n\n \\bibitem{Marinari}\nE.~Marinari, G.~Parisi and J.~J.~Ruiz-Lorenzo: {\\it Numerical\n\tSimulations of Spin Glass Systems}, in ``{\\it Spin Glasses and\n\tRandom Fields}'', ed. A.~P.~Young (World Scientific,\n\tSingapore, 1997), p.~59.\n\\bibitem{non-eq}\nY.~Ozeki and N.~Ito: J.~Phys.~A {\\bf 31} (1998) 5451.\n \\bibitem{Nishimori}\nH.~Hishimori: Prog.~Theor.~Phys. {\\bf 66} (1981) 1169. \n\\bibitem{DH}\nP.~Le~Doussal and A.~B.~Harris: \nPhys.~Rev.~Lett. {\\bf 61} (1988) 625. \n\\bibitem{DH2}\nP.~Le~Doussal and A.~B.~Harris: Phys.~Rev.~B {\\bf 40} (1989) 9249. \n\\bibitem{HTE}\nR.~R.~P.~Singh and J.~Adler: Phys.~Rev.~B {\\bf 54} (1996) 364. \n\\bibitem{HTE2}\nR.~R.~P.~Singh: Phys.~Rev.~Lett. {\\bf 67} (1991) 899.\n\\bibitem{ON}\nY.~Ozeki and H.~Nishimori: J.~Phys.~Soc.~Jpn. {\\bf 56} (1987) 1568.\n \\bibitem{non-eq2}\nN.~Ito, Y.~Ozeki and H.~Kitatani: J. Phys. Soc. Jpn. \n\t{\\bf 68} (1998) 803.\n \\bibitem{KY}\nN.~Kawashima and A.~P.~Young: Phys.~Rev.~B {\\bf 53} (1996) R484. \n\n \\bibitem{Nash}\nA.~E.~Nash, C.~A.~Ramos and V.~Jaccarino: Phys.~Rev.~B {\\bf 47} (1993) 5805. \n\\end{thebibliography}"
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cond-mat0002091
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On the Interplay of Disorder and Correlations
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I address here the question of the mutual interplay of strong correlations and disorder in the system. I consider random version of the Hubbard model. Diagonal randomness is introduced {via} random on-site energies and treated by the coherent potential approximation. Strong, short ranged, electron - electron interactions are described by the slave boson technique and found to induce additional disorder in the system. As an example I calculate the density of states of the random interacting binary alloy and compare it with that for non-interacting system. PACS numbers: 71.10.Fd, 71.23.-k, 71.55.Ak.
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"name": "Mos99.tex",
"string": "%% \n\\documentstyle[11pt,twoside,jltp,epsf]{article}\n\\title{On the Interplay of Disorder and Correlations}\n\\author{Karol I. Wysoki\\'nski \n\\address{Institute of Physics, M.\nCurie-Sk\\l{}odowska University, ul. Radziszewskiego 10, \nPl-20-031 Lublin, POLAND}}\n\\runninghead{K.I. Wysoki\\'nski}{Interplay of Disorder and\nCorrelations} \n\\begin{document} \n\\begin{abstract}\nI address here the question of the mutual interplay of strong\ncorrelations and disorder in the system. I consider random version\nof the Hubbard model. Diagonal randomness is introduced {\\it via}\nrandom on-site energies and treated by the coherent potential\napproximation. Strong, short ranged, electron - electron\ninteractions are described by the slave boson technique and found \nto induce additional disorder in the system. As an example I\ncalculate the density of states of the random interacting binary\nalloy and compare it with that for non-interacting system.\nPACS numbers: 71.10.Fd, 71.23.-k, 71.55.Ak.\n\\end{abstract}\n\\maketitle\n%%\n\\section{INTRODUCTION}\nThe list of materials in which carriers are strongly interacting\nwith each other and scatter by random impurities is quite long. It\ncomprises {\\it inter alia} high temperature superconducting\noxides \\cite{HTS} and various heavy fermion alloys \\cite{Kondo}.\nTwo models are usually used to study such system. It is either\nHubbard or Anderson model suitably extended to allow for the\ndescription of real materials. Here the single band Hubbard model\nhas been used, as it is the simplest model of correlated system. \nThe disorder is introduced into the model by allowing for\nfluctuations of the local on-site energies $\\varepsilon_{i}$. \nThe main purpose of this work is to study the interplay of disorder\nand correlations in the Hubbard model. I shall present analytical\nand numerical calculations indicating that correlations induce\nadditional disorder in the system. {\\it Mutatis mutandis} disorder\nin the system affects the parameters of the Mott-Hubbard \nmetal-insulator transition. In the recent studies of weakly\ninteracting disordered systems it has been found that due to\ndisorder the interactions scale to strong coupling limit \\cite{DB}.\nHere to treat many particle aspect of the problem we use the slave\nboson technique, which is known to be qualitatively valid for all \nstrength of interaction. In particular this method reproduces the\nresults of the Gutzwiller approximation at the saddle-point level\n\\cite{Br}. \n\n\\section{THE MODEL AND APPROACH}\n\nWe start here with random version of the Hubbard model \n\n\\begin{equation} \nH=\\sum_{ij\\sigma }t_{ij}c_{i\\sigma }^{+}c_{j\\sigma }+\\sum_{i\\sigma \n}(\\varepsilon_i -\\mu )c_{i\\sigma }^{+}c_{i\\sigma\n}+U\\sum_{i}n_{i\\uparrow }n_{i\\downarrow }\\,. \n\\end{equation} \nThe meaning of symbols is standard. The first term describes the hopping of \ncarriers through the crystal specified by lattice sites $i$, $j$. \n$\\varepsilon _{i}$ denotes the fluctuating site energy - it\nintroduces disorder into the model. $U$ is the repulsion between two\nopposite spin electrons occupying the same site. \nIn the slave boson technique\\cite{KotRuck} one introduces 4 auxiliary boson \nfields: $\\hat e_i$, $\\hat s_{i\\sigma}$, $\\hat d_i$ such that $\\hat e^+_i\\hat \ne_i$, $\\hat s^+_{i\\sigma}\\hat s_{i\\sigma}$, $\\hat d^+_i\\hat d_i$ project \nonto the empty, singly occupied by $\\sigma$ and doubly occupied site $i$. \nThe Hamiltonian (1) is in this enlarged Hilbert space written\nas \n\\begin{equation} \nH = \\sum_{ij\\sigma} t_{ij} \\hat z^+_{i\\sigma}\\hat z_{j\\sigma} \nc^+_{i\\sigma}c_{j\\sigma} + U\\sum_i \\hat d^+_i\\hat d_i +\n\\sum_{i\\sigma} (\\varepsilon_i - \\mu) c^+_{i\\sigma}c_{i\\sigma}\\,, \n\\end{equation} \nwhere $\\hat z_{i\\sigma} = (\\hat 1 - \\hat d^+_i\\hat d_i - \\hat s^+_{i\\sigma} \n\\hat s_{i\\sigma})^{-1/2} (\\hat e^+_i\\hat s_{i\\sigma} + \\hat s^+_{i-\\sigma} \n\\hat d_i) \\cdot (\\hat 1 - \\hat e^+_i\\hat e_i - \\hat s^+_{i-\\sigma} \\hat \ns_{i-\\sigma})^{-1/2}$. Equation (2) is strictly equivalent to (1), when the \nconstraints \n\\begin{eqnarray} \n&& \\hat e^+_i\\hat e_i + \\sum_\\sigma \\hat s^+_{i\\sigma}\\hat s_{i\\sigma} + \n\\hat d^+_i\\hat d_i = 1 \\nonumber \\\\ \n&& c^+_{i\\sigma}c_{i\\sigma} = \\hat p^+_{i\\sigma}\\hat p_{i\\sigma} + \\hat \nd^+_i \\hat d_i;~~~~~~\\sigma = \\uparrow, \\downarrow \n\\end{eqnarray} \nare fulfilled. \nIn disordered system and at $T$ = 0 K, the mean field approach to slave \nbosons can be formulated by suitably generalizing the clean limit. One \nintroduces the constraints into Hamiltonian with help of Langrange \nmultipliers $\\lambda _{i}^{(1)}$ and $\\lambda _{i\\sigma }^{(2)}$ and \nreplaces boson operators $\\hat{e}_{i}$, $\\hat{s}_{i\\sigma }$, $\\hat{d}_{i}$ \nby classical, site dependent, amplitudes $e_{i}$, $s_{i\\sigma }$, $d_{i}$. \nThese amplitudes are calculated from the, configuration dependent, ground \nstate energy $E_{GS}=\\langle H\\rangle $ by minimizing $E_{GS}$ with respect \nto all seven parameters $e_{i}$, $s_{i\\sigma }$, $d_{i}$, $\\lambda \n_{i}^{(1)} $, $\\lambda _{i\\sigma }^{(2)}$. As a result one gets three \nconstraints: $e_{i}^{2}+\\sum_{\\sigma }s_{i\\sigma }^{2}+d_{i}^{2}=1$, $ \n\\langle c_{i\\sigma }^{+}c_{i\\sigma }\\rangle =s_{i\\sigma }^{2}+d_{i}^{2}$ and \nfour additional equations which read \n\\begin{eqnarray} \n&&\\lambda _{i}^{(1)}e_{i}=-{\\frac{1}{\\xi _{i}}}{\\frac{\\partial \\xi _{i}}{\n\\partial e_{i}}}\\,{\\rm Re}\\left( \\sum_{j\\sigma }t_{ij}\\xi _{i}\\langle \nc_{i\\sigma }^{+}c_{j\\sigma }\\rangle \\xi _{i}\\right) \\nonumber \\\\ \n&&(\\lambda _{i}^{(1)}-\\lambda _{i\\sigma }^{(2)})s_{i\\sigma }=-{\\frac{1}{\\xi \n_{i}}}{\\frac{\\partial \\xi _{i}}{\\partial s_{i\\sigma }}}\\,{\\rm Re}\\left( \n\\sum_{j}t_{ij}\\xi _{i}\\langle c_{i\\sigma }^{+}c_{j\\sigma }\\rangle \\xi \n_{i}\\right) \\nonumber \\\\ \n&&\\left( U+\\lambda _{i}^{(1)}-\\sum_{\\sigma }\\lambda _{i\\sigma }^{(2)}\\right) \nd_{i}=-{\\frac{1}{\\xi _{i}}}{\\frac{\\partial \\xi _{i}}{\\partial d_{i}}}\\left( \n{\\rm Re}\\sum_{j\\sigma }t_{ij}\\xi _{i}\\langle c_{i\\sigma }^{+}c_{j\\sigma \n}\\rangle \\xi _{j}\\right) \n\\end{eqnarray} \nwhere $\\xi _{i}=(1-d_{i}^{2}-s_{i\\sigma }^{2})^{-1/2}(e_{i}s_{i\\sigma \n}+s_{i-\\sigma }d_{i})(1-e_{i}^{2}-s_{i-\\sigma }^{2})^{-1/2}$. \n \nThe important next step is connected with calculation of $\\langle c_{i\\sigma \n}^{+}c_{i\\sigma }\\rangle $ and $E_{i\\sigma }=\\sum_{j}t_{ij}\\xi _{i}\\langle \nc_{i\\sigma }^{+}c_{j\\sigma }\\rangle \\xi _{i}$ in the presence of disorder. \nIn principle these quantities do depend on the particular distribution of \nall impurities (i.e. distribution of site energies $\\varepsilon _{i}$). We \nshall calculate them from the corresponding Green's function. We use a \nversion\\cite{KIW} of the coherent potential approximation (CPA) to\ncalculate (averaged and conditionally averaged) Green's functions.\nTo this end we rewrite the mean field Hamiltonian in the form \n\\begin{equation} \n\\tilde{H}=\\sum_{ij\\sigma }t_{ij}\\xi _{i}\\xi _{j}c_{i\\sigma }^{+}c_{j\\sigma \n}+\\sum_{i\\sigma }(\\varepsilon _{i}-\\mu +\\lambda _{i\\sigma }^{(2)})c_{i\\sigma \n}^{+}c_{i\\sigma }+{\\rm const}\\,. \n\\end{equation} \nNote, that Hamiltonian (5) contains not only the site dependent\nparameters $ \\varepsilon _{i}$ but due to correlations there appear\nalso $\\lambda _{i\\sigma }^{(2)}$ which are the additional source\nof, diagonal and spin dependent, disorder. Besides that, the\nparameters $\\xi _{i}$, $\\xi _{j}$ which vary from site to site make\neffective hopings $\\tilde{t}_{ij}=t_{ij}\\xi _{i}\\xi _{j}$ random\nquantities. This off-diagonal (in Wannier space) disorder is\nparticularly important. Fortunately multiplicative dependence of\neffective hopping on the kind of atoms at sites $i$ and $j$ is\neasy to handle within CPA. \n\nThe procedure is standard\\cite{KIW} and one gets the following equation for \nthe density of states in paramagnetic phase \n\\begin{equation} \nD(\\varepsilon )=-{\\frac{1}{\\pi }}\\,{\\rm Im}\\langle {\\frac{\\xi \n_{i}^{2}F[\\Sigma (\\varepsilon ^{+})]}{1-\\left[ \\Sigma (\\varepsilon \n^{+})-(\\varepsilon -\\varepsilon _{i}+\\mu -\\lambda _{i}^{(2)})/\\xi _{i}^{2}\n\\right] \\,F[\\Sigma (\\varepsilon )]}}\\rangle _{{\\rm imp}} \n\\end{equation} \nand the coherent potential $\\Sigma (\\varepsilon )$ is determined as a \nsolution of equations \n\\begin{eqnarray} \n&&\\langle {\\frac{\\Sigma (z)-(z-\\varepsilon _{i}+\\mu -\\lambda _{i}^{(2)})/\\xi \n_{i}^{2}}{1-\\left[ \\Sigma (z)-(z-\\varepsilon _{i}+\\mu -\\lambda \n_{i}^{(2)})/\\xi _{i}^{2}\\right] \\,F[\\Sigma (z)]}}\\rangle _{{\\rm\nimp}}=0 \\nonumber \\\\ \n&&F[\\Sigma (z)]={\\frac{1}{N}}\\sum_{\\vec{k}}{\\frac{1}{\\Sigma (z)-\\varepsilon (\n\\vec{k})}}={\\frac{1}{N}}\\sum_{\\vec{k}}\\,\\bar{G}_{\\vec{k}}(z) \n\\end{eqnarray} \nHere $\\langle \\cdots \\rangle _{{\\rm imp}}$ means averaging over disorder and \n$\\varepsilon (\\vec{k})={\\frac{1}{N}}\\sum_{ij}t_{ij}\\,e^{-i\\vec{k}(\\vec{R} \n_{i}-\\vec{R}_{j})}$ is electron spectrum in host (clean) material. \n \nTo solve equations (4) one still has to calculate the quantities\n$E_{i\\sigma }=\\sum_{j}t_{ij}\\xi _{i}\\langle c_{i\\sigma\n}^{+}c_{j\\sigma }\\rangle \\xi _{j}$ , which in general depend on\nthe configuration of all impurities in the system. They can be\ncalculated from the knowledge of the CPA Green's functions in the\nfollowing way. We assume that site $i$ is described by actual \nparameters {\\it i.e.} $\\varepsilon _{i}$, $\\lambda _{i\\sigma}^{(2)}$,\n$\\xi _{i}$, while all other sites in the system are replaced by\neffective ones described by the coherent potential $\\Sigma (z)$ and\nthe Green's functions $\\bar{G}(z)$. Then it is easy to find that\n{\\it e.g}. \n \\begin{equation} E_{i\\sigma }=\\int d\\omega\n{\\frac{1}{e^{\\beta \\omega }+1}}\\sum_{j}t_{ij} \\left(\n-{\\frac{1}{\\pi }}\\right) \\,{\\rm Im}\\tilde{G}_{ij\\sigma }(\\omega\n+i0) \\end{equation} \n\\noindent and \\cite{Economou} \n\\begin{equation} \n\\tilde{G}_{ij\\sigma }(z)=\\bar{G}_{ij\\sigma }(z){\\frac{1}{1-\\tilde{\\varepsilon \n}_{i\\sigma }(z)\\bar{G}_{ii\\sigma }(z)}} \n\\end{equation} \nHere $\\tilde{\\varepsilon}_{i\\sigma }(z)=(z-\\varepsilon _{i}+\\mu -\\lambda \n_{i\\sigma }^{(2)})/\\xi _{i}^{2}$, and by definition $\\bar{G}_{ii\\sigma \n}(z)\\equiv F[\\Sigma (z)]$ and $\\bar{G}_{ij\\sigma }(z)=\\frac{1}{N}\\sum_{\\vec{k\n}}\\bar{G}_{\\vec k}(z)e^{i\\vec{k}(\\vec{R}_{i}-\\vec{R}_{j})}$ \\ This\ncloses the system of equations to be solved in a self-consistent\nmanner. In the next section we show numerical calculations of the\neffect of correlations on the density of states in disordered\nalloy. \n\n\\section{NUMERICAL RESULTS AND DISCUSSION} \n \nFor the purpose of numerical illustration of the general approach sketched \nin previous section we calculate here the density of states (DOS) of perfect \ninteracting, random noninteracting and random interacting systems. For the \npurpose of numerical analysis we shall assume $U=\\infty $ limit and {\\it bcc} \ncrystal structure, which leads to the following canonical\ntight-binding spectrum of noninteracting carriers in clean material \n \n\\begin{equation} \n\\varepsilon _{\\vec{k}}=-8t\\cos (k_{x}a)\\cos (k_{y}a)\\cos (k_{z}a). \n\\end{equation} \nUsing t=0.0625$eV$ leads to the noninteracting system bandwidth $W=1eV$. In \nthe following all energies and frequencies are expressed in $eV$. The \ndensity of states corresponding to this spectrum is well known. It possesses \nVan Hove singularity in the middle of the band, which extends from $-8t$ to $ \n+8t$. The spectrum $\\varepsilon _{\\vec{k}}$ of a clean but interacting \nsystem in the $U=\\infty $ limit is replaced by $\\xi ^{2}\\varepsilon _{\\vec{k} \n}-\\lambda ^{(2)}$. The corresponding density of states is thus of the same \nform but placed in the energy window $[-8t(1-n)-\\lambda \n^{(2)},+8t(1-n)-\\lambda ^{(2)}]$ and scaled by the factor $(1-n)^{-1}$. Note \nthat in this $U=\\infty $ limit the upper Hubbard band is pushed to infinity \nand not visible. \n \n\\begin{figure}[tbp] \n\\vspace{-1.0in}\n{\\epsfxsize=2.4in \\epsfbox{fig1a.eps}}\n\\hspace{0.12in}\n{\\epsfxsize=2.4in\\epsfbox{fig1b.eps}}\n\\caption{Comparison\nof the DOS of (a) disordered noninteracting $ A_{1-x}B_{x} $ alloy\nwith $x=0.5$, $\\protect\\varepsilon _{A}=0$, $\\protect \\varepsilon\n_{B}=-0.3eV$ with that calculated for the same alloy but with \ninteracting carriers (b). Carrier density $n=0.4$. } \n\\end{figure}\n\nThe changes of the spectrum due to correlations in disordered $A_{1-x}B_{x}$ \nalloy are illustrated in figure (1) where we show the averaged $D(E)$ and \nconditionally averaged densities of states $D_{A}(E)$ and $D_{B}(E)$ \\ ($\nx=0.5$, $\\varepsilon _{A}=0,\\varepsilon _{B}=-0.3eV$) without (Fig. 2a) and \nwith (Fig. 2b) electron correlations. The carrier concentration $n=0.4$. \nNoninteracting alloy DOS is symmetric. We observe strong asymmetry, the \nopening of the real gap in the spectrum of interacting carriers and the \nappreciable increase of the density of states at the fermi level (taken as \nE=0 in the figure). \n \nIn conclusion, we have shown that interplay of disorder and\ncorrelations leads to strong renormalisation of the electron\nspectrum. Let us stress that all other properties of such systems\nwill also be strongly affected by interaction induced disorder. The\ncalculations of {\\it dc} and {\\it ac} transport properties are in\nprogress. \n\\begin{thebibliography}{9} \n\\bibitem{HTS} K. Ishida {\\it et al.}, J.Phys.Soc. Jpn. {\\bf 62}, 2803 \n(1993); Gang Xiao {\\it et al}., Phys. Rev. {\\bf B42}, 8752 (1990); \nB.Beschoten{\\it \\ et al}., Phys. Rev. Lett. {\\bf 77}, 1837 (1996). \n \n\\bibitem{Kondo} B. Andraka and G. R. Stewart, Phys. Rev. {\\bf B47}, 3208 \n(1993); O.~O.~Bernal {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 2023 (1995); \nB.~Maple {\\it et al.}, J. Low Temp. Phys. {\\bf 99}, 223 (1995). \n \n\\bibitem{DB} D. Belitz and T. R. Kirkpatrick, Rev. Mod. Physics {\\bf 66}, \n261 (1994). \n \n\\bibitem{Br} W. F. Brinkman and T. M. Rice, Phys. Rev. B {\\bf 2}, 4302 \n(1970). \n \n\\bibitem{KotRuck} G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. {\\bf 57 \n}, 1362 (1986). \n \n\\bibitem{KIW} K. I. Wysoki\\'{n}ski, J. Phys. {\\bf C11}, 291 (1978). \n \n\\bibitem{Economou} E. N. Economou, {\\it Green's Function in Quantum Physics}\n, Springer Verlag, Berlin 1983, ch. 6. \n\\end{thebibliography} \n \n\\end{document} \n"
}
] |
[
{
"name": "cond-mat0002091.extracted_bib",
"string": "\\begin{thebibliography}{9} \n\\bibitem{HTS} K. Ishida {\\it et al.}, J.Phys.Soc. Jpn. {\\bf 62}, 2803 \n(1993); Gang Xiao {\\it et al}., Phys. Rev. {\\bf B42}, 8752 (1990); \nB.Beschoten{\\it \\ et al}., Phys. Rev. Lett. {\\bf 77}, 1837 (1996). \n \n\\bibitem{Kondo} B. Andraka and G. R. Stewart, Phys. Rev. {\\bf B47}, 3208 \n(1993); O.~O.~Bernal {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 2023 (1995); \nB.~Maple {\\it et al.}, J. Low Temp. Phys. {\\bf 99}, 223 (1995). \n \n\\bibitem{DB} D. Belitz and T. R. Kirkpatrick, Rev. Mod. Physics {\\bf 66}, \n261 (1994). \n \n\\bibitem{Br} W. F. Brinkman and T. M. Rice, Phys. Rev. B {\\bf 2}, 4302 \n(1970). \n \n\\bibitem{KotRuck} G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. {\\bf 57 \n}, 1362 (1986). \n \n\\bibitem{KIW} K. I. Wysoki\\'{n}ski, J. Phys. {\\bf C11}, 291 (1978). \n \n\\bibitem{Economou} E. N. Economou, {\\it Green's Function in Quantum Physics}\n, Springer Verlag, Berlin 1983, ch. 6. \n\\end{thebibliography}"
}
] |
cond-mat0002093
|
Statistical mechanics approach to the phase unwrapping problem
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[
{
"author": "Sebastiano Stramaglia"
}
] |
The use of Mean-Field theory to unwrap principal phase patterns has been recently proposed. In this paper we generalize the Mean-Field approach to process phase patterns with arbitrary degree of undersampling. The phase unwrapping problem is formulated as that of finding the ground state of a locally constrained, finite size, spin-L Ising model under a non-uniform magnetic field. The optimization problem is solved by the Mean-Field Annealing technique. Synthetic experiments show the effectiveness of the proposed algorithm. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RIEMPIRE
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[
{
"name": "mfaart.tex",
"string": "% Upper-case A B C D E F G H I J K L M N O P Q R S T U V W X Y Z\n% Lower-case a b c d e f g h i j k l m n o p q r s t u v w x y z\n% Digits 0 1 2 3 4 5 6 7 8 9\n% Exclamation ! Double quote \" Hash (number) #\n% Dollar $ Percent % Ampersand &\n% Acute accent ' Left paren ( Right paren )\n% Asterisk * Plus + Comma ,\n% Minus - Point . Solidus /\n% Colon : Semicolon ; Less than <\n% Equals = Greater than > Question mark ?\n% At @ Left bracket [ Backslash \\\n% Right bracket ] Circumflex ^ Underscore _\n% Grave accent ` Left brace { Vertical bar |\n% Right brace } Tilde ~\n\n\n\\documentclass{elsart}\n%\\usepackage[active]{srcltx} % SRC Specials: DVI [Inverse] Search\n\\usepackage{graphicx}\n\\usepackage{amsmath}\n\\newcommand{\\norm}[1]{\\left\\Vert#1\\right\\Vert}\n\\newcommand{\\abs}[1]{\\left\\vert#1\\right\\vert}\n\\newcommand{\\set}[1]{\\left\\{#1\\right\\}}\n\\newcommand{\\Real}{\\mathbb R}\n\\newcommand{\\eps}{\\varepsilon}\n\\newcommand{\\To}{\\longrightarrow}\n\\newcommand{\\BX}{\\mathbf{B}(X)}\n\\newcommand{\\A}{\\mathcal{A}}\n\\newcommand{\\grad}[1] {\\ensuremath{\\nabla\\!_{\\!#1}\\,}}\n\\newcommand{\\estgrad}[1] {\\ensuremath{\\widetilde{\\nabla}\\!_{\\!#1}\\,}}\n\\newcommand{\\for}[1]{\\textsl{#1}}\n\\newcommand{\\qt}[1]{``#1''}\n\\newcommand{\\qqt}[1]{`#1'}\n\\newcommand{\\avg}[1]{\\left\\langle#1\\right\\rangle}\n\\newcommand{\\keyw}[1]{{\\bf #1}}\n\\renewcommand{\\vec}[1]{\\ensuremath{\\mathbf{#1}}}\n\n\n\\begin{document}\n\\runauthor{Stramaglia et al.}\n\\begin{frontmatter}\n\\title{Statistical mechanics approach to the phase unwrapping problem\n}\n\\author[Dip]{Sebastiano Stramaglia}\n\\author[INFM,Dip]{Alberto Refice}\n\\author[Dip]{Luciano Guerriero}\n\n\\address[Dip]{Dipartimento Interateneo di Fisica, Via Amendola, 173, Bari (Italy)}\n\\address[INFM]{Istituto Nazionale di Fisica della Materia (INFM), Sez. G, Bari (Italy)}\n\n\n\n\\begin{abstract}\nThe use of Mean-Field theory to unwrap principal phase patterns has \nbeen recently proposed. In this paper we generalize the Mean-Field \napproach to process phase patterns with arbitrary degree of \nundersampling. The phase unwrapping problem is formulated as that of \nfinding the ground state of a locally constrained, finite size, spin-L \nIsing model under a non-uniform magnetic field. The optimization \nproblem is solved by the Mean-Field Annealing technique. Synthetic \nexperiments show the effectiveness of the proposed algorithm. \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RIEMPIRE\n\\end{abstract}\n\n\\begin{keyword}\n Interferometry, Phase Unwrapping, Non Convex Optimization, Mean \n Field Theory\n\\end{keyword}\n\n\n\\end{frontmatter}\n\n\\emph{PACS}: 95.75.Kk , 95.75.-z , 45.10.Db \n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe determination of the absolute phase from a fringe pattern is an\nimportant problem that finds applications in many areas: homomorphic\nsignal processing~\\cite{Oppenheim-75}, solid state\nphysics~\\cite{Hjalmarson-85}, holographic\ninterferometry~\\cite{Nakadate-85}, adaptive or compensated\noptics~\\cite{Fried-77}, magnetic resonance imaging~\\cite{Ching-92} and\nsynthetic aperture radar interferometry~\\cite{Zebker-86}.\n\nIn all these applications one obtains a two-dimensional fringe pattern\nwhose spatially-varying phase is related to the physical quantity to be\nmeasured. The computation of phase by any inverse trigonometric\nfunction (e.g.\\ arctangent) provides only principal phase values, which\nlie between $\\pm \\pi$ radians. The process of phase unwrapping (PU), \ni.e.\\ the addition of a proper integer multiple of $2 \\pi$ to all the \npixels, must be carried out before the physical quantity can be \nreconstructed from the phase distributions. Since many possible \nabsolute phase fields are compatible with a given fringe pattern, phase \nunwrapping is ill-posed in a mathematical sense: \nHadamard~\\cite{Hadamard-02} defined a mathematical problem to be \nwell-posed if a unique solution exists that depends continuously on the \ndata; in this case, the uniqueness requirement is violated. \n\nIll-posed problems arise frequently in many areas of science and \nengineering; well-known examples are analytic continuation, the Cauchy \nproblem for differential equations, computer tomography, and many \nproblems in image processing and machine vision that involve the \nreconstruction of images from noisy data. \n\nThe fact that a problem is not well-posed does not mean that it cannot \nbe solved: rather, in order to be solved it must be first \n\\emph{regularized} by introducing additional constraints (prior \nknowledge) about the behaviour of the solution. Variational \nregularization corresponds to modeling the physical constraints of the \nproblem by a suitable functional; the solution is then sought as the \nminimizer of this functional. \n\nIn the last years, an increasing interest has been devoted to adapt \nmethods from Statistical Mechanics to nonconvex optimization problems \narising from the variational regularization of ill-posed problems. \nGeman and Geman~\\cite{Geman-84} suggested that the Ising model is \napplicable to image restoration through the Bayesian formalism. This \nproblem corresponds to searching the ground state of a finite-size \nIsing model under a non-uniform external field. Geman and Geman applied \nthis to the recovery of corrupted images by using simulated annealing \nof a spin-S Ising model. After that, Gidas~\\cite{Gidas-89} proposed a \nnew method based on a combination of the renormalization group \ntechnique and the simulated annealing procedure; then, \nZhang~\\cite{Zhang-92} introduced Mean-Field Annealing to treat the \nimage reconstruction problem, while Tanaka and \nMorita~\\cite{Tanaka-95,Morita-96} applied the cluster variation method. \nMethods of statistical mechanics have also been used to study \ncombinatorial optimization problems (see, e.g.,~\\cite{Ray-97} and \nreferences therein). \n\nIn a couple of recent papers, the phase unwrapping problem was handled \nby methods of Statistical Mechanics. Simulated annealing was applied \nin~\\cite{Guerriero-98}. In~\\cite{Stramaglia-99} the problem is solved \nby the Mean-Field Annealing (MFA) technique; PU is formulated as a \nconstrained optimization problem for the field of integer corrections \nto be added to the wrapped phase gradient in order to recover the true \nphase gradient, with the cost function consisting of second order \ndifferences, and measuring the smoothness of the reconstructed phase \nfield. This is equivalent~\\cite{Stramaglia-99} to finding the ground \nstate of a locally-constrained ferromagnetic spin-1 Ising model under a \nnon-uniform external field. The optimization problem is then solved by \nMFA and consistent solutions are found in difficult situations \nresulting from noise and undersampling. \n\nMean Field Annealing is closely related to Simulated \nAnnealing~\\cite{Geman-84}. Both approaches formulate the optimization \nproblem in terms of minimizing a cost function and defining a \ncorresponding Gibbs distribution. Simulated Annealing then proceeds by \nsampling the Gibbs probability distribution as the temperature is \nreduced to zero, whereas MFA attempts to track an approximation of the \nmean of the same distribution. \n\nThe algorithm described in~\\cite{Stramaglia-99} was constructed under \nthe assumption that the possible values for the correction field were \nrestricted to belong to the set $\\{ -1, 0, 1 \\}$. In this paper we \ngeneralize the MFA algorithm to the case of a set $\\{-L, \\ldots, L\\}$, \nwith $L$ an arbitrary integer. The corresponding statistical system is \na locally-constrained spin-L Ising model. \n\nThe present generalization allows the processing of input phase\npatterns with arbitrary degree of undersampling; our experiments on\nsynthetic phase fields show the effectiveness of the proposed\nalgorithm.\n\nThe paper is organized as follows. In sect.~\\ref{sec:pu} the phase \nunwrapping problem is introduced and its ill-posedness is highlighted. \nIn sect.~\\ref{sec:alg} the deterministic MFA algorithm is described in \ndetail. Then, in sect.~\\ref{sec:exp} some experimental results on \nsimulated phase fields are presented. Some conclusions are then drawn \nin sect.~\\ref{sec:concl}. \n\n\n\\section{Phase Unwrapping Problem}\n\\label{sec:pu}\n\nWe briefly recall here the phase unwrapping terminology, and refer the\nreader to~\\cite{Ghiglia-98} for a complete discussion.\n\nGiven an absolute phase pattern $f(x,y)$ on a two-dimensional square \ngrid, what is actually measured is the wrapped phase field $g(x,y)$ \nwhich can be expressed in terms of the $f$ field through a wrapping \noperator, Wr, defined so that $g(x,y)$ always lies in the interval \n$[-\\pi,+\\pi)$: \n %\n \\begin{equation}\n\t g(x,y) = \\mathrm{Wr} [f(x,y)] = \\arg \\left\\{ \\exp [ \\mathrm{i}\n\t f(x,y)]\\right\\}.\n \\end{equation}\n %\n\nPhase unwrapping means recovering the absolute phase field $f$, which \nis usually related to the physical quantity to be measured, from the \nknowledge of the $g$ field. This can be done in practice by estimating \nthe absolute phase gradient from the wrapped phase field and \nintegrating it throughout the 2-D grid. This simple method is effective \nonly in absence of phase aliasing, i.e.\\ if the phase field is \ncorrectly sampled. \n\nIn fact, if the Nyquist condition:\n %\n \\begin{equation}\n\t \\abs{\\vec{\\nabla} f(x,y)} < \\pi,\n \\label{eq:CondNoAlias}\n \\end{equation}\n %\nwhere $\\nabla$ is the discrete gradient, is verified everywhere on the \ngrid, the absolute phase gradient is obtained by wrapping the gradient \nof the wrapped phase field, according to the formula: \n %\n \\begin{equation}\n\t \\vec{A}(x,y) = \\mathrm{Wr} [ \\vec{\\nabla} g (x,y)].\n \\end{equation}\n %\nAs mentioned, if condition~\\eqref{eq:CondNoAlias} is satisfied, one \nhas:\n %\n \\begin{equation}\n\t \\vec{\\nabla} f(x,y) = \\vec{A}(x,y),\n \\label{eq:noalias}\n \\end{equation}\n %\nand the $f$-pattern is obtained by integrating $\\vec{A}$ along any path\nconnecting all sites on the grid.\n\nThe Nyquist condition is often violated because of undersampling of the \nsignal from which the principal phase is extracted. This can result \neither from system noise, or from critical values of the slopes of the \nphysical surface which is analyzed through interferometry. For example, \nin the case of SAR interferometry, the surface is the portion of Earth \nimaged from the sensor (usually air- or satellite-borne), while noise \ncan arise from sensor thermal electronic motion, or from other sources \nof electronic signal disturbances. \n\nIf the Nyquist condition is not satisfied everywhere on the grid, then \nthe wrapped gradient $\\vec{A}$ of the wrapped phase field is not \nassured to equal the absolute phase gradient. In this case, a more \ngeneral relation must be written, rather than~\\eqref{eq:noalias}, i.e.: \n %\n \\begin{equation}\n\t \\vec{\\nabla} f(x,y) = \\vec{A}(x,y) + 2 \\pi \\vec{k}(x,y),\n \\label{eq:aliasgrad}\n \\end{equation}\n %\nwhere $\\vec{k}(x,y)$ is a vector field of integers. In this case, \nsolving PU amounts to finding the correct field $\\vec{k}$. \n\nPhase aliasing conditions imply that the integration of field $\\vec{A}$ \ndepends on the path. The sources of this nonconservative behaviour are \ndetectable by calculating the integral of the field $\\vec{A}$ over \nevery minimum closed path, i.e.\\ the 2$\\times$2 square having the site \n$(x,y)$ as a corner: \n %\n \\begin{equation}\n\t I(x,y) = \\frac{1}{2\\pi} \\bigl[ A_x(x,y) + A(y(x+1,y) - A_x(x,y+1) - A_y(x,y) \n\t \\bigr].\n \\label{eq:resdef}\n \\end{equation}\n %\nOne can show that the integral $I(x,y)$ will always have a value in the \nset $\\{-1, 0, 1\\}$. Locations with $I \\neq 0$ are called \\qt{residues}. \nIn presence of residues, the field $\\vec{A}$ is no more irrotational; \nthis causes the path-dependence of the integration step previously \ndescribed.\n\nTo restore the consistency of the phase gradient, then, the $\\vec{k}$ \nfield must satisfy the following consistency condition: \n %\n \\begin{equation}\n\t \\vec{\\nabla} \\times \\left[ \\vec{A}(x,y) + 2\\pi\n\t \\vec{k}(x,y) \\right] = 0,\n \\label{eq:consist}\n \\end{equation}\n %\nwhere $\\nabla \\times \\cdot$ is the discrete curl operator. Since there \nare many possible $\\vec{k}$ fields satisfying eq.~\\eqref{eq:consist}, \nPU is an ill-posed problem according to Hadamard's definition. \n\nOne of the most classical and widely-used algorithms for phase \nunwrapping is the Least Mean Squares (LMS) approach~\\cite{Ghiglia-94}, \nwhich consists in finding the scalar field $f$ whose gradient is closer \nto $\\vec{A}$ in the Least Squares sense, i.e.\\ the minimizer of: \n\t%\n\t\\begin{equation}\n\t\t\\sum ( \\vec{\\nabla} f - \\vec{A})^2.\n\t\\end{equation}\n\t%\n\nAs mentioned before, in~\\cite{Stramaglia-99} a variational approach has\nbeen used, and the field $\\vec{k}$ was \\qt{chosen} as the minimizer of\nthe following functional:\n %\n \\begin{eqnarray}\n R &=& {1\\over 4\\pi^2} \\sum \n\t\t \\left[\n\t\t \\nabla_{x} f(x+1,y) - \\nabla_{x} f(x,y)\n\t\t \\right]^2 \\nonumber \\\\ &\\qquad& +\n {1\\over 4\\pi^2} \\sum \n\t\t \\left[\n\t\t \\nabla_{y} f(x,y+1) - \\nabla_{y} f(x,y) \n\t\t \\right]^2 \\nonumber \\\\\n\t &\\qquad& + {1\\over 4\\pi^2}\\sum\n\t\t \\left[\n\t\t \\nabla_{x} f(x,y+1) - \\nabla_{x} f(x,y)\n\t\t \\right]^2 \\nonumber \\\\ &\\qquad& +\n\t {1\\over 4\\pi^2}\\sum\n\t\t \\left[\n\t\t \\nabla_{y} f(x+1,y) - \\nabla_{y} f(x,y)\n\t\t \\right]^2,\n \\label{eq:nostro}\n \\end{eqnarray}\n %\nsubject to constraint~\\eqref{eq:consist}. Due to~\\eqref{eq:aliasgrad}, \n$R$ is a functional of $\\vec{k}$, i.e.\\ $R=R[\\vec{k}]$, and it measures \nthe smoothness of the reconstructed phase surface. The optimization \nproblem was then solved by a MFA algorithm under the assumption that \nthe $\\vec{k}$ field be restricted to take values in $\\{-1, 0, 1\\}$. In \nthe next section we generalize the MFA algorithm to the case of \n$\\vec{k}$ fields belonging to $\\{-L, \\ldots, L\\}$, with $L$ an \narbitrary integer. \n\n\\section{The algorithm}\n\\label{sec:alg}\n\nAs explained in sect.~\\ref{sec:pu}, we assume the solution of PU to be \nthe minimizer of the functional~\\eqref{eq:nostro}, subject to \nconstraint~\\eqref{eq:consist}. Let us assume that the possible values \nof the $\\vec{k}$ field are restricted to belong to $\\{-L, \\ldots, L\\}$. \nThe field $\\vec{k}$ may then be regarded as a system of spin-L units. \nWe assume the Gibbs distribution: \n\t%\n\t\\begin{equation}\n\t\tP[\\vec{k}] = \\frac{\\exp\\left[ \\frac{-R[\\vec{k}]}{T} \\right]}\n\t\t{\\sum_{\\vec{k}'} \\exp \\left[ \\frac{-R[\\vec{k}']}{T} \n\t\t\\right]},\n\t\\end{equation}\n\t%\nwhere the sum is over the $\\vec{k}'$ fields \nsatisfying~\\eqref{eq:consist}, and $T$ is the statistical temperature. \nInconsistent fields $\\vec{k}'$ are assumed to have zero probability. \n\nFollowing Mean-Field theory~\\cite{Parisi-88}, we consider a probability \ndistribution for the correction field $\\vec{k}$ which treats all the \nvariables as independent, i.e.\\ it is the product of the marginal \ndistributions of each variable. \n\nLet $\\rho_x(x,y,\\alpha)$ be the probability that $k_x(x,y) = \\alpha$,\nwith $\\alpha = -L, \\ldots, L$, and $\\rho_y(x,y,\\alpha)$ be the\ncorresponding probability for $k_y(x,y)$. Normalization of these\nmarginal probabilities implies a penalty functional:\n %\n \\begin{equation}\n\t \\Theta[\\rho] = \\sum_{(x,y)} \\biggl[ V_x(x,y) \\left(1 -\n\t \\sum_{\\alpha} \\rho_x(x,y,\\alpha) \\right) +\n\t V_y(x,y) \\left(1 -\n\t \\sum_{\\alpha} \\rho_y(x,y,\\alpha) \\right) \\biggr],\n \\end{equation}\n %\nwhere $\\{V\\}$ are Lagrange multipliers.\n\nThe entropy of the system, in the mean field approximation, is:\n %\n \\begin{equation}\n\t S[\\rho] = - \\sum_{(x,y)} \\sum_{\\alpha = -L}^L \\bigl[\n\t \\rho_x(x,y,\\alpha) \\log \\rho_x(x,y,\\alpha) + \\rho_y(x,y,\\alpha)\n\t \\log \\rho_y(x,y,\\alpha) \\bigr].\n \\end{equation}\n %\n\nIt is useful to introduce the average fields $\\vec{m} =\n\\avg{\\vec{k}}_{\\vec{\\rho}}$ and $\\vec{Q} =\n\\avg{\\vec{k}^2}_{\\vec{\\rho}}$, defined by:\n %\n \\begin{eqnarray}\n\t m_x(x,y) = \\sum_{\\alpha = -L}^L \\alpha \\rho_x (x,y,\\alpha),\n\t \\qquad m_y(x,y) = \\sum_{\\alpha = -L}^L \\alpha \\rho_y\n\t (x,y,\\alpha);\n\t \\label{eq:avefields1}\\\\\n\t Q_x(x,y) = \\sum_{\\alpha = -L}^L \\alpha^2 \\rho_x (x,y,\\alpha),\n\t \\qquad Q_y(x,y) = \\sum_{\\alpha = -L}^L \\alpha^2 \\rho_y\n\t (x,y,\\alpha).\n\t \\label{eq:avefields2}\n \\end{eqnarray}\n %\nThe average $U$ of the cost functional $R$ is called \\emph{internal\nenergy}. It is easy to show that the internal energy depends only on\n$\\vec{m}$ and $\\vec{Q}$:\n %\n \\begin{equation}\n\t U[\\vec{m},\\vec{Q}] = \\avg{R[\\vec{A} + 2 \\pi\n\t \\vec{k}]}_{\\vec{\\rho}}.\n \\end{equation}\n %\n\nA penalty functional is introduced to enforce \nconstraints~\\eqref{eq:consist}:\n\t%\n\t\\begin{eqnarray}\n\t\t\\Gamma [ \\vec{m} ] &= \\sum_{(x,y)} \\lambda (x,y) \\bigl[ \n\t\tm_x(x,y) + m_y(x+1,y) \\nonumber \\\\\n\t\t&\\qquad - m_x(x,y+1) - m_y(x,y) + I(x,y) \n\t\t\\bigr],\n\t\\end{eqnarray}\n\t%\nwhere $\\{\\lambda\\}$ is another set of Lagrange multipliers.\n\nLet us now introduce an effective cost functional, the \\emph{free \nenergy}, which depends on $T$:\n\t%\n\t\\begin{equation}\n\t\tF[\\rho] = U[\\vec{m}, \\vec{Q}] - T S[\\vec{\\rho}] + \\Gamma[\\vec{m}] + \n\t\t\\Theta[\\vec{\\rho}]\n\t\\end{equation}\n\t% \n\nThe free energy is the weighted sum of the internal energy (the \noriginal cost function) and the entropy functional; $\\Gamma$ and \n$\\Theta$ are penalty functionals to enforce the constraints of the \nproblem. According to the variational principle of Statistical \nMechanics, the best approximation to the Gibbs distribution is the \nminimizer of the free energy~\\cite{Parisi-88}. Since $-TS$ is a convex \nfunctional, the free energy is convex at high temperature and the \nglobal minimum can be easily attained. The solution can then be \ncontinuated as temperature is lowered, so as to reach a minimum of $U$. \nThis procedure has shown to be less sensitive to local minima than \nconventional descent methods, and gives results close to the ones from \nSimulated Annealing, while requiring less computational \ntime~\\cite{Yuille-94}. \n\nThe equations for the minimum of the free energy are usually called \n\\qt{mean-field equations}:\n\t%\n\t\\begin{equation}\n\t\t\\frac{\\partial F}{\\partial \\rho_x (x,y,\\alpha)} = 0; \n\t\t\\quad \\frac{\\partial F}{\\partial \\rho_y (x,y,\\alpha)} = 0\n\t\\label{eq:mf}\n\t\\end{equation}\n\t% \n\nAfter simple calculations, the solution of eqs.~\\eqref{eq:mf} is found \nto have the following form:\n\t%\n\t\\begin{eqnarray}\n\t\t\\rho_x(x,y,\\alpha) &= \n\t\t\t\\frac{\\exp\\left\\{ -\\beta \\left[ \n\t\t\t\t\\frac{\\partial U}{\\partial \\rho_x(x,y,\\alpha)} + \n\t\t\t\t\\frac{\\partial \\Gamma}{\\partial \\rho_x(x,y,\\alpha)}\n\t\t\t\t\\right] \\right\\}}\n\t\t\t {\\sum_{\\alpha' = -L}^L \\exp \\left\\{ -\\beta \\left[ \n\t\t\t\t\\frac{\\partial U}{\\partial \\rho_x(x,y,\\alpha')} + \n\t\t\t\t\\frac{\\partial \\Gamma}{\\partial \\rho_x(x,y,\\alpha')}\n\t\t\t\t\\right] \\right\\}},\n\t\t\\label{eq:solutionrho1}\\\\\n\t\t\\rho_y(x,y,\\alpha) &= \n\t\t\t\\frac{\\exp\\left\\{ -\\beta \\left[ \n\t\t\t\t\\frac{\\partial U}{\\partial \\rho_y(x,y,\\alpha)} + \n\t\t\t\t\\frac{\\partial \\Gamma}{\\partial \\rho_y(x,y,\\alpha)}\n\t\t\t\t\\right] \\right\\}}\n\t\t\t {\\sum_{\\alpha' = -L}^L \\exp \\left\\{ -\\beta \\left[ \n\t\t\t\t\\frac{\\partial U}{\\partial \\rho_y(x,y,\\alpha')} + \n\t\t\t\t\\frac{\\partial \\Gamma}{\\partial \\rho_y(x,y,\\alpha')}\n\t\t\t\t\\right] \\right\\}}\n\t\t\\label{eq:solutionrho2}\n, \n\t\\end{eqnarray}\n\t%\nwhere the $\\{V\\}$ multipliers have been fixed to normalize the \ndistributions, and $\\beta = 1/T$ is the inverse temperature. Now we \nobserve that, for each site $(x,y)$ on the grid:\n\t%\n\t\\begin{equation}\n\t\t\\frac{\\partial U}{\\partial \\rho_x(\\alpha)} = \n\t\t\t\\frac{\\partial U}{\\partial m_x} \\frac{\\partial m_x}{\\partial \\rho_x(\\alpha)}\n\t\t\t+ \\frac{\\partial U}{\\partial Q_x} \\frac{\\partial Q_x}{\\partial \n\t\t\t\\rho_x(\\alpha)} =\n\t\t\t\\alpha \\frac{\\partial U}{\\partial m_x(x,y)} + \n\t\t\t\\alpha^2 \\frac{\\partial U}{\\partial Q_x(x,y)}.\n\t\\label{eq:partial1}\n\t\\end{equation}\n\t%\n\nAnalogously, one can easily find:\n\t%\n\t\\begin{eqnarray}\n\t\\label{eq:partial2}\n\t\t\\frac{\\partial U}{\\partial \\rho_y(\\alpha)} &=& \n\t\t\t\\alpha \\frac{\\partial U}{\\partial m_y(x,y)} + \n\t\t\t\\alpha^2 \\frac{\\partial U}{\\partial Q_y(x,y)},\\\\\n\t\t\\frac{\\partial \\Gamma}{\\partial \\rho_x(\\alpha)} &=&\n\t\t\t\\alpha \\left[ \\lambda(x,y) - \\lambda(x,y-1) \n\t\t\t\\right],\\\\\n\t\t\\frac{\\partial \\Gamma}{\\partial \\rho_y(\\alpha)} &=& \n\t\t\t\\alpha \\left[ - \\lambda(x,y) + \\lambda(x-1,y) \n\t\t\t\\right].\n\t\\end{eqnarray}\n\t%\n\t\nThe derivatives of $U$ with respect to the $\\{m\\}$ and $\\{Q\\}$ \nvariables are reported in Appendix A. From these expressions it is \nclear that the present formulation of PU is equivalent to finding the \nground state of a finite-size, spin-L Ising model with local \nconstraints, and under a non-uniform magnetic field. \n\nThe consistency constraints are written as equations for the \n$\\{\\lambda\\}$ field:\n\t%\n\t\\begin{eqnarray}\n\t\t\\lambda(x,y) &= \\lambda(x,y) - \n\t\t\tb \\bigl[ m_x(x,y) + m_y(x+1,y) \\nonumber \\\\\n\t\t\t&\\qquad - m_x(x,y+1) - m_y(x,y) + I(x,y) \n\t\t\t\\bigr],\n\t\\label{eq:solutionlambda}\n\t\\end{eqnarray}\n\t%\nwhere $b$ is a small constant.\n\nEquations~(\\ref{eq:solutionrho1}--\\ref{eq:solutionrho2}) \nand~\\eqref{eq:solutionlambda} are the mean-field equations for PU for \narbitrary $L$. As already explained, the MFA technique consists in \nsolving iteratively the mean-field equations at high temperature (low \n$\\beta$), and then track the solution as the temperature is lowered \n($\\beta$ grows). \n\nThe algorithm can be summarized as follows. The initial distributions \ngive the same probability to each value of the correction field, i.e.\\ \n$\\rho_x(x,y,\\alpha) = \\rho_y(x,y,\\alpha) = \\frac{1}{2L+1}$; the inverse \ntemperature is set to $\\beta_{\\mathrm{MIN}}$ ($\\beta_{\\mathrm{MAX}}$ is \nthe lowest temperature). Then:\n\t%\n\t\\begin{enumerate}\n\t\t\\item \\keyw{Evaluate} $\\{\\vec{m}\\}$ and $\\{\\vec{Q}\\}$ \n\t\tfields by \n\t\tEqs.~(\\ref{eq:avefields1}--\\ref{eq:avefields2});\n\t\t\\item \\keyw{Iterate} \n\t\tEqs.~(\\ref{eq:solutionrho1}--\\ref{eq:solutionrho2});\n\t\t\\item \\keyw{Iterate} \n\t\tEq.~(\\ref{eq:solutionlambda});\n\t\t\\item \\keyw{If} \n\t\tEqs.~(\\ref{eq:solutionrho1}--\\ref{eq:solutionrho2}) or \n\t\tEq.~(\\ref{eq:solutionlambda}) are not satisfied, \n\t\t\\keyw{goto} step 1;\n\t\t\\item \\keyw{If} $\\beta < \\beta_{\\mathrm{MAX}}$, increase \n\t\t$\\beta$ and \\keyw{goto} step 1.\n\t\\end{enumerate}\n\t%\n\nThe output of this algorithm is a field $\\{\\vec{m}_{\\mathrm{OUT}}\\}$ \nwhich approximates the average of $\\{\\vec{k}\\}$ over the global minima \nof the cost functional $R$.\n\nWe remark that the output of the algorithm described \nin~\\cite{Stramaglia-99} satisfies $m \\in [-1, 1]$ for each component of \n$\\{\\vec{m}_{\\mathrm{OUT}}\\}$, whereas the present algorithm satisfies \nthe weaker constraint $m \\in [-L, L]$ and therefore can be used also in \nthe case of high degree of undersampling. The estimate for the true \nphase gradient is $(\\vec{\\nabla} f)_{\\mathrm{est}} = \\vec{A} + 2\\pi \n\\vec{m}_{\\mathrm{OUT}}$; the phase pattern $f$ can then be \nreconstructed by $(\\vec{\\nabla} f)_{\\mathrm{est}}$ as described \nin~\\cite{Stramaglia-99}.\n\n\\section{Experiments}\n\\label{sec:exp}\n\nIn this section we describe some experiments we performed to test the \neffectiveness of the proposed algorithm.\n\nIn fig.~\\ref{fig:one}-(a) a synthetic phase pattern is shown, while in \nfig.~\\ref{fig:one}-(b) the wrapped phase pattern is depicted. This test \nphase pattern has been constructed by the following formula: \n\t%\n\t\\begin{equation}\n\t\t\\Phi(x,y) = 120 \\exp \\left[-\\half r^2(x,y) (\\mu_1 + \\mu_2 c(x,y)) \n\t\t\\right], \\quad 1 \\leq x,y \\leq 128\n\t\\label{eq:experim}\n\t\\end{equation}\n\t% \nwith:\n\t%\n\t\\begin{eqnarray}\n\t\tr(x,y) &=& \\sqrt{(x - 35.5)^2 + (y - 65.5)^2}, \\nonumber\\\\\n\t\tc(x,y) &=& \\frac{x - 35.5}{r}, \\nonumber\\\\\n\t\t\\mu_1 &=& 0.01, \\nonumber\\\\\n\t\t\\mu_2 &=& 0.0004, \\nonumber\n\t\\end{eqnarray}\n\t%\nwhere $\\Phi$ is in radians. \n\nBy construction, $\\Phi$ is undersampled: in fig.~\\ref{fig:two}-(a) \nblack pixels represent locations where the true correction field \n$\\vec{k}$ is such that \\makebox{$\\max \\{ k_x, k_y \\} = 2$}, while gray \npixels represent locations where $\\max \\{ k_x, k_y \\} = 1$. The residue \nmap is depicted in fig.~\\ref{fig:two}-(b). \n\nWe applied the proposed algorithm to unwrap this test pattern. We used \n$L=2, b=0.05$ and the annealing schedule was established as consisting \nof 25 temperature values, equally spaced in the interval \n$[\\beta_{\\mathrm{MIN}}=0.05, \\beta_{\\mathrm{MAX}}=1.5]$. The output \nphase pattern is depicted in fig.~\\ref{fig:three}. The computational \ntime was comparable to that corresponding to~\\cite{Stramaglia-99}. The \ninput phase surface was perfectly reconstructed. Let us now compare the \nperformance of the MFA algorithm with that from LMS~\\cite{Ghiglia-94}. \nIn fig.~\\ref{fig:threethree} we show the output of LMS applied to the \nsurface of fig.~\\ref{fig:one}-(a). Due to severe undersampling, the LMS \nperformance is poor. \n\nWe also investigated the robustness of the MFA algorithm with respect \nto noise. In fig.~\\ref{fig:four}-(a) the phase pattern obtained by \nadding unit-variance Gaussian noise to the surface of \nfig.~\\ref{fig:one}-(a) is shown. The wrapped phase pattern is depicted \nin fig.~\\ref{fig:four}-(b), while in fig.~\\ref{fig:four}-(c) the \ninconsistencies are shown. The output of the MFA algorithm is shown in \nfig.~\\ref{fig:five}-(a), while in fig.~\\ref{fig:five}-(b) we show the \nphase pattern obtained by re-wrapping the MFA output. The smoothing \ncapability of the proposed algorithm appears clearly by comparing \nfigs.~\\ref{fig:four}-(b) and~\\ref{fig:five}-(b). \n\n\\section{Conclusions}\n\\label{sec:concl}\n\nIn this paper we have generalized a previously presented MFA algorithm \nto unwrap phase patterns. This problem is formulated as that of finding \nthe ground state of a locally-constrained, spin-L Ising model under a \nnon-uniform external field. The present generalization allows \nprocessing of noisy and highly undersampled input phase fields. The \neffectiveness of this statistical approach to PU has been demonstrated \non simulated phase surfaces. Further work will be devoted to the \nestimation of the optimal value of $L$ from the observed wrapped phase \ndata. \n\n\n\n\\clearpage\n%%%%%%%%%% Figures\n\t%\n\t\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.6\\textwidth]{fig1a.eps}\n\t\t\\includegraphics[width=0.35\\textwidth]{fig1b.eps}\n\t\t\\\\\n\t\t(a) \\hspace{0.4\\textwidth} (b)\n\t\\end{center}\n\t\\caption{(a) Synthetic phase surface generated via eq.~\\eqref{eq:experim}; \n\t\t(b) wrapped phase pattern:\n\t\tprincipal phase values span the interval from $-\\pi$ (black pixels) \n\t\tto $+ \\pi$ (white pixels).}\n\t\\label{fig:one}\n\t\\end{figure}\n\t%\n\t%\n\t\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.35\\textwidth]{fig1c.eps}\n\t\t\\includegraphics[width=0.35\\textwidth]{resnn.eps}\n\t\t\\\\\n\t\t(a) \\hspace{0.4\\textwidth} (b)\n\t\\end{center}\n\t\\caption{(a) Degree of undersampling of the phase field depicted in \n\t\tfig.~\\ref{fig:one}:\n\t\tgray pixels represent locations where the absolute phase gradient differs from\n\t\tits estimate (wrapping of the principal phase gradient) by one \n\t\t$2\\pi$-cycle, black pixels represent locations where the difference is 2 cycles;\n\t\t(b) residue map: white pixels are positive residues ($I=1$), \n\t\tblack pixels are negative residues ($I=-1$), \n\t\tgray pixels correspond to irrotational locations ($I=0$).}\n\t\\label{fig:two}\n\t\\end{figure}\n\t%\n\t%\n\t\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.6\\textwidth]{fig2e.eps}\n\t\\end{center}\n\t\\caption{Unwrapped phase reconstructed by the proposed algorithm \n\t\tfrom the wrapped phase field shown in fig.~\\ref{fig:one}-(b).}\n\t\\label{fig:three}\n\t\\end{figure}\n\t%\n\t%\n\t\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.6\\textwidth]{fig1d.eps}\n\t\\end{center}\n\t\\caption{Unwrapped phase reconstructed by the LMS algorithm \n\t\tfrom the wrapped phase field shown in fig.~\\ref{fig:one}-(b).}\n\t\\label{fig:threethree}\n\t\\end{figure}\n\t%\n\t%\n\t\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.6\\textwidth]{fig2a.eps}\n\t\t\\includegraphics[width=0.35\\textwidth]{fig2b.eps}\n\t\t\\\\\n\t\t(a) \\hspace{0.4\\textwidth} (b)\\\\\n\t\t\\includegraphics[width=0.35\\textwidth]{resn1.eps}\\\\\n\t\t(c)\n\t\\end{center}\n\t\\caption{(a) Same synthetic surface as in fig.~\\ref{fig:one}, \n\t\twith added Gaussian noise with unit variance; (b) principal phase pattern;\n\t\t(c) residue map.}\n\t\\label{fig:four}\n\t\\end{figure}\n\t%\n\t%\n\t\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.6\\textwidth]{fig2c.eps}\n\t\t\\includegraphics[width=0.35\\textwidth]{fig2d.eps}\n\t\t\\\\\n\t\t(a) \\hspace{0.4\\textwidth} (b)\n\t\\end{center}\n\t\\caption{(a) Unwrapped surface reconstructed by the proposed algorithm \n\t\tfrom the wrapped phase field shown in fig.~\\ref{fig:four}-(b);\n\t\t(b) principal phase pattern obtained by re-wrapping the solution depicted in (a).}\n\t\\label{fig:five}\n\t\\end{figure}\n\t%\n%%%%%%%%%%%%%%%%%%\n\n\\clearpage\n\\appendix\n\\section{Appendix: Derivatives of the internal energy}\n\\label{sec:app} \n\nWe report here the expressions of the derivatives of the internal \nenergy in the Mean-field approximation. \n\nOne easily finds that:\n%\n\\begin{eqnarray}\n {\\partial U\\over \\partial Q_x(x,y)}=4 \\nonumber \\\\ \n {\\partial U\\over \\partial Q_y(x,y)}=4, \n\t\t\t\t\\label{s1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n {\\partial U\\over \\partial m_x(x,y)}&=& -2 m_x(x-1,y)+{1\\over \n \\pi}\\left[A_x(x,y)-A_x(x-1,y)\\right] + \\nonumber \\\\ && -2 \n m_x(x+1,y)+{1\\over \\pi}\\left[A_x(x,y)-A_x(x+1,y)\\right] + \\nonumber \n \\\\\n &&-2 m_x(x,y-1)+{1\\over \\pi}\\left[A_x(x,y)-A_x(x,y-1)\\right] +\\nonumber \n \\\\\n &&-2 m_x(x,y+1)+{1\\over \\pi}\\left[A_x(x,y)-A_x(x,y+1)\\right]. \n\t\t\t\\label{s2}\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n {\\partial U\\over \\partial m_y(x,y)}&=& -2 m_y(x,y-1)+{1\\over \n \\pi}\\left[A_y(x,y)-A_y(x,y-1)\\right] +\\nonumber \\\\ &&-2 \n m_y(x,y+1)+{1\\over \\pi}\\left[A_y(x,y)-A_y(x,y+1)\\right] +\\nonumber \n \\\\\n &&-2 m_y(x+1,y)+{1\\over \\pi}\\left[A_y(x,y)-A_y(x+1,y)\\right] +\\nonumber \n \\\\\n &&-2 m_y(x-1,y)+{1\\over \\pi}\\left[A_y(x,y)-A_y(x-1,y)\\right]. \n\t\t \\label{s3}\n\\end{eqnarray}\n\n\n\\begin{ack}\n\tThe authors thank Dr. G. Gonnella for useful discussions on \n\tMean-Field theory.\n\\end{ack}\n\n\\clearpage\n\n\\begin{thebibliography}{99}\n\\bibitem{Oppenheim-75}\n A. V. Oppenheim, R. W. Schafer, \\emph{Digital Signal processing},\n Prentice-Hall, Englewood Cliffs, N.J., 1975.\n\\bibitem{Hjalmarson-85}\n H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris,\n \\emph{Phys. Rev. B} 32, 4300 (1985).\n\\bibitem{Nakadate-85}\n S. Nakadate, H. Saito, \\emph{Applied Optics} 24, 2172 (1985).\n\\bibitem{Fried-77}\n D. L. Fried, \\emph{J. Opt. Soc. Am.} 67, 370 (1977).\n\\bibitem{Ching-92}\n N. H. Ching, D Rosenfeld, M. Braun, \\emph{IEEE Trans. Im. Proc.} 1, 355\n (1992).\n\\bibitem{Zebker-86}\n H. A. Zebker, R. M. Goldstein, \\emph{J. Geophis. Res.} 91, (B5), 4993\n (1986).\n\\bibitem{Hadamard-02}\n J. Hadamard, \\qt{Sur les probl\\'emes aux d\\'eriv\\'ees partielles et leur\n signification physique}, \\emph{Princeton University Bulletin} 13 (1902).\n\\bibitem{Geman-84}\n S. Geman, D. Geman, \\emph{IEEE Trans. Pattern Anal. Mach. Intell.} 6, 721\n (1984).\n\\bibitem{Gidas-89}\n B. Gidas, \\emph{IEEE Trans. Pattern Anal. Mach. Intell.} 11, 164 (1989).\n\\bibitem{Zhang-92}\n J. Zhang, \\emph{IEEE Trans. Signal Process.} 40, 2570 (1992).\n\\bibitem{Tanaka-95}\n K. Tanaka, T. Morita, \\emph{Phys. Lett. A} 203, 122 (1995).\n\\bibitem{Morita-96}\n T. Morita, K. Tanaka, \\emph{Physica A} 223, 244 (1996).\n\\bibitem{Ray-97}\n J. Ray, R. W. Harris, \\emph{Phys. Rev. E} 55, 5270 (1997).\n\\bibitem{Guerriero-98}\n L. Guerriero, G. Nico, G. Pasquariello, S. Stramaglia,\n \\emph{Applied Optics} 37, 3053 (1998).\n\\bibitem{Stramaglia-99}\n S. Stramaglia, L. Guerriero, G. Pasquariello, N. Veneziani,\n \\emph{Applied Optics} 38, 1377 (1999).\n\\bibitem{Ghiglia-98} \n D. C. Ghiglia, M. D. Pritt,\n \\emph{Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software},\n John Wiley \\& Sons, New York, (1998). \n\\bibitem{Ghiglia-94}\n D. C. Ghiglia, J. A. Romero, \n \\emph{J. Opt. Soc. Am. A} 11, 107--117, 1994.\n\\bibitem{Parisi-88}\n G. Parisi, \\emph{Statistical Field Theory}, Addison-Wesley, Reading\n MA, (1988).\n\\bibitem{Yuille-94}\n For a review on Mean-Field Annealing methods, see e.g.\\ A. L. \n Yuille, J. J. Kosowsky, \\emph{Neural Computation} 6, 341 (1994).\n\\end{thebibliography}\n\n\\clearpage\n\n\\listoffigures \n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002093.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{Oppenheim-75}\n A. V. Oppenheim, R. W. Schafer, \\emph{Digital Signal processing},\n Prentice-Hall, Englewood Cliffs, N.J., 1975.\n\\bibitem{Hjalmarson-85}\n H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, C. B. Norris,\n \\emph{Phys. Rev. B} 32, 4300 (1985).\n\\bibitem{Nakadate-85}\n S. Nakadate, H. Saito, \\emph{Applied Optics} 24, 2172 (1985).\n\\bibitem{Fried-77}\n D. L. Fried, \\emph{J. Opt. Soc. Am.} 67, 370 (1977).\n\\bibitem{Ching-92}\n N. H. Ching, D Rosenfeld, M. Braun, \\emph{IEEE Trans. Im. Proc.} 1, 355\n (1992).\n\\bibitem{Zebker-86}\n H. A. Zebker, R. M. Goldstein, \\emph{J. Geophis. Res.} 91, (B5), 4993\n (1986).\n\\bibitem{Hadamard-02}\n J. Hadamard, \\qt{Sur les probl\\'emes aux d\\'eriv\\'ees partielles et leur\n signification physique}, \\emph{Princeton University Bulletin} 13 (1902).\n\\bibitem{Geman-84}\n S. Geman, D. Geman, \\emph{IEEE Trans. Pattern Anal. Mach. Intell.} 6, 721\n (1984).\n\\bibitem{Gidas-89}\n B. Gidas, \\emph{IEEE Trans. Pattern Anal. Mach. Intell.} 11, 164 (1989).\n\\bibitem{Zhang-92}\n J. Zhang, \\emph{IEEE Trans. Signal Process.} 40, 2570 (1992).\n\\bibitem{Tanaka-95}\n K. Tanaka, T. Morita, \\emph{Phys. Lett. A} 203, 122 (1995).\n\\bibitem{Morita-96}\n T. Morita, K. Tanaka, \\emph{Physica A} 223, 244 (1996).\n\\bibitem{Ray-97}\n J. Ray, R. W. Harris, \\emph{Phys. Rev. E} 55, 5270 (1997).\n\\bibitem{Guerriero-98}\n L. Guerriero, G. Nico, G. Pasquariello, S. Stramaglia,\n \\emph{Applied Optics} 37, 3053 (1998).\n\\bibitem{Stramaglia-99}\n S. Stramaglia, L. Guerriero, G. Pasquariello, N. Veneziani,\n \\emph{Applied Optics} 38, 1377 (1999).\n\\bibitem{Ghiglia-98} \n D. C. Ghiglia, M. D. Pritt,\n \\emph{Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software},\n John Wiley \\& Sons, New York, (1998). \n\\bibitem{Ghiglia-94}\n D. C. Ghiglia, J. A. Romero, \n \\emph{J. Opt. Soc. Am. A} 11, 107--117, 1994.\n\\bibitem{Parisi-88}\n G. Parisi, \\emph{Statistical Field Theory}, Addison-Wesley, Reading\n MA, (1988).\n\\bibitem{Yuille-94}\n For a review on Mean-Field Annealing methods, see e.g.\\ A. L. \n Yuille, J. J. Kosowsky, \\emph{Neural Computation} 6, 341 (1994).\n\\end{thebibliography}"
}
] |
cond-mat0002094
|
[] |
[
{
"name": "lt22meso.tex",
"string": "\n\\documentclass[twocolumn,a4paper]{article}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\usepackage{epsfig}\n\n\\frenchspacing\n\\pagestyle{empty}\n\\addtolength{\\textheight}{1.6in}\\addtolength{\\voffset}{-0.8in}\n\\input tcilatex\n\\begin{document}\n\n\\twocolumn[\n\n\\begin{center}\n\n\\textbf{\\Large Intrinsic tunneling spectroscopy in small Bi2212 mesas}\n\n\\vskip1ex { Vladimir Krasnov$^{1,2}$, August Yurgens$^{1,3}$, Dag Winkler$^{4,1}$ and Per\nDelsing$^1$ }\n\n\\vskip0.5ex \\textsl{ $^1$ MINA, Chalmers University of Technology,\nS41296, G\\\"oteborg, Sweden\\\\ $^2$ Institute of Solid state\nPhysics, 142432 Chernogolovka, Russia\\\\ $^3$ P.L.Kapitsa Institute, 117334 Moscow, Russia\\\\\n$^4$ IMEGO Institute, Aschebergsgatan 46, S41133, G\\\"oteborg, Sweden}\n\n\\end{center}\n]\n\n%\\textbf{Abstract:} We present experimental study of small Bi$_2$Sr$_2$CaCu$_2 $O$_{8+x}$\n%mesa structures, containing few intrinsic Josephson junctions.\n%The mesas exhibit clear tunnel-type current-voltage characteristics. This\n%allows us to distinguish and simultaneously observe superconducting and\n%pseudo gaps. We show that the superconducting gap vanishes at the critical\n%temperature, while the pseudo-gap may sustain up to room temperature.\\\\\n\nTunneling spectroscopy of high-$T_c$ superconductors (HTSC)\nprovides an important information about quasiparticle density of\nstates (DOS), which is crucial for understanding HTSC mechanism.\nSurface tunneling experiments \\cite{Renner,Wilde} showed, that\nbesides the superconducting gap in the DOS, $ \\Delta $, there is a structure\nusually referred to as the ''pseudo-gap'', which exists well above $T_c $. Both the\nbehaviour of the superconducting gap and it's correlation with the\npseudo-gap are still a matter of controversy\\cite{Renner,Wilde}.\n\nTo avoid drawbacks of surface tunneling experiments, such as\ndependence on the surface deterioration, surface states and undefined\ngeometry\\cite{Wilde}, we used ''intrinsic'' tunneling\nspectroscopy. HTSC single crystals can be considered as stacks of\natomic scale intrinsic Josephson junctions (IJJ's). Using\nmicrofabrication, it is possible to make small HTSC mesa\nstructures with a well defined geometry\\cite{Yurgens}. Moreover,\nIJJ's far from the sample surface can be measured, and\ndeterioration of the sample surface becomes less important.\nCurrent-voltage characteristics (IVC's) of mesas exhibiting tunnel\njunction behavior and can be used for studying\nDOS\\cite{Suzuki}. On the other hand, intrinsic tunneling spectroscopy can suffer from\nohmic heating of the IJJ's and steps (defects)\non the surface of the crystal.\n\nIn this paper we present experimental data for small area Bi$_2$Sr$_2$CaCu$%\n_2 $O$_{8+x}$ (Bi2212) mesas containing few IJJ's. By decreasing\nthe mesa area, $S$, we minimize both the effect of overheating and\nthe probability of defects in the mesa. Mesas with dimensions from\n2 to 20 $\\mu $m were fabricated simultaneously on top of Bi2212\nsingle crystals. First a long and narrow mesa was fabricated using\nphotolithography and chemical etching. Next, insulating CaF$_2$\nlayer was deposited and lift-off was used to make an opening.\nFinally, Ag film was deposited and electrodes were formed on top\nof the initial mesa by photolithography and Ar-ion etching. After\netching, mesas beneath Ag electrodes remain.\n\nIn Fig.1, normalized IVC's of three mesas at $T$=4.2K and $T$=150\nK are shown. The vertical axis represents the current density,\n$J=I/S$, and the horizontal axis shows the voltage per junction,\n$V/N$. IVC's for different samples are plotted by\ndifferent colours. Parameters of the mesas are listed in Table 1.\nFrom Fig.1 it is seen, that the normalized IVC's merge quite well\ninto one curve. This indicates good reproducibility of the\nfabrication procedure. The $c$-axis normal resistivity was, $\\rho\n_N=R_NS/Ns=44\\pm 2.0$ $\\Omega $cm, where $s=15.5\\AA $ is the\nspacing periodicity of IJJ's and $R_N$ is the resistance at large bias current and $%\nT_c\\simeq 93$ K.\n\n\\begin{tabular}{lllll}\nTable 1: & mesa & $S$ ($\\mu $m$^2$) & $N$ & $R_N (\\Omega)$ \\\\\n& S251b & $6\\times 6$ & 12 & 229.9 \\\\\n& S255b & $5.5\\times 6$ & 12 & 256.4 \\\\\n& S211b & $4\\times 7.5$ & 12 & 272.5 \\\\\n& S216b & $4\\times 20$ & 10 & 87\n\\end{tabular}\n\nFrom Fig.1 it is seen that IVC's exhibit clear tunnel junction behavior: (i)\nAt low bias, multiple quasiparticle branches are seen, representing\none-by-one switching of IJJ's into the resistive state\\cite{Yurgens,Suzuki}.\nThe number of IJJ's in the mesa was obtained by counting those branches.\n(ii) At intermediate currents there is a pronounced knee in IVC's,\nrepresenting the sum-gap voltage, $V_g=2\\Delta /e$. (iii) At high currents,\nthere is a well defined normal resistance branch, $R_N$. As seen from Fig.1,\n$R_N$ is almost temperature independent, as may be expected for a tunnel\nresistance. In contrast, the zero bias resistance, $R(0)$, increases sharply\nwith decreasing $T$, as shown in inset b).\n\nInset a) in Fig.1 shows temperature dependence of the gap voltage, $V_g$,\nand the maximum spacing between multiple quasiparticle branches, $\\Delta V$,\nfor four mesas on two diffrerent chips. Solid and open symbols represent branches for $V>0$ and $V<0$%\n, respectively. $\\Delta V$ was determined at the ''maximum\ncritical current'', at which the last IJJ switches to the\nresistive state, e.g. in Fig.1 that would correspond to $J\\simeq\n1.3\\times 10^3$ A/cm$^2$. This current is fluctuating from run to\nrun, therefore, causing an uncertainty in determination of $\\Delta\nV$. From the inset a) it is seen, that $\\Delta V$ is approximately\ntwo times less than $V_g$. This is simply due to the fact that all\nthe IJJ's switch to the resistive state before they reach the gap\nvoltage, see Fig.1. However, it is seen that the temperature dependence of $%\n\\Delta V$ reflects that for $V_g$. The observed $V_g$ correspond\nto $\\Delta \\simeq 32$ meV at $T$=4.2 K, in agreement with\n\\cite{Wilde}.\n\n\\begin{figure}[hbt]\n\\centering \\epsfig{file=fig1.eps,width=0.9\\columnwidth}\n\\caption{$J$ vs. $V/N$ curves for three mesas at $T$=4.2K and\n$T$=150 K. Inset a) shows temperature dependencies of the gap\nvoltage, $V_g$, and the voltage separation of multiple\nquasiparticle branches, $\\Delta V$. Inset b) shows temperature\ndependencies of the zero-bias resistance.} \\label{fig1}\n\\end{figure}\n\nThe IVC's in Fig.1 suggest that there is no significant heating\neffect in our mesas. Indeed, overheating should cause the\nreduction of $V_g$. Instead, we observed that the normalized IVC's\nmerge into a single curve with\nidentical $V_g$, despite a considerable difference in the dissipated power, $%\nP\\propto S$. Moreover, we have checked that the voltages of the\nquasiparticle branches scale with their number, $V_n\\simeq \\frac\nnNV_N$, where $V_N$ is the top branch, with all IJJ's in the\nresistive state. Thus, switching of additional IJJ's does not\ncause visible overheating of the mesa. On the other hand, we did\nobserve a strong overheating for even\nsmaller mesas (2$\\times 4$ $\\mu $m$^2$), containing larger number of IJJ's $%\n(\\sim 100)$, so that a clear back-banding was seen at large\ncurrents, and significantly lower $V_g$ was obtained, similar to\nthat in Refs.\\cite {Yurgens,Latysh}. Therefore, a small number of\nIJJ's in the mesa decreases the risk of overheating.\n\nIn Fig.2, the conductance at different temperatures is shown for\none of the samples. The sharp peak at $V_g$ and the depletion of\nconductance at $\\left| V\\right| <V_g$ is seen at low $T$,\nrepresenting the superconducting gap in the DOS. The suppression\nof DOS below the gap results in strong temperature dependence of\n$R(0)$, see inset b) in Fig.1. With increasing temperature, the\npeak at $V_g$ shifts to lower voltages and decreases in magnitude.\nAt $T\\sim $80 K, the peak is smeared out completely and only\nsmooth\ndepletion of the conductance remains at $V$=0. With the further increase of $%\nT$, this depletion gradually decreases, but is still visible even at room\ntemperature, representing the pseudo-gap in DOS. In agreement with the\nsurface tunneling experiments\\cite{Renner,Wilde}, there is almost no changes\nin the conductance at $T_c$, which implies that the pseudo-gap coexists with\nthe superconductivity.\n\n\\begin{figure}[hbt]\n\\centering \\epsfig{file=fig2.eps,width=0.9\\columnwidth}\n\\caption{Conductance at different temperatures for $6 \\times 6$\n$\\mu$m$^2$ mesa.} \\label{fig2}\n\\end{figure}\n\nThere is a crucial difference between our, ''intrinsic'',\nand the surface tunneling experiments\\cite{Renner,Wilde}, which\nallows us to distinguish the superconducting gap from the pseudo\ngap. The difference is in existence of multiple quasiparticle\nbranches in IVC's, see Fig.1. Therefore, we have an additional\nquantity, the spacing between the quasiparticle branches, $\\Delta\nV$, which can be used for estimation of $\\Delta $. Even though the\npeak in conductance is smeared out\nat $T>$80 K, the quasiparticle branches in the IVC's remain well defined up to $%\nT_c$. The $\\Delta V$ continuously decreases with increasing $T$\nand vanishes exactly at $T_c$, as shown in inset a) in Fig.1.\n\nThis brings us to conclusion that the superconducting gap does\nclose at $T_c$, in contrast to the statement of Ref.\\cite{Renner}.\nOn the other hand, the pseudo gap is almost\nindependent of $T$, in agreement with \\cite {Renner,Wilde}. The pseudo\ngap can exist well above $T_c$ and, probably, can coexist with the\nsuperconducting gap even at $T<T_c$, see Fig.2. We can not\nconclude that the superconducting gap is developed from the pseudo\ngap nor that they are competing with each other. From our\nexperiment, it seems more natural to assume that those two gaps\nare independent or only weakly dependent, despite having the same\norder of magnitude. In Ref. \\cite{Halbritt} possible scenarios of\nthe pseudo gap were reviewed. One of the possible mechanisms is\ndue to Coulomb charging effect in IJJ's. Some experimental\nevidence for that was obtained in \\cite{Latysh}. For the smallest\nmesas, we have also seen certain features, such as a complete\nsuppression of the critical current and an offset voltage in\nIVC's, which may be explained in terms of the Coulomb charging\neffect. However, further study is necessary before making a\ndecisive conclusion about the origin of the pseudo-gap.\n\n\\begin{thebibliography}{9}\n\\bibitem{Renner} Ch.Renner, et al, Phys.Rev.Lett. \\textbf{80}(1998) 149\n\n\\bibitem{Wilde} Y.DeWilde, et al, Phys.Rev.Lett. \\textbf{80}(1998) 153\n\n\\bibitem{Yurgens} A.Yurgens,et al, Physica C \\textbf{235-240}(1994)3269\n\n\\bibitem{Suzuki} M.Itoh, et al, Phys.Rev.B \\textbf{55} (1997) R12001\n\n\\bibitem{Latysh} Yu.I.Latyshev, et al, JETP Letters \\textbf{69}(1999) 84\n\n\\bibitem{Halbritt} J.Halbritter, Proc.SPIE \\textbf{3480} (1998) 222\n\\end{thebibliography}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002094.extracted_bib",
"string": "\\begin{thebibliography}{9}\n\\bibitem{Renner} Ch.Renner, et al, Phys.Rev.Lett. \\textbf{80}(1998) 149\n\n\\bibitem{Wilde} Y.DeWilde, et al, Phys.Rev.Lett. \\textbf{80}(1998) 153\n\n\\bibitem{Yurgens} A.Yurgens,et al, Physica C \\textbf{235-240}(1994)3269\n\n\\bibitem{Suzuki} M.Itoh, et al, Phys.Rev.B \\textbf{55} (1997) R12001\n\n\\bibitem{Latysh} Yu.I.Latyshev, et al, JETP Letters \\textbf{69}(1999) 84\n\n\\bibitem{Halbritt} J.Halbritter, Proc.SPIE \\textbf{3480} (1998) 222\n\\end{thebibliography}"
}
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cond-mat0002095
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Influence of center-of-mass correlations on spontaneous emission and Lamb shift in dense atomic gases$^1$
|
[
{
"author": "M.~Fleischhauer\\qquad %"
},
{
"author": "Roger Temam\\inst{2} %"
},
{
"author": "Jeffrey Dean\\inst{2} %"
},
{
"author": "David Grove\\inst{1} %"
},
{
"author": "Craig Chambers\\inst{2} %"
},
{
"author": "Kim~B.~Bruce\\inst{2} %"
},
{
"author": "Elsa Bertino\\inst{1}"
}
] |
Local field effects on the rate of spontaneous emission and Lamb shift in a dense gas of atoms are discussed taking into account correlations of atomic center-of-mass coordinates. For this the exact retarded propagator in the medium is calculated in independent scattering approximation and employing a virtual-cavity model. The resulting changes of the atomic polarizability lead to modifications of the medium response which can be of the same order of magnitude but of opposite sign than those due to local field corrections of the dielectric function derived by Morice, Castin, and Dalibard [Phys.Rev.A {51}, 3896 (1995)]. \footnote{This paper is dedicated to the memory of Dan Walls}
|
[
{
"name": "Walls4.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% This is a sample input file for your contribution to a multi-\n% author book to be published by Springer-Verlag.\n%\n% Please use it as a template for your own input, and please\n% follow the instructions for the formal editing of your\n% manuscript as described in the file \"1readme\".\n%\n% Please send the Tex and figure files of your manuscript\n% together with any additional style files as well as the\n% PS file to the editor of your book.\n%\n% He or she will collect all contributions for the planned\n% book, possibly compile them all in one go and pass the\n% complete set of manuscripts on to Springer.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n%RECOMMENDED%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\documentclass[runningheads,fleqn]{cl2emult}\n\n\\usepackage{makeidx} % allows index generation\n\\usepackage{graphicx} % standard LaTeX graphics tool\n % for including eps-figure files\n\\usepackage{subeqnar} % subnumbers individual equations\n % within an array\n\\usepackage{multicol} % used for the two-column index\n\\usepackage{cropmark} % cropmarks for pages without\n % pagenumbers\n\\usepackage{phys} % flushleft layout of math and captions\n\\makeindex % used for the subject index\n % please use the style sprmidx.sty with\n % your makeindex program\n\n%upright Greek letters (example below: upright \"mu\")\n\\newcommand{\\euler}[1]{{\\usefont{U}{eur}{m}{n}#1}}\n\\newcommand{\\eulerbold}[1]{{\\usefont{U}{eur}{b}{n}#1}}\n\\newcommand{\\umu}{\\mbox{\\euler{\\char22}}}\n\\newcommand{\\umub}{\\mbox{\\eulerbold{\\char22}}}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n%OPTIONAL%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\n%\\usepackage{amstex} % useful for coding complex math\n%\\mathindent\\parindent % needed in case \"Amstex\" is used\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%AUTHOR_STYLES_AND_DEFINITIONS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\n%Please reduce your own definitions and macros to an absolute\n%minimum since otherwise the editor will find it rather\n%strenuous to compile all individual contributions to a\n%single book file\n%\n\\newcommand{\\E}{{\\mathrm e}}\n\\newcommand{\\I}{{\\mathrm i}}\n\\newcommand{\\D}{{\\mathrm d}}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{document}\n%\n\\title*{Influence of center-of-mass correlations on\nspontaneous emission and Lamb shift in dense atomic gases$^1$}\n% \n%\n%\\toctitle{All-order analytic solutions for a mirrorless\n% oscillator based on resonantly enhanced 4-wave mixing\n% \\protect\\newline}\n% allows explicit linebreak for the table of content\n%\n%\n\\titlerunning{Spontaneous emission in dense atomic gases }\n% allows abbreviation of title, if the full title is too long\n% to fit in the running head\n%\n\\author{M.~Fleischhauer\\qquad\n%\\and Roger Temam\\inst{2}\n%\\and Jeffrey Dean\\inst{2}\n%\\and David Grove\\inst{1}\n%\\and Craig Chambers\\inst{2}\n%\\and Kim~B.~Bruce\\inst{2}\n%\\and Elsa Bertino\\inst{1}}\n%\n%\\authorrunning{Ivar Ekeland et al.}\n% if there are more than two authors,\n% please abbreviate author list for running head\n%\n%\n\\institute{Sektion Physik, \nUniversit\\\"at M\\\"unchen, \nD-80333 M\\\"unchen, Germany}\n%\\and Universit\\'{e} de Paris-Sud,\n% Laboratoire d'Analyse Num\\'{e}rique,\n% B\\^{a}timent 425,\\\\\n% F-91405 Orsay Cedex, France \\\\\ne-mail: mfleisch@theorie.physik.uni-muenchen.de}\n\n\\maketitle % typesets the title of the contribution\n\n\n\n\n\\begin{abstract}\nLocal field effects on \nthe rate of spontaneous emission and Lamb shift in a dense gas \nof atoms are discussed taking into account correlations\nof atomic center-of-mass coordinates.\nFor this the exact retarded propagator in the medium is calculated\nin independent scattering approximation and employing a virtual-cavity\nmodel. \nThe resulting changes of the atomic polarizability \nlead to modifications of the\nmedium response which can be of the same order of magnitude \nbut of opposite sign\nthan those due to local field corrections\nof the dielectric function\nderived by Morice, Castin, and Dalibard [Phys.Rev.A {\\bf 51}, 3896 (1995)]. \n\\footnote{This paper is dedicated to the memory of Dan Walls}\n\\end{abstract}\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Introduction}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe experimental progress \nin cooling and trapping of atoms and the observation \nof Bose-Einstein condensation \\cite{BEC,BEC2} in atomic vapors\nhas lead to a growing theoretical interest in the\ninteraction of light with dense atomic gases \n\\cite{opt_prop1,opt_prop2,opt_prop3,opt_prop4}. \nIn the present paper it is analyzed how the spatial distribution of\nnearest neighbors in a dense gas,\ncharacterized by two-particle correlations, \naffects the interaction of an excited atom with the electromagnetic \nfield vacuum. In particular modifications\nof the rate of spontaneous emission and the Lamb \nshift are calculated and the resulting modifications of the\ndielectric function discussed. \n\nWhile the interaction of light with a dilute gas is well described\nin terms of macroscopic quantities \nwith the well-known Maxwell-Bloch equations,\nthis is no longer true when the gas becomes dense. \nHere two new types of effects \narise:\nWhen the density of atoms\nbecomes large enough such that the resonant absorption length is less than\nthe characteristic medium dimension, re-absorption and\nmultiple scattering of spontaneous photons need to be taken into account. \nSecondly with increasing density dipole-dipole interactions \nbetween nearest neighbors become important. Here the macroscopic \npicture of a homogeneous polarization breaks down and it is necessary \nto introduce local field corrections. \n\n\n\nThe most famous local-field correction\nleads to the Lorentz-Lorenz (LL) relation between atomic \npolarizability $\\alpha(\\omega)$ and dielectric function $\\varepsilon(\\omega)$ \n\\cite{LL1,LL2}.\nThe LL-correction removes the\nunphysical contact interaction of two atoms at the same position\nwhich arises in a continuum picture of a homogeneous polarization\nand is independent of any specifics of the atoms. Recently\nMaurice, Castin, and Dalibard \\cite{Maurice95} have derived a \ngeneralization of the LL relation that takes into account\ncenter-of-mass correlations for finite distances \nat which point specific properties of the atomic gas enter. \nThey showed in particular \nthat the tendency of bosonic atoms to bunch,\nwhen the critical temperature of condensation\nis approached, leads to a measurable change of \nthe complex refractive index. Even more pronounced\neffects such as a dramatic line-narrowing were recently predicted by\nRuostekoski and Javanainen for a Fermi gas in the\ncase ideal case \\cite{Ruostekoski99}\nor in the presence of a BEC transition\n\\cite{Ruostekoski99b}.\n\nOn the other hand, it is known \nsince the early work of Purcell \\cite{Purcell46}, that the\nmicroscopic environment of an excited atom can also change its interaction\nwith the field vacuum. Thus local field effects should\nlead to a modification of the atomic polarizability itself,\nin particular the spontaneous emission rate and the Lamb shift. Different \nmacroscopic models for Lorentz-Lorenz-type corrections of \nthe spontaneous emission rate of an atom\nembedded in a {\\it lossless dielectric} have been developed\n\\cite{Nienhuis76,Knoester89,Milonni95,Glauber91,deVries98} \nand experimentally tested \\cite{Rikken95,Schuurmans98}. In the presence\nof losses, as is the case in dense gases of the same kind of atoms, \nmacroscopic \\cite{Scheel99a,Scheel99b} and microscopic \napproaches \\cite{Fleischhauer99b} have indicated however that\nlocal-field effects due to\nnearest neighbors may be equally important as the LL correction\nof the contact interaction. \nIn the present paper I analyze these effects\n using a Greens function approach. \n\n\nAccording to Fermi's golden rule the rate of spontaneous \nemission $\\Gamma$ and the Lamb shift\n$\\Delta$ are given by the\n(regularized) exact retarded propagator ${\\bf D}$ \nof the electric field \nat the position ${\\vec r}_0$ of the probe atom \n\\cite{Fleischhauer99b,Fleischhauer99a,Barnett92}. \n%\n%\n\\begin{eqnarray}\n\\Gamma &=& \\frac{2}{\\hbar^2} \\,\n {\\vec d}\\cdot{\\rm Re}\\Bigl[ {\\bf D} \n ({\\vec r}_0,{\\vec r}_0;\\omega_0)\\Bigr]\\cdot{\\vec d},\\\\\n\\Delta\\, &=& \\frac{1}{\\hbar^2} \\,\n {\\vec d}\\cdot {\\rm Im} \\Bigl[ {\\bf D} \n ({\\vec r}_0,{\\vec r}_0;\\omega_0)\\Bigr]\\cdot{\\vec d}.\n\\end{eqnarray}\n%\n%\n${\\vec d}$ is the dipole vector and $\\omega_0$ the \ntrue transition frequency of the atom. \nFrom the known propagator in free space and after regularisation\none finds for an isolated atom\n%\n%\n\\begin{eqnarray}\n\\Gamma=\\Gamma_0=\\frac{d^2\\omega_0^3}{3\\pi\\hbar\\epsilon_0 c^3},\n\\qquad\\qquad \\Delta=0.\n\\end{eqnarray}\n%\n%\nThe exact retarded propagator in a medium can formally be obtained from a\nscattering series. This series is here calculated \nneglecting multiple scattering of photons by the same atoms and \ntaking into account only two-particle correlations\nof center-of-mass coordinates.\nIt is shown that the alterations of the atomic polarizability\n$\\alpha$ due to local-field corrections of spontaneous emission and\nLamb shift lead to modifications of $\\varepsilon$ which are of the same\norder of magnitude as those found by\nMaurice, Castin and Dalibard \\cite{Maurice95} and Ruostekoski and Javanainen \n\\cite{Ruostekoski99,Ruostekoski99b}.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Scattering series for the retarded propagator}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nA dense medium affects the interaction of an excited probe atom with the\nsurrounding electromagnetic vacuum by multiple scattering\nof virtual photons emitted and re-absorbed \nby that atom. These scattering processes can formally be described by the\nexact retarded Greens-function (GF) \n%\n%\n\\begin{equation}\n{\\bf D}({\\vec r}_1,{\\vec r}_2 ;\\tau)\n=\\theta(\\tau)\\bigl\\langle 0 \\bigr\\vert \n\\bigl[{\\hat{\\vec E}}(\\vec r_1,t_1),\n{\\hat{\\vec E}}(\\vec r_2,t_2)\\bigr] \n\\bigl\\vert 0\\bigr\\rangle,\n\\end{equation}\n%\n%\nwhere $\\tau=t_1-t_2$ and\n${\\hat{\\vec E}}$\nis the operator of the electric field interacting with all atoms. \nThe free-space or vacuum GF in the frequency domain is given by\n\\cite{Pauli,deVries98b}\n%\n%\n\\begin{eqnarray}\n{\\bf G}^0(\\vec x,\\omega) &=&\n-k^2\\,\n\\frac{{\\rm e}^{\\I (k+\\I 0) x}}{4\\pi x}\\left[\nP\\left(\\I k x\\right)\\, {\\bf 1} \n+Q\\left(\\I k x\\right)\\frac{{\\vec x}\\circ{\\vec x}}{x^2}\n\\right] +\\frac{1}{3} \\delta(\\vec x)\\, {\\bf 1},\n\\label{D0coord}\n\\end{eqnarray}\n%\n%\nwhere ${\\bf D}^0=\\I\\hbar {\\bf G}^0/\\epsilon_0$. \nHere $k=\\omega/c$, $x=|\\vec x|$ and\n%\n\\begin{eqnarray}\nP(z) = 1-\\frac{1}{z} +\\frac{1}{z^2},\\qquad\nQ(z) = -1 + \\frac{3}{z} -\\frac{3}{z^2}.\\label{P-Q}\n\\end{eqnarray}\n%\n%\n${\\bf G}^0({\\vec x},\\omega)$ diverges as $x\\to 0$ which is\nrelated to the large-q behavior in reciprocal space. \nThis will lead to corresponding divergences in the exact propagator\n${\\bf G}({\\vec x},\\omega)$ which can however be removed \nby introducing \na regularisation ${\\bf G}({\\vec q},\\omega)\\to {\\bf G}({\\vec q},\\omega) \n\\, f(\\Lambda,q)$ with a wave-number cut-off $\\Lambda$.\nFor the purpose of \nthe present paper I will assume that the large-q behavior\nof the Greens function is properly regularized and ignore all\ncontributions containing the regularisation parameter $\\Lambda$.\nThe subject of regularisation will be discussed in more detail at a\ndifferent place.\n\n\nThe net effect of all possible \nmultiple scattering events can be described by \na scattering series for the exact propagator. \nWe here assume not too large densities, such that\ndependent scattering can be neglected. I.e. there can be\nas many scattering events as possible but never twice from the same atom. \nIn this so-called independent\nscattering approximation (ISA), the scattering series \ncan be expressed in the form\n%\n%\n\\begin{eqnarray}\n&&{\\bf G}(0,\\omega) =\n{\\bf G}^0(0,\\omega) -\n\\sum_{i\\ne 0} {\\bf G}^0({\\vec x}_{0i},\\omega)\n\\cdot{\\bf a}_i(\\omega) \\cdot{\\bf G}^0({\\vec x}_{i0},\\omega)\n+\\nonumber\\\\\n&&\\quad +\\sum_{i\\ne j\\ne 0}\n{\\bf G}^0({\\vec x}_{0i},\\omega)\\cdot{\\bf a}_i(\\omega)\\cdot\n{\\bf G}^0({\\vec x}_{ij},\\omega)\\cdot {\\bf a}_j(\\omega)\\cdot \n{\\bf G}^0({\\vec x}_{j0},\\omega) +\\cdots,\n\\end{eqnarray}\n%\n%\nwhere ${\\bf a}_j(\\omega)$ is the polarizability tensor of the $j$th atom\nof the host material,\nthe summation is over all atomic positions, and \n${\\vec x}_{ij}={\\vec r}_i-{\\vec r}_j$.\nIt should be noted that the polarizability of the {\\it excited}\nprobe atom does not enter the scattering series. \nHowever, the probe atom can affect the\nspatial distribution of the surrounding scatterers and\nit is necessary to keep track of its presence.\n\n\nWe now assume a homogeneous medium of density $\\varrho$\nwith randomly oriented two-level atoms \nsuch that $a_i(\\omega)=\\alpha(\\omega)\\, {\\bf 1}$.\nIn this case we may replace the\nsums over atomic positions \nby integrals. For this we introduce normalized joint\nprobabilities $p_2({\\vec r}_0,{\\vec r}_1)$, \n$p_3({\\vec r}_0,{\\vec r}_1,{\\vec r}_2)$ etc. \nto find one atom at position ${\\vec r}_1$ and\nthe probe atom at ${\\vec r}_0$; to find two\natoms at positions ${\\vec r}_1$ and ${\\vec r}_2$\nand the probe atom at ${\\vec r}_0$ etc, with $p_1({\\vec r})=1$.\nThis leads to\n%\n%\n\\begin{eqnarray}\n&&{\\bf G}(0) = \n{\\bf G}^0(0) -\\varrho{\\alpha}\\int \\!\\D^3{\\vec r}_i\\enspace\np_2({\\vec r}_0,{\\vec r}_i)\\, {\\bf G}^0({\\vec x}_{0i})\n\\cdot{\\bf G}^0({\\vec x}_{i0})\n+\\nonumber\\\\\n&& + \\varrho^2{\\alpha}^2\\int\\!\\!\\int\\! \\D^3{\\vec r}_i\\, \\D^3{\\vec r}_j\\enspace\np_3({\\vec r}_0,{\\vec r}_i,{\\vec r}_j)\n\\,{\\bf G}^0({\\vec x}_{0i})\\cdot\n{\\bf G}^0({\\vec x}_{ij})\\cdot \n{\\bf G}^0({\\vec x}_{j0}) +\\cdots\\label{scatter}\n\\end{eqnarray}\n%\n%\nwhere we have suppressed the frequency argument for notational simplicity.\n\nFor a dilute gas the positions of the atoms can be treated as independent\nand one can factorize all particle correlations, which\namounts to $p_m\\equiv 1$. \nThe scattering series can then easily be solved.\nThe poles of ${\\bf G}({\\vec q},\\omega)=\n\\int \\!\\D^3{\\vec x}\\, \\E^{\\I {\\vec q}\\cdot{\\vec x}}\\, {\\bf G}({\\vec x},\\omega)$\ndetermine the dielectric function for which one finds the well-known \ndilute-medium result\n%\n%\n\\begin{eqnarray}\n\\varepsilon(k)=\\frac{q_0^2}{k^2}=1+\\varrho\\alpha(\\omega).\\qquad\\qquad\n(\\omega= kc)\n\\end{eqnarray}\n%\n%\nProperly regularizing ${\\bf G}({\\vec q},\\omega)$ and\ntransforming the result back to coordinate space eventually\nyields the spontaneous emission rate and the Lamb shift relative to\nthe vacuum\n%\n\\begin{eqnarray}\n\\Gamma=\\Gamma_0 \\,{\\rm Re}\\,\\bigl[\\sqrt{\\varepsilon}\\bigr],\n\\qquad\\Delta =\n\\frac{\\Gamma_0}{2}\\,{\\rm Im}\\,\\bigl[\\sqrt{\\varepsilon}\\bigr].\n\\end{eqnarray}\n%\nHere $\\varepsilon$ is the dielectric function of the gas at the\ntrue transition frequency.\nFor an atom embedded in a dielectric host with real dielectric function the \nresult for $\\Gamma$ is identical to that obtained by\nNienhuis and Alkemande based on a density-of-states argument \n\\cite{Nienhuis76}. \n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Local field effects and center-of-mass correlations}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn order to obtain a non-perturbative result for the retarded propagator\nin a dense gas, one has to sum all contributions of the \nscattering series (\\ref{scatter}) without factorizing the center-of-mass\ncorrelations. Apart from some special cases such as hard-sphere scatterers\n\\cite{Lagendijk97} it is not possible to bring the scattering series\nin an exact closed form and approximations are needed. \nThe first approximation I use here is to take into\naccount only two-particle correlations by using a Kirkwood-type\nfactorization \\cite{Hansen76} of higher-order contributions\n%\n%\n\\begin{eqnarray}\np_3(1,2,3)=p_2(1,2)\\, p_2(2,3)\\, p_2(1,3),\\quad {\\rm etc.}\n\\end{eqnarray}\n%\n%\nFurthermore it is assumed that only \ncorrelations between successive scatterers\nmatter, which is however correct up to second order in the density. \nWith this the scattering series (\\ref{scatter}) can be represented in\nthe form\n%\n%\n\\begin{eqnarray}\n{\\bf G}(0) &=& \n{\\bf G}^0(0) -\\varrho{\\alpha}\\, {\\bf H}^0({\\vec x}_{0i})\n\\cdot{\\bf G}^0({\\vec x}_{i0})+\\nonumber\\\\\n&&+\\varrho^2{\\alpha}^2 \n\\,{\\bf H}^0({\\vec x}_{0i})\\cdot\n{\\bf H}^0({\\vec x}_{ij})\\cdot \n{\\bf H}^0({\\vec x}_{j0}) -\\label{scatter2}\\\\\n&&-\\varrho^3\\alpha^3\\, {\\bf H}^0({\\vec x}_{0i})\\cdot{\\bf H}^0({\\vec x}_{ij})\\cdot {\\bf H}^0({\\vec x}_{jk})\\cdot {\\bf H}^0({\\vec x}_{k0})+\\cdots,\n\\nonumber\n\\end{eqnarray}\n%\n%\nwhere the spatial integration has been suppressed and\n%\n%\n\\begin{eqnarray}\n{\\bf H}^0({\\vec r}_1,{\\vec r}_2,\\omega) = \np_2({\\vec r}_1,{\\vec r}_2)\\,{\\bf G}^0({\\vec r}_1,{\\vec r}_2,\\omega)\n\\end{eqnarray}\n%\n%\nis the retarded propagator modified by the two-particle correlation\n$p_2$. Note that the first order term contains only a single function\n${\\bf H}^0$.\n\n\nSince two-particle correlations between atoms of the host\nmaterial can be different from correlations \nbetween the (excited) probe atom and a host atom, \nwe will distinguish these two in the following. \nThis also includes the case of an atomic impurity in\nan environment of a different species.\nThe scattering series can then be written in the form\n%\n%\n\\begin{eqnarray}\n&&{\\bf G}(0,\\omega)={\\bf G}^0(0,\\omega)-\n\\varrho\\alpha \\int\\!\\!\\!\\D^3{\\vec r}_1\n{\\bf H}^0_{e}({\\vec r}_0,{\\vec r}_1,\\omega)\\cdot \n{\\bf G}^0({\\vec r}_1,{\\vec r}_0,\\omega)+\n\\nonumber\\\\\n&&\\quad+\\int\\!\\!\\!\\int\\!\\!\\D^3{\\vec r}_1\\,\\D^3{\\vec r}_2\\,\n{\\bf H}^0_{e}({\\vec r}_0,{\\vec r}_1,\\omega)\\cdot \n{\\bf T}^{(2)}({\\vec r}_1,{\\vec r}_2,\\omega)\n\\cdot{\\bf H}^0_{e}({\\vec r}_2,{\\vec r}_0,\\omega),\\label{HDyson}\n\\end{eqnarray}\n%\n%\nwhere ${\\bf T}^{(2)}({\\vec r}_1,{\\vec r}_2,\\omega)$ is the \npart of the scattering matrix that contains at least two\nscattering processes in the host material.\n${\\bf H}^0_e$ is the free propagator modified by the correlation\nof the excited probe atom with an atom of the background gas. \nAlthough in ISA the ${\\bf T}$-matrix does not contain scattering events from\nthe probe atom, it would in general still depend on its presence through \nthe position correlations. With the earlier assumption that only \ncorrelations between successive scatterers matter, this dependence is lost.\n I.e. we treat the scattering of photons\nin the gas as if the place of the probe atom would be filled\nwith the host material, which is equivalent to the \n{\\it virtual cavity model} of Knoester and\nMukamel \\cite{Knoester89}.\n From the discussion of impurities \nin cubic dielectric host materials by deVries\nand Lagendijk \\cite{deVries98} it is however \nexpected that this approximation\ndoes not affect the results in leading order of the density. \nFinally we will restrict ourselves to the case of a\nhomogeneous and isotropic gas, such that the two-particle correlation \ndepends only on the distance between the atoms\n$p_2({\\vec r}_1,{\\vec r}_2)\n=p_2(x_{12})$ where $x_{12}=|{\\vec r}_1-{\\vec r}_2|$.\nIn this case the scattering matrix obeys a simple Dyson equation in\nreciprocal space\n%\n\\begin{eqnarray}\n{\\bf T}({\\vec q},\\omega)=\\varrho\\alpha(\\omega){\\bf H}^0_{g}({\\vec q},\\omega)\n\\varrho\\alpha(\\omega) - \\varrho\\alpha(\\omega){\\bf H}^0_{g}({\\vec q},\\omega)\n\\cdot{\\bf T}({\\vec q},\\omega)\n\\label{T}\n\\end{eqnarray}\n%\n%\n${\\bf H}^0_{g}$ is the free propagator modified by the\ntwo-particle correlations of the host material.\n\n\nIt is convenient at this point to introduce the\nirreducible correlation $h_2^\\mu$ according to $p_2^\\mu=1+h_2^\\mu$. \nOne then has\n%\n%\n\\begin{equation}\n{\\bf H}^0_\\mu({\\vec q},\\omega) = -\\frac{\\bigl(\\frac{1}{3} q^2 \n+\\frac{2}{3}k^2\\bigr)\\, {\\bf 1}\n-{\\vec q}\\circ{\\vec q}}\n{q^2-k^2-\\I 0}+f_1^\\mu(q,\\omega)\\, {\\bf 1} +f_2^\\mu(q,\\omega) \\, \n\\frac{{\\vec q}\\circ{\\vec q}}{q^2},\\label{H0}\n\\end{equation}\n%\n%\nwith\n%\n%\n\\begin{eqnarray}\nf_1^\\mu(q,\\omega) &=& -k^2\\int_0^\\infty\\!\\!\\D x\\, x \\E^{\\I k x}\\, h_2^\\mu(x)\n\\Bigl[j_0(qx) P(\\I kx) + \\frac{j_1(qx)}{qx} Q(\\I kx)\\Bigr],\\\\\nf_2^\\mu(q,\\omega) &=& k^2\\int_0^\\infty\\!\\!\\D x\\, x \\E^{\\I k x}\\, h_2^\\mu(x)\n\\, j_2(qx)\\, Q(\\I kx).\n\\end{eqnarray}\n%\n%\n$j_n(z)$ are spherical Bessel functions and $P$ and $Q$ have been defined\nin eq.(\\ref{P-Q}). \nIn the first term of eq.(\\ref{H0}) we have made use of \n$p_2^\\mu({\\vec x})\\, \\delta({\\vec x})=0$ which corresponds to the\nLL correction of the contact interaction.\nThe solution of eq. (\\ref{T}) is now easy to obtain.\nThe poles $q_0$ of ${\\bf T}$ which determine the dielectric function \nfollow from the equation\n%\n%\n\\begin{equation}\nq_0^2-k^2-\\varrho \n\\alpha(\\omega)\\left(\\frac{1}{3}q_0^2+\\frac{2}{3}k^2-f_1^g(q_0,\\omega)\n(q_0^2-k^2)\\right)-\\I 0=0.\n\\end{equation}\n%\n%\nSince in lowest order of $\\varrho\\alpha$ one has\n $q_0^2\\approx k^2+\\I 0$, we may replace $f_1^g(q_0,\\omega)$\nby $f_1^g(k,\\omega)$ which yields\n%\n%\n\\begin{equation}\n\\varepsilon(k)=\\frac{q_0^2}{k^2}=1+\\frac{\\varrho\\alpha(\\omega)}\n{1-\\varrho\\alpha(\\omega)/3+\\varrho\\alpha(\\omega)f_1^g(k,\\omega)}.\n\\label{epsilon}\n\\end{equation}\n%\n%\nThis result is identical to that \nof Maurice, Castin and Dalibard \\cite{Maurice95} in ISA.\nIt is interesting to note that although the free GF ${\\bf H}^0$\ncontained also longitudinal components (proportional to $f_2^g\n{\\vec q}\\circ{\\vec q}/\nq^2$), they exactly cancel in the expression for the dielectric function. \n\n\nTo obtain the spontaneous emission rate and Lamb shift, we\nconsider only the leading order corrections in the density \nwhere we can replace $f_{1,2}^\\mu(q,\\omega)$ by \n$f_{1,2}^\\mu(k,\\omega)$ and introduce a\nregularisation of the large-q behavior. This yields\nafter some algebra\nfor the orientation averaged retarded GF:\n%\n%\n\\begin{eqnarray}\nG({\\vec x}=0,\\omega)&=& -\\frac{\\I k^3}{6\\pi}\n\\, \\Biggl[1+\\varrho\\alpha\\,\\biggl(\n\\frac{7}{6} - f_1^e\\biggr)+\\nonumber\\\\\n&&\n+\\varrho^2\\alpha^2\\, \\biggl(\\frac{17}{24}-\\frac{7}{3} f_1^e\n+{f_1^e}^2-\\frac{7}{6} f_1^g \n+2f_1^ef_1^g\\biggr)+\\cdots\\Biggr]\n\\end{eqnarray}\n%\n%\nOne recognizes that there is again no \ncontribution from the longitudinal terms\n$f_2$ up to second order in $\\varrho$. \nFurthermore the two-particle correlation $f_1^g$ between ground-state\natoms enters only in second order of the density, while there\nis a first-order contribution from $f_1^e$. This is physically \nintuitive since correlations between ground-state atoms\nenter only after two scattering events, while correlations\ninvolving the probe atom are already important\nin first order. \nFrom the above results one finds in leading order of the density\n%\n\\begin{eqnarray}\n\\Gamma&=&\\Gamma_0\\,\\biggl[1+\\frac{7}{6}\\varrho\\alpha'\n- \\varrho (\\alpha'{f_1^e}'-\\alpha''{f_1^e}'')+\\cdots\\biggr],\\label{Gamma_f}\\\\\n\\Delta&=&\\frac{\\Gamma_0}{2}\n\\,\\biggl[\\frac{7}{6}\\varrho\\alpha''- \\varrho(\\alpha''{f_1^e}'+\n\\alpha'{f_1^e}'')+\\cdots\\biggr]\\label{Delta_f}\n\\end{eqnarray}\n%\nwhere we have introduced the real and imaginary parts of $f_1^e=\n{f_1^e}'+\\I {f_1^e}''$.\nEqs.(\\ref{Gamma_f}) and (\\ref{Delta_f}) are the main result of the\npresent paper. The first-order corrections to spontaneous emission\nrate and Lamb shift, which are independent on the $f$'s are the\nLorentz-Lorenz local field corrections derived in\n\\cite{Fleischhauer99b}.\n\nIn order to illustrate the implications \nof eqs.(\\ref{Gamma_f}) and (\\ref{Delta_f}) to \nthe medium response, I will discuss in the following section the\ndielectric function of a dense gas of two-level atoms using some\nsimple model functions for the center-of-mass correlations.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Modifications of medium response}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe have shown that local field corrections change not only\nthe relation between\natomic polarizability and dielectric function of the medium \nbut also \nthe atomic polarizability itself (with respect to the thin-medium case).\nTo illustrate the net effect of both corrections to the\ndielectric function, I now consider a classical, homogeneous gas of \nradiatively broadened (cold) two-level atoms\nwith randomly oriented dipole vectors. The dimensionless\natomic polarizability\nof such atoms is isotropic and has in free space the strength \n%\n%\n\\begin{eqnarray}\n\\overline{\\alpha}\\,\\bigl(\\overline{\\delta}\\, \\bigr)=\n\\alpha\\, k_0^3= \\frac{6\\pi}{\\overline{\\delta} -\\I}\n\\end{eqnarray}\n%\n%\nwhere $k_0$ is the resonance wavenumber, \n$\\overline{\\delta}=(\\omega_{ab}-\\omega)/\\gamma_{ab}$ is the normalized detuning\nfrom the {\\it true} resonance \nand $\\gamma_{ab}=\\Gamma_0/2$ is the free-space dipole decay rate.\nWith this we find in lowest order of the dimensionless \ndensity $\\bar\\varrho=\\varrho/k_0^3$ \n%\n%\n\\begin{eqnarray}\n\\Gamma=\\Gamma_0\\,\\biggl[1\n+6\\pi\\,\n\\bar{\\varrho}\\,{f_1^e}''+\\cdots\\biggr],\n\\qquad\\Delta=\\frac{\\Gamma_0}{2}\n\\,\\biggl[7\\pi\\, \\bar{\\varrho}-6\\pi\\,\\bar{\\varrho}\\, {f_1^e}'\n+\\cdots\\biggr],\\label{Delta_2NS}\n\\end{eqnarray}\n%\n%\nSubstituting these expressions into the dielectric function,\neq.(\\ref{epsilon}),\n one eventually\nfinds for the shift relative to the dilute-medium resonance\nand the effective linewidth up to first order in ${\\bar\\varrho}$:\n%\n%\n\\begin{eqnarray}\n\\gamma_{\\rm eff}&=&\\frac{\\Gamma_0}{2}\n\\Bigl[1- 6\\pi\\, \\bar{\\varrho}\\,{f_1^g}''+ 6\\pi \\bar{\\varrho}\\,{f_1^e}''\n+\\cdots\\Bigr],\\\\\n\\Delta_{\\rm eff} &=& \\frac{\\Gamma_0}{2}\n\\Bigl[-2\\pi\\,\\bar{\\varrho} + 6\\pi\\,\\bar{\\varrho}\\, {f_1^g}'\n +7\\pi\\, \\bar{\\varrho}\n- 6\\pi\\,\\bar{\\varrho}\\, {f_1^e}'\n+\\cdots\\Bigr]\\label{Delta_eff}\n\\end{eqnarray}\n%\n%\nThe second term in the \nexpression for the linewidth \nis due to local field corrections of the \ndielectric function and the third one due\nto the changed spontaneous emission rate. \nIt is interesting to note that both contributions\nare of the same order of magnitude but differ in sign. Thus \nlocal field effects to the vacuum interaction may compensate\nthe line-narrowing / broadening effects \nresulting from local field corrections\nof the dielectric function in lowest order of the density.\nIn the expression for the line-shift, eq.(\\ref{Delta_eff}), one recognizes the\nfamiliar Lorentz-Lorenz shift $-2\\pi\\bar\\rho$. The second \n term emerges again from local field corrections of the \ndielectric function and the two last terms are due\nto modifications of the Lamb shift.\n\nTo illustrate the effect of the center-of-mass correlations\nlet us consider a gas with repulsive interaction such that \nthe two-particle correlation $h_2$ is close to $-1$ over \nan effective correlation distance $z$ and then approaches \nzero. As simple model functions\nwe use a Gaussian and a hyper-Gaussian\n%\n\\begin{eqnarray}\nh_2^{(a)}(x)=-\\exp\\left[-\\bar{x}^2/\\bar{z}^2\\right]\\quad{\\rm and}\\quad\nh_2^{(b)}(x)=-\\exp\\left[-\\bar{x}^8/\\bar{z}^8\\right]\n\\end{eqnarray}\n%\nwhere the distance $\\bar{x}= x/\\lambda$ and \nthe correlation length $\\bar{z}=z/\\lambda$ are normalized to the resonance \nwavelength $\\lambda$.\nI have plotted in the following figures the real and imaginary parts\nof $f_1$ for both correlations. It is worth noting that for \nvalues of the correlation length \nlarger than the resonance wavelength, \n$f'\\rightarrow 7/12$, while $f''$ becomes a linear\nfunction of $\\bar{z}$. \n%\n%\n\\begin{figure}\n%\\sidecaption\n\\includegraphics[width=.95\\textwidth]{Walls_fig.eps}\n\\caption[width=.8\\textwidth]\n{Real ($f'$) and imaginary parts ($f''$) of $f$ \nas function of normalized correlation length\n$\\bar {z}=z/\\lambda$ for Gaussian\nand hyper-Gaussian correlation functions $h_2^{(a)}$ and\n$h_2^{(b)}$}\n\\end{figure}\n%\n%\n\nIf the correlation length \n$\\bar{z}$ scales with the density according to \n$\\bar{z}\\sim{\\bar\\rho}^{-1/3}$ \nthe above behavior can be associated to the density dependence\nof the correction terms.\nIn the following plots the real and imaginary parts of $\\bar{z}^{-3} f_1\n\\sim \\bar\\varrho f_1$ are shown as function of $\\bar{z}^{-3}\\sim\n\\bar\\varrho$.\n%\n%\n\\begin{figure}\n%\\sidecaption\n\\includegraphics[width=.95\\textwidth]{Walls_fig2.eps}\n\\caption[width=.8\\textwidth]\n{Real and imaginary parts of ${\\bar z}^{-3} f$ \nas function of \n$\\bar {z}^{-3}\\sim\\bar\\varrho$ for Gaussian\nand hyper-Gaussian correlation functions $h_2^{(a)}$ and\n$h_2^{(b)}$}\n\\end{figure}\n%\n%\nOne recognizes that $\\bar\\varrho f_1''$\nscales for small $\\bar\\varrho$ as $\\bar\\varrho^{2/3}$, while\n$\\bar\\varrho f_1'$ is approximately linear in the density.\nThus in the\nlow-density limit linewidth changes will dominate line shifts.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{summary}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn the present paper local field effects on spontaneous emission and\nLamb shift in a dense atomic gas have been discussed taking into account\ntwo-particle center-of-mass correlations. It has been shown that the\ncorresponding changes of the atomic polarizability can lead to\nmodifications of the medium response, which are of the same order as \nthose resulting from direct corrections of the dielectric function\nfound by Maurice, Castin and Dalibard \\cite{Maurice95}\nand Ruostekoski and Javanainen \\cite{Ruostekoski99}.\nThey are however of opposite sign\nand thus may compensate the leading order corrections in the density. \nThe present approach is based on an independent scattering approximation \nand a virtual cavity assumption. \nIt is thus only applicable for densities which are sufficiently \nsmaller than the cubic\nwavenumber, $\\bar\\varrho\\ll 1$.\nAlso the existence of a distinguished probe atom has been assumed, which\nis valid only for classical gases. Nevertheless the results indicate\nthat local field corrections to the atomic polarizability\nmay change or even reverse the predicted line-shifts and\nlinewidth modifications found for Bose gases near the\ncondensation temperature and for low-temperature Fermi gases. \n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\addcontentsline{toc}{section}{References}\n\n\\begin{thebibliography}{99}\n \n\n\\bibitem{BEC} M. H. Anderson {\\it et. al}, Science {\\bf 269}, 198 (1995);\nK. B. Davies {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 3969 (1995);\nC. C. Bradley {\\it et al.}, {\\it ibid}, {\\bf 75}, 1687 (1995).\n\n\n\\bibitem{BEC2} for a review of theoretical work on BEC in \natomic vapors\n see: A. S. Parkins and D. F. Walls, Phys. Rep. {\\bf 303}, 1 (1998).\n\n\\bibitem{opt_prop1} \nB. Svistunov and G. Shlyapnikov, Zh. Eksp. Teor. Fiz.\n{\\bf 97}, 821 (1990), {\\it ibid} {\\bf 98}, 129 (1990) [Sov. Phys.\nJETP {\\bf 70}, 460 (1990), Sov. Phys.\nJETP {\\bf 71}, 71 (1990)].\n\n\\bibitem{opt_prop2} H. Politzer, Phys. Rev. A {\\bf 43}, 6444 (1991).\n\n\\bibitem{opt_prop3} M. Lewenstein and Li You, Phys. Rev. Lett. \n{\\bf 71}, 1339 (1993); Li You, M. Lewenstein and J. Cooper,\nPhys. Rev. A {\\bf 50}, R3565 (1994);\nM. Lewenstein {\\it et al.}, Phys. Rev. A {\\bf 50}, 2207 (1994);\n\n\\bibitem{opt_prop4}\nJ. Javanainen, Phys. Rev. Lett. {\\bf 75}, 1927 (1995);\nJ. Javanainen and J. Ruostekoski, Phys. Rev. A. {\\bf 52}, 3033 (1995).\n\n\n\\bibitem{LL1} H.~A.~Lorentz, Wiedem. Ann. {\\bf 9}, 641 (1880); L. Lorenz,\nWiedem. Ann. {\\bf 11}, 70 (1881); \n\n\\bibitem{LL2} L. Onsager, J. Am. Chem. Soc.\n{\\bf 58}, 1486 (1936);\nC. J. F. B\\\"ottcher, {\\it Theory of electric polarization}\n(Elsevier, Amsterdam, 1973);\nM.~Born and E.~Wolf {\\it \nPrinciples of Optics}, (Wiley, New York, 1975);\nJ. van Kronendonk and J. E. Sipe, in {\\it Progress in Optics},\ned. by E. Wolf (North-Holland, Amsterdam, 1977) Vol. XV;\nJ. T. Manassah, Phys. Rep. {\\bf 101}, 359 (1983);\n C.~M.~Bowden and J.~Dowling, Phys.~Rev.~A {\\bf 47}, \n1247 (1993); {\\it ibid} {\\bf 49}, 1514 (1994). \n\n\n\\bibitem{Maurice95} O. Maurice, Y. Castin, and J. Dalibard, \nPhys. Rev. A {\\bf 51}, 3896 (1995).\n\n\\bibitem{Ruostekoski99} J. Ruostekoski and J. Javanainen,\nPhys. Rev. Lett. {\\bf 82}, 4741 (1999).\n\n\\bibitem{Ruostekoski99b} J. Ruostekoski, preprint: cond-mat/9908276\n\n\\bibitem{Purcell46} E. M. Purcell, Phys. Rev. {\\bf 69}, 681 (1946).\n\n\\bibitem{Nienhuis76} G. Nienhuis and C. Th. J. Alkemade, Physica\n{\\bf 81C}, 181 (1976).\n\n\\bibitem{Knoester89} J. Knoester and S. Mukamel, Phys. Rev. A {\\bf 40},\n7065 (1989).\n\n\n\n\\bibitem{Milonni95} P. W. Milonni, J. Mod. Optics {\\bf 42}, 1991 (1995).\n\n\n\\bibitem{Glauber91} R. J. Glauber and M. Lewenstein, \nPhys. Rev. A {\\bf 43}, 467 (1991).\n\n\n\\bibitem{deVries98} P. de Vries and A. Lagendijk, Phys. Rev. Lett.\n{\\bf 81}, 1381 (1998).\n\n\n\\bibitem{Rikken95} G. L. J. A. Rikken, and Y. A. R. R. Kessener,\nPhys. Rev. Lett. {\\bf 74}, 880 (1995).\n\n\\bibitem{Schuurmans98} Frank J. P. Schuurmans, D. T. H. de Lang,\nG. H. Wegdam, R. Spirk, and A. Lagendijk, Phys. Rev. Lett. {\\bf 80},\n5077 (1998).\n\n\n\n\\bibitem{Scheel99a} S. Scheel, L. Kn\\\"oll, D.-G. Welsch, S.M. Barnett: \nPhys. Rev. A {\\bf 60} (1999) 1590.\n\n\\bibitem{Scheel99b} S. Scheel, L. Kn\\\"oll, D.-G. Welsch:\nPhys. Rev. A, {\\bf 60} (1999) 4094.\n\n\\bibitem{Fleischhauer99b} M. Fleischhauer, Phys. Rev. A \n{\\bf 60}, \n2534 (1999).\n\n\\bibitem{Fleischhauer99a} M. Fleischhauer and S. F. Yelin, \nPhys. Rev. A {\\bf 59}, 2427 (1999).\n\n\n\\bibitem{Barnett92} S. M. Barnett, B. Huttner, and\nR. Loudon, Phys. Rev. Lett. {\\bf 68}, 3698 (1992).\n\n\\bibitem{Pauli} P.~Jordan and W.~Pauli, Zeit.~f\\\"ur Physik {\\bf 47}, 151\n(1928)\n\n\\bibitem{deVries98b} P. de Vries, D. V. van Coevorden, and A. Lagendijk,\nRew. Mod. Phys. {\\bf 70}, 447 (1998).\n\n\\bibitem{Lagendijk97} Ad Lagendiijk, B. Nienhuis, B. A. van Tiggelen, and\nP. de Vries, Phys. Rev. Lett. {\\bf 79}, 657 (1997).\n\n\\bibitem{Hansen76} J. P. Hansen and I. R. McDonald {\\it Theory of Simple\nLiquids}, (Academic Press, London, 1976).\n\n\n\n\\end{thebibliography}\n\n%INDEX%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\clearpage\n\\addcontentsline{toc}{section}{Index}\n\\flushbottom\n\\printindex\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\\end{document}\n\n\n"
}
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[
{
"name": "cond-mat0002095.extracted_bib",
"string": "\\begin{thebibliography}{99}\n \n\n\\bibitem{BEC} M. H. Anderson {\\it et. al}, Science {\\bf 269}, 198 (1995);\nK. B. Davies {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 3969 (1995);\nC. C. Bradley {\\it et al.}, {\\it ibid}, {\\bf 75}, 1687 (1995).\n\n\n\\bibitem{BEC2} for a review of theoretical work on BEC in \natomic vapors\n see: A. S. Parkins and D. F. Walls, Phys. Rep. {\\bf 303}, 1 (1998).\n\n\\bibitem{opt_prop1} \nB. Svistunov and G. Shlyapnikov, Zh. Eksp. Teor. Fiz.\n{\\bf 97}, 821 (1990), {\\it ibid} {\\bf 98}, 129 (1990) [Sov. Phys.\nJETP {\\bf 70}, 460 (1990), Sov. Phys.\nJETP {\\bf 71}, 71 (1990)].\n\n\\bibitem{opt_prop2} H. Politzer, Phys. Rev. A {\\bf 43}, 6444 (1991).\n\n\\bibitem{opt_prop3} M. Lewenstein and Li You, Phys. Rev. Lett. \n{\\bf 71}, 1339 (1993); Li You, M. Lewenstein and J. Cooper,\nPhys. Rev. A {\\bf 50}, R3565 (1994);\nM. Lewenstein {\\it et al.}, Phys. Rev. A {\\bf 50}, 2207 (1994);\n\n\\bibitem{opt_prop4}\nJ. Javanainen, Phys. Rev. Lett. {\\bf 75}, 1927 (1995);\nJ. Javanainen and J. Ruostekoski, Phys. Rev. A. {\\bf 52}, 3033 (1995).\n\n\n\\bibitem{LL1} H.~A.~Lorentz, Wiedem. Ann. {\\bf 9}, 641 (1880); L. Lorenz,\nWiedem. Ann. {\\bf 11}, 70 (1881); \n\n\\bibitem{LL2} L. Onsager, J. Am. Chem. Soc.\n{\\bf 58}, 1486 (1936);\nC. J. F. B\\\"ottcher, {\\it Theory of electric polarization}\n(Elsevier, Amsterdam, 1973);\nM.~Born and E.~Wolf {\\it \nPrinciples of Optics}, (Wiley, New York, 1975);\nJ. van Kronendonk and J. E. Sipe, in {\\it Progress in Optics},\ned. by E. Wolf (North-Holland, Amsterdam, 1977) Vol. XV;\nJ. T. Manassah, Phys. Rep. {\\bf 101}, 359 (1983);\n C.~M.~Bowden and J.~Dowling, Phys.~Rev.~A {\\bf 47}, \n1247 (1993); {\\it ibid} {\\bf 49}, 1514 (1994). \n\n\n\\bibitem{Maurice95} O. Maurice, Y. Castin, and J. Dalibard, \nPhys. Rev. A {\\bf 51}, 3896 (1995).\n\n\\bibitem{Ruostekoski99} J. Ruostekoski and J. Javanainen,\nPhys. Rev. Lett. {\\bf 82}, 4741 (1999).\n\n\\bibitem{Ruostekoski99b} J. Ruostekoski, preprint: cond-mat/9908276\n\n\\bibitem{Purcell46} E. M. Purcell, Phys. Rev. {\\bf 69}, 681 (1946).\n\n\\bibitem{Nienhuis76} G. Nienhuis and C. Th. J. Alkemade, Physica\n{\\bf 81C}, 181 (1976).\n\n\\bibitem{Knoester89} J. Knoester and S. Mukamel, Phys. Rev. A {\\bf 40},\n7065 (1989).\n\n\n\n\\bibitem{Milonni95} P. W. Milonni, J. Mod. Optics {\\bf 42}, 1991 (1995).\n\n\n\\bibitem{Glauber91} R. J. Glauber and M. Lewenstein, \nPhys. Rev. A {\\bf 43}, 467 (1991).\n\n\n\\bibitem{deVries98} P. de Vries and A. Lagendijk, Phys. Rev. Lett.\n{\\bf 81}, 1381 (1998).\n\n\n\\bibitem{Rikken95} G. L. J. A. Rikken, and Y. A. R. R. Kessener,\nPhys. Rev. Lett. {\\bf 74}, 880 (1995).\n\n\\bibitem{Schuurmans98} Frank J. P. Schuurmans, D. T. H. de Lang,\nG. H. Wegdam, R. Spirk, and A. Lagendijk, Phys. Rev. Lett. {\\bf 80},\n5077 (1998).\n\n\n\n\\bibitem{Scheel99a} S. Scheel, L. Kn\\\"oll, D.-G. Welsch, S.M. Barnett: \nPhys. Rev. A {\\bf 60} (1999) 1590.\n\n\\bibitem{Scheel99b} S. Scheel, L. Kn\\\"oll, D.-G. Welsch:\nPhys. Rev. A, {\\bf 60} (1999) 4094.\n\n\\bibitem{Fleischhauer99b} M. Fleischhauer, Phys. Rev. A \n{\\bf 60}, \n2534 (1999).\n\n\\bibitem{Fleischhauer99a} M. Fleischhauer and S. F. Yelin, \nPhys. Rev. A {\\bf 59}, 2427 (1999).\n\n\n\\bibitem{Barnett92} S. M. Barnett, B. Huttner, and\nR. Loudon, Phys. Rev. Lett. {\\bf 68}, 3698 (1992).\n\n\\bibitem{Pauli} P.~Jordan and W.~Pauli, Zeit.~f\\\"ur Physik {\\bf 47}, 151\n(1928)\n\n\\bibitem{deVries98b} P. de Vries, D. V. van Coevorden, and A. Lagendijk,\nRew. Mod. Phys. {\\bf 70}, 447 (1998).\n\n\\bibitem{Lagendijk97} Ad Lagendiijk, B. Nienhuis, B. A. van Tiggelen, and\nP. de Vries, Phys. Rev. Lett. {\\bf 79}, 657 (1997).\n\n\\bibitem{Hansen76} J. P. Hansen and I. R. McDonald {\\it Theory of Simple\nLiquids}, (Academic Press, London, 1976).\n\n\n\n\\end{thebibliography}"
}
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cond-mat0002096
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Circulating electrons, superconductivity, and the Darwin-Breit interaction
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[
{
"author": "Circulating electrons"
},
{
"author": "superconductivity"
},
{
"author": "and the Darwin-Breit interaction"
}
] |
The importance of the Darwin-Breit interaction between electrons in solids at low temperatures is investigated. The model problem of particles on a circle is used and applied to mesoscopic metal rings in their normal state. The London moment formula for a rotating superconducting body is used to calculate the number, $N$, of superconducting electrons in the body. This number is found to be equal to the size, $R$, of the system divided by the classical electron radius, i.e.\ $N=Rmc^2/e^2$. The Darwin-Breit interaction gives a natural explanation for this relation from first principles. It also is capable of electron pairing. Collective effects of this interaction require a minimum of two dimensions but electron pairing is enhanced in one-dimensional systems.
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[
{
"name": "cond-mat0002096.tex",
"string": "\n\\newcommand{\\dfd}{{\\rm d}}\n\\newcommand{\\vecr}{\\bbox{r}}\n\\newcommand{\\vecs}{\\bbox{s}}\n\\newcommand{\\vecv}{\\bbox{v}}\n\\newcommand{\\vece}{\\bbox{e}}\n\\newcommand{\\vecA}{\\bbox{A}}\n\\newcommand{\\vecB}{\\bbox{B}}\n\\newcommand{\\vecOmega}{\\bbox{\\Omega}}\n\\newcommand{\\ZF}{Z_{\\mbox{\\tiny F}}}\n\\newcommand{\\EF}{{\\cal E}_{\\mbox{\\tiny F}}}\n\\newcommand{\\vF}{v_{\\mbox{\\tiny F}}}\n\n% ****** Start of file template.aps ****** %\n\\documentstyle[aps]{revtex}\n\\begin{document}\n% \\draft command makes pacs numbers print\n\\draft\n% repeat the \\author\\address pair as needed\n\\title{Circulating electrons, superconductivity, and the\nDarwin-Breit interaction} \n\\author{Hanno Ess\\'en}\n\\address{Department of Mechanics \\\\Royal Institute\nof Technology \\\\ S-100 44 Stockholm, Sweden} \n\\date{August 1999} \n\\maketitle \n\\begin{abstract}\nThe importance of the Darwin-Breit interaction between electrons\nin solids at low temperatures is investigated. The model\nproblem of particles on a circle is used and applied to\nmesoscopic metal rings in their normal state. The London moment\nformula for a rotating superconducting body is used to calculate\nthe number, $N$, of superconducting electrons in the body. This\nnumber is found to be equal to the size, $R$, of the system\ndivided by the classical electron radius, i.e.\\ $N=Rmc^2/e^2$.\nThe Darwin-Breit interaction gives a natural explanation for this\nrelation from first principles. It also is capable of electron\npairing. Collective effects of this interaction require a\nminimum of two dimensions but electron pairing is enhanced in\none-dimensional systems.\n\\end{abstract}\n% insert suggested PACS numbers in braces on next line\n\\pacs{74.20.-z, 74.20.Hi}\n\n% body of paper here\n\n\\section{Introduction}\nArguments and results will be presented that hopefully convince\nthe open-minded reader that superconductivity is caused by the\nDarwin-Breit (magnetic) interaction between semiclassical\nelectrons. The starting point is a careful study of the model\nproblem of electrons on a circle. This simple model is chosen\nsince it allows accurate treatment of the notoriously difficult\nproblem of relativistic and magnetic effects in many-electron\nsystems. Since classical ideas are closer to our intuition the\nclassical picture is taken as far as possible before quantum\nmechanics is reluctantly adopted. The semiclassical point of\nview is an extremely powerful one\n\\cite{brack_bhaduri,gutzwiller} and the reader will find further\nexamples of this below.\n\nRelativistic quantities, to a first approximation, have a\nmagnitude $(v/c)^2$ times those of non-relativistic quantities.\nWhile this always is small in everyday life, in the atomic world\nthis parameter is $\\sim 10^{-4}$, which is fairly small, but\nrarely negligible. A striking example of this is the energy gap\nin superconductors which typically is order of magnitude\n$10^{-4}$ of the Fermi energy. Any study of this phenomenon that\ndoes not take relativistic effects into account must consequently\nremain inconclusive \n\nThe Darwin-Breit interaction \\cite{darwin,breit0,breit} is the\nfirst order relativistic correction, \n\\begin{equation}\n\\label{eq.V1.velocity.form}\nV_1=-\\sum_{i<j}^N \\frac{e^2}{c^2}\\frac{\\vecv_i\\cdot\\vecv_j\n+ (\\vecv_i\\cdot\\vece_{ij}) (\\vecv_j\\cdot\\vece_{ij})}{2 r_{ij} },\n\\end{equation}\nto the Coulomb potential. Sucher \\cite{sucher} in a recent review\n({\\em What is the force between two electrons?}) gives a\nthorough discussion of its origin in QED. While well known as an\nimportant perturbation in accurate atomic calculations\n\\cite{strange,cook} it has until recently (Ess\\'en\n\\cite{essen95,essen96,essen97,essen99}) usually been taken for \ngranted, without proof or justification, that it is negligible in\nlarger systems. Welker \\cite{welker} suggested in 1939 that\nmagnetic attraction of parallel currents might cause\nsuperconductivity, but after that the idea seems to have been\nforgotten. Other types of magnetic interaction have been\nsuggested though \\cite{mathur}. Some efforts to include the\nDarwin-Breit interaction in density functional approaches to\nsolids are reviewed in Strange \\cite{strange}. Capelle and Gross\n\\cite{capelle} have also made efforts towards a relativistic\ntheory of superconductivity.\n\nIn section \\ref{sec.rings.flux} we introduce the analytical\nmechanics of particles on a circle and apply it to mesoscopic\nrings. This serves to introduce the mathematical model and also\nthrows some light of the theory behind the persistent currents\nfound in these. We later find that, though these rings are not\nsuperconducting, electron pairing might be relevant to understand\ntheir physics.\n\nSection \\ref{sec.london.moment} closes in upon the main subject\nof superconductivity. The London moment formula connecting the\nangular velocity of a superconducting body and the magnetic\nfield it produces is introduced and motivated. The formula,\ntogether with classical electromagnetism can be used to\ncalculate the number of superconducting electrons present. This\nnumber is found to be determined entirely by fundamental\nconstants and the size of the body.\n\nFinally in section \\ref{sec.darwin.breit} the importance of the\nDarwin-Breit interaction is investigated. We show how it can lead\nto electron pairing and calculate the relevant temperatures at\nwhich these form. We also investigate when the interaction\nmight become dominating and find that exactly the combination of\nnumber, size, and fundamental constants that followed from the\nLondon moment is the condition for this. When the condition is\nfulfilled the particles no longer move individually, or in pairs,\nbut collectively. The behavior of this condition as a function\nof spatial dimension is investigated. Interestingly it is found\nthat the one-dimensionality of the ring enhances pair-formation\nbut suppresses collective behavior (superconductivity). \nAfter that the conclusions are summarized.\n\n\n\n\\section{Rings, persistent currents, and flux periodicity}\n\\label{sec.rings.flux}\nIn solid state physics cold mesoscopic metal rings have attracted\na lot of attention. In particular since theoretical predictions \n\\cite{buttiker,bloch} that an external magnetic flux through the\nring causes a persistent current round it, have been\nexperimentally verified \\cite{webb,levy,chandrasekhar}. The\nagreement between theory and experiment is, however, still far\nfrom perfect \\cite{johnson}, for reviews see\n\\cite{imry,imry_book}. One normally assumes that it is\ncorrect to treat the conduction electrons semiclassically, one\nspeaks about ballistic electrons \\cite{brack_bhaduri,imry_book},\nand we will do so here. Superconductivity is not treated in this\nsection, but we assume that the rings are perfect conductors\n(have zero resistance). \n\n\n\\subsection{Charged particles on a circle}\nWe now set up the model problem of charged particles constrained\nto move on a circle. Assuming that the circle has radius $R$,\npositions and velocities are given by\n\\begin{equation}\n\\label{eq.pos.vel.vectors.meso}\n\\vecr_i(\\varphi_i)=R \\vece_{\\rho}(\\varphi_i), \\hskip 0.3cm\n\\mbox{and} \\hskip 0.3cm\n\\vecv_i(\\varphi_i,\\dot\\varphi_i)=\nR \\dot\\varphi_i \\vece_{\\varphi}(\\varphi_i),\n\\end{equation}\nwhere $\\vece_{\\rho}(\\varphi)= \\cos\\varphi\\, \\vece_x\n+\\sin\\varphi\\,\\vece_y$ and $\\dot{\\vece}_{\\rho}\n=\\dot\\varphi\\vece_{\\varphi}$, as usual. We take the zeroth order\nLagrangian to be\n\\begin{equation}\n\\label{eq.zeroth.circle.lagrangian.V0}\nL_0=T_0-V_0=\\frac{1}{2} \\sum_{i=1}^N m_i R^2\\dot\\varphi_i^2 -\nV_0(\\varphi_1,\\ldots,\\varphi_N). \n\\end{equation}\nSince we will have metallic conduction electrons in mind the\npotential $V_0$ does not necessarily represent the Coulomb\ninteractions, but rather interactions with the lattice plus,\npossibly, Debye screened two particle interactions. The\ngeneralized (angular) momenta are\n$J_i=\\partial L_0/\\partial\\dot\\varphi_i = mR^2\\dot\\varphi_i$ so \nthe Hamiltonian is\n\\begin{equation}\n\\label{eq.basic.ring.hamiltonian}\nH_0=\\sum_{i=1}^N \\frac{J_i^2}{2m_i R^2} + V_0.\n\\end{equation}\nIf there is a magnetic flux $\\Phi =\\int \\vecB\\cdot\\dfd\\vecs\n=\\oint \\vecA\\cdot\\dfd\\vecr =2\\pi R A_{\\varphi}$ through the ring\nthe Hamiltonian changes to\n\\begin{equation}\n\\label{eq.ring.hamiltonian.ext.A}\nH_0=\\sum_{i=1}^N \\frac{1}{2m_i}\\left(\n\\frac{J_i}{R} -\\frac{e_i}{c}A_{\\varphi} \\right)^2 + V_0=\n\\sum_{i=1}^N \\frac{1}{2m_i R^2}\\left(\nJ_i -\\frac{e_i}{2\\pi c} \\Phi \\right)^2 + V_0,\n\\end{equation}\nsince $A_{\\varphi}=\\Phi/(2\\pi R)$.\n\nWe find the equations of motion\n\\begin{eqnarray}\n\\dot J_i = -\\frac{\\partial H_0}{\\partial \\varphi_i}\n=-\\frac{\\partial V_0}{\\partial \\varphi_i},\\\\\n\\dot \\varphi_i = \\frac{\\partial H_0}{\\partial\nJ_i}=\\frac{J_i}{m_i R^2}-\\frac{e_i \\Phi}{m_i R^2 2\\pi c}.\n\\end{eqnarray}\nThe current round the ring is by definition\n\\begin{equation}\n\\label{eq.ring.current.ext.A}\nI=\\sum_{i=1}^N e_i \\frac{\\dot\\varphi_i}{2\\pi} =\\frac{1}{2\\pi}\n\\sum_{i=1}^N\\left( \\frac{e_i J_i}{m_i R^2}-\\frac{e_i^2 \\Phi}{m_i\nR^2 2\\pi c} \\right) \\equiv I_0 + I_{\\Phi}.\n\\end{equation}\nOne notes that the relation\n\\begin{equation}\n\\label{eq.ring.current.dH.dA}\nI=-c\\frac{\\partial H_0}{\\partial \\Phi}\n\\end{equation}\nholds.\n\nFor non-interacting particles on the ring we have \n\\begin{equation}\nV_0=\\sum_{i=1}^N U_0(\\varphi_i). \n\\end{equation}\nThen $H_0=\\sum_i H_i(J_i,\\varphi_i)$ where $H_i$ are constants\nof the motion, $H_i=E_i$, whether there is a flux or not. There are\nthen the adiabatic invariants \\cite{landau1}\n\\begin{equation}\nI_{\\varphi_{i}} \\equiv \\frac{1}{2\\pi} \\oint J_i(\\varphi_i\n;E_i,\\Phi) \\dfd\\varphi_i = \\overline{J_i},\n\\end{equation}\nthe averages, $\\overline{J_i}$, of the $J_i$ round the ring.\nIf the flux is turned on slowly they will retain their zero\nflux values. The zero flux average current \n\\begin{equation}\n\\label{eq.ring.current.A.0}\n\\overline{ I_0} =\\frac{1}{2\\pi R^2} \\sum_{i=1}^N \\frac{e_i\n\\overline{J_i}}{m_i}\n\\end{equation}\nis thus also an adiabatic invariant, and remains constant. This\nmeans that slowly turning on a flux $\\Phi$ through the ring\nresults in the extra diamagnetic circulating current\n\\begin{equation}\n\\label{eq.ring.current.due.A}\nI_{\\Phi} = -\\frac{\\Phi}{4\\pi^2 R^2} \\sum_{i=1}^N\n\\frac{e_i^2}{m_i c} \n\\end{equation}\nindependently of any pre-existing current. Below we will find that\nthe above result can be found using Larmor's theorem and thus, in\nfact, is independent of electron interactions provided other\nconditions are fulfilled.\n\n\n\\subsection{Two types of current}\nWe find that there are two different types of current\npossible in these rings. The `ballistic' current $I_0$, which\nshould be, at most \\cite{geller}, order of magnitude a few $ e\n\\vF/(2\\pi R)$, where $\\vF$ is the Fermi velocity, and the Larmor\ncurrent $I_{\\Phi}$ induced by the flux. Assuming that only\nelectrons contribute (\\ref{eq.ring.current.due.A}) becomes\n\\begin{equation}\n\\label{eq.ring.current.due.A.elect}\nI_{\\Phi} = \\frac{\\Phi}{4\\pi^2 R^2} N\\frac{e^2}{mc} .\n\\end{equation}\nPutting\n\\begin{equation}\n\\label{eq.flux.unit}\n\\Phi = n_{\\phi} \\frac{hc}{|e|} \\equiv n_{\\phi} \\Phi_0 ,\n\\end{equation}\nwhere $n_{\\phi}$ is dimensionless and $\\Phi_0=hc/|e|$ is the flux\nquantum, we get the expression\n$ I_{\\Phi} \\pi R^2 = - N n_{\\phi} \\mu_{\\rm B}$.\nHere $\\mu_{\\rm B}=|e|\\hbar/(2m)$ is the Bohr magneton. Gaussian\nunits are used in most formulas; to get equation\n(\\ref{eq.ring.current.due.A.elect}) in SI-units we simply delete\n$c$. If the flux is $\\Phi=B \\pi R^2$ we can then rewrite it in\nthe form\n\\begin{equation}\n\\label{eq.ring.current.due.A.SI}\nI_{\\Phi} = -N\\cdot B\\cdot 2.242 \\,{\\rm nA/T}.\n\\end{equation}\nTo get a number out of this formula we must estimate the number\n$N$ of semiclassical electrons and know the magnetic field in\nteslas. The speed corresponding to the Larmor current is, in atomic\nunits, $v_{\\Phi}=n_{\\phi}/R \\ll \\vF=1.92/r_s$, where $r_s$ is the\nradius parameter. On the other hand all semiclassical electrons\ncontribute to $I_{\\Phi}$, whereas the number contributing to $I_0$\nnecessarily is small.\n\nLevy et al.\\ \\cite{levy} found an average current of\n$I_{\\rm av}=3\\cdot 10^{-3} \\cdot e \\vF/\\ell= 0.36\\,$nA in their\nCu-rings, of circumference $\\ell=2.2\\,\\mu$m. If this is\ninterpreted as a Larmor-current we can calculate $N$. At the\nmagnetic field $B_0=1.3 \\cdot 10^{-2}\\,$T corresponding to the\nflux quantum $\\Phi_0$ this gives the reasonable result $N \\approx\n100$ for the number of semiclassical electrons in the system.\nChandrasekhar et al.\\ \\cite{chandrasekhar}, on the other hand,\nfound currents $I=$(0.3 -- 2.0) $e\\vF/(2\\pi R)$ in a single gold\nring. These can thus only be interpreted as due to ballistic\ncurrents. They might be due to electron pairs, which may form\neven in the normal state, as we will see below.\n\n\n\n\\subsection{Larmor's theorem}\nConsider a system of particles, all of the same charge to mass\nratio $e/m$. Assume that they move in a common external potential,\n$U_e(\\rho,z)$, that is axially symmetric, i.e.\\ independent of\n$\\varphi$, under the influence of arbitrary interparticle\ninteractions. Now place this system in a weak magnetic\nfield, $B_z$, along the $z$-axis. One can then apply Larmor's\ntheorem \\cite{strange,essen89} to show that the response of the\nsystem to this field is a rotation with angular velocity \n\\begin{equation}\n\\label{eq.larmor.frequency}\n\\Omega_z = -\\frac{e}{2mc} B_z \n\\end{equation}\ngiven by the Larmor frequency.\n\nThis means that there will be a circulating Larmor current\n\\begin{equation}\n\\label{eq.larmor.frequency.current}\nI_L=Ne\\frac{\\Omega_z}{2\\pi}= -\\frac{B_z}{4\\pi} N \\frac{e^2}{mc} \n\\end{equation}\nwhere $Ne$ is the total amount of charge on the particles ($N$\nis not necessarily the number of particles). If we insert\n$B_z=\\Phi/( \\pi R^2)$ we recover essentially equation\n(\\ref{eq.ring.current.due.A.elect}). This is why we called\n$I_{\\Phi}$ the Larmor current. Note that we derived\n(\\ref{eq.ring.current.due.A.elect}) under the assumption of\narbitrary charge to mass ratios $e_i/m_i$ but no interparticle\ninteraction. Here we need identical charge to mass ratios $e/m$\nand an axially symmetric external field but can have arbitrary\ninteractions between the particles. The general results\n(\\ref{eq.ring.current.due.A.elect}) and\n(\\ref{eq.ring.current.due.A.SI}) for semiclassical electrons (or\nelectron pairs or groups) in cold metal rings thus seem fairly\nreliable.\n\nIt is noteworthy that the result of equation\n(\\ref{eq.ring.current.due.A}) is not necessarily due to any\nmagnetic field affecting the particles. The flux $\\Phi$ could\nvery well go through a smaller surface completely inside the ring\nmaterial. This means that the current in\n(\\ref{eq.ring.current.due.A}) is a classical Aharonov-Bohm effect\n\\cite{aharonov}. That is, an effect due to the vector\npotential at zero magnetic field. By contrast the\nLarmor result (\\ref{eq.larmor.frequency.current}) is derived\nassuming that the magnetic field penetrates the ring.\n\n\n\n\\subsection{Quantizing the electron on the circle and flux\nperiodicity} \nThe above results are purely classical. When we quantize them we\nwill find that physical properties must be periodic in (half?)\nthe flux quantum, as will now be shown show. Our previous\nclassical results for currents must be thought of as averages\nover these quantum periods (beats). Flux quantization was\noriginally suggested by London \\cite{london}, for a thorough\ndiscussion see Thouless \\cite{thouless}.\n\nThe classical Hamiltonian of an electron moving freely on\na circle of radius\n$R$ threaded by a flux $\\Phi$ is, according to equation\n(\\ref{eq.ring.hamiltonian.ext.A}),\n\\begin{equation}\n\\label{eq.Ham.free.elec.circ}\nH=\\frac{1}{2mR^2}\\left(J +\\frac{|e|}{2\\pi c}\\Phi \\right)^2.\n\\end{equation}\nWe quantize this by letting $J \\rightarrow \\hat{J}=-{\\rm\ni}\\hbar \\partial /\\partial \\varphi$ and thus get the \nSchr\\\"odinger equation\n\\begin{equation}\n\\label{eq.schrod.eq.free.elec.circ}\n\\frac{\\hbar^2}{2mR^2}\\left(-{\\rm\ni}\\frac{\\partial}{\\partial\\varphi} +n_{\\phi} \\right)^2\n\\psi(\\varphi)=E\\psi(\\varphi),\n\\end{equation}\nwhere we have used equation (\\ref{eq.flux.unit}). Putting\n\\begin{equation}\n\\label{eq.gauge.transf}\n\\psi(\\varphi)=\\exp(-{\\rm i}n_{\\phi}\\varphi)\\,\\psi'(\\varphi)\n\\end{equation}\nwe get \n\\begin{equation}\n\\label{eq.schrod.eq.free.elec.circ.gt}\n-\\frac{\\hbar^2}{2mR^2}\\frac{\\partial^2\n}{\\partial\\varphi^2}\\psi' =E \\psi' \n\\end{equation}\nfor the gauge transformed wave function. It is now frequently\nargued \\cite{byers,bloch} that the wave function must be single\nvalued and that therefore\n\\begin{equation}\n\\label{eq.single.value.cond}\n\\psi(\\varphi+2\\pi)=\\psi(\\varphi).\n\\end{equation}\nVia (\\ref{eq.gauge.transf}) this leads to the\nphysical condition \n\\begin{equation}\n\\label{eq.single.value.cond.gt}\n\\psi'(\\varphi+2\\pi)=\\exp({\\rm i}n_{\\phi}2\\pi) \\psi'(\\varphi)\n\\end{equation}\non the solutions of (\\ref{eq.schrod.eq.free.elec.circ.gt}),\nwhere the flux has been transformed away. This boundary\ncondition is unchanged if $n_{\\phi}$ changes by unity.\nThis implies that physical quantities must be periodic in the\nflux with period $\\Phi_0$. \n\nThe above argument is not necessarily reliable, however. The\ncorrect wave function for an electron is a spinor (in the\nnon-relativistic case a two component spinor). A spinor is\nwell known to change sign when rotated by $2\\pi$. The\nquestion is then: will the spinor rotate as the electron\ntravels round the circle? A free electron is known to have\nconserved helicity, the projection of the spin on the\nmomentum. As the ring radius is large compared to atomic\ndimensions the electron momentum turns slowly and it seems\nreasonable that the helicity will remain conserved (as an\nadiabatic invariant). This, of course, means that the spinor\nmust rotate with the momentum. The conclusion of all this is\nthat the correct condition on the spinor wave function, for a\nsingle electron, should be\n\\begin{equation}\n\\label{eq.double.value.cond}\n\\psi(\\varphi+4\\pi)=\\psi(\\varphi),\n\\end{equation}\nand thus that\n\\begin{equation}\n\\label{eq.double.value.cond.gt}\n\\psi'(\\varphi+4\\pi)=\\exp({\\rm i}n_{\\phi}4\\pi)\\psi'(\\varphi).\n\\end{equation}\nThis condition is unchanged whenever $n_{\\phi}$ changes by\none half. I.e.\\ physical quantities must be periodic in the\nflux with period $\\Phi_0/2$. Note that the same result is\nobtained if $|e|$ in equation (\\ref{eq.Ham.free.elec.circ}) is\nchanged to $2|e|$. The $n_{\\phi}$ in\n(\\ref{eq.schrod.eq.free.elec.circ}) changes to $2 n_{\\phi}$ and\nequation (\\ref{eq.single.value.cond.gt}) becomes identical to \n(\\ref{eq.double.value.cond.gt}).\n\nIn conclusion the observation of the $\\Phi_0/2$ periodicity does\nnot necessarily imply electron pairs. It might be due to single\nelectrons going round the ring with conserved helicity. Both the\n$\\Phi_0$ and the $\\Phi_0/2$ periodicities have been experimentally\nobserved \\cite{deaver,doll,gough,webb,levy,chandrasekhar}.\n\n\n\\section{Rotating superconductors and the number of\nsuperconducting electrons}\n\\label{sec.london.moment}\n There is another surprising result concerning circulating\nelectrons that is easily explained by Larmor's theorem\n(\\ref{eq.larmor.frequency}). London \\cite{london} showed (see\nalso \\cite{brady,cabrera,liu}), using his phenomenological theory\nof superconductivity, that a superconducting sphere that rotates\nwith angular velocity\n$\\vecOmega$ will have an induced magnetic field (Gaussian units)\n\\begin{equation}\n\\label{eq.london.mag.field}\n\\vecB = \\frac{2 mc}{|e|} \\vecOmega\n\\end{equation}\nin its interior. Here $m$ and $e$ are the mass and charge of\nthe electron. This prediction has been experimentally verified\nwith considerable accuracy and is equally true for high\ntemperature and heavy fermion superconductors\n\\cite{tate,sanzari}. With minor modifications it is also valid for\nother axially symmetric shapes of the body, for example cylinders\nor rings.\n\n\n\\subsection{Understanding the London moment}\nThe London field, or `moment', (\\ref{eq.london.mag.field}) can\nbe thought of as follows. Assume that the superconducting body\ncan be viewed as a system of interacting particles with the\nelectronic charge to mass ratio confined by an axially symmetric\nexternal potential. When the body rotates we can transform the\nequations of motion to a co-rotating system, in which it is at\nrest, but in this system the particles will be affected by a\nCoriolis force $-m2\\vecOmega\\times \\vecv$. Larmor's theorem\nteaches us that such a Coriolis force is equivalent to an\nexternal magnetic field. Magnetic fields are, however, not\nallowed inside superconductors according to the Meissner effect.\nTo get rid of the Coriolis forces the rotation induces surface\nsupercurrents that produce a suitable compensating magnetic field\n$\\vecB$. The Lorentz force of this field is $-(|e|/c)\\vecv\\times\n\\vecB$. Provided the relation between $\\vecB$ and $\\vecOmega$ is\ngiven by (\\ref{eq.london.mag.field}) the two forces cancel. The\nequations of motion in the rotating system are then the same, in\nthe interior, as if the system did not rotate. The disturbance\nfrom the rotation on the dynamics is minimized.\n\nThe above explanation may sound compelling, but the most direct\nway of understanding formula (\\ref{eq.london.mag.field}) is,\nin fact, much simpler. The superconducting electrons, which are\nalways found just inside the surface \\cite{london}, are not\ndragged by the positive ion lattice so when it starts to rotate\nthe superconducting electrons ignore this and remain in whatever\nmotion they prefer. This, however, means that there will be an\nuncompensated motion of positive charge density on the surface\nof the body. This surface charge density, $\\sigma$, will, of\ncourse, be the same as the density of superconducting\nelectrons, but of opposite sign, and will produce the magnetic\nfield. Using this we can calculate the number, $N$, of\nsuperconducting electrons.\n\n\n\\subsection{The number of superconducting electrons}\nIt is well known that a rotating uniform surface charge density\nwill produce a uniform interior magnetic field in a sphere. If\nthis rotating surface charge density is $\\sigma$, then the total\ncharge $Q$ is given by\n\\begin{equation}\nQ=N |e|= 4\\pi R^2 \\sigma, \n\\end{equation}\nand the resulting magnetic field in the interior is\n\\begin{equation}\n\\label{eq.int.mag.field.surface.dens}\n\\vecB = \\frac{2}{3}\\frac{Q}{cR} \\vecOmega =\n\\frac{8\\pi}{3}\\frac{\\sigma R}{c} \\vecOmega ,\n\\end{equation}\nwhere $R$ is the radius of the sphere (relevant formulas for the\ncalculation can be found in Ess\\'en \\cite{essen89}). Putting\n$Q=N|e|$ and comparing this equation with\n(\\ref{eq.london.mag.field}) one finds that the number $N$ must be\ngiven by $N=R m c^2/e^2 =R/r_{\\rm e}$. We thus find that the\nrelationship\n\\begin{equation}\n\\label{eq.N.re.R.sup.cond.sphere}\n\\frac{N r_{\\rm e}}{R} =1 ,\n\\end{equation}\nwhere $r_{\\rm e}$ is the classical electron radius, and $N$ the\nnumber of electrons contributing to the supercurrent,\ncharacterizes the superconductivity on a sphere of radius $R$.\n\nThe corresponding calculation for a cylinder, long\nenough for edge effects to be negligible, is elementary and gives\n$Nr_{\\rm e}/\\ell =1$, where $\\ell$ is the length of the cylinder.\nWe will return to the crucial significance of the dimensionless\ncombination $N r_{\\rm e}/R$ below. It is noteworthy that the \nnumber $N$ depends only on the geometry (size) and fundamental\nconstants ($r_{\\rm e}$). How can this be if superconductivity is\ncaused by some effective interaction with the lattice?\n\n\n\n\\section{Pairing and collective effects due to the Darwin-Breit\ninteraction}\n\\label{sec.darwin.breit}\nWe now continue the study of the semiclassical (ballistic)\nelectrons in the ring using the model of charged particles\nconstrained to move on a circle. Now we further assume that the\nelectrons are free particles to zeroth order and investigate how\nthis is affected by the first order Darwin-Breit term. The\nrelativistic mass-velocity correction is probably not of much\ninterest here. \n\n\\subsection{The Darwin-Breit term on the ring} \nFor the positions and velocities of equation\n(\\ref{eq.pos.vel.vectors.meso}) the Darwin-Breit term\n(\\ref{eq.V1.velocity.form}) becomes\n\\begin{equation}\n\\label{eq.darwin.breit.meso}\nV_1=-\\frac{e^2 }{R c^2} \\sum_{i<j}^N R^2\n\\dot\\varphi_i\\dot\\varphi_j\n\\frac{1}{4}\\frac{1+3\\cos(\\varphi_i-\\varphi_j)}{\\sqrt{\n2[1-\\cos(\\varphi_i-\\varphi_j)]}} \\equiv -\\frac{e^2 R}{c^2}\n\\sum_{i<j}^N\n\\dot\\varphi_i\\dot\\varphi_j V_{\\varphi}(\\varphi_i-\\varphi_j),\n\\end{equation}\nand the first order Lagrangian $L=T_0-V_1$, with $T_0$ given in\nequation (\\ref{eq.zeroth.circle.lagrangian.V0}), is\n\\begin{equation}\n\\label{eq.lagrangian.darwin.breit.meso}\nL=\\frac{1}{2}mR^2 \\sum_{i=1}^N \\dot\\varphi_i^2 + \n\\frac{e^2 R}{c^2} \\sum_{i<j} \n\\dot\\varphi_i \\dot\\varphi_j V_{\\varphi}(\\varphi_i-\\varphi_j).\n\\end{equation}\nThe nature of the\nfunction $V_{\\varphi}$ is indicated in equation\n(\\ref{eq.series.V.phi}) below. \nIf we introduce (note that the electron has charge $e=-|e|$)\n\\begin{equation}\n\\label{eq.int.vec.pot.meso}\nA_i= \\frac{e}{c} \\sum_{j(\\neq i)}^N \\dot\\varphi_j\nV_{\\varphi}(\\varphi_i-\\varphi_j)\n\\end{equation}\nwe can write this\n\\begin{equation}\n\\label{eq.hamilton.darwin.breit.meso.A}\nL= \\sum_{i=1}^N \\left( \\frac{1}{2}mR^2 \n\\dot\\varphi_i^2 + \\frac{e}{2c} R\\dot\\varphi_i A_i \\right).\n\\end{equation}\nIt is easy to show that for real electrons distributed\nround a (one-dimensional) ring of real atoms the Darwin-Breit\nterm will always be a small perturbation \\cite{geller}.\nIndividual terms in the interaction may still be large if some\npair of interparticle distances is very small. This would\ncorrespond to pair formation and is treated in the next\nsubsection. In the real world of two and three dimensions the \nDarwin-Breit term as a whole can become large. This means that\nindividually moving particles is no longer a good first\napproximation. This is shown in the following subsection.\n\n\n\\subsection{The one-dimensional hydrogen atom}\nThe Darwin-Breit term represents an interaction which is\nattractive for parallel currents. For small relative velocities\nof the electrons it seems possible that it could lead to bound\nstates (for the relative motion of the particles). Let us\ninvestigate this. Most conduction electrons in the metal ring\nwill be inside the (one-dimensional) Fermi surface and they will\noccur in pairs of opposite momentum with no net current. Assume\nthat only two electrons have unpaired momenta and move in the\nsame direction around the ring approximately with the Fermi\nvelocity. The Lagrangian of these two is then\n\\begin{equation}\n\\label{eq.Lagrangian.two.part.ring.darw}\nL= \\frac{mR^2}{2}(\\dot\\varphi_1^2+\\dot\\varphi_2^2)+\\frac{e^2 R}{\nc^2}\\dot\\varphi_1 \\dot\\varphi_2 V_{\\varphi}(\\varphi_1-\\varphi_2).\n\\end{equation}\nWe now make the coordinate transformation\n\\begin{equation}\n\\label{eq.two.part.ring.coord.cm.rel}\n\\varphi_{\\mbox{\\tiny C}}=\\frac{1}{2}(\\varphi_1 +\\varphi_2),\\hskip 0.5cm\n\\varphi=\\varphi_1-\\varphi_2\n\\end{equation}\nto center of mass angle $\\varphi_{\\mbox{\\tiny C}}$ and relative angle\n$\\varphi$. The inverse transformation is\n\\begin{equation}\n\\label{eq.two.part.ring.coord.cm.rel.inv}\n\\varphi_1=\\varphi_{\\mbox{\\tiny C}}+\\frac{1}{2}\\varphi ,\\hskip 0.5cm\n\\varphi_2=\\varphi_{\\mbox{\\tiny C}}-\\frac{1}{2}\\varphi,\n\\end{equation}\nand the Lagrangian becomes\n\\begin{equation}\n\\label{eq.Lagrangian.two.part.ring.darw.cm.rel}\nL= \\frac{mR^2}{2}\\left(2\\dot\\varphi_{\\mbox{\\tiny C}}^2+ \\frac{1}{2}\n\\dot\\varphi^2\\right) + \\frac{e^2 R}{ c^2}\\left(\\dot\\varphi_{\\mbox{\\tiny C}}^2\n-\\frac{1}{4}\\dot\\varphi^2\\right) V_{\\varphi}(\\varphi).\n\\end{equation}\nWe define $J_{\\mbox{\\tiny C}}\\equiv \\partial L/\\partial\\dot\\varphi_{\\mbox{\\tiny C}}$ and \n$J\\equiv \\partial L/\\partial\\dot\\varphi$ and get the\n(exact) Hamiltonian\n\\begin{equation}\n\\label{eq.Hamiltonian.two.part.ring.darw.cm.rel}\nH= J_{\\mbox{\\tiny C}} \\dot\\varphi_{\\mbox{\\tiny C}} +J \\dot\\varphi - L= \\frac{1}{4}\n\\frac{J_{\\mbox{\\tiny C}}^2}{mR^2\\left(1+ \\frac{e^2 V_{\\varphi}(\\varphi)}{m c^2 R}\n\\right)} +\\frac{J^2}{ mR^2 \\left(1 - \\frac{e^2\nV_{\\varphi}(\\varphi)}{m c^2 R}\n\\right)}.\n\\end{equation}\nClearly $\\dot J_{\\mbox{\\tiny C}} = -\\partial\nH/\\partial\\dot\\varphi_{\\mbox{\\tiny C}} =0$ so the center of mass (angular)\nmomentum $J_{\\mbox{\\tiny C}}$ is conserved. We put\n\\begin{equation}\n\\label{eq.def.J.F}\n|J_{\\mbox{\\tiny C}}| \\equiv 2 J_{\\mbox{\\tiny F}} = \\mbox{const.}\n\\end{equation}\nand expand to first order in the parameter $\\frac{e^2/R}{m\nc^2 }=r_{\\rm e}/R$. Throwing away a constant we end up with the\nfollowing Hamiltonian for the relative motion of the electrons\n\\begin{equation}\n\\label{eq.Hamiltonian.two.part.ring.darw.rel.1}\nH= \\frac{J^2}{mR^2}- \\frac{J_{\\mbox{\\tiny F}}^2 -J^2}{mR^2}\n\\frac{e^2}{m c^2}\n\\frac{V_{\\varphi}(\\varphi)}{R} .\n\\end{equation}\nConsistency with our original assumptions requires that\n$J^2 \\ll J_{\\mbox{\\tiny F}}^2$ and thus we neglect the $J^2$ in\nthe second term. Series expansion of $V_{\\varphi}$ gives\n\\begin{equation}\n\\label{eq.series.V.phi}\nV_{\\varphi}(\\varphi)=\\frac{1}{4}\\frac{1+3\\cos\\varphi}{\\sqrt{\n2(1-\\cos\\varphi)}} = \\frac{1}{|\\varphi|} -\n\\frac{1}{3}|\\varphi|+\\frac{97}{5760}|\\varphi|^3 +\\ldots,\n\\end{equation}\nfor the angular potential energy, so near $\\varphi=0$ this is\nessentially a (one-dimensional) Coulomb potential. We keep the\nfirst term and introduce \n\\begin{equation}\n\\label{eq.def.one.dim.hydrogen}\np \\equiv J/R,\\;\\; \\mu \\equiv m/2,\\;\\; r \\equiv R\\varphi ,\\;\\;\n\\ZF\\equiv \\frac{J_{\\mbox{\\tiny F}}^2/(mR^2)}{m c^2} =\\frac{\\EF}{m\nc^2},\n\\end{equation}\nwhere $\\EF$ is the Fermi energy. The Hamiltonian for the relative\nmotion then becomes the well known Hamiltonian,\n\\begin{equation}\n\\label{eq.Hamiltonian.two.part.ring.darw.rel.2}\nH= \\frac{p^2}{2\\mu}- \\frac{\\ZF e^2}{|r|},\n\\end{equation}\nfor a (one dimensional) one electron atom with reduced mass\n$\\mu$ and nuclear charge $\\ZF$.\n\nThe analysis above for two electrons on a circle can be done in\nan almost identical way in three dimensions\n\\cite{essen95,essen96,essen99} and shows that the Breit\ninteraction can bind two electrons in their relative motion while\ntheir center of mass moves through the metal at\nthe Fermi speed. The ground state energy in that case corresponds\nto a temperature of $\\sim 0.1$~mK. In the present one-dimensional\ncase all parameters are the same except the dimensionality of the\nspace. The one-dimensional hydrogen atom is treated in the\nliterature \\cite{haines,rau} and the ground state energy is known\nto go logarithmically to minus infinity when the dimension\napproaches one. To get a finite result we must therefore take\naccount of the thickness, $a$, of our ring and change the potential\nto\n\\begin{equation}\n\\label{eq.trunc.coulomb}\nV_1(r) =-\\frac{\\ZF e^2}{|r|+a}.\n\\end{equation}\nIn the three dimensional case the Bohr radius of the Hamiltonian\n(\\ref{eq.Hamiltonian.two.part.ring.darw.rel.2}) is $a_m=\n2/\\ZF\\approx 1.52 \\cdot 10^4 \\; r_s^2/a_0$, where $a_0$ is\nthe ordinary Bohr radius and $r_s$ the radius parameter. The\nthree-dimensional ground state energy is\n$E_{3d}=-1/a_m^2=-\\ZF^2/4$. The corresponding result for the\none-dimensional potential (\\ref{eq.trunc.coulomb}) is\n\\cite{haines,rau}\n\\begin{equation}\n\\label{eq.trunc.coulomb.one.dim.energy}\nE_{1d}=-\\frac{1}{a_m^2}[2 \\ln(a_m/a)]^2 .\n\\end{equation}\nThe condition for this is that $a\\ll a_m$. For the gold ring\nof Chandrasekhar et al. \\cite{chandrasekhar} with $a \\approx\n80\\,$nm one finds that $a_m/a \\approx 10^2$ using standard\nvalues for the Fermi energy of Au. One gets similar values for\nthe Cu rings of Levy et al.\\ \\cite{levy}. The $1d$-condition is\nthus clearly satisfied in both experiments. We thus get that the\nground state energy of the Darwin-Breit bound electron pairs\ncorresponds to a temperature of roughly 1 -- 2 mK. This is a bit\nbelow the temperatures (7 mK) at which the persistent current\ngold ring experiments in \\cite{chandrasekhar} were performed,\nbut the order of magnitude agreement is noteworthy. In\nthe $10^7$ Cu-rings experiment of Levy et al.\\ \\cite{levy} the\ntemperature range 7 -- 400 mK was used. Physicists working\nwith the theory of these phenomena can certainly not ignore the\nDarwin-Breit interaction and the possibility of pairing.\n\n\n\n\n\\subsection{When does the Darwin-Breit term become large?}\nIn the previous subsection we saw that the Darwin-Breit\ninteraction, though small, can have important qualitative effect\nand lead to pairing of electrons. This effect is enhanced by\none-dimensionality because of the logarithmic divergence of\nthe $1/r$-interaction in one dimension. Let us now investigate\nthe possibility of collective effects due to this term. \n\nWe return to the Lagrangian\n(\\ref{eq.lagrangian.darwin.breit.meso}) and try to get the\nHamiltonian without approximation. The generalized momentum is\n\\begin{equation}\n\\label{eq.generalized.mom.ring.meso}\nJ_i \\equiv \\frac{\\partial L}{\\partial \\dot\\varphi_i} = mR^2\n\\dot\\varphi_i +\\frac{e^2R}{c^2} \\sum_{j(\\neq i)}^N \\dot\\varphi_j\nV_{\\varphi}(\\varphi_i-\\varphi_j).\n\\end{equation}\nIn order to get an exact Hamiltonian we must solve for the\n$\\dot\\varphi_i$ in terms of the $J_i$. If we introduce the\nabbreviation $V_{ij}\\equiv V_{\\varphi}(\\varphi_i-\\varphi_j)$ we\ncan write the $N$ equations (\\ref{eq.generalized.mom.ring.meso})\n\\begin{equation}\n\\label{eq.generalized.mom.ring.meso.2}\nJ_i = mR^2 \\left( \\dot\\varphi_i +\\frac{r_{\\rm e}}{R} \\sum_{j(\\neq\ni)}^N V_{ij} \\dot\\varphi_j \\right), \\hskip 0.5cm i=1,\\ldots,N,\n\\end{equation}\n($r_{\\rm e}$=classical electron radius). As long as the sum here\nis negligible we have $J_i \\approx mR^2 \\dot\\varphi_i$ and\neasily find an approximate Hamiltonian. For few particles, small\n$N$, the sum will, in practice, never exceed the small number $N\nr_{\\rm e}/R$ by much, since in quantum mechanics the uncertainty\nprinciple prevents the $V_{ij}$ from becoming to large. If,\nhowever, $N$ is very large, the sum can still be small if the\nvelocities $\\dot\\varphi_j$ have random signs. \n\nWe see that the condition for breakdown of the approximation $J_i\n\\approx mR^2 \\dot\\varphi_i$, and thus for important collective\neffects of the Darwin-Breit term, is that $N r_{\\rm e}/R$ no\nlonger is small. A three dimensional estimate in\n\\cite{essen97} shows that, in fact, magnetic energy is\nminimized when\n\\begin{equation}\n\\label{eq.cond.big.collective.DB}\n\\frac{N r_{\\rm e}}{R} \\sim 1\n\\end{equation}\nwhere $N$ is the number of correlated velocities. If we put\n$\\varepsilon_e \\equiv r_{\\rm e}/R$ we can write equation\n(\\ref{eq.generalized.mom.ring.meso.2}) in the matrix form \n\\begin{equation}\n\\label{eq.generalized.mom.ring.meso.3}\n\\left( \\begin{array}{c} J_1 \\\\J_2 \\\\ \\vdots \\\\ J_N \\end{array}\n\\right) = \n mR^2 \\left( \\begin{array}{cccc}\n1 & \\varepsilon_e V_{12} & \\cdots & \\varepsilon_e V_{1N} \\\\\n \\varepsilon_e V_{21} & 1 & \\cdots & \\varepsilon_e V_{2N} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\n\\varepsilon_e V_{N1} & \\varepsilon_e V_{N2} & \\cdots & 1\n\\end{array} \\right) \n\\left( \\begin{array}{c} \\dot\\varphi_1 \\\\\\dot\\varphi_2 \\\\\n\\vdots \\\\ \\dot\\varphi_N \\end{array} \\right).\n\\end{equation}\nThis shows that collective Darwin-Breit behavior is due to\n\"off-diagonal long range order\", a concept invented by C.\\ N.\nYang \\cite{yang}. Here the concept reappears in a classical\ncontext and arises in the Legendre transformation from the\nLagrangian, with a Darwin-Breit interaction, to the Hamiltonian.\n\n\nIn a real one-dimensional ring of atoms with electrons this\ncannot happen, as will be shown below. The algebra, however, is,\nbarring notational and other irrelevant details, the same in two\nand three dimensions \\cite{essen96,essen97}. We have already seen,\nin equation (\\ref{eq.N.re.R.sup.cond.sphere}), that this\nparameter, $Nr_{\\rm e}/R$, can be unity in three dimensions when\nthe system is superconducting. Everything thus falls nicely\ninto place. The Darwin-Breit term can lead to pairing of\nelectrons at sufficiently low temperatures. Provided one has\nlong range correlation of velocities it can also lead to a large\ncollective effect, which, in fact, seems to be superconductivity.\n\nThe condition (\\ref{eq.cond.big.collective.DB}) will imply\ndifferent physics for different spatial dimension $d$. The\nnumber $N$ of ballistic, or semiclassical, or superconducting,\nor velocity-momentum correlated, electrons will be limited by\nthe fact that there will be at most one contributed per atom,\nusually much less. Assume, for definiteness, the maximum number.\nFor a sample of spatial dimension $d$ and side length $R$ this\ngives, very roughly,\n\\begin{equation}\n\\label{eq.max.N.R}\nN_{\\rm max}(d) = R^d / a_0^d,\n\\end{equation}\nwhere $a_0$ is the Bohr-radius. If we put this in equation\n(\\ref{eq.cond.big.collective.DB}) we get\n$\\frac{R^d r_{\\rm e}}{a_0^d R} \\sim 1$ which implies that\n\\begin{equation}\n\\label{eq.max.R.of.d}\nR^{d-1} \\sim a_0^d / r_{\\rm e}.\n\\end{equation}\nThis gives the following (minimum) sizes $R$ of superconducting\nstructures in spatial dimension $d$\n\\begin{equation}\nd\\rightarrow 1+ \\hskip 0.5cm \\Rightarrow\\hskip 0.5cm R\n\\rightarrow \\infty, \\hskip 0.9cm\n\\end{equation}\n\\begin{equation}\nd=2 \\hskip 0.5cm \\Rightarrow\\hskip 0.5cm R \\sim a_0^2 /\nr_{\\rm e}\n\\approx 19 000\\, a_0 \\approx 1 \\,\\mu{\\rm m} , \n\\end{equation}\n\\begin{equation}\nd=3 \\hskip 0.5cm \\Rightarrow\\hskip 0.5cm R \\sim a_0\n\\sqrt{a_0/r_{\\rm e}}\n\\approx 140\\, a_0 \\approx 10\\,{\\rm nm} . \n\\end{equation}\nAs stated above, we see that $d=1$ does not permit long range\ncorrelation. We saw that this does not mean that electron\npairs do not form. It only means that no long range collective\nphenomenon (phase transition?) will be possible. Two dimensions\ndiffer from three in that structures (samples) must be at least two\norders of magnitude larger in (linear) size. \n\n\n\\section{Conclusions}\nThe experienced theoretical physicist, should, just by looking\nat formula (\\ref{eq.V1.velocity.form}), see that there is trouble\nwith the thermodynamics ahead, since the interaction is long\nrange ($\\sim 1/r$) and there is no natural screening mechanism\nsimilar to that which limits the range of the Coulomb\ninteraction. This trouble is here identified with\nsuperconductivity. The main new point here, compared to the\nprevious investigations by the author, is the discovery that the\nparameter $Nr_{\\rm e}/R$, of equation\n(\\ref{eq.cond.big.collective.DB}), which has appeared again and\nagain in my study of the Darwin Hamiltonian (the exact\nHamiltonian corresponding to the Lagrangian with the Darwin-Breit\nterm), also miraculously appears in an estimate of the number of\nsuperconducting electrons, equation\n(\\ref{eq.N.re.R.sup.cond.sphere}). This gives a direct connection\nto the heart of superconductivity that was missing before. The\npainful but only conclusion must be that the Darwin-Breit\ninteraction is {\\em the} interaction between electrons that\ncauses superconductivity.\n\n\n\n% now the references. delete or change fake bibitem. delete next\n% three\n% lines and directly read in your .bbl file if you use bibtex.\n\n\\begin{references}\n\n\\bibitem{brack_bhaduri}\nM. Brack and R.~K. Bhaduri, {\\em Semiclassical Physics} (Addison Wesley,\n Reading, Massachusetts, 1997).\n\n\\bibitem{gutzwiller}\nM.~C. Gutzwiller, {\\em Chaos in Classical and Quantum Mechanics}\n (Springer-Verlag, New York, 1990).\n\n\\bibitem{darwin}\nC.~G. Darwin, Phil. Mag. {\\bf 39}, 537 (1920).\n\n\\bibitem{breit0}\nG. Breit, Phys. Rev. {\\bf 34}, 553 (1929).\n\n\\bibitem{breit}\nG. Breit, Phys. Rev. {\\bf 39}, 616 (1932).\n\n\\bibitem{sucher}\nJ. Sucher, Advances in Quantum Chemistry {\\bf 30}, 433 (1998).\n\n\\bibitem{strange}\nP. Strange, {\\em Relativistic Quantum Mechanics, with applications in condensed\n matter and atomic physics} (Cambridge University Press, Cambridge, 1998).\n\n\\bibitem{cook}\nA.~H. Cook, Proc. R. Soc. Lond. A {\\bf 415}, 35 (1988).\n\n\\bibitem{essen95}\nH. Ess\\'en, Physica Scripta {\\bf 52}, 388 (1995).\n\n\\bibitem{essen96}\nH. Ess\\'en, Phys. Rev. E {\\bf 53}, 5228 (1996).\n\n\\bibitem{essen97}\nH. Ess\\'en, Phys. Rev. E {\\bf 56}, 5858 (1997).\n\n\\bibitem{essen99}\nH. Ess\\'en, J. Phys. A: Math. Gen. {\\bf 32}, 2297 (1999).\n\n\\bibitem{welker}\nH. Welker, Z. Physik {\\bf 114}, 525 (1939).\n\n\\bibitem{mathur}\nN.~D. Mathur {\\it et~al.}, Nature {\\bf 394}, 39 (1998).\n\n\\bibitem{capelle}\nK. Capelle and E.~K.~U. Gross, Physics Letters A {\\bf 198}, 261 (1995).\n\n\\bibitem{buttiker}\nM. B\\\"uttiker, Y. Imry, and R. Landauer, Phys. Lett. {\\bf 96A}, 365 (1983).\n\n\\bibitem{bloch}\nF. Bloch, Phys. Rev. B {\\bf 2}, 109 (1970).\n\n\\bibitem{webb}\nR.~A. Webb, S. Washburn, C.~P. Umbach, and R.~B. Laibowitz, Phys. Rev. Lett.\n {\\bf 54}, 2696 (1985).\n\n\\bibitem{levy}\nL.~P. Levy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. {\\bf 64},\n 2074 (1990).\n\n\\bibitem{chandrasekhar}\nV. Chandrasekhar {\\it et~al.}, Phys. Rev. Lett. {\\bf 67}, 3578 (1991).\n\n\\bibitem{johnson}\nB.~L. Johnson and G. Kirczenow, Can.\\ J.\\ Phys. {\\bf 76}, 173 (1998).\n\n\\bibitem{imry}\nY. Imry and M. Peshkin, in {\\em Electron, a centenary volume}, edited by M.\n Springford (Cambrige University Press, Cambridge, 1997), pp.\\ 208--236.\n\n\\bibitem{imry_book}\nY. Imry, {\\em Introduction to mesoscopic physics} (Oxford, New York, 1997).\n\n\\bibitem{landau1}\nL.~D. Landau and E.~M. Lifshitz, {\\em Mechanics}, 3rd ed. (Pergamon, Oxford,\n 1976).\n\n\\bibitem{geller}\nM.~R. Geller, Phys. Rev. B {\\bf 53}, 9550 (1996).\n\n\\bibitem{essen89}\nH. Ess\\'en, Physica Scripta {\\bf 40}, 761 (1989).\n\n\\bibitem{aharonov}\nY. Aharonov and D. Bohm, Phys. Rev. {\\bf 115}, 485 (1959).\n\n\\bibitem{london}\nF. London, {\\em Superfluids, Volume 1, Macroscopic Theory of\n Superconductivity}, 2nd ed. (Dover, New York, 1961).\n\n\\bibitem{thouless}\nD.~J. Thouless, {\\em Topological quantum numbers in nonrelativistic physics}\n (World Scientific, Singapore, 1998).\n\n\\bibitem{byers}\nN. Byers and C.~N. Yang, Phys. Rev. Lett. {\\bf 7}, 46 (1961).\n\n\\bibitem{deaver}\nB.~S. Deaver, Jr. and W.~M. Fairbank, Phys. Rev. Lett. {\\bf 7}, 43 (1961).\n\n\\bibitem{doll}\nR. Doll and M. N\\\"abauer, Phys. Rev. Lett. {\\bf 7}, 51 (1961).\n\n\\bibitem{gough}\nC.~E. Gough {\\it et~al.}, Nature {\\bf 326}, 855 (1987).\n\n\\bibitem{brady}\nR.~M. Brady, Journal of Low Temperature Physics {\\bf 49}, 1 (1982).\n\n\\bibitem{cabrera}\nB. Cabrera and M.~E. Peskin, Phys. Rev. B {\\bf 39}, 6425 (1989).\n\n\\bibitem{liu}\nM. Liu, Phys. Rev. Lett. {\\bf 81}, 3223 (1998).\n\n\\bibitem{tate}\nJ. Tate, B. Cabrera, S. Felch, and J.~T. Anderson, Phys. Rev. Lett. {\\bf 62},\n 845 (1989).\n\n\\bibitem{sanzari}\nM.~A. Sanzari, H.~L. Cui, and F. Karwacki, Appl. Phys. Lett. {\\bf 68}, 3802\n (1996).\n\n\\bibitem{haines}\nL.~K. Haines and D.~H. Roberts, Am. J. Phys. {\\bf 37}, 1145 (1969).\n\n\\bibitem{rau}\nA.~R.~P. Rau, Am. J. Phys. {\\bf 53}, 1183 (1985).\n\n\\bibitem{yang}\nC.~N. Yang, Rev. Mod. Phys. {\\bf 34}, 694 (1962).\n\n\\end{references}\n%\\bibliographystyle{prsty}\n%\\bibliography{darwin,supcond}\n\n\n\\end{document}\n%\n% ****** End of file template.aps ******\n\n"
}
] |
[
{
"name": "cond-mat0002096.extracted_bib",
"string": "\\bibitem{brack_bhaduri}\nM. Brack and R.~K. Bhaduri, {\\em Semiclassical Physics} (Addison Wesley,\n Reading, Massachusetts, 1997).\n\n\n\\bibitem{gutzwiller}\nM.~C. Gutzwiller, {\\em Chaos in Classical and Quantum Mechanics}\n (Springer-Verlag, New York, 1990).\n\n\n\\bibitem{darwin}\nC.~G. Darwin, Phil. Mag. {\\bf 39}, 537 (1920).\n\n\n\\bibitem{breit0}\nG. Breit, Phys. Rev. {\\bf 34}, 553 (1929).\n\n\n\\bibitem{breit}\nG. Breit, Phys. Rev. {\\bf 39}, 616 (1932).\n\n\n\\bibitem{sucher}\nJ. Sucher, Advances in Quantum Chemistry {\\bf 30}, 433 (1998).\n\n\n\\bibitem{strange}\nP. Strange, {\\em Relativistic Quantum Mechanics, with applications in condensed\n matter and atomic physics} (Cambridge University Press, Cambridge, 1998).\n\n\n\\bibitem{cook}\nA.~H. Cook, Proc. R. Soc. Lond. A {\\bf 415}, 35 (1988).\n\n\n\\bibitem{essen95}\nH. Ess\\'en, Physica Scripta {\\bf 52}, 388 (1995).\n\n\n\\bibitem{essen96}\nH. Ess\\'en, Phys. Rev. E {\\bf 53}, 5228 (1996).\n\n\n\\bibitem{essen97}\nH. Ess\\'en, Phys. Rev. E {\\bf 56}, 5858 (1997).\n\n\n\\bibitem{essen99}\nH. Ess\\'en, J. Phys. A: Math. Gen. {\\bf 32}, 2297 (1999).\n\n\n\\bibitem{welker}\nH. Welker, Z. Physik {\\bf 114}, 525 (1939).\n\n\n\\bibitem{mathur}\nN.~D. Mathur {\\it et~al.}, Nature {\\bf 394}, 39 (1998).\n\n\n\\bibitem{capelle}\nK. Capelle and E.~K.~U. Gross, Physics Letters A {\\bf 198}, 261 (1995).\n\n\n\\bibitem{buttiker}\nM. B\\\"uttiker, Y. Imry, and R. Landauer, Phys. Lett. {\\bf 96A}, 365 (1983).\n\n\n\\bibitem{bloch}\nF. Bloch, Phys. Rev. B {\\bf 2}, 109 (1970).\n\n\n\\bibitem{webb}\nR.~A. Webb, S. Washburn, C.~P. Umbach, and R.~B. Laibowitz, Phys. Rev. Lett.\n {\\bf 54}, 2696 (1985).\n\n\n\\bibitem{levy}\nL.~P. Levy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. {\\bf 64},\n 2074 (1990).\n\n\n\\bibitem{chandrasekhar}\nV. Chandrasekhar {\\it et~al.}, Phys. Rev. Lett. {\\bf 67}, 3578 (1991).\n\n\n\\bibitem{johnson}\nB.~L. Johnson and G. Kirczenow, Can.\\ J.\\ Phys. {\\bf 76}, 173 (1998).\n\n\n\\bibitem{imry}\nY. Imry and M. Peshkin, in {\\em Electron, a centenary volume}, edited by M.\n Springford (Cambrige University Press, Cambridge, 1997), pp.\\ 208--236.\n\n\n\\bibitem{imry_book}\nY. Imry, {\\em Introduction to mesoscopic physics} (Oxford, New York, 1997).\n\n\n\\bibitem{landau1}\nL.~D. Landau and E.~M. Lifshitz, {\\em Mechanics}, 3rd ed. (Pergamon, Oxford,\n 1976).\n\n\n\\bibitem{geller}\nM.~R. Geller, Phys. Rev. B {\\bf 53}, 9550 (1996).\n\n\n\\bibitem{essen89}\nH. Ess\\'en, Physica Scripta {\\bf 40}, 761 (1989).\n\n\n\\bibitem{aharonov}\nY. Aharonov and D. Bohm, Phys. Rev. {\\bf 115}, 485 (1959).\n\n\n\\bibitem{london}\nF. London, {\\em Superfluids, Volume 1, Macroscopic Theory of\n Superconductivity}, 2nd ed. (Dover, New York, 1961).\n\n\n\\bibitem{thouless}\nD.~J. Thouless, {\\em Topological quantum numbers in nonrelativistic physics}\n (World Scientific, Singapore, 1998).\n\n\n\\bibitem{byers}\nN. Byers and C.~N. Yang, Phys. Rev. Lett. {\\bf 7}, 46 (1961).\n\n\n\\bibitem{deaver}\nB.~S. Deaver, Jr. and W.~M. Fairbank, Phys. Rev. Lett. {\\bf 7}, 43 (1961).\n\n\n\\bibitem{doll}\nR. Doll and M. N\\\"abauer, Phys. Rev. Lett. {\\bf 7}, 51 (1961).\n\n\n\\bibitem{gough}\nC.~E. Gough {\\it et~al.}, Nature {\\bf 326}, 855 (1987).\n\n\n\\bibitem{brady}\nR.~M. Brady, Journal of Low Temperature Physics {\\bf 49}, 1 (1982).\n\n\n\\bibitem{cabrera}\nB. Cabrera and M.~E. Peskin, Phys. Rev. B {\\bf 39}, 6425 (1989).\n\n\n\\bibitem{liu}\nM. Liu, Phys. Rev. Lett. {\\bf 81}, 3223 (1998).\n\n\n\\bibitem{tate}\nJ. Tate, B. Cabrera, S. Felch, and J.~T. Anderson, Phys. Rev. Lett. {\\bf 62},\n 845 (1989).\n\n\n\\bibitem{sanzari}\nM.~A. Sanzari, H.~L. Cui, and F. Karwacki, Appl. Phys. Lett. {\\bf 68}, 3802\n (1996).\n\n\n\\bibitem{haines}\nL.~K. Haines and D.~H. Roberts, Am. J. Phys. {\\bf 37}, 1145 (1969).\n\n\n\\bibitem{rau}\nA.~R.~P. Rau, Am. J. Phys. {\\bf 53}, 1183 (1985).\n\n\n\\bibitem{yang}\nC.~N. Yang, Rev. Mod. Phys. {\\bf 34}, 694 (1962).\n\n"
}
] |
cond-mat0002097
|
Charge localization and phonon spectra in hole doped La$_{2}$NiO$_{4}$
|
[
{
"author": "R. J. McQueeney"
},
{
"author": "A. R. Bishop"
},
{
"author": "and Ya-Sha Yi"
}
] |
The in-plane oxygen vibrations in La$_{2}$NiO$_{4}$ are investigated for several hole-doping concentrations both theoretically and experimentally via inelastic neutron scattering. Using an inhomogeneous Hartree-Fock plus RPA numerical method in a two-dimensional Peierls-Hubbard model, it is found that the doping induces stripe ordering of localized charges, and that the strong electron-lattice coupling causes the in-plane oxygen modes to split into two subbands. This result agrees with the phonon band splitting observed by inelastic neutron scattering in La$_{2-x}$Sr$_{x}$NiO$_{4}$. Predictions of strong electron-lattice coupling in La$_{2}$NiO$_{4}$, the proximity of both oxygen-centered and nickel-centered charge ordering, and the relation between charged stripe ordering and the splitting of the in-plane phonon band upon doping are emphasized.
|
[
{
"name": "Yashas_PRL_eprint.tex",
"string": "\\documentstyle[aps,prl,epsfig]{revtex}\n\n\\begin{document}\n\n\\draft\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname\n@twocolumnfalse\\endcsname\n\n\\title{Charge localization and phonon spectra in hole doped La$_{2}$NiO$_{4}$}\n\n\\author{R. J. McQueeney, A. R. Bishop, and Ya-Sha Yi}\n\\address{Los Alamos National Laboratory, Los Alamos, New Mexico 87545}\n\n\n\\author{Z. G. Yu}\n\\address{Department of Chemistry, Iowa State University, Ames, Iowa 50011}\n\n\\date{Received on January 13, 2000}\n\n\\maketitle\n\n\\begin{abstract}\nThe in-plane oxygen vibrations in La$_{2}$NiO$_{4}$ are investigated for \nseveral hole-doping concentrations both theoretically and experimentally via \ninelastic neutron scattering. Using an inhomogeneous Hartree-Fock plus RPA \nnumerical method in a two-dimensional Peierls-Hubbard model, it is\nfound that the doping induces stripe ordering of localized charges,\nand that the strong electron-lattice coupling causes the in-plane \noxygen modes to split into two subbands. This result\nagrees with the phonon band splitting observed by inelastic neutron \nscattering in La$_{2-x}$Sr$_{x}$NiO$_{4}$.\nPredictions of strong electron-lattice coupling in La$_{2}$NiO$_{4}$,\nthe proximity of both oxygen-centered and nickel-centered charge\nordering, and the relation between charged stripe ordering and the\nsplitting of the in-plane phonon band upon doping are emphasized. \n\\end{abstract}\n\\smallskip\n\\pacs{PACS numbers: 75.60.Ch, 74.25.Kc, 71.45.Lr, 71.38.+i}\n]\n\n\nThere is currently great interest in the importance of charge\nlocalization and ordering tendencies in a variety of doped transition\nmetal oxides: including nickelates, bismuthates, cuprates, and\nmanganites\\cite{tranquada94,tranquada96,chen93,cheong9497,nakajima97,tranquada9597,ramirez97}.\nRecent experiments have suggested nanoscale coexistence of charge and\nspin ordering, as well as related multiscale dynamics\n\\cite{tranquada94,tranquada96,chen93,cheong9497,nakajima97,tranquada9597,ramirez97,blumberg98,katsufuji96,yamamoto98}.\nThe cuprates have been widely investigated, both theoretically and\nexperimentally, as this inhomogeneity may be related to\nhigh-temperature\nsuperconductivity\\cite{theoryblock,emery93,emery95,castroneto96,white97,krotov97,zaanen96,bianconi96}.\n\nThe nickelates are considered strong electron-lattice (e-l) coupling\nsystems, which helps stabilize charge-ordering in the form of \"stripe\"\nphases \\cite{wochner98,zaanen94}. For the commensurate 1/3 doping\ncase of La$_{1.67}$Sr$_{0.33}$NiO$_{4}$, it has been shown in optical\nabsorption and Raman scattering experiments that new phonon modes\nappear when the temperature is lowered below the stripe-ordering\ntemperature ($T_{so}$=240 K); this is a signature of the stripe\nformation \\cite{blumberg98,katsufuji96,yamamoto98}. Until now, only\nthe temperature dependence and the apical oxygen (Ni-O(2)) vibrations\nhave been investigated. However, the doping dependence and the\nin-plane oxygen vibrations (Ni-O(1) stretching modes) are also very\nimportant for the properties of the quasi-two dimensional nickelate\nmaterials. In this study, we use an inhomogeneous Hartree-Fock (HF)\nplus random-phase approximation (RPA) numerical method for a\ntwo-dimensional (2D), four-band Peierls-Hubbard model, to interpret\nthe inelastic neutron scattering spectra. This reveals specific\nsignatures of the stripe patterns in the in-plane oxygen phonons.\n\nOur main results are: (i) There is agreement between the results from\nour multiband model including electron-electron and e-l interactions\nand the inelastic neutron scattering spectra for the in-plane oxygen\nvibrations with various commensurate hole-doping concentrations; (ii)\nThe theoretical results predict new vibrational modes (``edge modes'')\nwhich are associated with oxygen motions near localized holes or in\nthe vicinity of stripes; (iii) The e-l coupling strength, at which the\nbest agreement between our model and the inelastic neutron scattering\ndata is achieved, is close to the transition from an oxygen-centered\nstripe phase to a nickel-centered one. This suggests that the\nnickelates may be in a mixed state of both stripe phases, and\nsensitive to temperature, pressure and magnetic field.\n\nThe inelastic neutron scattering spectra were measured on polycrystalline \nLa$_{2-x}$Sr$_{x}$NiO$_{4}$ for various doping concentrations, \n$x=$ 0, 1/8, 1/4, 1/3, 1/2. Time-of-flight neutron\nscattering measurements were performed on the Low Resolution Medium\nEnergy Chopper Spectrometer at Argonne National Laboratory's Intense\nPulsed Neutron Source. For all measurements, an incident neutron\nenergy of 120 meV was chosen and data were summed over all scattering\nangles from $2^{\\circ}-120^{\\circ}$. Detailed information\non the experiment, as well as the preparation of the samples used, can\nbe found in Ref. \\cite{mcqueeney99}. The focus is on the\n(neutron-scattering-weighted) generalized density-of-states (GDOS) of\nphonons and particular attention is given to the in-plane oxygen\nvibrations, i.e., the Ni-O(1) stretching modes. \n\nFigure \\ref{fig1} shows the experimental GDOS for several hole\nconcentrations at $T$=10 K for phonon modes in the range from 50-100\nmeV\\@. From the analysis of lattice dynamical shell model\ncalculations, it is known that the in-plane Ni-O(1) oxygen stretching\nmodes are separated in frequency from other vibrations, and the phonon\nintensity above 65 meV is associated entirely with these vibrations.\nFor the low doping samples (\\(x= 1/8\\) and \\(1/4\\), not shown), there\nis little change in the $\\sim$81 meV phonon band \\cite{mcqueeney99}, but\nfor the \\(x=1/3\\) and \\(x=1/2\\) samples, a new peak appears around 75\nmeV along with a slight hardening of the main band\\@. This feature is\ninterpreted as a splitting of the 81 meV Ni-O(1) stretching band into\ntwo phonon bands centered at approximately 83 meV and 75 meV.\n\n\\begin{figure}\n\\centering\n\\epsfxsize\\columnwidth\n\\epsffile{mcq_fig1.eps}\n\\caption{The generalized phonon density-of-states for three\nconcentrations of La$_{2-x}$Sr$_{x}$NiO$_{4}$. The frequency region\nshown (above 65 meV) consists of in-plane polarized oxygen modes\n(breathing modes) which are well-separated from other types of\nphonons. Data are offset vertically for clarity.}\n\\label{fig1}\n\\end{figure}\n\nIn order understand this dependence of the Ni-O(1) stretching modes on\nhole concentration and its possible reflection of stripe ordering, we\nhave performed a calculation of the phonon spectrum in a minimal\nPeierls-Hubbard model in 2D\\@. Due to the strong e-l coupling expected\nin the nickelates, we resort to modeling with an inhomogeneous HF plus\nRPA numerical approach \\cite{batistic92,yonemitsu92,yonemitsu93}.\nThis has proven to be a very robust method for studying charge\nlocalization and stripe formation, especially when electron-lattice\ncoupling is strong, obviating subtle many-body effects and quantum\nfluctuations \\cite{yi98}.\n\nWe use a 2D four-band extended Peierls-Hubbard model of a doped\nNiO$_{2}$ plane, which includes both electron-electron and e-l\ninteractions \\cite{yonemitsu92,yonemitsu93}. Here, for nickelate\noxides, besides the \\(d_{x^{2}-y^{2}}\\) orbital used in the cuprate\noxide models \\cite{yonemitsu92}, the \\(d_{3z^{2}-1}\\) Ni $d$ orbitals\nmust be included to account for the higher spin state (\\(S=1\\)) at\nhalf-filling (i.e. undoped). Our model Hamiltonian is \\cite{yi98}:\n\n\\begin{eqnarray}\nH & = & \\sum_{\\langle ij\n\\rangle,m,n,\\sigma}t_{im,jn}(u_{ij})(c^{\\dagger}_{im\\sigma}\nc_{jn\\sigma}+{\\rm H.c.})\\nonumber \\\\\n& + & \\sum_{i,m,\\sigma}\\epsilon_m\nc^{\\dagger}_{im\\sigma} c_{im\\sigma}+\\sum_{\\langle ij\n\\rangle}\\frac{1}{2}K_{ij}u^2_{ij}+H_c,\n\\end{eqnarray}\nwhere $c^{\\dagger}_{im\\sigma}$ creates a hole with spin $\\sigma$ at\nsite $i$ in orbital $m$ (Ni $d_{x^2-y^2}$, $d_{3z^2-1}$, or O $p$).\nThe Ni-O hopping $t_{im,jn}$ has two values: $t_{pd}$ between\n$d_{x^2-y^2}$ and $p$ and $\\pm t_{pd}/\\sqrt{3}$ between $d_{3z^2-1}$\nand $p$. The O-site electronic energy is \\(\\epsilon_{p}\\), and\nNi-site energies are \\(\\epsilon_{d}\\) and \\(\\epsilon_{d}+E_{z}\\) for\n\\(\\epsilon_{m}\\), with \\(E_{z}\\) the crystal-field splitting on the Ni\nsite. $H_c$ describes the electron correlations in the Ni orbitals,\n\n\\begin{eqnarray}\nH_c & = &\\sum_{im}(U+2J)n_{im\\uparrow}n_{im\\downarrow}\n-\\sum_{i,m\\ne n} 2J {\\rm\\bf S}_{im}\\cdot {\\rm\\bf S}_{in}\\nonumber\\\\ \n& + &\\sum_{i,m\\ne n,\\sigma,\\sigma^{\\prime}}(U-J/2)n_{im\\sigma}\nn_{in\\sigma^{\\prime}}\\nonumber\\\\\n& + &\\sum_{i,m,n}Jc^{\\dagger}_{im\\uparrow}\nc^{\\dagger}_{im\\downarrow}c_{in\\downarrow}c_{in\\uparrow}\n\\end{eqnarray} \nThe electron-electron interactions include the on-site Ni Coulomb\nrepulsions ($U$) as well as the Hund interaction ($J$) at the same Ni\nsite to account for the high spin situation. (The interplay of the\ntwo orbitals can also lead to pseudo Jahn-Teller distortions, but\nthese are not our focus here). We emphasize that, due to the large\nspin at the nickel site, Hund's rule leads to ferromagnetic exchange\ncoupling $-2J$, and \\({\\bf\nS_{im}}=\\frac{1}{2}\\sum_{\\tau\\tau'}c_{im\\tau}^{\\dagger}\n\\mbox{\\boldmath $\\sigma_{\\tau\\tau'}$}c_{im,\\tau'}\\),with\n\\(\\mbox{\\boldmath $\\sigma$}\\) the Pauli matrix. For the e-l\ninteraction, we consider that the Ni-O hopping is modified linearly by\nthe O-ion displacement $u_{ij}=u_{\\rm O}$ as\n$t_{im,jn}(u_{ij})=t_{im,jn}(1\\pm \\alpha u_{\\rm O})$, where the $+$\n($-$) applies if the bond shrinks (stretches) with positive $u_{\\rm\nO}$. For the lattice terms, we study only the motion of O ions along\nthe Ni-O bonds---other oxygen (or Ni) distortion modes can readily be\nincluded if necessary. It is known that for the nickelate oxides the\ne-l coupling is stronger than in cuprate oxides\n\\cite{wochner98,zaanen94}, and is therefore likely to play an even\nmore decisive role in the formation, localization, and nature of\nstripe phases. We adopt the following representative paremeters for\nthe nickelate materials \\cite{zaanen94}: \\(t_{pd}=1\\),\n\\(\\Delta=\\epsilon_{p}-\\epsilon_{d}=9\\), \\(U=4\\), \\(J=1\\), \\(E_{z}=1\\)\nand \\(K=32t_{pd}/\\AA^{2}\\) (all in units of \\(t_{pd}\\)). In real\noxides \\cite{zaanen94,vanelp92}, \\(t_{pd}\\) is estimated to be in the\nrange 1.3 eV $\\sim$ 1.5 eV\\@. The electron-lattice coupling strength\nis varied to achieve a best fit to the neutron scattering data; we\nfind \\(\\alpha \\approx 3.0\\). The commensurate doping cases are\nexamined in a 4x4 unit supercell for \\(x=0\\), 1/2 and a 3x3 unit supercell\nfor \\(x=1/3\\). Periodic boundary conditions are used.\n\nThe densities-of-states (DOS) of in-plane phonons were calculated from\nour model at $x=0$, 1/3, 1/2 and are shown in Fig. \\ref{fig2}. For the\nundoped case, as the ground state is spatially homogeneous, only one\noxygen phonon band appears, centered around 80.5 meV\\@. When holes\nare added into the NiO$_{2}$ plane at \\(x=1/3\\), the ground state is\nfound to be a stripe pattern with more holes accumulating along the\n(1,1,0) direction, forming an antiphase domain wall within the\noriginal antiferromagnetic background. This is consistent with many\nneutron, optical and Raman scattering experiments\n\\cite{cheong9497,blumberg98,katsufuji96,yamamoto98}. Interestingly,\nwe find that a new phonon band appears centered at 75 meV\\@. In\naddition, there is a hardening of the main phonon band (to 83 meV)\ncorresponding to an overall splitting of 8 meV\\@. From examination of\nthe eigenvectors, the main character of this band is local oxygen\nvibrations in the vicinity of the stripe, i.e. having the nature of\nlocalized \"edge modes\" (see \\cite{yi98} for more details). For the\nhigher doping \\(x=1/2\\), the charge-ordering takes on a commensurate\nchecker-board pattern. A similar splitting of the 85 meV phonon band\ninto two phonon bands around 83 and 75 meV is again found. At this\nhalf doping, the checker-board ground state is in essence a\ncommensurate charge-density-wave (CDW) system with the nature of an\nordered binary alloy.\n\n\\begin{figure}\n\\centering\n\\epsfxsize\\columnwidth\n\\epsffile{yi_fig2.eps}\n\\caption{The calculated densities-of-states of the oxygen breathing\nmodes for various doping concentrations. The results have been\nbroadened with a lorentzian of width 2 meV for $x=0$ and 1 meV for\n$x=1/3$ and 1/2. The electron-lattice coupling constant used\nis \\(\\alpha=3.0\\).}\n\\label{fig2}\n\\end{figure}\n\nThe above results are in agreement with the GDOS data obtained from\ninelastic neutron scattering, which are shown in Fig. \\ref{fig1}. In\naddition to the splitting energy, even the slight hardening of the\nmain band observed experimentally is accounted for in the model.\nBesides the new phonon modes centered at 75 meV appearing for the 1/3\nand 1/2 doping, another low intensity phonon mode around 65 meV is\nalso predicted in our model at \\(x=1/2\\) (Fig. \\ref{fig2}). The\nsignature for these modes are weak, however, so that they may be\ndifficult to detect in the current experiment. In so far as we have\nincluded only a small subset of the possible oxygen displacement\npatterns and wavevectors in the model (whereas the neutron scattering\nexperiment samples all wavevectors and polarizations), the relative\nintensities and widths of the bands obtained by experiment and theory\ncannot be usefully compared.\n\n\\begin{figure}\n\\centering\n\\epsfxsize\\columnwidth\n\\epsffile{yi_fig3.eps}\n\\caption{The energy dependence on the electron-lattice coupling\n$\\alpha$ for the 1/3 doped nickelates (solid line). For \n\\(\\alpha \\leq 2.2\\), the ground state is in an O-centered stripe\nphase. While for \\(\\alpha \\geq 3.0\\), the Ni-centered state is found as the\nground state. The sensitive transition region is from \\(\\alpha=2.2\\)\nto \\(\\alpha=3.0\\).}\n\\label{fig3}\n\\end{figure}\n\nWe emphasize that the excellent agreement between our model and the\nGDOS experimental data is achieved by varying the e-l coupling\nstrength to match the positions of the phonon bands. As noted, the\nchoice of \\(\\alpha \\approx 3\\) best fits the data; our model\ncalculations predict a variation of the phonon splitting with $\\alpha$\nwhich is quite large (\\(\\Delta\\omega_{split}/\\Delta\\alpha \\approx\\) 7\nmeV). Most strikingly, as illustrated in Fig. \\ref{fig3}, an\nO-centered stripe is found as the ground state at small\n\\(\\alpha \\alt 2.2\\), while for a larger \\(\\alpha \\agt\n3.0\\), a Ni-centered stripe is found as the ground state\\cite{yi98}.\nThe transition region includes \\(\\alpha \\approx 3.0\\), where the best\nagreement between Fig. \\ref{fig2} based on our model and\nFig. \\ref{fig1} on the inelastic neutron scattering spectra is\nachieved. It has been suggested from various experimental data that\nstripe formation for $x=1/3$ can not be simply assigned as Ni-centered\nor O-centered \\cite{wochner98}, but is also dependent on temperature.\nOur comparison of theory and experiment provides a possible\nexplanation on the sensitivity of stripe formation; they suggest that\nLa$_{1.67}$Sr$_{0.33}$NiO$_{4}$ may be in a mixed stripe phase state,\nand also in a region of sensitivity to temperature, pressure, magnetic\nfield, etc.\n\nIn conclusion, we have made a study of oxygen breathing lattice\nvibrations in La$_{2-x}$Sr$_{x}$NiO$_{4}$ via inelastic neutron\nscattering compared with predictions of a 2D four-band model,\nincluding both electron-lattice and electron-electron interactions.\nThe in-plane oxygen vibrations above 65 meV were thoroughly\ninvestigated. The splitting of the in-plane 81 meV band upon doping\ninto two subbands centered around 75 meV and 83 meV is observed\nexperimentally and predicted theoretically, and interpreted in terms\nof new localized phonon modes (``edge modes'' at charge localized\nstripes). The excellent agreement between the experiment and the\nmodel strongly supports the view that strong electron-lattice coupling\nin this kind of material plays a decisive role on the charge\nlocalization and mesoscopic stripe formation. For the doping at\n$x=1/3$, at which stripes are found both in experiments and our model,\nour results suggest that there may be a mixed state of O- and\nNi-centered stripe phases, and sensitivity to temperature, pressure,\nand magnetic field. Our model calculations also predict distinctive\ndispersion of the phonon bands, as well as inhomogeneous\nmagnetoelastic coupling along the boundaries between charge-rich and\nmagnetic nanophase domains \\cite{yi98}. These predictions require\nadditional experiments for their confirmation and consequences.\n\nWe have benefitted from valuable discussions with\nDr. J. T. Gammel. This work is supported (in part) by the U.~S.\\\nDepartment of Energy under contract W-7405-Eng-36 with the\nUniversity of California. This work has benefited from the use of the\nIntense Pulsed Neutron Source at Argonne National Laboratory. This\nfacility is funded by the U.~S.\\ Department of Energy, BES-Materials\nScience, under contract W-31-109-Eng-38.\n\n%\\appendix\n%\\section{Appendixes}\n\n\\begin{references}\n\\bibitem{tranquada94}J. M. Tranquada, D. J. Buttrey, V. Sachan, and\n J. E. Lorenzo, Phys.\\ Rev.\\ Lett. {\\bf 73}, 1003 (1994).\n\\bibitem{tranquada96}J. M. Tranquada {\\it et al.}, Phys.\\ Rev.\\ B\n {\\bf 54}, 7489 (1996). \n\\bibitem{chen93}C. H. Chen, S.-W. Cheong and A. S. Cooper, Phys.\\\n Rev.\\ Lett. {\\bf 71}, 2461 (1994).\n\\bibitem{cheong9497}S.-W. 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"name": "cond-mat0002097.extracted_bib",
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cond-mat0002098
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Partition Function Zeros and Finite Size Scaling\\ of Helix-Coil Transitions in a Polypeptide
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"author": "\\wideabs{ \\title{Partition Function Zeros and Finite Size Scaling"
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We report on multicanonical simulations of the helix-coil transition of a polypeptide. The nature of this transition was studied by calculating partition function zeros and the finite-size scaling of various quantities. New estimates for critical exponents are presented.
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"name": "ah99b.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% %\n% A LaTeX file with some definitions at the top. %\n% %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\n% MTU-PHY-HA-99/2\n% MAY 3, 1999\n% REVISED AUG 16. 1999\n% REVISED DEC 14. 1999\n\n\\documentstyle[aps,prl,twocolumn,graphicx,floats]{revtex}\n\n\\begin{document}\n\n\\pagenumbering{arabic}\n\\def\\E{{\\rm e}}\n\\newcommand{\\be}{\\begin{enumerate}}\n\\newcommand{\\ee}{\\end{enumerate}}\n\\newcommand{\\beq}{\\begin{equation}}\n\\newcommand{\\eeq}{\\end{equation}}\n\\newcommand{\\uu}{\\underline}\n\\renewcommand{\\thesubsection}{\\arabic{subsection}}\n\n% \\draft command makes pacs numbers print\n\\draft\n% repeat the \\author\\address pair as needed\n{\\wideabs{ \n\\title{Partition Function Zeros and Finite Size Scaling\\\\ of Helix-Coil \n Transitions in a Polypeptide}\n\\author{Nelson A. Alves\\footnote{alves@quark.ffclrp.usp.br}}\n\\address{Departamento de F\\'{\\i}sica e Matem\\'atica, FFCLRP \n Universidade de S\\~ao Paulo. Av. Bandeirantes 3900. CEP 014040-901 \n Ribeir\\~ao Preto, SP, Brazil}\n\\author{Ulrich H.E. Hansmann \\footnote{hansmann@mtu.edu}}\n\\address{Department of Physics, Michigan Technological University,\n Houghton, MI 49931-1291, USA}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nWe report on multicanonical simulations of the helix-coil transition\nof a polypeptide. The nature of this transition was studied by calculating\npartition function zeros and the finite-size scaling of various\nquantities. New estimates for critical exponents are presented. \n\\end{abstract}\n\\pacs{87.15.He, 87.15-v, 64.70Cn, 02.50.Ng}\n}}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nA common, ordered structure in proteins is the \\hbox{$\\alpha$-helix} and it is \nconjectured that formation of $\\alpha$-helices is a key factor in the early \nstages of protein-folding \\cite{BSG}. It is long known that $\\alpha$-helices \nundergo a sharp transition towards a random coil state when the temperature is\nincreased. The characteristics of this so-called helix-coil transition have \nbeen studied extensively \\cite{Poland}, \nmost recently in Refs.~\\cite{Jeff,HO98c}.\nThey are usually described in the framework of Zimm-Bragg-type\ntheories \\cite{ZB} in which the homopolymers are approximated by a\none-dimensional Ising model with the residues as ``spins'' taking values \n``helix'' or ``coil'', and solely local interactions. Hence, in such theories\nthermodynamic phase transitions are not possible. However, \nin preliminary work \\cite{HO98c} it was shown that our all-atom model of \npoly-alanine exhibits \na phase transition between the ordered helical state and the disordered \nrandom-coil state. It was conjectured that this transition \nis due to long range interactions in our\nmodel and the fact that it is not one-dimensional:\nit is known that the 1D Ising model with long-range interactions\nalso exhibits a phase transition at finite $T$ if the interactions\ndecay like $1/r^{\\sigma}$ with $1\\le \\sigma < 2$ \\cite{1dlr}.\nOur aim now is to investigate this transition in the frame work of a critical\ntheory by means of the finite size scaling (FSS) analysis of partition function \nzeros. Analysis of partition function zeros is a well-known tool in the \nstudy of phase transitions, but was to our knowledge never used before to \nstudy biopolymers.\n\n\nFor our project, the use of the multicanonical algorithm \\cite{MU}\nwas crucial. The various competing interactions within the polymer lead\nto an energy landscape characterized by a multitude of local minima.\nHence, in the low-temperature region, canonical \nsimulations will tend to get trapped in one of these\nminima and the simulation will not thermalize within the available\nCPU time. One standard way to overcome this problem is the application\nof the {\\it multicanonical algorithm} \\cite{MU} and other \n{\\it generalized-ensemble} techniques \\cite{Review} to the protein folding \nproblem \\cite{HO}. For poly-alanine, \nboth the failure of standard Monte Carlo techniques \nand the superior performance of the\nmulticanonical algorithm are extensively documented in \nearlier work \\cite{OH95b}. \n\nIn the multicanonical algorithm \\cite{MU} \nconformations with energy $E$ are assigned a weight\n$ w_{mu} (E)\\propto 1/n(E)$. Here, $n(E)$ is the density of states.\nA simulation with this weight \nwill lead to a uniform distribution of energy:\n\\begin{equation}\n P_{mu}(E) \\, \\propto \\, n(E)~w_{mu}(E) = {\\rm const}~.\n\\label{eqmu}\n\\end{equation}\nThis is because the simulation generates a 1D random walk in the energy,\nallowing itself to escape from any local minimum. \nSince a large range of energies are sampled, one can\nuse the reweighting techniques \\cite{FS} to calculate thermodynamic\nquantities over a wide range of temperatures by\n\\begin{equation}\n<{\\cal{A}}>_T ~=~ \\frac{\\displaystyle{\\int dx~{\\cal{A}}(x)~w^{-1}(E(x))~\n e^{-\\beta E(x)}}}\n {\\displaystyle{\\int dx~w^{-1}(E(x))~e^{-\\beta E(x)}}}~,\n\\label{eqrw}\n\\end{equation}\nwhere $x$ stands for configurations. \n\nIt follows from Eq.~\\ref{eqmu} that the multicanonical algorithm allows us\nto calculate estimates for the spectral density:\n\\begin{equation} \n n(E) = P_{mu} (E) w^{-1}_{mu} (E)~.\n\\end{equation}\nWe can therefore construct\nthe partition function from these estimates by\n\\begin{equation}\n Z(\\beta) = \\sum_{E} n(E) u^{E} , \\label{eq:r1}\n\\end{equation}\nwhere $u=e^{-\\beta}$ with $\\beta$ the inverse temperature, $\\beta = 1/k_B T$.\nThe complex solutions of the partition function \ndetermine the critical behavior of the model. \nThey are the so-called Fisher zeros \\cite{fisher,itzykson}, and\ncorrespond to the complex extension of the temperature variable.\n\nOur investigation of the helix-coil transition for poly-alanine is\nbased on a detailed, all-atom representation of that homopolymer, and\ngoes beyond the approximations of the Zimm-Bragg model \\cite{ZB}.\nThe interaction between the atoms was\ndescribed by a standard force field, ECEPP/2,\\cite{EC} (as implemented \nin the KONF90 program \\cite{Konf}) and is given by:\n\\begin{eqnarray}\nE_{tot} & = & E_{C} + E_{LJ} + E_{HB} + E_{tor},\\\\\nE_{C} & = & \\sum_{(i,j)} \\frac{332q_i q_j}{\\epsilon r_{ij}},\\\\\nE_{LJ} & = & \\sum_{(i,j)} \\left( \\frac{A_{ij}}{r^{12}_{ij}}\n - \\frac{B_{ij}}{r^6_{ij}} \\right),\\\\\nE_{HB} & = & \\sum_{(i,j)} \\left( \\frac{C_{ij}}{r^{12}_{ij}}\n - \\frac{D_{ij}}{r^{10}_{ij}} \\right),\\\\\nE_{tor}& = & \\sum_l U_l \\left( 1 \\pm \\cos (n_l \\chi_l ) \\right).\n\\end{eqnarray}\nHere, $r_{ij}$ (in \\AA) is the distance between the atoms $i$ and $j$, and\n$\\chi_l$ is the $l$-th torsion angle. Note that with the electrostatic\nenergy term $E_{C}$ our model contains a long range interaction neglected\nin the Zimm-Bragg theory \\cite{ZB}.\nSince one can avoid the\ncomplications of electrostatic and hydrogen-bond interactions of\nside chains with the solvent for alanine (a nonpolar amino acid), explicit\nsolvent molecules were neglected. \nChains of up to $N=30$ monomers were considered. \nWe needed between 40,000 sweeps ($N=10$) and 500,000 sweeps ($N=30$) for\nthe weight factor calculations by the iterative procedure described in \nRefs.~\\cite{MU,HO94c}. \nAll thermodynamic quantities were estimated from one\nproduction run of $N_{sw}$ Monte Carlo sweeps starting from a random\ninitial conformation, i.e. without introducing any bias.\nWe chose $N_{sw}$=400,000, 500,000, 1,000,000, and 3,000,000 sweeps\nfor $N=10$, 15, 20, and 30, respectively.\n\nFor our analysis of the partition function zeros we first divide the\nenergy range into intervals of lengths $0.5$ kcal/mol. Equation~\\ref{eq:r1}\nbecomes now a polynomial in the variable $u$ and can be easily solved\nwith MATHEMATICA to obtain all complex zeros $u_j^0$ ($j=1,2, ...$).\nFor the case of $N=10$ we also repeated the calculation of the zeros for\nenergy bin sizes $1.0$ kcal/mol and $0.25$ kcal/mol. The changes in\nthe zeros were smaller than the statistical errors. The effect of\nthe energy bin size on the zeros is also discussed in Ref.~\\cite{Kar}.\nFigure 1 shows the distribution of the zeros for $N=30$ and provides \nalready strong evidence for a singularity on the real axis: \nin the case of the (analytic) Zimm-Bragg theory \nthe zeros would be located solely on the negative real $u$-axis \\cite{Shrock}. \nWe summarize in Table I the\nleading zeros for each of the four chain lengths, where we have used the\nmapping $u=e^{-\\beta/2}$ due to our binning procedure.\n\n% TABLE 1\n\\begin{table}[t]\n\\begin{tabular}{lllll} \n \\\\[-0.4cm] \n ~$N$ & ~~${\\rm Re}\\,(u_1^0)$ & ~~${\\rm Im}\\,(u_1^0)$ & \n ~~~~${\\rm Re}\\,(\\beta_1^0)$ & ~~${\\rm Im}\\,(\\beta_1^0)$ \\\\\n\\\\[-0.45cm]\n\\hline\n\\\\[-0.4cm]\n ~10 & ~$0.5620(60)$ & ~$0.0702(33)$ & ~~~$1.138(21)$ & ~$0.248(11)$ \\\\\n ~15 & ~$0.6015(23)$ & ~$0.0472(21)$ & ~~~$1.0104(77)$ & ~$0.1566(67)$ \\\\\n ~20 & ~$0.6105(29)$ & ~$0.03275(88)$ & ~~~$0.9842(94)$ & ~$0.1072(26)$ \\\\\n ~30 & ~$0.6159(19)$ & ~$0.02200(78)$ & ~~~$0.9681(63)$ & ~$0.0714(25)$ \\\\\n\\end{tabular}\n\\vspace{0.1cm}\n\\caption{\\baselineskip=0.8cm First partition function zeros for \n poly-alanine chains of various chain lengths.}\n\\end{table}\n\n The FSS relation by Itzykson {\\it et al.} \\cite{itzykson} for the leading\nzero $u_1^0(N)$, \n\\beq\n u_1^0(N) = u_c + A N^{-1/ d\\nu}[1+O(N^{y/d})]~, ~~~~~~ y<0 \\label{eq:r2}\n\\eeq\nshows that the distance from the closest zero $u_1^0$, to the\ninfinite-chain critical point $u_c = e^{-\\beta_c/2}$ on Re($u$)\naxis, scales with a relevant linear length $L$, which we translated as \n$N^{1/d}$ in the above equation. Here, $\\beta_c$ is the inverse critical\ntemperature of the infinite long polymer chain and $y$ is the correction\nto scaling exponent.\n We remark that, unlike in the Zimm-Bragg model, we have no \ntheoretical indication to assume $d$ as a \nparticular integer geometrical dimension and report therefore \n estimates for the quantity $d\\nu$.\n\n\n\n For sufficiently large $N$, the exponent $d\\nu$ can be obtained\nfrom the linear regression\n\\beq\n-\\,{\\rm ln}\\, |u_1^0(N)-u_c| = \\frac{1}{d\\nu}\\,{\\rm ln} (N) + a~. \\label{eq:r3}\n\\eeq\n This relation requires an accurate estimate for $u_c$. Therefore,\n we prefer to calculate our estimates for $d\\nu$ from the corresponding\nrelation with $| u_1^0 - u_c|$ replaced by its imaginary part \n${\\rm Im}\\, u_1^0$.\nIncluding chains of all lengths, $N=10-30$, \nthis approach leads to $d\\nu = 0.93(5)$, with a goodness of fit $Q=0.48$. \nFigure~2 displays the corresponding fit. \nOmitting the smallest chain, i.e. restricting the fit to the range\n$N=15-30$, does not change the above result. We obtain now $d\\nu=0.93(7)$, \nwith $Q=0.22$. This indicates that the $d\\nu$ determination is stable \nover the studied chains and therefore, the correction exponent $y$ can be\ndisregarded in face of the present statistical error.\n\nConsidering the real part of the leading zeros given in Table I,\n${\\rm Re}\\, (\\beta_1^0(N)) =\n - {\\rm ln}\\{ [{\\rm Re}\\,u_1^0(N)]^2\\, + [{\\rm Im}\\,u_1^0(N)]^2 \\}$, \nwe can \nderive the critical temperature through the following FSS fit \n\\cite{fukugita},\n\\beq {\\rm Re}\\,(\\beta_1^0(N)) = \\beta_c + b N^{-1/d\\nu} ~. \\label{eq:r4}\n\\eeq\nWe obtain $\\beta_c = 0.906(12)$ ($Q=0.005$)\nfor the range $N=10-30$, and $\\beta_c =0.929(14)$ ($Q=0.63$) for\n$N=15-30$.\nThe last and more acceptable estimate, corresponds to \n$T_c(\\infty)=541(8)$ K.\n\n\n A stronger version for the relation (\\ref{eq:r2}) considers that\nthe next zeros $u_j^0(N)$, should also satisfy a scaling relation \n\\cite{itzykson},\n\\beq\n |u_j^0(N) - u_c| \\sim \\left( \\frac{j}{N} \\right)^{1/d\\nu}~,\n \\label{eq:r6}\n\\eeq\nwhere $j$ labels the zeros in order of increasing distance from $u_c$.\nThis relation is expected to be satisfied for large $j$ and allows for an\nindependent check of the estimate for our exponent $d\\nu$. The scaling plot \nin Fig. 3 for the \nroots closest to the critical point $u_c$ demonstrates that the\nassumed scaling relation is indeed observed for our data as $N$ increases\nand consistent with our estimate of the exponent $d\\nu$.\n\n\n% TABLE 2\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{llllll} \n\\\\[-0.4cm] \n ~$N$ & ~~~$T_c$ & ~~~$C^{max}$ & ~~$\\Gamma_{C}$ & ~$T_{min}$ & ~$b(T_{min},N)$\\\\\n\\\\[-0.45cm]\n\\hline\n\\\\[-0.4cm]\n ~10& ~427(7)& ~~8.9(3) & ~150(7) & ~~298 & ~~-0.48(4)\\\\\n ~15& ~492(5)& ~~12.3(4) & ~119(5) & ~~429 & ~~-0.59(10)\\\\\n ~20& ~508(5)& ~~16.0(8) & ~88(5) & ~~469 & ~~-0.55(8)\\\\\n ~30& ~518(7)& ~~22.8(1.2)& ~58(4) & ~~500 & ~~-0.20(4)\\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\baselineskip=0.8cm\n Numerical results for poly-alanine chains of various lengths:\n critical temperature $T_c$ defined by the maximum of specific heat $C_{max}$,\n width $\\Gamma_{C}$ of peak in specific heat and\n temperature $T_{min}$ where the Binder cumulant $b(T,N)$ \n has its minimum, $b(T_{min},N)$.}\n\\end{table}\n Our results for the critical temperature and critical exponent can be\ncompared with independent estimates obtained from FSS of the specific heat:\n \\begin{equation}\n C_N (T) = {\\beta}^2 \\ (<E^2(T)> - {<E(T)>}^2)/N~.\n\\label{eqsh}\n\\end{equation}\nDefining the critical temperature $T_c(N)$ as the\nposition where the specific heat $C_N(T)$ has its maximum,\nwe can again calculate the critical temperature by means of Eq.~\\ref{eq:r4}.\nWith the values in Table~II we obtain \n$T_c(\\infty) = 544(12)$ K, which is consistent with \nthe value obtained from the partition function zeros analysis,\n$T_c(\\infty) =541(8)$ K.\nChoosing $T_1(N)$ and $T_2(N)$ \n such that $C(T_1) = 1/2 \\,C(T_c) = C(T_2)$, we have the \nfollowing scaling relation\nfor the width $\\Gamma_C (N)$ of the specific heat \\cite{fukugita},\n\\begin{equation}\n\\Gamma_C (N) = T_2(N) - T_1(N) \\propto N^{-1/d\\nu}.\n\\label{nu1}\n\\end{equation}\nUsing the above equation and the values given in Table~II,\nwe obtain $d\\nu = 0.98(11)$ $(Q=0.9)$\nfor chains of length $N=15$ to $N=30$, i.e. omitting the shortest \nchain.\nThis value is in agreement with \nour estimate $d\\nu=0.93(5)$, \n obtained from the partition function zero analysis. \nIncluding $N=10$ leads to $d\\nu=1.19(10)$, but with a\nless acceptable fit ($Q=0.1$). \nThe analysis of partition function zeros seems\nalso to be more stable than one relying on Eq.~\\ref{nu1}. \nNo significant change in $d\\nu$ was observed\nwhen the data from ref.~\\cite{HO98c} (which relied on much smaller\nnumber of Monte Carlo sweeps) were used in the partition function zeros \nanalysis, while Eq.~\\ref{nu1} leads for this reduced\nstatistics to an estimate for $d\\nu=1.9$. \n\n\n\nThrough the scaling relation for the peak of the specific heat, we can \nevaluate yet another critical exponent, the specific heat exponent \n$\\alpha$, by: \n\\begin{equation}\nC_N^{max} \\propto N^{\\alpha/d\\nu}~.\n\\label{alpha}\n\\end{equation}\nIn particular, with the values for $C_N^{max}$ as given in Table~II,\nwe obtain $\\alpha = 0.86(10)$. \nThe scaling plot for the specific heat is shown in Fig.~4: curves for all\nlengths of the poly-alanine chains nicely collapse on each other indicating\nthe scaling of the specific heat and the reliability of our exponents.\n It worths to note that our estimates for $d\\nu$ and $\\alpha$,\nas obtained from the finite size scaling of the specific heat,\nobey within the errorbars the hyperscaling relation\n$ d\\nu = 2 - \\alpha $.\n\nIt is well-known that renormalization-group fixed point picture \nleads to a critical exponent $d\\nu=1$, $\\alpha=1$ and $\\gamma =1$\n for a first-order phase transition \n\\cite{fukugita,fisher_nu,alves_nu}. \nOur estimate $d\\nu = 0.93(5)$ for the correlation exponent deviates from unity\nand rather indicates that the `helix-coil-transition'' is\na strong second order transition. However, the errorbars are such that\na first order phase transition cannot be excluded. Our values \nfor the specific heat exponent $\\alpha=0.86(10)$ and the\nsusceptibility exponent $\\gamma=1.06(14)$ (data not shown) are \nconsistent with a first order phase transition, but also not conclusive.\nA common way to evaluate the order of a phase transition is by means of\nthe Binder energy cumulant \\cite{binder1},\n\\begin{equation}\n b(T,N) = 1 - \\frac{<E^4(T,N)>}{3 <E^2(T,N)>^2}~.\n\\end{equation}\nFor a second order phase transition one would expect that the \nminimum of this quantity $b(T_{min},\\infty)$ approaches $2/3$.\n Here $T_{min}$ defines the temperature where the cumulant \n reaches its minimum value and\n$ b(T_{min},\\infty) = \\lim_{N \\rightarrow \\infty} b(T,N) $.\nWith the present values of Table II we find the infinite \nvolume extrapolation\n$ b(T_{min},\\infty) = 0.23(13)$ $(Q=0.12)$, for the range $N=15-30$,\nwhich is consistent with a first order phase transition.\n However, we cannot exclude the possibility \nof a second order phase transition because the energy cumulant scales with the\nmaximum of specific heat \\cite{berg1}, \n$ b(T,N) \\sim N^{\\alpha/d\\nu -1} $, \nand the true asymptotic limit is reached only for \nrather large chains due to the value of $\\alpha/d\\nu$. \nIn fact,\nthe straight line fit for the range $N=10-30$ is less consistent with\n our data ($Q\\simeq 0.001$). \n%Hence, we conclude that both values for the critical exponents and the\n%minimum of the Binder energy cumulant seems to favor a\nHence, we conclude that our results seem to favor a \n(weak) first order phase transition, but are not precise enough to exclude\nthe possibility of a second order phase transition.\n\nTo summarize, we have used a common technique for\ninvestigation of phase transitions, analysis of the finite size scaling\nof partition function zeros, to evaluate the helix-coil \ntransition in an all-atom model of poly-alanine. \n Although our results are due to the complexity of the simulated model \nnot precise enough to determine the order of the phase transition, we\n have demonstrated that\nthe transition can be described by a set of critical exponents. \n Hence, we have shown for this example that structural transitions \nin biological molecules can be described within \nthe frame work of a critical theory. \n\n\\noindent\n{\\bf Acknowledgements}: \\\\\nFinancial supports from FAPESP and a Research Excellence\nFund of the State of Michigan\nare gratefully acknowledged. \n\n%\\end{multicols}\n\n%\\begin{thebibliography}{99}\n\\begin{references}\n\\bibitem{BSG} R.M.~Ballew, J.~Sabelko, and M.~Gruebele, \n Proc.~Nat.~Acad.~Sci. USA {\\bf 93}, 5759 (1996).\n\\bibitem{Poland} D.~Poland and H.A.~Scheraga, {\\it Theory of Helix-Coil\n Transitions in Biopolymers} (Academic Press, New York, 1970).\n\\bibitem{Jeff} J.P.~Kemp and Z.Y. Chen, Phys.~Rev.~Lett. \n {\\bf 81}, 3880 (1998).\n\\bibitem{HO98c} U.H.E.~Hansmann and Y.~Okamoto, J. Chem.~Phys. {\\bf 110},\n 1267 (1999); {\\bf 111} 1339(E) (1999).\n\\bibitem{ZB} B.H. Zimm and J.K. Bragg, J. Chem. Phys. {\\bf 31}, 526 (1959).\n\\bibitem{1dlr} F.J.~Dyson, Commun.~Math.~Phys. {\\bf 12}, 212 (1969);\n J.F.~Nagle and J.C.~Bonner, J.~Phys.~C {\\bf 3}, 352 (1970);\n E.~Bayong, H.T.~Diep and V.~Dotsenko, Phys.~Rev.~Lett. {\\bf 83},\n 14 (1999).\n\\bibitem{MU} B.A.~Berg and T.~Neuhaus, \n Phys. Lett. {\\bf B 267}, 249 (1991).\n\\bibitem{Review} U.H.E.~Hansmann and Y.~Okamoto, \n in: Stauffer, D. (ed.) ``{\\it Annual Reviews in Computational \n Physics VI}''\n (Singapore: World Scientific), p.129. (1998).\n\\bibitem{HO} U.H.E. Hansmann and Y. Okamoto, J.~Comp.~Chem.\n {\\bf 14}, 1333 (1993).\n\\bibitem{OH95b} Y.~Okamoto and U.H.E.~Hansmann,\\ J.~Phys.~Chem.\n {\\bf 99}, 11276 (1995).\n\\bibitem{FS} A.M. Ferrenberg and R.H. Swendsen, Phys.\\ Rev.\\ Lett.\n {\\bf 61}, 2635 (1988); Phys. Rev. Lett. {\\bf 63},\n1658(E) (1989), and\n references given in the erratum.\n\\bibitem{fisher} M.E. Fisher, in {\\it Lectures in Theoretical Physics}, \n vol. 7c p.1 (University of Colorado Press, Boulder, 1965).\n\\bibitem{itzykson} C. Itzykson, R.B. Pearson, and J.B. Zuber,\n Nucl. Phys. B {\\bf 220} [FS8], 415 (1983).\n\\bibitem{EC} M.J. Sippl, G. N{\\'e}methy, and H.A. Scheraga,\n{\\it J. Phys. Chem.} {\\bf 88}, 6231~(1984), and references therein.\n\\bibitem{Konf} H.~Kawai, Y.~Okamoto, M.~Fukugita, T.~Nakazawa, and\n T.~Kikuchi, Chem. Lett. {\\bf 1991}, 213 (1991);\n Y.~Okamoto, M.~Fukugita, T.~Nakazawa, and H.~Kawai,\n Protein Engineering {\\bf 4}, 639 (1991).\n\\bibitem{HO94c} U.H.E.~Hansmann and Y.~Okamoto, \n Physica A {\\bf 212}, 415 (1994).\n\\bibitem{Kar} M. Karliner,S.R. Sharpe and Y.F. Chang, \n Nucl. Phys. {\\bf B302}, 204 (1988).\n\\bibitem{Shrock} V. Matveev and R. Shrock, Phys. Lett. {\\bf A204}, 353 (1995).\n\\bibitem{fukugita} M. Fukugita, H. Mino, M. Okawa and A. Ukawa,\n J. Stat. Phys. {\\bf 59}, 1397 (1990),\n and references given therein.\n\\bibitem{fisher_nu} M.E. Fisher and A.N. Berker, Phys. Rev. B {\\bf 26}, 2507 \n (1982).\n\\bibitem{alves_nu} N.A. Alves, B.A. Berg, and R. Villanova, \n Phys. Rev. B {\\bf 43}, 5846 (1991). \n\\bibitem{binder1} K. Binder, Phys. Rev. Lett. {\\bf 47}, 693 (1981).\n\\bibitem{berg1} N.A. Alves, B.A. Berg, and S. Sanielevici, Phys. Lett. \n B {\\bf 241}, 557 (1990).\n\\end{references}\n%\\end{thebibliography}\n\n\n\\newpage\n{\\Large Figure Captions:}\\\\\n\\begin{enumerate}\n\\item Partition function zeros in the complex $u$ plane for $N=30$.\n For the Zimm-Bragg model the zeros would be located solely on\n the negative real $u$ axis.\n\\item Linear regression for -ln\\,(Im$\\,u^0_1(N)$) in the range $N=10-30$.\n\\item Scaling behavior of the first $j$ complex zeros closest\n to $u_c = 0.6284$, for chain lengths $N = 10,15,20$ and 30.\n\\item Scaling plot for the specific heat $C_N(T)$ as a function of \n temperature $T$, for poly-alanine molecules of chain lengths \n $N=10, 15, 20,$ and $30$.\n\\end{enumerate}\n\n\n\n\\newpage\n\\cleardoublepage\n\n%FIGURE 1\n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}[t]{0.95\\textwidth}\n\\centering\n\\includegraphics[angle=-90,width=0.72\\textwidth]{ah99_fig1.eps}\n\\renewcommand{\\figurename}{FIG.}\n\\caption{Partition function zeros in the complex $u$ plane for $N=30$.\n For the Zimm-Bragg model the zeros would be located solely on\n the negative real $u$ axis.}\n\\label{fig1}\n\\end{minipage}\n\\end{center}\n\\end{figure}\n\n\n\\newpage\n\\cleardoublepage\n\n%FIGURE 2\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{minipage}[t]{0.95\\textwidth}\n\\centering\n\\includegraphics[angle=-90,width=0.72\\textwidth]{ah99_fig2.eps}\n\\renewcommand{\\figurename}{FIG.}\n\\caption{Linear regression for -ln\\,(Im$\\,u^0_1(N)$) in the range $N=10-30$.}\n\\label{fig2}\n\\end{minipage}\n\\end{center}\n\\end{figure}\n\n\\newpage\n\\cleardoublepage\n\n\n%FIGURE 3\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{minipage}[t]{0.95\\textwidth}\n\\centering\n\\includegraphics[angle=-90,width=0.72\\textwidth]{ah99_fig3.eps}\n\\renewcommand{\\figurename}{FIG.}\n\\caption{Scaling behavior of the first $j$ complex zeros closest\n to $u_c = 0.6284$, for chain lengths $N = 10,15,20$ and 30.}\n\\label{fig3}\n\\end{minipage}\n\\end{center}\n\\end{figure}\n\n\n\\newpage\n\\cleardoublepage\n\n%FIGURE 4\n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}[t]{0.95\\textwidth}\n\\centering\n\\includegraphics[angle=-90,width=0.72\\textwidth]{ah99_fig4.eps}\n\\renewcommand{\\figurename}{FIG.}\n\\caption{Scaling plot for the specific heat $C_N(T)$ \n as a function of temperature $T$, for poly-alanine \n molecules of chain lengths $N = 10, 15, 20$ and 30.}\n\\label{fig4}\n\\end{minipage}\n\\end{center}\n\\end{figure}\n\n\n\\end{document}\n\n\n"
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[
{
"name": "cond-mat0002098.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\begin{references}\n\\bibitem{BSG} R.M.~Ballew, J.~Sabelko, and M.~Gruebele, \n Proc.~Nat.~Acad.~Sci. USA {\\bf 93}, 5759 (1996).\n\\bibitem{Poland} D.~Poland and H.A.~Scheraga, {\\it Theory of Helix-Coil\n Transitions in Biopolymers} (Academic Press, New York, 1970).\n\\bibitem{Jeff} J.P.~Kemp and Z.Y. Chen, Phys.~Rev.~Lett. \n {\\bf 81}, 3880 (1998).\n\\bibitem{HO98c} U.H.E.~Hansmann and Y.~Okamoto, J. Chem.~Phys. {\\bf 110},\n 1267 (1999); {\\bf 111} 1339(E) (1999).\n\\bibitem{ZB} B.H. Zimm and J.K. Bragg, J. Chem. Phys. {\\bf 31}, 526 (1959).\n\\bibitem{1dlr} F.J.~Dyson, Commun.~Math.~Phys. {\\bf 12}, 212 (1969);\n J.F.~Nagle and J.C.~Bonner, J.~Phys.~C {\\bf 3}, 352 (1970);\n E.~Bayong, H.T.~Diep and V.~Dotsenko, Phys.~Rev.~Lett. {\\bf 83},\n 14 (1999).\n\\bibitem{MU} B.A.~Berg and T.~Neuhaus, \n Phys. Lett. {\\bf B 267}, 249 (1991).\n\\bibitem{Review} U.H.E.~Hansmann and Y.~Okamoto, \n in: Stauffer, D. (ed.) ``{\\it Annual Reviews in Computational \n Physics VI}''\n (Singapore: World Scientific), p.129. (1998).\n\\bibitem{HO} U.H.E. Hansmann and Y. Okamoto, J.~Comp.~Chem.\n {\\bf 14}, 1333 (1993).\n\\bibitem{OH95b} Y.~Okamoto and U.H.E.~Hansmann,\\ J.~Phys.~Chem.\n {\\bf 99}, 11276 (1995).\n\\bibitem{FS} A.M. Ferrenberg and R.H. Swendsen, Phys.\\ Rev.\\ Lett.\n {\\bf 61}, 2635 (1988); Phys. Rev. Lett. {\\bf 63},\n1658(E) (1989), and\n references given in the erratum.\n\\bibitem{fisher} M.E. Fisher, in {\\it Lectures in Theoretical Physics}, \n vol. 7c p.1 (University of Colorado Press, Boulder, 1965).\n\\bibitem{itzykson} C. Itzykson, R.B. Pearson, and J.B. Zuber,\n Nucl. Phys. B {\\bf 220} [FS8], 415 (1983).\n\\bibitem{EC} M.J. Sippl, G. N{\\'e}methy, and H.A. Scheraga,\n{\\it J. Phys. Chem.} {\\bf 88}, 6231~(1984), and references therein.\n\\bibitem{Konf} H.~Kawai, Y.~Okamoto, M.~Fukugita, T.~Nakazawa, and\n T.~Kikuchi, Chem. Lett. {\\bf 1991}, 213 (1991);\n Y.~Okamoto, M.~Fukugita, T.~Nakazawa, and H.~Kawai,\n Protein Engineering {\\bf 4}, 639 (1991).\n\\bibitem{HO94c} U.H.E.~Hansmann and Y.~Okamoto, \n Physica A {\\bf 212}, 415 (1994).\n\\bibitem{Kar} M. Karliner,S.R. Sharpe and Y.F. Chang, \n Nucl. Phys. {\\bf B302}, 204 (1988).\n\\bibitem{Shrock} V. Matveev and R. Shrock, Phys. Lett. {\\bf A204}, 353 (1995).\n\\bibitem{fukugita} M. Fukugita, H. Mino, M. Okawa and A. Ukawa,\n J. Stat. Phys. {\\bf 59}, 1397 (1990),\n and references given therein.\n\\bibitem{fisher_nu} M.E. Fisher and A.N. Berker, Phys. Rev. B {\\bf 26}, 2507 \n (1982).\n\\bibitem{alves_nu} N.A. Alves, B.A. Berg, and R. Villanova, \n Phys. Rev. B {\\bf 43}, 5846 (1991). \n\\bibitem{binder1} K. Binder, Phys. Rev. Lett. {\\bf 47}, 693 (1981).\n\\bibitem{berg1} N.A. Alves, B.A. Berg, and S. Sanielevici, Phys. Lett. \n B {\\bf 241}, 557 (1990).\n\\end{references}\n%\\end{thebibliography}"
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cond-mat0002099
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Motivated by recent work on Heisenberg antiferromagnetic spin systems on various lattices made up of triangles, we examine the low-energy properties of a chain of antiferromagnetically coupled triangles of half-odd-integer spins. We derive the low-energy effective Hamiltonian to second order in the ratio of the coupling $J_2$ between triangles to the coupling $J_1$ within each triangle. The effective Hamiltonian contains four states for each triangle which are given by the products of spin-$1/2$ states with the states of a pseudospin-$1/2$. We compare the results obtained by exact diagonalization of the effective Hamiltonian with those obtained for the full Hamiltonian using exact diagonalization and the density-matrix renormalization group method. It is found that the effective Hamiltonian is accurate only for the ground state for rather low values of the ratio $J_2 / J_1$ and that too for the spin-$1/2$ case with linear topology. The chain of spin-$1/2$ triangles shows interesting properties like spontaneous dimerization and several singlet and triplet excited states lying close to the ground state.
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{
"name": "triangle.tex",
"string": "\\documentstyle[epsfig,aps,prb]{revtex}\n\\setlength{\\topmargin}{-2cm}\n\\raggedbottom\n\\abovedisplayskip=3mm\n\\belowdisplayskip=3mm\n\\abovedisplayshortskip=0mm\n\\belowdisplayshortskip=2mm\n\\setlength{\\baselineskip}{24pt}\n\\setlength{\\evensidemargin}{0pt}\n\\setlength{\\oddsidemargin}{-0.2cm}\n\\setlength{\\parskip}{0.13cm}\n\\setlength{\\textwidth}{16truecm}\n\\setlength{\\textheight}{21cm}\n\\baselineskip=24pt\n\n\\newcommand\\beq{\\begin{equation}}\n\\newcommand\\eeq{\\end{equation}}\n\\newcommand\\bea{\\begin{eqnarray}}\n\\newcommand\\eea{\\end{eqnarray}}\n\n\\begin{document}\n\n\\begin{center}\n{\\Large Evaluation of low-energy effective Hamiltonian techniques for\ncoupled spin triangles}\n\\end{center}\n\n\\vskip .5 true cm\n\\centerline{\\bf C. Raghu$^1$, Indranil Rudra$^1$, S. Ramasesha$^1$}\n\\centerline{\\bf and Diptiman Sen$^2$} \n\\vskip .5 true cm\n\n\\centerline{\\it $^1$ Solid State and Structural Chemistry Unit} \n\\centerline{\\it $^2$ Centre for Theoretical Studies} \n\\centerline{\\it Indian Institute of Science, Bangalore 560012, India} \n\\vskip .5 true cm\n\n\\begin{abstract}\n\nMotivated by recent work on Heisenberg antiferromagnetic\nspin systems on various lattices made up of triangles, we examine the \nlow-energy properties of a chain of antiferromagnetically coupled triangles \nof half-odd-integer spins. We derive the low-energy effective Hamiltonian to \nsecond order in the ratio of the coupling $J_2$ between triangles to the \ncoupling $J_1$ within each triangle. The effective Hamiltonian contains \nfour states for each triangle which are given by the products of spin-$1/2$ \nstates with the states of a pseudospin-$1/2$. We compare the\nresults obtained by exact diagonalization of the effective Hamiltonian with \nthose obtained for the full Hamiltonian using exact diagonalization\nand the density-matrix renormalization group method. It is found that the\neffective Hamiltonian is accurate only for the ground state for rather low \nvalues of the ratio $J_2 / J_1$ and that too for the spin-$1/2$ case with \nlinear topology. The chain of spin-$1/2$ triangles shows interesting \nproperties like spontaneous dimerization and several singlet and triplet \nexcited states lying close to the ground state.\n\n\\end{abstract}\n\\vskip .5 true cm\n\n~~~~~~ PACS number: ~75.10.Jm, ~75.50.Ee\n\n\\newpage\n\n\\section{Introduction}\n\nLow-dimensional quantum spin systems with frustration (i.e., competing \nantiferromagnetic interactions) have been studied extensively in recent \nyears. A particularly interesting class of such systems have lattice \nstructures built out of triangles of spins. Many such systems\nhave been realized experimentally in one and two dimensions, such\nas the sawtooth chain \\cite{saw} and the Kagome lattice \\cite{kag1}.\nA variety of theoretical techniques, both analytical and numerical, are \navailable to study the relevant models \\cite{kag2,kag3,lech,kagstrip}. Due to \nlarge quantum fluctuations, such systems often have several unusual \nproperties. For instance, there may be \na gap between a singlet ground state and the nonsinglet excited states; this \nleads to a magnetic susceptibility which vanishes exponentially at \nlow temperatures. Secondly, the two-spin correlation function may\ndecay rapidly with distance indicating that the magnetic correlation\nlength is not much bigger than the lattice spacing. Finally, there is \nsometimes no gap to a large number of singlet excited states; this produces \nan interesting structure for the specific heat at low temperature. The\nabsence of a singlet excitation gap also suggests that there may be a \nnonmagnetic long-range order, but the nature of this order\nis not well-understood.\n\nClassically, these systems often have an enormous ground state degeneracy\narising from local degrees of freedom which cost no energy; this leads to an\nextensive entropy at zero temperature. Quantum mechanically, this\ndegeneracy is lifted, but one might still expect a remnant of the \nclassical degeneracy in the form of a large number of low-energy excitations.\nIt is therefore useful to develop simple ways of understanding the\nlow-energy quantum excitations. In this paper, we will critically examine\none such way which is to consider the system as being made out of triangles \nwhich are weakly coupled to each other \\cite{pert1,pert2,pert3}. We\nfirst find the ground states of a single triangle assuming all the\nthree couplings to be antiferromagnetic with a strength $J_1$. We then use \ndegenerate perturbation theory to study what happens when different triangles \nare coupled to each other with antiferromagnetic bonds of strength\n$J_2$, where $J_2 \\ll J_1$. Since experimental systems typically\nhave $J_2 = J_1$, we must finally study how accurate the perturbation \ntheory is when the ratio $J_2 / J_1$ approaches $1$.\n\nIn Sec. II, we discuss the low-energy states of a single triangle of \nequal spins coupled antiferromagnetically to each other. If the spin at\neach site is an integer, then there is a unique ground state for the \ntriangle and the problem is not very interesting. But if the site spin $S$\nis a half-odd-integer, then the ground state has a four-fold\ndegeneracy. This is because the ground state has total spin-$1/2$,\nand there is an additional factor of two which can be interpreted\nas the chirality. The chirality can be expressed as the eigenvalue of a\npseudospin operator which also has spin-$1/2$.\nWe then discuss the low-energy effective Hamiltonian (LEH) for \na system in which each pair of triangles is coupled to each other by not \nmore than one antiferromagnetic bond. To first order in $J_2 /J_1$,\nthe spins and pseudospins of two triangles get coupled to each other.\nTo second order, up to three triangles can get coupled to each other.\nWe have obtained the first order LEH (LEH1) for all values of $S$, and\nthe second order LEH (LEH2) for $S$ equal to $1/2$ and $3/2$.\nWe have not gone beyond second order because the LEHs rapidly\nbecomes more complicated and longer range as we go to higher orders. For\nsmall systems, we numerically diagonalize the LEH1 and LEH2 \nto obtain the energies of the ground state and the first excited state \nas functions of $J_2 / J_1$.\n\nIn Sec. III, we use both exact diagonalization for small systems and the \ndensity-matrix renormalization group method (DMRG) for larger systems to \nobtain the low-lying energies for the complete Hamiltonian. The DMRG \nis currently the most accurate numerical method known for obtaining the\nlow-energy properties of quantum systems in one-dimension \\cite{whit1}. We\nbriefly describe the DMRG procedure for our system, and then compare the \nenergies obtained from the LEHs and DMRG. Depending on the quantities of \ninterest, we find that the results from the LEHs start deviating \nsignificantly from the DMRG values once $J_2 /J_1$ exceeds $0.2 ~-~ 0.4$. \nThis is true even for the LEH2 which is significantly more \naccurate than the LEH1 for small values of $J_2 /J_1$. \n\nWe also examine a simple system of five triangles\ncoupled to each other in such a way as to form a small fragment of the \ntwo-dimensional Kagome lattice; we find that the LEHs \nstarts deviating from the exact results at even smaller values of $J_2 /J_1$. \nThus the LEH approach becomes less accurate the more two-dimensional the\ngeometry is, i.e., the larger the coordination number is for each triangle. \n\n\\section{Low-energy Hamiltonians for a chain of triangles}\n\nWe are interested in the system of spins shown in Fig. 1. The sites\nare labeled as $(n,a)$, where $n$ labels the triangle and $a=1,2,3$ labels \nthe three sites of a triangle as indicated. The spin at each site has the \nvalue $S$, and all the couplings are antiferromagnetic. The Hamiltonian is\n\\beq\n{\\hat H} ~=~ \\sum_n ~\\Bigl[~ J_1 ~(~ {\\hat {\\bf S}}_{n,1} \\cdot {\\hat {\\bf \nS}}_{n,2} ~+~ {\\hat {\\bf S}}_{n,2} \\cdot {\\hat {\\bf S}}_{n,3} ~+~ {\\hat {\\bf \nS}}_{n,3} \\cdot {\\hat {\\bf S}}_{n,1} ~)~ +~ J_2 ~{\\hat {\\bf S}}_{n,2} \\cdot {\n\\hat {\\bf S}}_{n+1,3} ~]~.\n\\label{ham1}\n\\eeq\nIt is convenient to set $J_1 =1$ and consider only the parameter $J_2$.\n \nLet us first examine a single triangle and drop the label $n$. The \nHamiltonian ${\\hat h}_{\\Delta}$ is proportional to the square of the total \nspin operator, ignoring a shift in the zero of energy,\n\\beq\n{\\hat h}_{\\Delta} ~=~ {\\hat {\\bf S}}_1 \\cdot {\\hat {\\bf S}}_2 ~+~ {\\hat {\\bf \nS}}_2 \\cdot {\\bf S}_3 ~+~ {\\hat {\\bf S}}_3 \\cdot {\\hat {\\bf S}}_1 ~=~ \n\\frac{1}{2} ~ (~ {\\hat {\\bf S}}_1 ~+~ {\\hat {\\bf S}}_2 ~+~ {\\hat {\\bf S}}_3 ~\n)^2 ~ -~ \\frac{3}{2} ~S (S+1) ~.\n\\label{ham2}\n\\eeq\nIf $S$ is an integer ($1,2,...$), then the ground state of this triangle\nis unique and is a singlet. Thus the low-energy sector of the entire system \nconsists of only one state when different triangles are coupled to each \nother. The situation is much more interesting if $S$ is a half-odd-integer, \ni.e., $1/2,3/2,...~$. Then the ground state of each triangle is four-fold \ndegenerate because it must have total spin $1/2$ and there are two ways of \nobtaining total spin-$1/2$ by combining three equal spins. For instance, \nsites $2$ and $3$ can first combine to give total spin $S_{23}$ equal to\neither $S+1/2$ or $S-1/2$; $S_{23}$ can then combine with the first spin to \nproduce a total spin-$1/2$. Since the total spin operator ${\\hat {\\bf S}} = \n{\\hat {\\bf S}}_1 + {\\hat {\\bf S}}_2 + {\\hat {\\bf S}}_3$ commutes with all the \nsix permutations of the three spins, it is clear that the two ground states \nof the Hamiltonian in (\\ref{ham2}) (with a given value of the total spin \ncomponent $S^z = \\pm 1/2$) must form a two-dimensional representation of the \npermutation group. This representation can be made explicit by introducing \npseudospin-$1/2$ operators ${\\hat {\\bf \\tau}}$ as follows. \n\nWe will assume henceforth that all the site spins are identical and have the \nhalf-odd-integer value $S$. To obtain a total spin-$1/2$ for a triangle of \nthree spins, the total spin of any two sites should be one of the integers \n$S \\pm 1/2$. Under an exchange of the two spins $2$ and $3$, a state \nwith their total spin $S_{23}$ transforms by the phase $(-1)^{S_{23}+1}$;\nthus it is symmetric or antisymmetric depending on whether $S_{23}$ is odd or \neven. Let us define the pseudospin operator ${\\hat \\tau}^x$ such that the \nsymmetric states have the eigenvalue of ${\\hat \\tau}^x = 1$, while the \nantisymmetric states have the eigenvalue of ${\\hat \\tau}^x = -1$.\nSimilarly, we can introduce an operator ${\\hat \\tau}^y$ such that states\nwhich are symmetric (antisymmetric) with respect to exchange of spins $1$ and \n$2$ have eigenvalues of $(-{\\hat \\tau}^x + {\\sqrt 3} {\\hat \\tau}^y )/2$ equal \nto $1$ ($-1$). For spins $1$ and $3$, the operator $(-{\\hat \\tau}^x - {\\sqrt \n3} {\\hat \\tau}^y )/2$ will have the same property. We define the \nthird pseudospin operator ${\\hat \\tau}^z = - (i/2) [{\\hat \\tau}^x , {\\hat \n\\tau}^y ]$. From these statements, it follows that within the space of the \nfour ground states of a triangle, we have the operator identities\n\\bea\n{\\hat {\\bf S}}_2 \\cdot {\\hat {\\bf S}}_3 ~&=& ~A ~-~ \\frac{(-1)^{S+1/2}}{4} ~\n(2S+1) ~ {\\hat \\tau}^x ~, \\nonumber \\\\\n{\\hat {\\bf S}}_3 \\cdot {\\hat {\\bf S}}_1 ~&=& ~A ~-~ \\frac{(-1)^{S+1/2}}{4} ~\n(2S+1) ~\\Bigl[ ~- ~\\frac{1}{2} {\\hat \\tau}^x ~-~ \\frac{\\sqrt 3}{2} {\\hat \n\\tau}^y ~\\Bigr] ~, \\nonumber \\\\\n{\\hat {\\bf S}}_1 \\cdot {\\hat {\\bf S}}_2 ~&=& ~A ~-~ \\frac{(-1)^{S+1/2}}{4} ~\n(2S+1) ~\\Bigl[ ~ - ~\\frac{1}{2} {\\hat \\tau}^x ~+~ \\frac{\\sqrt 3}{2} {\\hat\n\\tau}^y ~\\Bigr] ~, \n\\label{iden}\n\\eea\nwhere $A = -S^2 /2 - S/2 +1/8$. On taking the commutator of any two\nof the equations in (\\ref{iden}), we find that the three-spin chirality\noperator is given by\n\\beq\n{\\hat {\\bf S}}_1 \\cdot {\\hat {\\bf S}}_2 \\times {\\hat {\\bf S}}_3 ~=~ \\frac{\n\\sqrt 3}{4} ~(~ S ~+~ \\frac{1}{2} ~)^2 ~{\\hat \\tau}^z ~.\n\\label{chiral}\n\\eeq\n\nBefore proceeding further, we should mention that the pseudospin operators \nhave been discussed earlier in Refs. 7 and 8 for the spin-$1/2$ \ncase. In those papers, the two ground states (with a given \nvalue of $S^z$) are written in terms of a spin wave running around the \ntriangle with momenta $\\pm 2\\pi /3$; these are called right and left moving \nrespectively, and they are eigenstates of the operator ${\\hat \\tau}^z$. We \nhave instead chosen to describe the states in terms of two of the spins \nforming total spin $S+1/2$ or $S-1/2$. The two descriptions are clearly\nrelated to each other by an unitary transformation.\n\nTo continue, the Wigner-Eckart theorem says that the matrix elements of any of \nthe site spin operators ${\\hat {\\bf S}}_a$ (where $a=1,2,3$) between the four \nground states of a triangle must be proportional to the matrix elements of \nthe total spin operator ${\\hat {\\bf s}}$. The proportionality \nfactors must be independent of the spin component (i.e., $x$, $y$ or $z$), but \nthey will involve the pseudospin operators. Let us introduce the operators\n\\beq\n{\\hat \\tau}_a ~=~ \\cos \\frac{2\\pi}{3} (1-a) ~{\\hat \\tau}^x ~+~ \\sin \n\\frac{2\\pi}{3} (1-a) ~ {\\hat \\tau}^y ~,\n\\label{taua}\n\\eeq\nwhere $a$ can take the values $1,2,3$. Using the spin wave functions, we can \nthen show that the matrix elements of ${\\hat {\\bf S}}_a$ and ${\\hat {\\bf s}}$ \nare related as\n\\beq\n\\langle \\sigma , \\tau \\vert {\\hat {\\bf S}}_a \\vert \\sigma^\\prime , \\tau^\\prime \n\\rangle ~=~ \\frac{1}{3} ~\\langle \\sigma \\vert {\\hat {\\bf s}} \\vert \n\\sigma^\\prime \\rangle ~ \\Bigl[ ~\\delta_{\\tau , \\tau^\\prime} ~+~ (-1)^{S+1/2} ~\n(2S+1) ~ \\langle \\tau \\vert {\\hat \\tau}_a \\vert \\tau^\\prime \\rangle ~ \\Bigr] ~,\n\\label{wigeck}\n\\eeq\nwhere $\\sigma , \\sigma^\\prime$ are the eigenstates of ${\\hat \\sigma}^z$ and\n$\\tau , \\tau^\\prime$ are the eigenstates of ${\\hat \\tau}^z$.\n\nWe can now derive the LEHs when two or more triangles are coupled together\nas in Eq. (\\ref{ham1}). We write that Hamiltonian as ${\\hat H}= {\\hat H}_0 + \n{\\hat V}$, where ${\\hat H}_0$ consists of the interactions within the \ntriangles as in Eq. (\\ref{ham2}), while $\\hat V$ consists of the interactions \nbetween\ntriangles and is proportional to $J_2$. The LEH is a perturbative expansion in \nthe parameter $J_2$. For $J_2 =0$, the low-energy sector for $N$ triangles\ncontains $4^N$ states, all of which have the same energy $E_0 = Ne_0$, where\n\\beq\ne_0 ~=~ \\frac{3}{8} ~-~ \\frac{3}{2} ~S(S+1) ~.\n\\eeq\nis the ground state energy of (\\ref{ham2}). Following Ref. 9,\nlet us denote the different\nlow-energy states of the system as $p_i$ and the high-energy states\nof the system as $q_{\\alpha}$ (these are states in which at least one of the \ntriangles is in a state with total spin $\\ge 3/2$). The high-energy states \nhave energies $E_{\\alpha}$ according to the exactly solvable Hamiltonian \n${\\hat H}_0$. Then the LEH1 is given by degenerate perturbation theory, \n\\beq\n{\\hat H}_{eff}^{(1)} ~=~ \\sum_{ij} ~\\vert p_i \\rangle ~\\langle p_i \\vert \n{\\hat V} \\vert p_j \\rangle ~\\langle p_j \\vert ~.\n\\label{ham3}\n\\eeq\nThe LEH2 is given by\n\\beq\n{\\hat H}_{eff}^{(2)} ~=~ \\sum_{ij} ~\\sum_{\\alpha} ~\\vert p_i \\rangle ~\\frac{\n\\langle p_i \\vert {\\hat V} \\vert q_{\\alpha} \\rangle ~\\langle q_{\\alpha} \n\\vert {\\hat V} \\vert p_j \\rangle}{E_0 ~-~ E_{\\alpha}} ~\\langle p_j \\vert ~.\n\\label{ham4}\n\\eeq\nThe total effective Hamiltonian to this order is given by\n\\beq\n{\\hat H}_{LEH2} ~=~ E_0 ~+~ {\\hat H}_{eff}^{(1)} ~+~ {\\hat H}_{eff}^{(2)} ~.\n\\eeq\n\nThe LEH1 can now be directly read off from Eq. (\\ref{wigeck}) after adding the \ntriangle label $n$; thus the total spin operator of triangle $n$ is denoted \nas ${\\hat {\\bf s}}_n$ and the pseudospin operator as ${\\hat {\\bf \\tau}}_n$. \nIn general, if site number $a$ of triangle $l$ is connected to site number $b$ \nof triangle $m$ by a bond of strength $J_2$ (see Fig. 2), the contribution\nof that bond to the LEH1 is given by \n\\beq\n{\\hat h}_{eff}^{(1)} ~=~ \\frac{J_2}{9} ~{\\hat {\\bf s}}_l \\cdot {\\hat {\\bf \ns}}_m ~ \\Bigl[ ~1 ~+~ (-1)^{S+1/2} ~ (2S+1) ~ {\\hat \\tau}_l^a ~\\Bigl] ~\n\\Bigl[ ~1 ~+~ (-1)^{S+1/2} ~ (2S+1) ~{\\hat \\tau}_m^b ~ \\Bigl] ~.\n\\label{ham5}\n\\eeq\nFor the particular form of couplings shown in Fig. 1, the total LEH1 takes the\nform\n\\beq\n{\\hat H}_{LEH1} ~= - ~ Ne_0 + \\frac{J_2}{9} ~\\sum_n ~{\\hat {\\bf s}}_n \n\\cdot {\\hat {\\bf s}}_{n+1} ~\\Bigl[ 1 ~+~ (-1)^{S+1/2} ~ (2S+1) ~{\\hat \n\\tau}_n^2 ~\\Bigl] ~\\Bigl[ 1 ~+~ (-1)^{S+1/2} ~(2S+1) ~{\\hat \\tau}_{n+1}^3 ~ \n\\Bigl] .\n\\label{ham6}\n\\eeq\n\nThe derivation of the LEH2 in (\\ref{ham4}) requires a much \nlonger calculation since we have to first compute matrix elements of the\nform $\\langle q_{\\alpha} \\vert {\\hat {\\bf S}}_a \\vert p_i \\rangle$ for all the\nstates within each triangle, and we then have to take products of these\nto obtain the matrix elements of ${\\hat {\\bf s}}_{n,2} \\cdot {\\hat {\\bf \ns}}_{n+1,3}$. The second-order terms in the LEH can arise from either\n(i) a single bond connecting site $a$ of triangle $l$ to site $b$ of triangle\n$m$, or (ii) a bond connecting site $a$ of triangle $l$ to site $b$ of\ntriangle $m$ and another bond connecting site $c$ of triangle $m$ to site\n$d$ of triangle $n$, where $a,b,c,d$ take values from $1,2,3$ (here $b$ may or\nmay not be equal to $c$). The notation for these two types is shown in Fig. 2.\n\nLet us first consider the simplest situation in which all the site spins are \nequal to $S=1/2$. The contribution of type (i) to the LEH2 is then given by\n\\beq\n{\\hat h}_{eff,(i)}^{(2)} ~=~ - \\frac{J_2^2}{54} ~\\Bigl[ ~(1- {\\hat \n\\tau}_{l,a} { \\hat \\tau}_{m,b} )~(3 +4 {\\hat {\\bf s}}_l \\cdot {\\hat {\\bf \ns}}_m ) ~+~ (1+ { \\hat \\tau}_{l,a} )(1+ {\\hat \\tau}_{m,b} )~\\Bigr] ~,\n\\label{ham7}\n\\eeq\nwhere we have used the notation of Eq. (\\ref{taua}). The contribution of \ntype (ii) is\n\\bea\n{\\hat h}_{eff,(ii)}^{(2)} ~=~ - \\frac{4J_2^2}{243} ~(1- 2 {\\hat \\tau}_{l,a} \n)(1- 2 {\\hat \\tau}_{n,d} ) ~\\Bigl[ ~& & \\Bigl( ~\\cos \\frac{2\\pi}{3} (b-c) ~+~ \n{\\hat \\tau}_{m,-b-c} ~\\Bigr) ~{\\hat {\\bf s}}_l \\cdot {\\hat {\\bf s}}_n \n\\nonumber \\\\\n& & +~ \\sin \\frac{2\\pi}{3} (b-c) ~{\\hat \\tau}_m^z ~{\\hat {\\bf s}}_l \\cdot \n{\\hat {\\bf s}}_m \\times {\\hat {\\bf s}}_n ~\\Bigr] ~.\n\\label{ham8}\n\\eea\nPutting all this together for the system in Fig. 1, we find that for\n$S=1/2$, the total LEH2 is given by\n\\bea\n{\\hat H}_{LEH2} ~=~ & & - \\frac{3N}{4} ~+~ \\frac{J_2}{9} ~\\sum_n ~{\\hat \n{\\bf s}}_n \\cdot {\\hat {\\bf s}}_{n+1} ~( 1 - 2 {\\hat \\tau}_{n,2} )~( 1 - 2 \n{\\hat \\tau}_{n+1,3} ) \\nonumber \\\\\n& & - \\frac{J_2^2}{54} ~\\sum_n ~\\Bigl[ ~(1- {\\hat \\tau}_{n,2} {\\hat \n\\tau}_{n+1,3} )~(3 +4 {\\hat {\\bf s}}_n \\cdot {\\hat {\\bf s}}_{n+1} ) ~+~ (1+ \n{\\hat \\tau}_{n,2} ) ~(1+ {\\hat \\tau}_{n+1,3} )~\\Bigr] \\nonumber \\\\\n& & - \\frac{4J_2^2}{243} ~\\sum_n ~(1- 2 {\\hat \\tau}_{n,2} ) ~(1- 2 {\\hat \n\\tau}_{n+1,3} )~ \\Bigl[ ( - \\frac{1}{2} + {\\hat \\tau}_{n+1,1} ) ~{\\hat {\\bf \ns}}_n \\cdot {\\hat {\\bf s}}_{n+2} - \\frac{\\sqrt 3}{2} ~{\\hat \\tau}_{n+1}^z ~\n{\\hat {\\bf s}}_n \\cdot {\\hat {\\bf s}}_{n+1} \\times {\\hat {\\bf s}}_{n+2} \n\\Bigr] ~. \\nonumber \\\\\n& &\n\\label{ham9}\n\\eea\n\nIf the site spins are equal to $S=3/2$, the contribution of type (i) \nto the LEH2 is given by\n\\bea\n{\\hat h}_{eff,(i)}^{(2)} ~=~ - \\frac{J_2^2}{27} ~\\Bigl[ ~ & & 56 ~+~ \n42 {\\hat {\\bf s}}_l \\cdot {\\hat {\\bf s}}_m ~+~ ({\\hat \\tau}_{l,a} ~+~ {\\hat \n\\tau}_{m,b} ) ~(-1 ~+~ 8 {\\hat {\\bf s}}_l \\cdot {\\hat {\\bf s}}_m ) \n\\nonumber \\\\\n& & - {\\hat \\tau}_{l,a} {\\hat \\tau}_{m,b} (4 ~+~ 8 {\\hat {\\bf s}}_l \\cdot \n{\\hat {\\bf s}}_m ) ~.\n\\label{ham10}\n\\eea\nThe contribution of type (ii) is\n\\bea\n{\\hat h}_{eff,(ii)}^{(2)} ~=~ - \\frac{4J_2^2}{243} ~(1+4 {\\hat \\tau}_{l,a} \n)(1+4 {\\hat \\tau}_{n,d} ) ~ \\Bigl[ ~& & \\Bigl( ~7 ~\\cos \\frac{2\\pi}{3} (b-\nc) ~-~ 2~ { \\hat \\tau}_{m,-b-c} ~ \\Bigr) ~{\\hat {\\bf s}}_l \\cdot {\\hat {\\bf \ns}}_n \\nonumber \\\\\n& & +~ 4 ~\\sin \\frac{2\\pi}{3} (b-c) ~{\\hat \\tau}_m^z ~{\\hat {\\bf s}}_l \\cdot \n{\\hat {\\bf s}}_m \\times {\\hat {\\bf s}}_n ~\\Bigr] ~.\n\\label{ham12}\n\\eea\nFor the system in Fig. 1, the total LEH2 for $S=3/2$ is therefore\n\\bea\n{\\hat H}_{LEH2} ~=~ & & - \\frac{21N}{4} ~+~ \\frac{J_2}{9} ~\\sum_n ~{\\hat \n{\\bf s}}_n \\cdot {\\hat {\\bf s}}_{n+1} ~( 1 + 4 {\\hat \\tau_{n,2}} )~( 1 + 4 \n{\\hat \\tau}_{n+1,3} ) \\nonumber \\\\\n& & - \\frac{J_2^2}{27} ~\\sum_n ~\\Bigl[ 56 + 42 {\\hat {\\bf s}}_n \\cdot {\\hat \n{\\bf s}}_{n+1} + ({\\hat \\tau}_{n,2} + {\\hat \\tau}_{n+1,3}) (-1 + 8 {\\hat {\\bf \ns}}_n \\cdot {\\hat {\\bf s}}_{n+1}) - {\\hat \\tau}_{n,2} {\\hat \\tau}_{n+1,3} (4 \n+8 {\\hat {\\bf s}}_n \\cdot {\\hat {\\bf s}}_{n+1} )\\Bigr] \n\\nonumber \\\\\n& & - \\frac{4J_2^2}{243} ~\\sum_n ~(1+ 4 {\\hat \\tau}_{n,2} ) ~(1+ 4 {\\hat \n\\tau}_{n+1,3} )~ \\Bigl[ ( - \\frac{7}{2} + {\\hat \\tau}_{n+1,1} ) ~{\\hat {\\bf \ns}}_n \\cdot {\\hat {\\bf s}}_{n+2} - 2 {\\sqrt 3} ~{\\hat \\tau}_{n+1}^z ~{\\hat \n{\\bf s}}_n \\cdot {\\hat {\\bf s}}_{n+1} \\times {\\hat {\\bf s}}_{n+2} \\Bigr] . \n\\nonumber \\\\\n& &\n\\label{ham13}\n\\eea\n\nWe have carried out exact diagonalization studies of the LEH1 and LEH2 for \nsystems up to $10$ triangles for both the spin-$1/2$ and spin-$3/2$ cases.\n\n\\section{Density-matrix renormalization group study of the chain of triangles}\n\nWe have numerically studied the system described by Eq. (\\ref{ham1})\nusing exact diagonalization for small systems and the DMRG method for larger\nsystems. The number of triangles is $N$ and the number of sites is $3N$.\n\nFor small systems, we have performed exact diagonalization with periodic \nboundary conditions. For larger systems, we have done \nDMRG calculations (using the infinite system algorithm \\cite{whit1}) with \nopen boundary conditions. For exact diagonalization, we have gone up to $30$ \nsites ($10$ triangles). With DMRG, we have gone up to $50$ triangles and\nin some cases up to $100$ triangles, after checking that the DMRG \nand exact results match for $10$ triangles for the spin-$1/2$ system.\nThe number of dominant density matrix eigenstates, corresponding to\nthe $m$ largest eigenvalues of the density matrix, that we retained \nat each DMRG iteration was $64$. In fact, we varied the value of $m$ \nfrom $64$ to $100$ and found that $m=64$ gives satisfactory results in \nterms of agreement with exact diagonalization for small systems and good \nnumerical convergence for large systems. The system is grown by adding two \nnew triangles at each iteration; we found that this gives more accurate \nresults than either adding two new sites at each iteration (in which case \nwe would have obtained the triangle structure only after every third \niteration) or adding one triangle at each iteration. \n\nIn Fig. 3, we show the DMRG values of the ground state energy per site \nversus $1/N$ for a few illustrative values of $J_2$ (in units of $J_1$) \nfor the case in which each site has spin-$1/2$. After extrapolating\nto the thermodynamics limit $N \\rightarrow \\infty$, we show the ground state\nenergy per site as a function of $J_2$ in Fig. 4. In that figure, we\nalso show the numerical results obtained by exact diagonalization of the\nLEH1 and LEH2 respectively. As expected, the LEH2 is more accurate than \nthe LEH1 up to a larger value of $J_2$. However, even the LEH2 becomes \nfairly inaccurate beyond about $J_2 = 0.4$. \n\nA more detailed comparison of the LEH1 and LEH2 with the exact results for \nchains with up to $10$ triangles is given in Table 1. We see that the LEH2 \nagrees better with the exact results than the LEH1, although the agreement \nbecomes rather poor for large $J_2$. We also see \nthat the accuracy of the LEHs is poorer for the gap than it is\nfor the ground state energy. For instance, the LEH2 results for the\ngap become relatively inaccurate even for $J_2 = 0.2$.\n\nFig. 5 shows the low-lying triplet and singlet gaps above the ground state for \nthe same system at different values of $J_2$. The figure compares the results \nobtained from the LEH2 with those obtained after extrapolating the gaps from \nexact diagonalization of the full Hamiltonian for different values of $N$. \nBoth the triplet (Fig. 5 (a)) and singlet (Fig. 5 (b)) gaps deviate \nconsiderably from the \"exact\" results beyond $J_2 \\sim 0.4$. In fact, we see \nfrom Fig. 5 (a) that the ground state of the LEH2 switches from a singlet to \na triplet for $J_2 > 0.7$.\n\nIn Fig. 6, we show a larger number of $S=0$ and $S=1$ states which have very \nsmall gaps above the ground state. It is possible that some of these actually \nbecome degenerate with the ground state in the thermodynamic limit. The lowest \nspin sector with an appreciable gap (which is likely to remain non-zero in the \nthermodynamic limit) seems to be $S=2$. However it is possible that there\nare many more singlet and triplet states lying below the lowest $S=2$ state\nthan we have shown in the figure; it is very difficult to find more than\na few low-lying states in each spin sector using the DMRG method. For the\nsame reason, we cannot rule out the possibility of a finite gap to a \nhigher singlet or a higher triplet (than what we have targetted)\nlying below the quintet. Fig. 7 shows an even larger number of \nlow-lying states for the case $J_2 = 1.0$. The lowest three \nstates with $S^z =0$ (including the ground state), the lowest six states with \n$S^z =0$ and the lowest state with $S^z =2$ are shown there. To\nsummarize, we have found an unexpectedly large number of low-lying (and\npossibly gapless) singlet and triplet excitations in this system. We will \nprovide an explanation for some of these states below. We note that\nthe low-energy spectrum has a resemblance to that found in the \nKagome lattice in two dimensions \\cite{lech}. However there is an important \ndifference between the two cases; the gapless band of excitations in \nthe Kagome problem consists only of singlets, and the first gap\nis to a triplet state.\n\nIn Fig. 8 (a), we show the DMRG values of the bond order \nfor the middle two bonds, namely, the ground state expectation values\n$\\langle {\\hat {\\bf s}}_{p,2} \\cdot {\\hat {\\bf s}}_{p+1,3} \\rangle$ for \n$p = N/2 -1$ and $N$, as a function of $N$ at $J_2 =1.0$ for the spin-$1/2$ \nsystem. In Fig. 8 (b), we plot the bond order alternation (or spontaneous \ndimerization), defined to be the magnitude of the difference of the two \nneighbouring bond orders in Fig. 8 (a) extrapolated to the \nlimit $N \\rightarrow \\infty$, as a function of $J_2$. We observe \nthat the alternation is quite large for small values of $J_2$ and that it\nremains non-zero even at a large value of $J_2 =J_1$. Following Mila \n\\cite{pert2}, we can qualitatively understand the dimerization occurring \nat small values of $J_2$ as follows. Since the spin and pseudospin degrees of \nfreedom appear very asymmetrically in the LEHs, we perform a mean-field\ndecoupling of these two. We assume that the pseudospin variables take some \nfixed values and find the quantum ground state of the spin variables in that \nfixed background. For two triangles coupled together by the LEH1 \nin Eq. (\\ref{ham5}), we see that the lowest energy is attained if the \npseudospin variables satisfy $\\tau_{l,a} = \\tau_{m,b} = -1$ (for the\ncase $S=1/2$), and the effective spin-$1/2$ of the two triangles then form a\nsinglet. In the same way, the mean-field ground state of the chain is\ngiven by the dimerized configuration in which $\\tau_{2n,2} = \\tau_{2n+1,3} =\n-1$ and the effective spin-$1/2$ of triangles $2n$ and $2n+1$ form a \nsinglet. Such a dimerized ground state can also be obtained by translating \nthe above state by one triangle; it is therefore two-fold degenerate as in a\nspin-Peierls system.\n\nIt is instructive to contrast this system with the antiferromagnetic \nspin-$1/2$ chain with a nearest-neighbor coupling $J_1$ and a \nnext-nearest-neighbor coupling $J_2$ \\cite{whit2}. For $J_2 /J_1 > 0.241...$,\nthis is known to spontaneously dimerize, and there is also a finite gap to \nexcitations in the bulk and a finite correlation length $\\xi$ in the \nthermodynamic limit. In the dimerized phase, an open chain with an even \nnumber of sites has five ground states corresponding to two singlets and \none triplet. One singlet and one triplet arise from the free \nspin-$1/2$ degrees of freedom which reside at the two ends of the chain. \nThe finite correlation length $\\xi$ means that the splitting between \nthese states vanishes exponentially with the size of the system.\n\nSimilarly, for our system with a chain of an even number of triangles, we\nmay expect at least some of the low-energy states to arise from the effective\nspin-$1/2$ degrees of freedom residing on the two end triangles.\nDue to the presence of the pseudospin-$1/2$ degrees of freedom in each\ntriangle, we expect that there will be a total of $4^2 =16$ low-energy \nstates forming four triplets and four singlets.\nThe number of low-energy singlets and triplets that we actually find is more\nthan this; this implies that there are some additional low-energy degrees\nof freedom (probably associated with the bulk) which we do not yet understand.\n\nIn Table 2, we show the ground state energy per site versus $J_2$ for the case \nin which each site has spin-$3/2$. The numerical results obtained by exact \ndiagonalization of the LEH1 and LEH2 are also shown. A comparison with Table 1\nfor the spin-$1/2$ case shows that the LEH2 starts deviating from the exact \nresults for smaller values of $J_2$ as the site spin is increased.\n\nFinally, in Fig. 9, we show the ground state energy per site versus $J_2$ for \na group of five triangles forming a sub-system of a two-dimensional Kagome \nlattice. The reason for studying this system is to test how well the LEHs do\nas the coordination number of the triangles is increased from two, thereby\nmaking the system more two-dimensional. We find that the LEH2 fails even\nfaster with increasing $J_2$ than it does for the chains studied above.\n\n\\section{Summary and Outlook}\n\nWe have studied a chain of coupled triangles of half-odd-integer spins using \nboth the DMRG method and the LEH approaches. We find that the LEH approach is\naccurate for the ground state energy only for small values of the ratio of the \ncoupling between triangles and the coupling within each triangle; the accuracy\nfor the low-energy gaps is less than that for the ground state energy. The \nrange of accuracy of the LEHs is also smaller for larger values of the site \nspin as well as for a larger number of neighbors coupled to each triangle.\nWe therefore conclude that the LEH approaches may not be very reliable for \nthe low-energy properties of the currently existing experimental systems in \nwhich all couplings are of the same order and the geometry is two-dimensional\n\\cite{pert2}.\n\n\\vskip .7 true cm\n\\noindent {\\bf Acknowledgments}\n\nWe thank Kunj Tandon and Swapan Pati for useful discussions. SR thanks the\nDepartment of Science and Technology, Govt. of India for financial support \nunder project SP/S1/H-07/96.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{saw} T. Nakamura and K. Kubo, Phys. 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Rev. Lett. {\\bf 81}, 2356 (1999); V. \nSubrahmanyam, Phys. Rev. B {\\bf 52}, 1133 (1995).\n\n\\bibitem{pert3} K. Tandon, S. Lal, S. K. Pati, S. Ramasesha and D. Sen, Phys.\nRev. B {\\bf 59}, 396 (1999).\n\n\\bibitem{whit1} S. R. White, Phys. Rev. Lett. {\\bf 69}, 2863 (1992); Phys. \nRev. B {\\bf 48}, 10345 (1993).\n\n\\bibitem{whit2} S. R. White and I. Affleck, Phys. Rev. B {\\bf 54}, 9862 \n(1996).\n\n\\end{thebibliography}\n\n\\newpage\n\n\\begin{center}\n\\begin{tabular}{c c}\n\\hspace{1.8cm}{\\bf{Ground State}}\\hspace{3.0cm} {\\bf{First Excited State}} \\\\\n\\end{tabular}\n%this is for spin 1/2\n\\begin{tabular}{|c|c|c|c|c||c|c|c|}\n\\hline ${\\bf{J_{2}}}$ & {\\bf{N}} & {\\bf{LEH1}} & {\\bf{LEH2}} &\n{\\bf{Exact }} & {\\bf{LEH1}} & {\\bf{LEH2}} &\n{\\bf{Exact }} \\\\ \\hline \n&\\hspace{0.1cm} 4 \\hspace{0.1cm}&\\hspace{0.2cm}-3.15843\\hspace{0.2cm} &\n\\hspace{0.2cm} -3.16139\\hspace{0.2cm} &\\hspace{0.2cm} -3.16135\\hspace{0.2cm} &\n\\hspace{0.2cm}-3.14912\\hspace{0.2cm} &\\hspace{0.2cm}-3.15131\\hspace{0.2cm} &\n\\hspace{0.2cm}-3.15130\\hspace{0.2cm} \\\\ \\cline{2-8} \n0.1 &\\hspace{0.1cm} 6\\hspace{0.1cm} &\\hspace{0.2cm} -4.73740\\hspace{0.2cm} &\n\\hspace{0.2cm} -4.73772\\hspace{0.2cm} &\\hspace{0.2cm} -4.73768\\hspace{0.2cm} &\n\\hspace{0.2cm} -4.72959\\hspace{0.2cm} &\\hspace{0.2cm} -4.73334\\hspace{0.2cm} &\n\\hspace{0.2cm} -4.73331\\hspace{0.2cm} \\\\ \\cline{2-8}\n&\\hspace{0.1cm} 8\\hspace{0.1cm} &\\hspace{0.2cm} -6.30976\\hspace{0.2cm} &\n\\hspace{0.2cm} -6.31541\\hspace{0.2cm} &\\hspace{0.2cm} -6.31536\\hspace{0.2cm} &\n\\hspace{0.2cm} -6.30795\\hspace{0.2cm} &\\hspace{0.2cm} -6.31319\\hspace{0.2cm} &\n\\hspace{0.2cm} -6.31315\\hspace{0.2cm} \\\\ \\hline \\hline\n&\\hspace{0.1cm} 4\\hspace{0.1cm} &\\hspace{0.2cm} -3.31686\\hspace{0.2cm} &\n\\hspace{0.2cm} -3.32892\\hspace{0.2cm} &\\hspace{0.2cm} -3.32864\\hspace{0.2cm} &\n\\hspace{0.2cm} -3.29825\\hspace{0.2cm} &\\hspace{0.2cm} -3.30728\\hspace{0.2cm} &\n\\hspace{0.2cm} -3.30713\\hspace{0.2cm} \\\\ \\cline{2-8}\n0.2 &\\hspace{0.1cm} 6\\hspace{0.1cm} &\\hspace{0.2cm} -4.96681\\hspace{0.2cm} &\n\\hspace{0.2cm} -4.98455\\hspace{0.2cm} &\\hspace{0.2cm} -4.98417\\hspace{0.2cm} &\n\\hspace{0.2cm} -4.95918\\hspace{0.2cm} &\\hspace{0.2cm} -4.97455\\hspace{0.2cm} &\n\\hspace{0.2cm} -4.97427\\hspace{0.2cm} \\\\ \\cline{2-8}\n&\\hspace{0.1cm} 8\\hspace{0.1cm} &\\hspace{0.2cm} -6.61952\\hspace{0.2cm} &\n\\hspace{0.2cm} -6.64278\\hspace{0.2cm} &\\hspace{0.2cm} -6.64229\\hspace{0.2cm} &\n\\hspace{0.2cm} -6.61590\\hspace{0.2cm} &\\hspace{0.2cm} -6.63740\\hspace{0.2cm} &\n\\hspace{0.2cm} -6.63699\\hspace{0.2cm} \\\\ \\hline \\hline\n&\\hspace{0.1cm} 4\\hspace{0.1cm} &\\hspace{0.2cm} -3.79215\\hspace{0.2cm} &\n\\hspace{0.2cm} -3.87213\\hspace{0.2cm} &\\hspace{0.2cm} -3.86669\\hspace{0.2cm} &\n\\hspace{0.2cm} -3.74562\\hspace{0.2cm} &\\hspace{0.2cm} -3.80782\\hspace{0.2cm} &\n\\hspace{0.2cm} -3.80463\\hspace{0.2cm} \\\\ \\cline{2-8}\n0.5 &\\hspace{0.1cm} 6\\hspace{0.1cm} &\\hspace{0.2cm} -5.66702\\hspace{0.2cm} &\n\\hspace{0.2cm} -5.78669\\hspace{0.2cm} &\\hspace{0.2cm} -5.77885\\hspace{0.2cm} &\n\\hspace{0.2cm} -5.64794\\hspace{0.2cm} &\\hspace{0.2cm} -5.75145\\hspace{0.2cm} &\n\\hspace{0.2cm} -5.74604\\hspace{0.2cm} \\\\ \\cline{2-8}\n&\\hspace{0.1cm} 8\\hspace{0.1cm} &\\hspace{0.2cm} -7.54881\\hspace{0.2cm} &\n\\hspace{0.2cm} -7.70679\\hspace{0.2cm} &\\hspace{0.2cm} -7.69665\\hspace{0.2cm} &\n\\hspace{0.2cm} -7.53975\\hspace{0.2cm} &\\hspace{0.2cm} -7.68454\\hspace{0.2cm} &\n\\hspace{0.2cm} -7.67639\\hspace{0.2cm} \\\\ \\hline \\hline\n&\\hspace{0.1cm} 4\\hspace{0.1cm} &\\hspace{0.2cm} -4.58430\\hspace{0.2cm} &\n\\hspace{0.2cm} -4.93506\\hspace{0.2cm} &\\hspace{0.2cm} -4.88000\\hspace{0.2cm} &\n\\hspace{0.2cm} -4.49125\\hspace{0.2cm} & \\hspace{0.2cm}-4.78367 &\n\\hspace{0.2cm} -4.74638\\hspace{0.2cm} \\\\ \\cline{2-8}\n1.0 &\\hspace{0.1cm} 6\\hspace{0.1cm} &\\hspace{0.2cm} -6.83404\\hspace{0.2cm} &\n\\hspace{0.2cm} -7.02622\\hspace{0.2cm} &\\hspace{0.2cm} -7.28614\\hspace{0.2cm} &\n\\hspace{0.2cm} -6.79589\\hspace{0.2cm} &\\hspace{0.2cm} -7.26277\\hspace{0.2cm} &\n\\hspace{0.2cm} -7.20489\\hspace{0.2cm} \\\\ \\cline{2-8}\n&\\hspace{0.1cm} 8\\hspace{0.1cm} &\\hspace{0.2cm} -9.09761\\hspace{0.2cm} &\n\\hspace{0.2cm} -9.81245\\hspace{0.2cm} &\\hspace{0.2cm} -9.70079\\hspace{0.2cm} &\n\\hspace{0.2cm} -9.07950\\hspace{0.2cm} &\\hspace{0.2cm} -9.73152\\hspace{0.2cm} &\n\\hspace{0.2cm} -9.64368\\hspace{0.2cm} \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\vskip 1.2 true cm\n\n\\noindent Table 1. Ground state and first excited state energies in units of\n$J_1$ obtained for the spin-$1/2$ system using exact diagonalization of the \nfull Hamiltonian and the LEH1 and LEH2.\n\n\\vskip 2.8 true cm\n\n\\begin{center}\n\\hspace{-3cm}\n\\begin{tabular}{c c}\n\\hspace{4.0cm}{\\bf{N=3}} & \\hspace{5.0cm}{\\bf{N=4}} \\\\ \n\\end{tabular}\n% this is for spin 3/2\n\\begin{tabular}{|c|c|c|c||c|c|c|}\n\\hline ${\\bf{J_{2}}}$ & {\\bf{LEH1}} & {\\bf{LEH2}} &\n{\\bf{Exact }} & {\\bf{LEH1}} & {\\bf{LEH2}} &\n{\\bf{Exact }} \\\\ \\hline\n\\hspace{0.2cm}0.1\\hspace{0.2cm} &\\hspace{0.2cm} -16.0462\\hspace{0.2cm} &\n\\hspace{0.2cm} -16.1044\\hspace{0.2cm} &\\hspace{0.2cm} -16.1071\\hspace{0.2cm} &\n\\hspace{0.2cm} -21.4577\\hspace{0.2cm} &\\hspace{0.2cm} -21.5211\n\\hspace{0.2cm} & \\hspace{0.2cm} -21.5224\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}0.2\\hspace{0.2cm} &\\hspace{0.2cm} -16.3424\\hspace{0.2cm} &\n\\hspace{0.2cm} -16.5793\\hspace{0.2cm} &\\hspace{0.2cm} -16.5692 \n\\hspace{0.2cm} &\\hspace{0.2cm} -21.9154\\hspace{0.2cm} &\n\\hspace{0.2cm} -22.1775\\hspace{0.2cm} &\\hspace{0.2cm} -22.1611\n\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}0.3\\hspace{0.2cm} & \\hspace{0.2cm} -16.6386\\hspace{0.2cm} &\n\\hspace{0.2cm} -17.1799\\hspace{0.2cm} &\\hspace{0.2cm} -17.1132\\hspace{0.2cm} &\n\\hspace{0.2cm} -22.3731\\hspace{0.2cm} &\\hspace{0.2cm} -22.9829\\hspace{0.2cm} &\n\\hspace{0.2cm} -22.8961\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}0.4 \\hspace{0.2cm} & \\hspace{0.2cm} -16.9348\\hspace{0.2cm} &\n\\hspace{0.2cm} -17.9107\\hspace{0.2cm} &\\hspace{0.2cm} -17.7212\\hspace{0.2cm} &\n\\hspace{0.2cm} -22.8308\\hspace{0.2cm} &\\hspace{0.2cm} -23.9508\\hspace{0.2cm} &\n\\hspace{0.2cm} -23.7107\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}0.5\\hspace{0.2cm} & \\hspace{0.2cm} -17.2309\\hspace{0.2cm} &\n\\hspace{0.2cm} -18.7747\\hspace{0.2cm} &\\hspace{0.2cm} -18.3807\\hspace{0.2cm} & \n\\hspace{0.2cm} -23.2884\\hspace{0.2cm} &\\hspace{0.2cm} -25.0933\\hspace{0.2cm} &\n\\hspace{0.2cm} -24.5916\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}0.6\\hspace{0.2cm} & \\hspace{0.2cm} -17.5271\\hspace{0.2cm} &\n\\hspace{0.2cm} -19.7746\\hspace{0.2cm} &\\hspace{0.2cm} -19.0834\\hspace{0.2cm} &\n\\hspace{0.2cm} -23.7461\\hspace{0.2cm} &\\hspace{0.2cm} -26.4200\\hspace{0.2cm} & \n\\hspace{0.2cm}-25.5294\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}0.7\\hspace{0.2cm} & \\hspace{0.2cm} -17.8233\\hspace{0.2cm} &\n\\hspace{0.2cm} -20.9118\\hspace{0.2cm} & \\hspace{0.2cm} -19.8233 \n\\hspace{0.2cm} & \\hspace{0.2cm} -24.2038\\hspace{0.2cm} &\n\\hspace{0.2cm} -27.9539\\hspace{0.2cm} &\\hspace{0.2cm} -26.5170\n\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}0.8\\hspace{0.2cm} & \\hspace{0.2cm} -18.1195\\hspace{0.2cm} &\n\\hspace{0.2cm} -22.1878\\hspace{0.2cm} &\\hspace{0.2cm} -20.5966 \n\\hspace{0.2cm} & \\hspace{0.2cm} -24.6615\\hspace{0.2cm} &\n\\hspace{0.2cm} -29.7650\\hspace{0.2cm} &\\hspace{0.2cm} -27.5488\n\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}0.9 \\hspace{0.2cm} & \\hspace{0.2cm} -18.4157\\hspace{0.2cm} &\n\\hspace{0.2cm} -23.6032\\hspace{0.2cm} &\\hspace{0.2cm} -21.4006\n\\hspace{0.2cm} & \\hspace{0.2cm} -25.1192\\hspace{0.2cm} &\n\\hspace{0.2cm} -31.8070\\hspace{0.2cm} &\\hspace{0.2cm} -28.6206\n\\hspace{0.2cm} \\\\ \\hline\n\\hspace{0.2cm}1.0\\hspace{0.2cm} &\\hspace{0.2cm} -18.7119\\hspace{0.2cm} &\n\\hspace{0.2cm} -25.1586\\hspace{0.2cm} &\\hspace{0.2cm} -22.2325\\hspace{0.2cm} &\n\\hspace{0.2cm} -25.5769\\hspace{0.2cm} & \\hspace{0.2cm} -34.0823\n\\hspace{0.2cm} & \\hspace{0.2cm} -29.7285\\hspace{0.2cm} \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\vskip 1.2 true cm\n\n\\noindent Table 2. Ground state energies obtained for the spin-$3/2$ system \nusing exact diagonalization of the full Hamiltonian and the LEH1 and LEH2 for \n$3$ and $4$ triangles.\n\n\\newpage\n\n\\noindent {\\bf Figure Captions}\n\\vskip .5 true cm\n\n\\noindent {1.} The chain of triangles showing the labeling of the vertices\nof each triangle and the antiferromagnetic couplings $J_1 = 1$ and $J_2$.\n\n\\noindent{2.} The triangle and site labels used in deriving the LEHs. \n\n\\noindent\n{3.} The ground state energy/site in units of $J_1$ vs $1/N$, for \na few values of $J_2$ with spin-$1/2$ at each site. \n\n\\noindent\n{4.} The ground state energy/site in units of $J_1$, for values of $J_2 = \n0.1$ to $1.0$. The three sets of points show the results using DMRG for the\nfull Hamiltonian and the results from the LEH1 and LEH2.\n\n\\noindent\n{5.} The energy gaps from the ground state in units of $J_1$, for values of \n$J_2 = 0.1$ to $1.0$ for the spin-$1/2$ system. (a) shows the triplet gap and \n(b) shows the singlet gap. The two sets of points show the results obtained\nfrom the LEH2 and DMRG.\n\n\\noindent\n{6.} Some low-lying energy gaps for $J_2 = 0.0$ to $1.0$ for the spin-$1/2$ \nsystem obtained by DMRG. The two lowest $S^z =0$ states (including the \nground state at the bottom of the figure), the two lowest $S^z =1$ states \nand the lowest $S^z =2$ state are shown.\n\n\\noindent\n{7.} A larger number of low-lying energy gaps for $J_2 = 1.0$ for the \nspin-$1/2$ system obtained by DMRG. The two lowest $S^z =0$ states (including \nthe ground state at the bottom), the six lowest $S^z =1$ states and the \nlowest $S^z =2$ state are shown.\n\n\\noindent\n{8.} The spin bond orders for the spin-$1/2$ system obtained by DMRG. \n(a) shows the bond orders for two neighbouring bonds in the middle\nof the chain as a function of $N$ for $J_2 =1.0$. \n(b) shows the difference of the two bond orders (called the bond alternation \nor dimerization) for various values of $J_2$.\n\n\\noindent\n{9.} The ground state energy in units of $J_1 = 1$, for values of $J_2 =\n0.1$ to $1.0$ for spin-$1/2$ at each site of a system of five triangles\nas shown in the inset. The two sets of points show the results from the\nexact diagonalization of the LEH2 and of the full Hamiltonian.\n\n\\newpage\n\n\\begin{figure}[ht]\n\\vspace*{3cm}\n\\begin{center}\n\\epsfig{figure=fig1.ps,width=15cm}\n\\end{center}\n\\vspace*{1cm}\n\\centerline{Fig. 1}\n\\label{fig1}\n\\end{figure}\n\n\\vspace*{5cm}\n\\begin{figure}[hp]\n\\begin{center}\n\\epsfig{figure=fig2.ps,width=15cm}\n\\end{center}\n\\vspace*{1cm}\n\\centerline{Fig. 2}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\begin{center}\n\\epsfig{figure=fig3.ps,bbllx=50,bblly=0,bburx=500,bbury=800,height=20cm}\n\\end{center}\n\\vspace*{-1cm}\n\\centerline{Fig. 3}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\begin{center}\n\\epsfig{figure=fig4.ps,bbllx=50,bblly=0,bburx=500,bbury=800,height=20cm}\n\\end{center}\n\\vspace*{-1cm}\n\\centerline{Fig. 4}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\begin{center}\n%\\epsfig{figure=fig5a.ps,bbllx=-100,bblly=0,bburx=800,bbury=800,height=10cm}\n\\epsfig{figure=fig5a.ps,bbllx=50,bblly=0,bburx=500,bbury=800,height=20cm}\n\\end{center}\n\\vspace*{-1cm}\n\\centerline{Fig. 5 (a)}\n\\newpage\n\\begin{center}\n%\\epsfig{figure=fig5b.ps,bbllx=-100,bblly=-50,bburx=800,bbury=750,height=10cm}\n\\epsfig{figure=fig5b.ps,bbllx=50,bblly=0,bburx=500,bbury=800,height=20cm}\n\\end{center}\n\\vspace*{-1cm}\n\\centerline{Fig. 5 (b)}\n\\label{fig5}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\begin{center}\n\\epsfig{figure=fig6.ps,bbllx=50,bblly=0,bburx=500,bbury=800,height=20cm}\n\\end{center}\n\\vspace*{-1cm}\n\\centerline{Fig. 6}\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\begin{center}\n\\epsfig{figure=fig7.ps,bbllx=50,bblly=0,bburx=500,bbury=800,height=20cm}\n\\end{center}\n\\vspace*{-1cm}\n\\centerline{Fig. 7}\n\\label{fig7}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\begin{center}\n%\\epsfig{figure=fig8a.ps,bbllx=-200,bblly=0,bburx=350,bbury=800,height=10cm}\n\\epsfig{figure=fig8a.ps,bbllx=50,bblly=0,bburx=500,bbury=800,height=20cm}\n\\end{center}\n\\vspace*{-1cm}\n\\centerline{Fig. 8 (a)}\n\\newpage\n\\begin{center}\n%\\epsfig{figure=fig8b.ps,bbllx=350,bblly=800,bburx=900,bbury=1600,height=10cm}\n\\epsfig{figure=fig8b.ps,bbllx=50,bblly=0,bburx=500,bbury=800,height=20cm}\n\\end{center}\n\\vspace*{-1cm}\n\\centerline{Fig. 8 (b)}\n\\vspace*{1cm}\n\\label{fig8}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\begin{center}\n\\epsfig{figure=fig9.ps,bbllx=50,bblly=-100,bburx=500,bbury=700,height=20cm,\nangle=-90}\n\\end{center}\n\\vspace*{2cm}\n\\centerline{Fig. 9}\n\\label{fig9}\n\\end{figure}\n\n\\end{document}\n\n"
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{
"name": "cond-mat0002099.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{saw} T. Nakamura and K. Kubo, Phys. Rev. B {\\bf 53}, 6393 (1996);\nD. Sen, B. S. Shastry, R. E. Walstedt and R. Cava, Phys. Rev. B {\\bf 53}, \n6401 (1996); K. Kubo, Phys. Rev. B {\\bf 48}, 10552 (1993).\n\n\\bibitem{kag1} X. Obradors {\\it et al}, Sol. Stat. Comm. {\\bf 65}, 189 \n(1988); A. P. Ramirez, G. P. Espinosa and A. S. Cooper, Phys. Rev. Lett. \n{\\bf 64}, 2070 (1990); Phys. Rev. B {\\bf 45}, 2505 (1992); C. Broholm,\nG. Aeppli, G. P. Espinosa and A. S. Cooper, Phys. Rev. Lett. {\\bf 65}, 3173\n(1990).\n\n\\bibitem{kag2} J. T. Chalker, P. C. W. Holdsworth and E. S. Shender,\nPhys. Rev. Lett. {\\bf 68}, 855 (1992); A. J. Harris, C. Kallin and A. J.\nBerlinsky, Phys. Rev. B {\\bf 45}, 2899 (1992); A. Chubukov, Phys. Rev. Lett.\n{\\bf 69}, 832 (1992); P. Chandra, P. Coleman and I. Ritchey, J. de Phys. I \n{\\bf 3}, 591 (1993); S. Sachdev, Phys. Rev. B {\\bf 45}, 12377 (1992);\nR. Shankar and D. Shubashree, cond-mat/9907389.\n\n\\bibitem{kag3} C. Zeng and V. Elser, Phys. Rev. B {\\bf 42}, 8436 (1990);\nJ. T. Chalker and J. F. G. Eastmond, Phys. Rev. B {\\bf 46}, 14201 (1992);\nR. R. P. Singh and D. A. Huse, Phys. Rev. Lett. {\\bf 68}, 1766 (1992). \n\n\\bibitem{lech} P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre and P.\nSindzingre, Phys. Rev. B {\\bf 56}, 2521 (1997); C. Waldtmann {\\it et al},\nEur. Phys. J. B {\\bf 2}, 501 (1998); P. W. Leung and V. Elser, \nPhys. Rev. B {\\bf 47}, 5459 (1993).\n\n\\bibitem{kagstrip} P. Azaria, C. Hooley, P. Lecheminant, C. Lhuillier and A. \nM. Tsvelik, Phys. Rev. Lett. {\\bf 81}, 1694 (1998); S. K. Pati and R. R. P. \nSingh, Phys. Rev. B {\\bf 60}, 7695 (1999).\n\n\\bibitem{pert1} K. Kawano and M. Takahashi, J. Phys. Soc. Jpn. {\\bf 66},\n4001 (1997); H. J. Schulz, in {\\it Strongly Correlated Magnetic and\nSuperconducting Systems}, edited by G. Sierra and M. A. Martin-Delgado, \nLecture Notes in Physics 478 (Springer, Berlin, 1997); D. C. Cabra, A.\nHonecker and P. Pujol, Phys. Rev. B {\\bf 58}, 6241 (1998).\n\n\\bibitem{pert2} F. Mila, Phys. Rev. Lett. {\\bf 81}, 2356 (1999); V. \nSubrahmanyam, Phys. Rev. B {\\bf 52}, 1133 (1995).\n\n\\bibitem{pert3} K. Tandon, S. Lal, S. K. Pati, S. Ramasesha and D. Sen, Phys.\nRev. B {\\bf 59}, 396 (1999).\n\n\\bibitem{whit1} S. R. White, Phys. Rev. Lett. {\\bf 69}, 2863 (1992); Phys. \nRev. B {\\bf 48}, 10345 (1993).\n\n\\bibitem{whit2} S. R. White and I. Affleck, Phys. Rev. B {\\bf 54}, 9862 \n(1996).\n\n\\end{thebibliography}"
}
] |
|
cond-mat0002100
|
Regularities in football goal distributions
|
[
{
"author": "L. C. Malacarne $\\&$ R. S. Mendes"
}
] |
Besides of complexities concerning to football championships, it is identified some regularities in them. These regularities refer to goal distributions by goal-players and by games. In particular, the goal distribution by goal-players it well adjusted by the Zipf-Mandelbrot law, suggesting a conection with an anomalous decay.
|
[
{
"name": "futfig.tex",
"string": "\n%\\documentstyle[12pt]{article}\n%\\documentstyle[preprint,prl,aps]{revtex}\n\\documentstyle[graphicx,prl,aps,twocolumn]{revtex}\n%\\usepackage[dvips]{graphicx}\n\\begin{document}\n%\\draft\n\\title{ Regularities in football goal distributions}\n\\author{ L. C. Malacarne $\\&$ R. S. Mendes}\n%\\thanks{ e-mail address: rsmendes@dfi.uem.br, fax number: 55 44 2634242}}\n\\address{Departamento de F\\'\\i sica, Universidade Estadual de Maring\\'a,\nAvenida Colombo 5790, 87020-900,\n Maring\\'a-PR, Brazil}\n%\\date{\\today }\n\\maketitle\n\n\n\\begin{abstract}\nBesides of complexities concerning to football championships, it\nis identified some regularities in them. These regularities refer\nto goal distributions by goal-players and by games. In particular,\nthe goal distribution by goal-players it well adjusted by the\nZipf-Mandelbrot law, suggesting a conection with an anomalous\ndecay.\n\\end{abstract}\n\\pacs{PACS number(s): 05.20.-y, 05.90.+m, 89.90.+n}\n\\date{\\today }\n\n Regularity in some complex systems can sometimes be\nidentified and expressed in terms of simple laws. Typical\nexamples of such situations are found in a wide range of contexts\nas the frequency of words in a long text\\cite{Zipf}, the\npopulation distribution in big cities\\cite{h1,2,3}, forest\nfires\\cite{5}, the distribution of species lifetimes for North\nAmerican breeding bird populations\\cite{h2}, scientific\ncitations\\cite{6,7}, www surfing\\cite{8}, ecology\\cite{8a}, solar\nflares\\cite{8b}, economic index\\cite{9}, epidemics in isolated\npopulations\\cite{Rhodes}, among others. Here, universal behaviours\nin the most popular sport, the football, are discussed. More\nprecisely, this work focuses on regularities in goal distribution\nby goal-players and by games in championships. Furthermore, the\ngoal distribution by goal-players is connected with an anomalous\ndecay related to the Zipf-Mandelbrot\\cite{Zipf,Mandelbrot} law and\nwith Tsallis nonextensive statistical\nmechanics\\cite{Tsallis,Curado,Denisov,Tsallis2}.\n\n\nIn many contexts, it is common that few phenomena with high\nintensity arise, and so do many ones with low intensity. For\ninstance, a long text generally contains many words that are\nemployed in few opportunities and a small number that occurs\nlargely\\cite{Zipf}. The above mentioned systems are good examples\ntoo. In particular, this kind of behaviour usually occurs in\nfootball championships, because there are many players that make\nfew goals in contrast with the topscorers.\n\nA detailed visualization of this behaviour can be well illustrated\nby considering some of the most competitive and traditional\nchampionships of the world. Our particular choice of championships\nhas been done guided by the criterion of easy accessibility of the\ncorresponding data to anyone\\cite{data1,data2}. Therefore, we\nconsider, here, some of the main league football championships\nfrom Italy, England, Spain and Brazil\\cite{obs1}. Each of these\nchampionships has the participation of about twenty teams,\ncontains around three hundred games, and approximately eight\nhundred goals\\cite{obs2}. In Fig. \\ref{f2} we exhibit data of\nthese championships. In these graphics, the abscissa presents the\nnumber of goals $x$ divided by an average of goals $m$ (total\nnumber of goals per total number of goal-players), and the\nordinate indicates the quantity $N(x)$ of players with $x$ goals\ndivided by the quantity of players with one goal, $N(1)$. The\nregular shape of the graphics presented in Fig. \\ref{f2} suggests\na general law to describe the distribution of goals.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n \\centering\n \\DeclareGraphicsRule{ps}{eps}{*}{}\n \\includegraphics*[width=10cm, height=7cm,trim=2.5cm 2.35cm 0cm 2cm]{fig1.eps}\n \\caption{Scaled distribution of goals in main league football\n championships from Italy, England, Spain and Brazil.\n The ordinate $N(x)$ is the number of players\n with $x$ goals divided by the number of players with one goal, $N(1)$,\n and the abscissa is the number of goals divided by the average of goals\n $m$. These scaled data indicate a regularity in the goal\n distribution for goal-players. The European championships\n start in one year and finish in the subsequent year,\n and the Brazilian championships start and finish in the same\n year.\n }\\label{f2}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn the study of the majority of the previously cited systems, the\nZipf's law\\cite{Zipf}, $N(x)=a/x^b$, arises naturally, at least in\npart of the analysis. In the Zipf's law, $a$ and $b$ are constants\nand $x$ is the independent variable. In order to give a better\nadjustment to a large part of the data, and based on information\ntheory, Mandelbrot\\cite{Mandelbrot} proposed $N(x)=a/(c+x)^b$ as a\ngeneralization of the Zipf's law, with $a$, $b$, and $c$ all being\nconstants. This Zipf-Mandelbrot's distribution also arises in the\ncontext of a generalized statistical mechanics proposed some years\nago\\cite{Tsallis,Curado,Tsallis2}, equivalently rewritten as\n\\begin{equation}\\label{m}\nN(x)= N_0 [1-(1-q)\\lambda x]^{\\frac{1}{1-q}}\\; ,\n\\end{equation}\nwhere $N_0$, $\\lambda$, a $q$ are real parameters. In addition,\nthis function satisfies an anomalous decay equation,\n\\begin{equation}\\label{1}\n\\frac{d}{d x} \\left(\\frac{N(x)}{N_0}\\right) =-\\lambda\n\\left(\\frac{N(x)}{N_0}\\right)^q \\; .\n\\end{equation}\n The parameter $q$ can be considered as a measure of how\nanomalous the decay is. In particular, equation (\\ref{m}) is\nreduced to the usual exponential decay, $N(x)= N_0 \\exp (-\\lambda\nx)$, in the limit $q\\rightarrow 1$.\n\nMotivated by these physical connections, we employ the\ndistribution (\\ref{m}) to adjust the goals data. Following the\nconstruction of Fig. \\ref{f2}, we use the number of goalplayers\nwith one goal, $N(1)$, and the average goal number by goalplayer,\n$m$, to eliminate $N_0$ and $\\lambda$. Furthermore, it is a good\napproximation to replace the discrete average with a continuous\none in the present analysis, {\\it i.e.},\n\\begin{equation}\\label{3}\n m= \\frac{\\int_0^{\\infty} x N(x)}{\\int_0^{\\infty} N(x)}=\n \\frac{1}{\\lambda (3-2 q)}\\;\\;\\;\\;\\;\\;\\; (q<3/2) .\n\\end{equation}\nThus, the distribution of goals dictated by equation (\\ref{m})\ncan be rewritten as\n\\begin{equation}\\label{4}\n N(x)=N(1)\\frac{ \\left[1-\\frac{(1-q)}{(3-2q)m} \\right]^{\\frac{1}{q-1}}}\n {\\left[1-\\frac{(1-q)}{(3-2q)m}x \\right]^{\\frac{1}{q-1}}}\\; ,\n\\end{equation}\nwhere $q$ becomes the unique parameter that remains to be\nadjusted, since $N(1)$ and $m$ are obtained directly from the\ndata. Fig. \\ref{f3} illustrates applications of equation (\\ref{4})\nfor four championships, indicating therefore the goodness of the\nformula (\\ref{4}). The same conclusion is obtained in the other\nchampionships showed in Fig. \\ref{f2}. Here, $q= 1.33$ was\nemployed as approximated value, leading to the Zipf-Mandelbrot's\nexponent $b \\approx 3$. In this way, $q\\approx 1.33$ can be\ninterpreted as the universal parameter for this kind of\nchampionships. Also, it is interesting to remark that $b\\approx 3$\noccurs in the distribution of scientific citations \\cite{6,7}.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n \\centering\n \\DeclareGraphicsRule{ps}{eps}{*}{}\n \\includegraphics*[width=9.5cm, height=7cm,trim=2cm 1.8cm 0cm 1.5cm]{fig2.eps}\n \\caption{Fit of the goal distribution for goal-players in Italian ({\\bf a}),\n English ({\\bf b}), Spanish ({\\bf c}), and Brazilian ({\\bf d}) championships.\n The black circles are the goal data and the solid line is the fitting with the $N(x)$\n distribution being given in equation (\\ref{4}) with $q=1.33$. } \\label{f3}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nOther kinds of regularities in goal distributions can be\nidentified, but with different behaviours. This fact can be\nverified in the distribution of goals per game. Proceeding in a\nsimilar way as done in Fig. \\ref{f2}, it is considered normalized\nscale distribution of games and goals. In this case, the abscissa\nis the number $x$ of goal divided by $M$, the mean goal per game\nof a championship (the number of goals of a championship divided\nby the corresponding number of games). In addition, the ordinate\nis given by the number of games with $x$ goals of a championship\ndivided by the number of games of the corresponding championship.\nFig. \\ref{f4} contains this kind of graphics illustrating this\nregular behaviour by considering, again, the main league football\nchampionships from Italy, England, Spain and Brazil. As one can\nsee, this figure strongly suggests a regularity in the\ndistribution of goals for distinct championships.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n \\centering\n \\DeclareGraphicsRule{ps}{eps}{*}{}\n \\includegraphics*[width=10cm, height=7cm,trim=2cm 1cm 0cm 1.5cm]{fig3.eps}\n \\caption{Semi-log graphic of scaled distributions of goals per game\n is illustrated by considering championships from Italy, England,\n Spain and Brazil. For each championship, $x$ is the number\n of goals per game, $M$ is the average of goals per game,\n $G(x)$ is the number of games with $x$ goals, and\n $G_{t}$ is the total number of games. }\\label{f4}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nBesides the numberless factors, including fluctuation due to\nrelatively small number of teams and games that are present in a\nfootball championship, regularities arise in the goal\ndistributions. In particular, the goal distribution for players\nthat make goals are well adjusted by a Zipf-Mandelbrot law,\nsuggesting a connection with ubiquitous phenomena such as\nanomalous diffusion.\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{thebibliography}{10}\n\\bibitem{Zipf} G. K. Zipf,\n{\\it Human Behavior and the Principle of Least Effort}\n(Addison-Wesley, Cambridge, MA, 1949).\n\\bibitem{h1} A. H. Makse, S. Havlin, H. E. Stanley,\n%{\\it Modelling urban growth patterns}.\n {\\it Nature} {\\bf 377}, 608 (1995).\n\\bibitem{2} D. H. Zanette, S. C. Manrubia,\n% {\\it Role of Intermittency in\n% Urban Development: A Model of Large-Scale City Formation}.\n{\\it Phys. Rev. Lett.} {\\bf 79}, 523 (1997).\n\\bibitem{3} M. Marsilli, Y.-C. Zhang,\n% {\\it Interacting Individual Leading to Zipf's Law}.\n {\\it Phys. Rev. Lett.} {\\bf 80}, 2741 (1998).\n\\bibitem{5} B. D. Malamud, G. Morein, D. L. Turcotte,\n%{\\it Forest Fires: An Example of Self-Organized Critical Behavior}.\n{\\it Science} {\\bf 281}, 1840 (1998).\n\\bibitem{h2} T. H. Keit, H. E. Stanley,\n%{\\it Dynamics of North American breeding bird populations}.\n {\\it Nature} {\\bf 393}, 257 (1998).\n\\bibitem{6} S. Redner,\n%{\\it How popular is your paper: An empirical study\n%of the citation distribution}.\n{\\it Eur. Phys. J. B} {\\bf 4}, 131 (1998).\n\\bibitem{7}C. Tsallis, M. P. Albuquerque,\n% {\\it Are citations of scientific papers a case of nonextensivity?}\n{\\it To be published in Eur. Phys. J. B (cond-mat/9903433)}.\n\\bibitem{8}B. A. Huberman, P. L. T. Pirolli, J. E. Pitkow, R. M.\nLukose,\n%{\\it Strong Regularities in Word Wide Web Surfing}.\n{\\it Science }{\\bf 280}, 95 (1998).\n\\bibitem{8a} J. R. Banavar, J. L. Green, J. Harte, and A. Maritan,\nPhys. Rev. Lett. {\\bf 83}, 4212 (1999).\n\\bibitem{8b} G. Boffetta, V. Carbone, P. Giuliani, P. Veltri, and\nA. Vulpiani, Phys. Rev. Lett. {\\bf 83}, 4662 (1999).\n\\bibitem{9} R. N. Mantegna, H. E. Stanley,\n%{\\it Scaling behaviour in the dynamics of an economic index}.\n{\\it Nature} {\\bf 376}, 46 (1995).\n\\bibitem{Rhodes} C. J. Rhodes, M. Anderson,\n% {\\it Power laws governing epidemics in isolated populations}.\n{\\it Nature} {\\bf 381}, 600 (1996).\n\\bibitem{Mandelbrot} B. B. Mandelbrot, {\\it The Fractal Geometry of\nNature} (Freeman, New York, 1977).\n\\bibitem{Tsallis}C. Tsallis,\n% {\\it Possible Generalization of Boltzmann-Gibbs Statistics}.\n{\\it J. Stat. Phys.} {\\bf 52}, 479 (1988).\n\\bibitem{Curado} E. M. F. Curado, C. Tsallis,\n% {\\it Generalized Statistical Mechanics Connection with Thermodinamics}.\n{\\it J. Phys. A: Math. Gen.} {\\bf 24}, L69 (1991); Corrigenda:\n{\\bf 24}, 3187 (1991) and {\\bf 25}, 1019 (1992).\n\\bibitem{Denisov} S. Denisov,\n% {\\it Fractal binary sequences: Tsallis thermodynamics and the Zipf Law}.\n{\\it Phys. Lett. A} {\\bf 235}, 447(1997).\n\\bibitem{Tsallis2}\nC. Tsallis, R. S. Mendes, A. R. Plastino,\n% {\\it The role of constraints within generalized nonextensive statistics}.\n{\\it Physica A} {\\bf 261}, 534 (1998).\n\\bibitem{data1}\nhttp://www.risc.uni-linz.ac.at/non-official/rsssf/ .\n\\bibitem{data2} http://www.globosat.com.br/sportv/ .\n\\bibitem{obs1}\nSince the complete set of data for all championships of these\ncountries is not easely avaliable, we have used data that are both\nrencent and easely accessible.\n\\bibitem{obs2}\nIn the European championships, the leagues are organized in such\nway that each team plays two games with each of the other teams.\nIn the Brazilian championships, each team plays only once with\neach one of the others. After this, the eight best teams go to the\nplayoff. Nevertheless, the number of games of the Brazilian\nchampionships in the playoff is much smaller than in the\nclassification phase, and it doesn't affect the analysis when\ncompared with European ones.\n\n\n\\end{thebibliography}\n\n\\acknowledgements One of us, Mendes, R. S., thanks partial\nfinantial support by CNPq (Brazilian Agency).\n\n\n%\\newpage\n\n\n\n\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002100.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\\bibitem{Zipf} G. K. Zipf,\n{\\it Human Behavior and the Principle of Least Effort}\n(Addison-Wesley, Cambridge, MA, 1949).\n\\bibitem{h1} A. H. Makse, S. Havlin, H. E. Stanley,\n%{\\it Modelling urban growth patterns}.\n {\\it Nature} {\\bf 377}, 608 (1995).\n\\bibitem{2} D. H. Zanette, S. C. Manrubia,\n% {\\it Role of Intermittency in\n% Urban Development: A Model of Large-Scale City Formation}.\n{\\it Phys. Rev. Lett.} {\\bf 79}, 523 (1997).\n\\bibitem{3} M. Marsilli, Y.-C. Zhang,\n% {\\it Interacting Individual Leading to Zipf's Law}.\n {\\it Phys. Rev. Lett.} {\\bf 80}, 2741 (1998).\n\\bibitem{5} B. D. Malamud, G. Morein, D. L. Turcotte,\n%{\\it Forest Fires: An Example of Self-Organized Critical Behavior}.\n{\\it Science} {\\bf 281}, 1840 (1998).\n\\bibitem{h2} T. H. Keit, H. E. Stanley,\n%{\\it Dynamics of North American breeding bird populations}.\n {\\it Nature} {\\bf 393}, 257 (1998).\n\\bibitem{6} S. Redner,\n%{\\it How popular is your paper: An empirical study\n%of the citation distribution}.\n{\\it Eur. Phys. J. B} {\\bf 4}, 131 (1998).\n\\bibitem{7}C. Tsallis, M. P. Albuquerque,\n% {\\it Are citations of scientific papers a case of nonextensivity?}\n{\\it To be published in Eur. Phys. J. B (cond-mat/9903433)}.\n\\bibitem{8}B. A. Huberman, P. L. T. Pirolli, J. E. Pitkow, R. M.\nLukose,\n%{\\it Strong Regularities in Word Wide Web Surfing}.\n{\\it Science }{\\bf 280}, 95 (1998).\n\\bibitem{8a} J. R. Banavar, J. L. Green, J. Harte, and A. Maritan,\nPhys. Rev. Lett. {\\bf 83}, 4212 (1999).\n\\bibitem{8b} G. Boffetta, V. Carbone, P. Giuliani, P. Veltri, and\nA. Vulpiani, Phys. Rev. Lett. {\\bf 83}, 4662 (1999).\n\\bibitem{9} R. N. Mantegna, H. E. Stanley,\n%{\\it Scaling behaviour in the dynamics of an economic index}.\n{\\it Nature} {\\bf 376}, 46 (1995).\n\\bibitem{Rhodes} C. J. Rhodes, M. Anderson,\n% {\\it Power laws governing epidemics in isolated populations}.\n{\\it Nature} {\\bf 381}, 600 (1996).\n\\bibitem{Mandelbrot} B. B. Mandelbrot, {\\it The Fractal Geometry of\nNature} (Freeman, New York, 1977).\n\\bibitem{Tsallis}C. Tsallis,\n% {\\it Possible Generalization of Boltzmann-Gibbs Statistics}.\n{\\it J. Stat. Phys.} {\\bf 52}, 479 (1988).\n\\bibitem{Curado} E. M. F. Curado, C. Tsallis,\n% {\\it Generalized Statistical Mechanics Connection with Thermodinamics}.\n{\\it J. Phys. A: Math. Gen.} {\\bf 24}, L69 (1991); Corrigenda:\n{\\bf 24}, 3187 (1991) and {\\bf 25}, 1019 (1992).\n\\bibitem{Denisov} S. Denisov,\n% {\\it Fractal binary sequences: Tsallis thermodynamics and the Zipf Law}.\n{\\it Phys. Lett. A} {\\bf 235}, 447(1997).\n\\bibitem{Tsallis2}\nC. Tsallis, R. S. Mendes, A. R. Plastino,\n% {\\it The role of constraints within generalized nonextensive statistics}.\n{\\it Physica A} {\\bf 261}, 534 (1998).\n\\bibitem{data1}\nhttp://www.risc.uni-linz.ac.at/non-official/rsssf/ .\n\\bibitem{data2} http://www.globosat.com.br/sportv/ .\n\\bibitem{obs1}\nSince the complete set of data for all championships of these\ncountries is not easely avaliable, we have used data that are both\nrencent and easely accessible.\n\\bibitem{obs2}\nIn the European championships, the leagues are organized in such\nway that each team plays two games with each of the other teams.\nIn the Brazilian championships, each team plays only once with\neach one of the others. After this, the eight best teams go to the\nplayoff. Nevertheless, the number of games of the Brazilian\nchampionships in the playoff is much smaller than in the\nclassification phase, and it doesn't affect the analysis when\ncompared with European ones.\n\n\n\\end{thebibliography}"
}
] |
cond-mat0002101
|
Gap deformation and classical wave localization in disordered two-dimensional photonic band gap materials
|
[
{
"author": "E.~Lidorikis$^1$"
},
{
"author": "M.~M.~Sigalas$^1$"
},
{
"author": "E.~N.~Economou$^2$ and C.~M.~Soukoulis$^{1,2}$"
}
] |
[
{
"name": "disordpapar_main.tex",
"string": "%lidorikis_main.tex\n%\\documentstyle[twocolumn,aps,psfig]{revtex}\n%\\documentstyle[preprint,aps,psfig]{revtex}\n\\documentstyle[prb,aps,psfig]{revtex}\n%\\documentclass[12pt]{article}\n%c for double space, comment out the \\tightlines\n%\\tightenlines\n%\\hoffset=-1.65in\n%\\textwidth=39pc\n%\\math-with-secnums\n\\begin{document}\n%\\draft\n%\\preprint{Ames Laboratory-U.S. DOE Preprint }\n\\title{Gap deformation and classical wave localization in disordered\ntwo-dimensional photonic band gap materials}\n\\author{\nE.~Lidorikis$^1$, M.~M.~Sigalas$^1$, E.~N.~Economou$^2$ and \nC.~M.~Soukoulis$^{1,2}$\n}\n\\address{\n$^1$Ames Laboratory-U.S.~DOE and Department of Physics and Astronomy, \nIowa State University, Ames, Iowa 50011 \\\\\n}\n\\address{\n$^2$Research Center of Crete-FORTH and Department of Physics, \nUniversity of Crete,Heraklion, Crete 71110, Greece \\\\\n}\n\\author{\\parbox[t]{5.5in}{\\small\nBy using two {\\em ab initio} numerical methods we study the effects\nthat disorder has on the spectral gaps and on wave localization in \ntwo-dimensional photonic band gap materials. We find that there are\nbasically two different responses depending on the lattice realization \n(solid dielectric cylinders in air or {\\em vise versa}), the \n wave polarization, and the particular form under which disorder is \nintroduced. \nTwo different pictures for the photonic states are employed, \n the ``nearly free'' photon and the ``strongly localized'' photon.\nThese originate from the two different mechanisms responsible for the\nformation of the spectral gaps, ie. multiple scattering and single\nscatterer resonances, and they qualitatively explain our results.\n\\\\ \\\\\nPACS numbers: }}\n\\maketitle\n\\normalsize \n%\\tightenlines\n%\\newpage\n\n\n\\section{Introduction}\n\nElectromagnetic waves traveling in periodic dielectric structures\nwill undergo multiple scattering. For the proper structural parameters \nand wave frequencies, all waves may backscatter coherently;\nthe result is total inhibition of propagation inside the structure. \nSuch structures are called photonic band gap (PBG) materials \n\\cite{soukoulis,joannopoulos} or\nphotonic crystals, and the corresponding frequency ranges, for which \n propagation is not allowed, photonic band gaps or stop bands.\nPBG materials can be artificially made in one, two, or three dimensions.\nFor example, a periodic lattice of dielectric spheres embedded in a \ndifferent dielectric medium would work as a three-dimensional PBG\nmaterial, for the proper choice of lattice symmetry, dielectric contrast,\nand sphere volume filling ratio. In two dimensions, a periodic array\nof parallel, infinitely long, dielectric cylinders could work as a \ntwo-dimensional PBG material, prohibiting propagation in a direction \nperpendicular to the cylinders' axis for some frequency range(s).\nThe absence of optical modes in a photonic band gap is often considered\nas analogous to the absence of electronic energy eigenstates in the \nsemiconductor energy gap. The ability of PBG materials to modulate\nelectromagnetic wave propagation, in a similar way semiconductors \nmodulate the electric current flow, can have a profound impact in many \nareas in pure and applied physics. It is then of fundamental importance\nto study the effects of disorder \\cite{fan,sigalas} \non the transmission properties of such materials.\n\nBesides the non-resonant, macroscopic Bragg-like multiple scattering, \nthere is also a second, resonant mechanism, that contributes to the\nformation of the spectral gaps. This is \\cite{ho,kafesaki,lidorikis}\n the excitation\nof single scatterer Mie resonances \\cite{mie}. \nIn a previous publication \\cite{lidorikis} it was \nshown that for two-dimensional PBG materials, for the $E$ polarization \nscalar wave case (electric field parallel to the cylinders' axis),\nthese Mie resonances are analogous to the electronic orbitals in\nsemiconductors. The idea of the linear combination of atomic orbitals \n(LCAO) method was extended to the classical wave case as a linear\ncombination of Mie resonances (LCMR), leading to a successful tight-binding\n(TB) parameterization for photonic band gap materials. \nThis moves the picture for the photon states, from a one analogous\nto the nearly free electron model, to the one analogous to the strongly\nlocalized electron whose transport is achieved only by hopping\n(tunneling) from atom to atom. Depending then on which mechanism\nis dominant for the formation of the photonic gaps, we expect\ndifferent changes to the system's properties when disorder is introduced.\nIf the Bragg-like multiple scattering mechanism is the dominant one,\nthe photonic gaps should close quickly with increasing disorder, while\nif it is the excitation of Mie resonances, the photonic gaps should\nsurvive large amounts of disorder, in a similar way the electronic \nenergy gap survives in amorphous silicon. \n\nIn this paper we will use two {\\em ab initio} numerical methods to study the\neffects of disorder on photonic gap formation and wave localization\nin two-dimensional PBG materials. \nThe first is the finite difference time domain (FDTD) spectral method\n\\cite{yee,taflove},\nfrom which we obtain the photonic density of states for an infinite,\ndisordered PBG material, and the second is the transfer matrix technique\n\\cite{pendry}, \nfrom which we obtain the transmission coefficient for a wave incident\nonto a finite slab of the disordered PBG material. From the transmission \ncoefficient we can obtain the localization length for the photonic states\nof the disordered material \\cite{sigalas}. The study will be on both \nPBG material realizations (solid high dielectric cylinders \nin air and cylindrical air holes\n in high dielectric), for both wave polarizations, and it will \nincorporate three different disorder realizations: disorder in position,\nradius, and dielectric constant (these systems, though, will still be \nperiodic on the average). We will find that only the case \nof solid dielectric cylinders in air with the wave $E_z$-polarized \nexhibits the \nbehavior\nexpected from the strongly localized photon picture, while for\nall other cases, the nearly free photon picture seems to be the dominant one.\n\n\n\\section{Numerical methods}\n\nElectromagnetic wave propagation in lossless composite dielectric\nmedia is described by Maxwell's equations\n\\begin{eqnarray}\n\\label{disord_eq1}\n\\mu \\frac{\\partial \\vec{H}}{\\partial t} = \n -\\vec{\\nabla} \\times \\vec{E}, & \\hspace{1.5cm} & \n\\epsilon(\\vec{r})\\frac{\\partial \\vec{E}}{\\partial t} = \n\\vec{\\nabla} \\times \\vec{H},\n\\end{eqnarray}\nwhere the dielectric constant $\\epsilon(\\vec{r})$ is a function of position.\nIn two dimensions, the two independent wave polarizations are decoupled.\nWe assume the variation of the dielectric constant, as well as the\npropagation direction, along the $xy$ plane, and so, the cylinders along\nthe $z$ axis. One of the polarizations \nis with the electric field parallel to the $z$ axis and the magnetic\nfield on the $xy$ plane ($E_z$ or TM polarized) \nand obeys a scalar wave equation. The other one with the magnetic field \nparallel to the $z$ axis and the electric field on the $xy$ plane \n($H_z$ or TE polarized) and obeys a vector wave equation. \n\nThe first method we will use, to study disordered PBG materials, is\nthe FDTD spectral method \\cite{chan,sakoda}.\nIn our FDTD scheme, we first discretize the $xy$ plane into a fine \nuniform grid. Each grid point is centered in a unit cell which is further\ndiscretized into a 10$\\times$10 subgrid, on which an arithmetic average \nof the dielectric constant is performed. In our problem we will assume\ndispersionless and lossless materials. For the $E_z$ polarization case we\n define the electric field on this grid and the magnetic field \non two additional grids, one tilted by $(d/2,0)$,\n on which $H_y$ is defined,\nand one tilted by $(0,d/2)$, on which we define $H_x$.\n $d$ is the side of the \ngrid cell.\nThe corresponding finite-difference equations for the space\nderivatives that are used in the curl operators are then central-difference\nin nature and second-order accurate. The electric and magnetic fields are\nalso displaced in time by a half time step $\\Delta t/2$, \nresulting in a ``leapfrog'' arrangement and central-difference equations \nfor the time derivatives as well. If one initialize the electric and\nmagnetic fields at $t=t_0$ and $t=t_0+\\Delta t/2$ respectively, then\nupdating the values of the electric field for each grid point $(i,j)$ at\n$t=t_0+\\Delta t$ is done by\n\\begin{eqnarray}\n\\label{disord_eq2}\nE_z\\vert_{i,j}^{t_0+\\Delta t} = E_z\\vert_{i,j}^{t_0} & + & \\frac{\\Delta t}\n{d\\ \\epsilon_{i,j}} ( H_y\\vert_{i+1/2,j}^{t_0+\\Delta t/2} -\nH_y\\vert_{i-1/2,j}^{t_0+\\Delta t/2}- \\nonumber \\\\ \n & - & H_x\\vert_{i,j+1/2}^{t_0+\\Delta t/2}+H_x\\vert_{i,j-1/2}^\n{t_0+\\Delta t/2} )\n\\end{eqnarray}\nwhere $\\epsilon_{i,j}$ is the averaged dielectric constant for the grid\npoint $(i,j)$. Similar equations follow for updating the magnetic field\ncomponents at $t=t_0+3\\Delta t/2$, then again Eq.~(\\ref{disord_eq2})\nfor $E_z$ at $t=t_0+2\\Delta t$ etc. This way the time evolution of\nthe system can be recorded. For numerical stability and good convergence \nthe number of grid points per wavelength $\\lambda/d$ must be at least 20,\n and also $\\Delta t \\leq d/\\sqrt{2}c$, where $c$ the speed of light in vacuum. \nSimilar equations, with the roles of the \nelectric and magnetic fields interchanged, apply for the $H_z$ polarization\ncase.\n \nIn order to find the eigenmodes of a particular periodic (or disordered) \n system, we first initialize the electric and magnetic fields in the\nunit cell (or a suitable supercell) using periodic boundary \nconditions: $\\vec{E}(\\vec{r}+\\vec{a})=e^{i\\vec{k}\\vec{a}}\\vec{E}(\\vec{r})$ \nand similarly\nfor $\\vec{H}(\\vec{r})$, where $\\vec{k}$ is the corresponding \nBloch wave vector\nand $\\vec{a}$ the lattice vector. These fields must have nonzero \nprojections with the modes in search. We choose a superposition of\nBloch waves for the magnetic field and set zero the electric field:\n\\begin{eqnarray}\n\\label{disord_eq3}\n\\vec{H}(\\vec{r})=\\sum_{\\vec{g}} \\hat{v}_{\\vec{g}}\ne^{i(\\vec{k}+\\vec{g})\\vec{r}+i\\phi_{\\vec{g}}}, & \\hspace{1cm} &\n\\vec{E}(\\vec{r})=0,\n\\end{eqnarray}\nwhere $\\phi_{\\vec{g}}$ is just a random phase and the unit vector $\\hat{v}$ \nis perpendicular to both $\\vec{E}$ and $(\\vec{k}+\\vec{g})$, ensuring that\n$\\vec{H}$ is transverse and that $\\vec{\\nabla}\\vec{H}=0$. \nOnce the initial fields are \ndefined, we can evolve them in time using the ``leapfrog'' difference\nequations, while recording the field values as a time series for some\nsampling points. As the electric fields ``builds'' up, some particular\nmodes dominate while most are depressed, reflecting the underline\nlattice symmetries. Here we record only the $E_z$ field for the $E_z$ \npolarization case, and the $H_z$ field for the $H_z$ polarization. \nAt the end of the simulation, the time series are Fourier \ntransformed back into frequency space, and the eigenmodes \n$\\omega(\\vec{k})$ of the system appear as sharp peaks. \nThe length of the simulation determines the \nfrequency resolution while the time difference between successive recordings\ndetermines the maximum frequency considered. This method scales\nlinearly with size: a larger system will still need the same number of\ntime steps for the same frequency resolution, thus sometimes referred\nto also as an ``order-N'' method \\cite{chan}. \n\nHere we will use this method to obtain the system's density of states\n(DOS). If one chooses a large supercell instead of the unit cell, then\nfor each $\\vec{k}$ point inside it's first Brillouin zone,\nthe Fourier transformed time series will consist of a number of peaks.\nAdding all contributions from all $\\vec{k}$'s will result to a smooth\nfunction for the DOS. This is in contrast to older methods that where\nusing random fields as initial boundary conditions \\cite{sigalas}. \nRandom initial fields \nwill ensure the condition for nonzero projections to all of the system's \neigenmodes, but in order to get coupled with them during ``built'' up,\na large simulation time is\nrequired. Furthermore, the produced DOS is not a smooth function of frequency,\nstill consisting of a large collection of peaks, and thus being useful only\nas an indication for the existence of spectral gaps. \nIn our method, the underline symmetries of the modes are already in the \ninitial fields and so they couple easier with them. Also, the larger the \nsupercell, the smaller is its first Brillouin zone, and so the smaller the \nfrequencies we initialize through the various $\\vec{k}$. This is why \nwe can get smooth results even for very low frequencies.\nIn Figs.~1 and 2, we show the calculated density of states for the case\nof solid dielectric cylinders in air and cylindrical air holes\nin dielectric respectively, both for a square lattice arrangement, and\n for both polarizations. Along with\nthem we also plot the corresponding band structure as obtained with\nthe plane wave expansion method. Our study is going to be based on these two \nphotonic structures. \n\n\nThe second method we will use is the transfer matrix technique in\norder to obtain the transmission coefficient for a wave incident\nalong the $xy$ plane on a slab (or a slice) of the photonic material. \nThe slice is assumed uniform along the $z$ axis, and periodic along the $x$ \ndirection through\napplication of periodic boundary conditions, while in the $y$ direction\nit has a finite width $L$.\nIn this method one first constructs the transmitted waves at one side\nof the slice and then integrates numerically the time-independent \nMaxwell's equations to the other side. There, the waves are projected\ninto incident and reflected waves, and so a value for the transmission\ncoefficient $T$ can be obtained. Here, we are interested in the \nwave localization in disordered photonic band gap materials, and in \nparticularly in the localization length $\\ell \\sim -2L/\\ln T$.\n\nA few remarks about the results of this\nmethod are in order. Waves with different\nincidence directions will have different reflection and transmission\ncoefficients, so if one is looking for an average transmission, \nall directions should be included. It is shown, however, that there\nis also a large dependence on the surface plane along which the\nstructure is cut. More specifically, a wave normally incident on \na (1,0) surface will have different transmission characteristics\nthan a wave incident with 45$^{o}$ on a (1,1) surface. This is\nbecause certain modes can not always get coupled with the incident wave.\nOne should then also average for the two different surface cuts,\notherwise it will not be a true average. This is shown in Figs.~3 and 4 \nwhere we plot the (1,0) and the (1,1) cuts, each with both incidence \ndirections (normal and 45$^{o}$ with respect to the surface) averaged.\nWe see that taken individually, none of them corresponds to the\ntrue gaps as shown in Figs.~1 and 2, but rather, to wider and generally\ndisplaced gaps. For example, in the $E_z$ polarization case in the \nfirst spectral gap (Figs. 3a and 4a), with the (1,0) cut, the incident waves \nfail to couple with the the {\\bf M} modes of the first band, while \nwith the (1,1) cut, the incident waves fail to couple with \nthe {\\bf X} modes of the second band. \n\nThis is expected to be lifted once disorder is introduced into our\nsystem, since the sense of direction will be somehow lost. \n Disorder can be introduced as a random displacement, a random\nchange in the radius, or, a random change in the dielectric constant\nof the cylinders. It is not clear however what amount of disorder\nwould be needed for this. We repeated the calculations for small enough \namounts of disorder so that the spectral gaps, as found from the\nFDTD method, remain almost unchanged, for all three different \ndisorder mechanisms. As seen in Figs.~3 and 4, indeed, in some cases the\ncoupling is achieved. For example, for the first gap in the \n$E_z$-polarization case, with the (1,0) cut, the {\\bf M} modes of the \nfirst band are now coupled\nwith the incident waves and appear in the transmission diagram. These\ncould be easily mistaken for disorder-induced localized states\nentering the gap, but they are not, since for the values of disorder used, \nthe first gap is virtually unchanged. On the other hand, with the (1,1)\ncut, the coupling to {\\bf X} modes of the second band is not yet achieved, \nstill yielding a wrong picture\nfor the gap. Increasing the disorder further will eventually destroy\nany sense of direction and there will be no distinction between the\ntwo cases. Figs.~3 and 4 will be useful as a guide of which results can\nbe trusted and which can not, if one uses only one surface cut and\nsmall values of disorder. As a general rule, we can deduce that the \n(1,0) cut should be used for the $E_z$ polarization case, while the\n(1,1) cut would be better for the $H_z$ polarization case.\n\n\n\\section{Results and discussion}\n\nWe first looked into the spectral gaps' dependence on disorder\nusing the FDTD spectral method. Our system consisted of a 8$\\times$8\nsupercell, each cell discretized into a 32$\\times$32 grid. We studied \ntwo systems: a square lattice array of solid cylinders, with dielectric \nconstant $\\epsilon_a$=10, in air ($\\epsilon_b$=1) with a filling\nratio $f$=0.28\\%, and a square lattice array of air cylinders ($\\epsilon_a$=1)\nin dielectric material $\\epsilon_b$=10, with air filling ratio\n$f$=0.71\\%, as described in Figs.~1 and 2. We divided the \nsupercell's first Brillouin zone into 10$\\times$10 grid, which for\nthe irreducible part yields 66 different $\\vec{k}$ points. \nFor each particular disorder realization (i.e. disorder type) \nand disorder strength, \nwe run the simulation for all these 66 $\\vec{k}$'s. At each $\\vec{k}$\nhowever we use a different disordered configuration \n(i.e. a different seed in the random number generator), and so a \nlarge statistical sample is automatically included in our result.\nIn each case the effective disorder is measured by the rms error\nof the dielectric constant $<\\epsilon>$, which is defined as \n\\cite{chan,sigalas}\n\\begin{eqnarray}\n\\label{disord_eq4}\n\\epsilon^2=\\frac{1}{N}\\sum_{i=1}^{N}(\\epsilon_i^d-\\epsilon_i^p)^2,\n\\end{eqnarray}\nwhere the sum goes over all $N=8\\times8\\times32\\times32=65536$ grid points,\n$\\epsilon_i^d$ and $\\epsilon_i^p$ are the dielectric constants at site\n$i$ in the disordered and periodic case respectively, and $< \\dots >$\nmeans the average over different configurations (different $\\vec{k}$'s\nin our case). In both settings (dielectric cylinders in air and \n{\\em vise versa}) the filling ratio of the high dielectric material \nis similar, and so $<\\epsilon>$ is expected to have the \nsame meaning and weight. \n\n\nFour different disorder realizations are studied: 1) disorder in\nposition, without though allowing any cylinders to overlap\nwith each other, 2) disorder in position allowing \ncylinder overlapping to occur, 3) disorder in radius, and 4) \ndisorder in dielectric constant (the last one only in the solid\ncylinder case). For each different realization we consider various \ndisorder strengths, and thus different effective disorders \n$<\\epsilon>$,\nfor which we record the upper \nand lower gap edges for the first two photonic band gaps (if they exist).\nResults are summarized in Figs.~5 and 6, for the solid and air\ncylinder cases respectively. We note that \n the $E_z$ polarization case for the solid cylinders is very different from\nall other cases: the gaps survive very large amounts of positional\ndisorder, especially if no overlaps are allowed. In fact, once the\ndisorder becomes large enough for overlaps to be possible,\nthe gap quickly closes, as shown in Fig.~5. The actual DOS graphs \nfor the two different realizations of the positional disorder \nare shown in Fig.~7, for three different values of the effective\ndisorder. On the other hand, if the disorder is of the third or fourth \nkind, the gaps \nclose very quickly, even for modest values of the effective disorder.\n\nThe picture is very different in the other cases, as seen in Fig.~6.\nThe effect of the positional disorder is the same, independent\nof whether overlaps are allowed or not. This is most clearly seen in Fig.~8, \nwhere the actual DOS graphs are plotted for the air\ncylinder case for both polarizations, and for both positional disorder\nrealizations. Allowing the air cylinders to overlap,\nthough, means that the connectivity of the background material will break. \nOur results, thus, indicate that there is no connection between the \nconnectivity of the background material and the formation of the spectral\ngaps in this 2D case.\n Most importantly, however, we note that the disorder in radius \nhas a similar effect with that of the positional disorder in closing the gaps. \nIn fact, it is also similar to the effect of the disorder in radius for \nthe $E_z$-solid-cylinder case. \nSo, in the case of air cylinders, the type of the disorder that is \nintroduced into the system does not play a significant role, but rather,\nit is only the effective disorder (measured through the dielectric constant's\nerror function) that determines the effect on the spectral gaps.\nOn the other hand, for the $E_z$-solid-cylinder case, the type of disorder\nplays a profound role: if the ``shape'' of the individual scatterer is\npreserved, the gaps can sustain large amounts of disorder, while if it is not\npreserved, the gaps collapse in a manner similar to the air cylinder \ncase.\n\nWe next go over the localization length results, which were obtained with\nthe transfer matrix technique. Here, our system consisted of a 3$\\times$7\nsupercell (3 along the $x$ axis), with each cell discretized into a \n18$\\times$18 grid \n(a small supercell\nwas used in order to ease the computation burden). In the $x$\ndirection we applied periodic boundary conditions, while in the $y$\ndirection the supercell was repeated 4 times, to provide a total length $L$\nfor the slab of $L$=28 unit cells. The structures studied are exactly the\nsame as described before. The lattice was cut along one only symmetry\ndirection, the (1,0), since for large disorders we expect all ``hidden''\nmodes to be coupled with the incident wave (in any case, we know from \nFigs.~3 and 4 which results can be completely trusted and which can\nnot). For each disorder realization and strength, we used 11 different\n$\\vec{k}$ values uniformly distributed between normal and 45$^o$ angle\nincidence, and for each $\\vec{k}$ we used a different disordered \nconfiguration, so these will constitute our statistical sample.\nFor each $\\vec{k}$ we find the minimum transmission coefficient\ninside each gap, from which we find the minimum localization length,\n and then average over all $\\vec{k}$'s, \nie. $\\ell\\sim-2L/<\\ln T>$ (in the periodic case we first averaged over $T$\nin order to correctly account for different propagation directions,\nbut in the highly disordered case it is not so much important any more, \nand so we just average over the localization lengths). \n\n\nOur results are shown in Figs.~9 and 10 (because of the small statistical\nsample and the small supercell used, the data points appear very\n``noisy'', especially for large disorders). We note here, as well, the distinct\ndifference between the $E_z$-solid-cylinder case for positional\ndisorder and all other cases. Especially for the first \nspectral gap, the localization length not only remains\nunaffected by the disorder, but it even decreases (this is not\nan artifact of the averaging procedure). The first\nconclusion from this, is that the mechanisms responsible\nfor the gap formation in this case are unaffected by the presence \nof positional disorder, and so they are definitely not macroscopic \n(long-range) in nature.\nThe fact that the localization length decreases, is attributed to the \ncoupling of more [1,1] symmetry modes with the incident wave \nas the disorder increases (they provide a smaller $\\ell$ to\nthe average, as seen in Fig.~3a). This decrease should not be mistaken\nfor additional localization induced by the disorder (the\nclassical analog of Anderson localization in electrons), since the latter \nis macroscopic in nature, and does not apply for strongly \nlocalized waves.\nThe decrease in the localization length continues until a fairly\nlarge disorder value, and then it increases to a saturation value\n(the dielectric error function can reach only up to some value for\npositional disorder).\nThis saturation value is higher for the case where overlaps are allowed,\nbut still is very small compared to other cases, so waves remain\nstrongly localized.\n\n\nAll other cases, on the other hand, show a common pattern of \nbehavior: photon states become quickly de-localized with increasing\ndisorder. The localization length is increased, until the point where\nthe localization induced by the disorder becomes dominant. After \nthis it starts decreasing, until finally it reaches some saturation point.\nNote also that there is an almost quantitative agreement between \nsome cases that was not really expected, eg. \nfor the disorder in radius in the first gap with the wave $E_z$-polarized,\nfor both lattice settings, as seen in Figs.~9a and 10a. Only the \ncase of disorder in the dielectric constant seems to deviate,\nhaving very quickly a very large effect, with the localization length \n directly saturating to some constant value. So, for air\ncylinders in dielectric with any type of disorder, and for the \n$E_z$-solid-cylinder case with disorder that does not preserve the\nscatterer's ``shape'', the behavior under disorder is similar.\n\nAll these results can be understood if we adopt two different ``pictures'' for \nthe photon states, depending on which is the dominant \n mechanism that is responsible \nfor the formation of the spectral gaps in each case. The first is \n the ``nearly free'' photon\npicture, in which the gap forming mechanism is the non-resonant \nmacroscopic Bragg-like\nmultiple scattering, while the second is the \n``strongly localized'' photon picture, in which the gap forming \nmechanism is the microscopic (short-range) excitation of single scatterer Mie \nresonances. \n\n\nSharp Mie resonances appear only for the solid cylinder \ncase, and they can be thought as analogous to\nthe atomic orbitals in semiconductors.\nUsing this analogy, a tight-binding model, based on a linear \ncombination of Mie resonances, was recently developed for the photonic\nstates in the $E_z$-solid-cylinder case \\cite{lidorikis}. \nBut if a tight-binding model\ncan give a satisfactory description of the photonic states, then it is\nexpected that certain behavioral patterns found in semiconductors \nshould apply in our case too. So, positional disorder should have\na small effect on the gaps, in a similar way the energy gap survives\nin amorphous silicon. Also, changing the scatterer should have a \nsimilar effect as changing the atoms in the semiconductor, yielding \na large amount of impurity modes that quickly destroys the gap. \nThis pattern is definitely confirmed here for the $E_z$-solid-cylinder case.\nIn this case, multiple scattering and interference can \nonly help to make the gaps wider, but are definitely not decisive on \nthe existence of a gap. \n\nFor the macroscopic Bragg-like multiple scattering mechanism, \nthe lattice periodicity is a very important factor for the existence \nof a spectral gap. If it is destroyed, then coherence in the backscattered\nwaves will be destroyed, and so will the spectral gaps. It is of small \nconsequence the exact way that the periodicity is destroyed, and so \ndifferent disorder realizations will have similar effects. Also, since\nthe gaps close more easily, it will be easier to observe the localization\ninduced on the waves by the disorder itself, ie. the classical \nanalog of Anderson localization in electrons. All these are recognized\nin the case of air cylinders in dielectric. \n\nFinally, in the $H_z$-solid-cylinder case, there were no gaps to\nbegin with, and so we can have no results about it. However, sharp\nMie resonances appear for this case as well, and if their excitation\nwas the dominant scattering mechanism, a gap would be expected here as well.\nThe difference with the $E_z$ is that the former is described by a\nvector wave equation, while the latter by a scalar one (and thus\ncloser to the electronic case). The form of the wave equation must, \nthen, be an important factor\nin determining the relative strength of the two gap forming mechanisms.\n\n\\section{Conclusions}\nWe have shown that several results in periodic and random photonic\nband gap materials can be understood in terms of two distinct photonic\nstates: (a) The ``local'' states, based on a single scatterer Mie\nresonance, with the multiple scattering playing a minor role; these states\nare more conveniently described in terms of an LCAO-type of approach and\nare the analog of the $d$-states in transition metals. ``Local'' \nphotonic states appear in the case of high dielectric cylinders \nsurrounded by a low-dielectric host and for $E$-polarized waves.\n(b) The ``nearly free'' photonic states, where Bragg-like multiple \nscattering is the dominant mechanism responsible for their appearance;\nthese states are more conveniently described in terms of a \npseudopotential-type of approach and are the analog of $s$ (or $p$)\nstates in simple metals.\n\nEach type of photonic states responds differently to the presence of \ndisorder: For the ``local'' states case, the gap is robust as the \nperiodicity is destroyed, and it is hardly affected by the disorder\nas long as the identity of each individual scatterer is preserved;\nhowever, if the shape, or other characteristics influencing the scattering\ncross section of each individual scatterer, is altered by disorder, the \ngap tends to disappear. On the other hand, for the ``nearly free''\nstates case, the gap is very sensitive and tends to disappear easily\nas the periodicity is destroyed. \n\n\n\\section{Acknowledgments} \nAmes Laboratory is operated for the U. S. Department of Energy by Iowa \nState University under contract No. W-7405-ENG-82. This work was supported\nby the Director of Energy Research office of Basic Energy Science and \nAdvanced Energy Projects. It was also supported by the Army Research Office, \na E.U. grant, and a NATO\ngrant.\n\n\n\\begin{references}\n%\\begin{thebibliography}{99}\n%\\begin{thebibnopage}{99}\n%\\addcontentsline{toc}{section}{Bibliography}\n\n\\bibitem{soukoulis} For excellent reviews on photonic band gap materials\nsee the proceedings of the NATO ARW, {\\em Photonic Band Gaps and \nLocalization}, ed. by C.~M.~Soukoulis, (Plenum, N.Y., 1993); \n{\\em Photonic Band Gap Materials}, ed. by C.~M.~Soukoulis, NATO ASI, \nSeries E, vol. 315. \n\n\\bibitem{joannopoulos} J.~D.~Joannopoulos, R.~D.~Meade, and J.~N.~Winn \n{\\em Photonic Crystals} (Princeton University Press, Princeton, 1995).\n\n\\bibitem{fan} S.~Fan, P.~R.~Villeneuve, \nand J.~D.~Joannopoulos, J. Appl. Phys. {\\bf 78}, 1415 (1995).\n\n\\bibitem{sigalas} M.~M.~Sigalas, C.~M.~Soukoulis, C.~T.~Chan, and D.~Turner,\nPhys. Rev. B {\\bf 53}, 8340 (1996); M.~M.~Sigalas, C.~M.~Soukoulis, \nC.~T.~Chan, and K.~M.~Ho \nin {\\em Photonic Band Gap Materials}, p. 563, ed. by C.~M.~Soukoulis \n(Kluwer, Dordrecht, 1996).\n\n\\bibitem{ho} S.~Datta, C.~T.~Chan, K.~M.~Ho, C.~M.~Soukoulis, and\nE.~N.~Economou, in {\\em Photonic Band Gaps and \nLocalization} [Ref. [1](a)], p. 289.\n\n\\bibitem{kafesaki} M.~Kafesaki, E.~N.~Economou, and M.~M.~Sigalas,\nin {\\em Photonic Band Gap Materials} [Ref. [1](c)], p. 143.\n\n\\bibitem{lidorikis} E.~Lidorikis, M.~M.~Sigalas, E.~N.~Economou, and\nC.~M.~Soukoulis, Phys. Rev. Lett. {\\bf 81}, 1405 (1998).\n\n\n\\bibitem{mie} G.~Mie, Ann. Phys. (Leipzig) {\\bf 25}, 377 (1908);\nC.~F.~Bohren and D.~R.~Huffman, {\\em Absorption and Scattering of \nLight by Small Particles} (J.~Wiley, New York, 1983). \n\n\\bibitem{yee} K.~S.~Yee, IEEE Trans. Antennas and Propagation {\\bf 14}, \n302 (1966).\n\n\\bibitem{taflove} Allen Taflove, {\\em Computational Electrodynamics: The \nFinite-Difference Time-Domain Method} (Artech House, Boston, 1995).\n\n\\bibitem{pendry} J.~B.~Pendry and A.~MacKinnon, Phys. Rev. Lett. {\\bf 69},\n2772 (1992).\n\n\n\\bibitem{chan} C.~T.~Chan, Q.~L.~Yu, and K.~M.~Ho, Phys. Rev. B \n{\\bf 51}, 16635 (1995).\n\n\n\\bibitem{sakoda} Kazuaki Sakoda and Hitomi Shiroma, Phys. Rev. B {\\bf 56},\n4830 (1997).\n\n\\end{references}\n%\\end{thebibliography}\n%\\end{thebibnopage}\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig1.eps,width=16cm,height=16cm,angle=270}\n\\caption{Band structure (obtained with a plane wave method) and density\nof states (obtained with the FDTD spectral method) for a two-dimensional\nsquare lattice array of dielectric cylinders $\\epsilon_a$=10 in\nair $\\epsilon_b$=1, with a filling ratio $f\\simeq 28$\\%.}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig2.eps,width=16cm,height=16cm,angle=270}\n\\caption{Band structure (obtained with a plane wave method) and density\nof states (obtained with the FDTD spectral method) for a two-dimensional\nsquare lattice array of air cylinders $\\epsilon_a$=1 in\ndielectric $\\epsilon_b$=10, with air filling ratio $f\\simeq 71$\\%.}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig3.eps,width=16cm,height=16cm,angle=270}\n\\caption{Transmission coefficient for the periodic, and weakly disorder,\nsystem described in Fig.~1 (obtained with the transfer matrix technique).\nCalculations are for two different surfaces along which the \nsample is cut. Effective disorders used (look Eq.~(\\ref{disord_eq4})):\nin position$\\sim$1.3, in radius$\\sim$0.5, and in dielectric$\\sim$0.3. \n}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig4.eps,width=16cm,height=16cm,angle=270}\n\\caption{Transmission coefficient for the periodic, and weakly disorder,\nsystem described in Fig.~2 (obtained with the transfer matrix technique).\nCalculations are for two different surfaces along which the \nsample is cut. Effective disorders used:\nin position$\\sim$0.35, in radius$\\sim$0.25}\n%\\label{fjkdf}\n\\end{figure}\n\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig5.eps,width=16cm,height=16cm,angle=270}\n\\caption{The edges of the photonic band gaps as a function of the\neffective disorder $<disorder> \\equiv <\\epsilon>$\n(as was defined in Eq.~(\\ref{disord_eq4})), \nfor the system described in Fig.~1. Four different disorder \nrealizations are studied.}\n%\\label{fjkdf}\n\\end{figure}\n\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig6.eps,width=16cm,height=16cm,angle=270}\n\\caption{The edges of the photonic band gaps as a function of the\neffective disorder $<disorder> \\equiv < \\epsilon>$\n(as was defined in Eq.~(\\ref{disord_eq4})), \nfor the system described in Fig.~2. Three different disorder \nrealizations are studied.}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig7.eps,width=16cm,height=16cm,angle=0}\n\\caption{The density of states for the system of Fig.~1 with the $E_z$\npolarization, for three different positional disorder strengths. The solid\nline is when no scatterer overlaps are allowed, while the dotted line is\nwhen scatterer overlaps are allowed. $<overlaps>$ is the average number \nof overlapping cylinders.}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig8.eps,width=16cm,height=16cm,angle=0}\n\\caption{The density of states for the system of Fig.~2 for both field \npolarizations, for two different positional disorder strengths. The solid\nline is when no scatterer overlaps are allowed, while the dotted line is\nwhen scatterer overlaps are allowed.}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig9_b.eps,width=16cm,height=16cm,angle=270}\n\\caption{The localization length as a function of the effective disorder\nfor the system described in Fig.~1 with the $E_z$ polarization, for four\ndifferent disorder realizations.}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=disordpapar_fig10_b.eps,width=16cm,height=16cm,angle=270}\n\\caption{The localization length as a function of the effective disorder\nfor the system described in Fig.~2 for both field polarizations, for two\ndifferent disorder realizations.}\n%\\label{fjkdf}\n\\end{figure}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002101.extracted_bib",
"string": "\\begin{thebibliography}{99}\n%\\begin{thebibnopage}{99}\n%\\addcontentsline{toc}{section}{Bibliography}\n\n\\bibitem{soukoulis} For excellent reviews on photonic band gap materials\nsee the proceedings of the NATO ARW, {\\em Photonic Band Gaps and \nLocalization}, ed. by C.~M.~Soukoulis, (Plenum, N.Y., 1993); \n{\\em Photonic Band Gap Materials}, ed. by C.~M.~Soukoulis, NATO ASI, \nSeries E, vol. 315. \n\n\\bibitem{joannopoulos} J.~D.~Joannopoulos, R.~D.~Meade, and J.~N.~Winn \n{\\em Photonic Crystals} (Princeton University Press, Princeton, 1995).\n\n\\bibitem{fan} S.~Fan, P.~R.~Villeneuve, \nand J.~D.~Joannopoulos, J. Appl. Phys. {\\bf 78}, 1415 (1995).\n\n\\bibitem{sigalas} M.~M.~Sigalas, C.~M.~Soukoulis, C.~T.~Chan, and D.~Turner,\nPhys. Rev. B {\\bf 53}, 8340 (1996); M.~M.~Sigalas, C.~M.~Soukoulis, \nC.~T.~Chan, and K.~M.~Ho \nin {\\em Photonic Band Gap Materials}, p. 563, ed. by C.~M.~Soukoulis \n(Kluwer, Dordrecht, 1996).\n\n\\bibitem{ho} S.~Datta, C.~T.~Chan, K.~M.~Ho, C.~M.~Soukoulis, and\nE.~N.~Economou, in {\\em Photonic Band Gaps and \nLocalization} [Ref. [1](a)], p. 289.\n\n\\bibitem{kafesaki} M.~Kafesaki, E.~N.~Economou, and M.~M.~Sigalas,\nin {\\em Photonic Band Gap Materials} [Ref. [1](c)], p. 143.\n\n\\bibitem{lidorikis} E.~Lidorikis, M.~M.~Sigalas, E.~N.~Economou, and\nC.~M.~Soukoulis, Phys. Rev. Lett. {\\bf 81}, 1405 (1998).\n\n\n\\bibitem{mie} G.~Mie, Ann. Phys. (Leipzig) {\\bf 25}, 377 (1908);\nC.~F.~Bohren and D.~R.~Huffman, {\\em Absorption and Scattering of \nLight by Small Particles} (J.~Wiley, New York, 1983). \n\n\\bibitem{yee} K.~S.~Yee, IEEE Trans. Antennas and Propagation {\\bf 14}, \n302 (1966).\n\n\\bibitem{taflove} Allen Taflove, {\\em Computational Electrodynamics: The \nFinite-Difference Time-Domain Method} (Artech House, Boston, 1995).\n\n\\bibitem{pendry} J.~B.~Pendry and A.~MacKinnon, Phys. Rev. Lett. {\\bf 69},\n2772 (1992).\n\n\n\\bibitem{chan} C.~T.~Chan, Q.~L.~Yu, and K.~M.~Ho, Phys. Rev. B \n{\\bf 51}, 16635 (1995).\n\n\n\\bibitem{sakoda} Kazuaki Sakoda and Hitomi Shiroma, Phys. Rev. B {\\bf 56},\n4830 (1997).\n\n\\end{references}\n%\\end{thebibliography}"
}
] |
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cond-mat0002102
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Packing of Compressible Granular Materials
|
[
{
"author": "Hern\\'an A. Makse"
},
{
"author": "David L. Johnson"
},
{
"author": "and Lawrence M. Schwartz"
}
] |
3D Computer simulations and experiments are employed to study random packings of compressible spherical grains under external confining stress. Of particular interest is the rigid ball limit, which we describe as a continuous transition in which the applied stress vanishes as $(\phi-\phi_c)^\beta$, where $\phi$ is the (solid phase) volume density. This transition coincides with the onset of shear rigidity. The value of $\phi_c$ depends, for example, on whether the grains interact via only normal forces (giving rise to random close packings) or by a combination of normal and friction generated transverse forces (producing random loose packings). In both cases, near the transition, the system's response is controlled by localized force chains. As the stress increases, we characterize the system's evolution in terms of (1) the participation number, (2) the average force distribution, and (3) visualization techniques.
|
[
{
"name": "fc.tex",
"string": "%\\documentstyle[aps,multicol,epsf,rotate]{revtex}\n%\\documentstyle[prl,twocolumn,aps]{revtex}\n\\documentstyle[aps,multicol,epsf]{revtex}\n%\\documentstyle[aps]{revtex}\n%\\documentstyle[aps,preprint,epsf]{revtex}\n\n\\begin{document}\n\\draft\n\n\\title{Packing of Compressible Granular Materials} \n\n\\author{Hern\\'an A. Makse, David L.\nJohnson, and Lawrence M. Schwartz}\n\n\\address{ Schlumberger-Doll Research, Old Quarry Road, Ridgefield, CT\n06877} \n\\date{\\today} \n\\maketitle\n\\begin{abstract}\n\n3D Computer simulations and experiments are employed to study random\npackings of compressible spherical grains under external confining\nstress. Of particular interest is the rigid ball limit, which we\ndescribe as a continuous transition in which the applied stress\nvanishes as $(\\phi-\\phi_c)^\\beta$, where $\\phi$ is the (solid phase)\nvolume density. This transition coincides with the onset of shear\nrigidity. The value of $\\phi_c$ depends, for example, on whether the\ngrains interact via only normal forces (giving rise to random close\npackings) or by a combination of normal and friction generated\ntransverse forces (producing random loose packings). In both cases,\nnear the transition, the system's response is controlled by localized\nforce chains. As the stress increases, we characterize the system's\nevolution in terms of (1) the participation number, (2) the average\nforce distribution, and (3) visualization techniques.\n\\end{abstract}\n\\pacs{PACS: 81.06.Rm}\n%81.06.Rm: porous materials, granular materials\n\n%\\newpage\n\n\\begin{multicols}{2}\n\nDense packings of spherical particles are an important starting point\nfor the study of physical systems as diverse as simple liquids,\nmetallic glasses, colloidal suspensions, biological systems, and\ngranular matter \\cite{bernal,finney,berryman,ibm,torquato}. In the\ncase of liquids and glasses, finite temperature molecular dynamics\n(MD) studies of hard sphere models have been particularly important.\nHere one finds a first order liquid-solid phase transition as the\nsolid phase volume fraction, $\\phi$, increases. Above the freezing\npoint, a metastable disordered state can persist until\n$\\phi\\to\\phi_{\\mbox{\\scriptsize RCP}}^-$ \\cite{torquato}, where\n$\\phi_{\\mbox{\\scriptsize RCP}}$ is the density of random close packing\n(RCP)--- the densest possible random packing of hard spheres.\n\n\n\nThis Letter is concerned with the non-linear elastic properties of\ngranular packings. Unlike glasses and amorphous solids, this is a\nzero temperature system in which the interparticle forces are both\nnon-linear, and path (i.e., history) dependent. [Because these forces\nare purely repulsive, mechanical stability is achieved only by imposing\nexternal stress.] The structure of packing depends in detail on the\nforces acting between the grains during rearrangement of grains;\nindeed, different rearrangement protocols can lead to either RCP or\nrandom loose packed (RLP) systems.\n\n\nIn the conventional continuum approach to this problem, the granular\nmaterial is treated as an elasto-plastic medium \\cite{nedderman}. \nHowever, this\napproach has been challenged by recent authors \\cite{bouchaud}\nwho argue that granular\npackings represents a new kind of {\\it fragile} matter and that more\nexotic methods, e.g., the fixed principal axis ansatz, are required to\ndescribe their internal stress distributions. These new continuum\nmethods are complemented by microscopic studies based on either \ncontact dynamics simulations of {\\it rigid} spheres or statistical\nmodels, such as the q-model, which makes no attempt to take account of\nthe character of the inter-grain forces \\cite{chicago,radjai}.\n\nIn our view, a proper description of the stress state in granular\nsystems must take account of the fact that the individual grains are\ndeformable. We report here on a 3D study of deformable spheres\ninteracting via Hertz-Mindlin contact forces. Our simulations cover\nfour decades in the applied pressure and allow us to understand the\nregimes in which the different theoretical approaches described above\nare valid. Since the grains in our simulations are deformable, the\nvolume fraction can be increased above the hard sphere limit and we\nare able to study the approach to the RCP and RLP\nstates from this realistic perspective.\n Within this framework, the rigid\ngrain limit is described as a continuous phase transition where the\norder parameter is the applied stress, $\\sigma$, which vanishes\ncontinuously as $(\\phi-\\phi_c)^\\beta$. Here $\\phi_c$ is the critical\nvolume density, and $\\beta$ is the corresponding critical exponent.\nWe emphasize that the fragile state corresponding\nto rigid grains is reached by looking at the limit $\\phi\\to\\phi^+_c$ from \nabove.\n\n%We describe the rigid ball limit, as a continuous phase transition\n%where the order parameter is the stress, $\\sigma$, which vanishes\n%continuously as $(\\phi-\\phi_c)^\\beta$, where $\\phi_c$ is the critical\n%volume density, and $\\beta$ is the corresponding critical exponent.\n%Thus, we characterize the hard sphere limit\n%%state where deformations are negligible by\n%by looking at the limit $\\phi\\to\\phi^+_c$ from above.\n\n\nOf particular importance is the fact that $\\phi_c$ depends on the type\nof interaction between the grains. If the grains interact via normal\nforces only \\cite{bubbles}, they slide and rotate freely mimicking the\nrearrangements of grains during shaking in experiments\n\\cite{bernal,finney,berryman}. We then obtain the RCP value $\\phi_c=\n0.634(4)(\\approx\\phi_{\\mbox{\\scriptsize RCP}})$. By contrast, if the\ngrains interact by combined normal and friction generated transverse\nforces, we get RLP \\cite{ibm} at the critical point with\n$\\phi_c=0.6284(2)<\\phi_{\\mbox{\\scriptsize RCP}}$. The power-law\nexponents characterizing the approach to $\\phi_c$ are not universal\nand depend on the strength of friction generated shear forces.\n\n%In both cases we show that the system at the critical density is at\n%the minimal average coordination number, $Z_c$, required for\n%mechanical stability. \nOur results indicate that the transitions at both RCP or RLP are\ndriven by localized force chains. Near the critical\ndensity there is a percolative fragile structure which we characterize\nby the participation number (which measures localization of force\nchains), the probability distribution of forces, and also by\nvisualization techniques. \nA subset of our results are experimentally\nverified using carbon paper measurements to study force\ndistributions in the granular assembly. We also consider in some\ndetail the relationship between our work and recent experiments in\n2D Couette geometries \\cite{behringer}.\n\n%Part of these results are experimentally\n%verified using standard carbon paper experiments to study force\n%distributions in the granular assembly, and we also compare with\n%recent experiments in 2D Couette geometries relevant to the present\n%study \\cite{behringer}. Our results provide further test to which competing\n%theories of granular materials could be confronted to.\n\n\n\n\\begin{figure}\n\\centerline{\n\\hbox{ \n\\epsfxsize=4.cm\n\\epsfbox{fig1a.ps}\n\\epsfxsize=4.cm\n\\epsfbox{fig1b.ps}\n}}\n%\\vspace{1.cm}\n\\narrowtext\n\\caption{(a) Confining stress and (b)\naverage coordination number\nas a function of volume fraction for friction and frictionless balls.}\n\\label{pressure-Z-pi}\n\\end{figure}\n\n{\\it Numerical Simulations}: To better understand the behavior of real\ngranular materials, we perform granular dynamics simulations of\nunconsolidated packings of deformable spherical glass beads using the\nDiscrete Element Method developed by Cundall and Strack\n\\cite{cundall}. Unlike previous work on rigid grains, we work with a\nsystem of deformable elastic grains interacting via normal and\ntangential Hertz-Mindlin forces plus viscous dissipative forces\n\\cite{johnson}. The grains have shear modulus 29 GPa,\nPoisson's ratio 0.2 and radius 0.1 mm.\n%At each grain contact\n%the normal force is $f_n = \\frac{2}{3} ~ C_n R^{1/2}w^{3/2}$, and \n%tangential \n%force\n%is $\\Delta f_t = C_t (R w)^{1/2} \\Delta s$ \\cite{johnson}, where\n%the normal overlap is $2 w= (R_1+R_2) - |\\vec{x}_1 - \\vec{x}_2|\n%>0$, and $\\vec{x}_1$, $\\vec{x}_2$ are the center positions of the\n%contacting spheres. The shear displacement is\n%$s(t)=\\int_{t_0}^t v_s(t') dt'$, where $v_s$ is the relative shear\n%velocity between the spheres, and $t_0$ is the initial time when the\n%contact is established. $R_1$ and $R_2$ are the radii of the \n%two spheres and $R= 2 R_1 R_2/(R_1+R_2)$. The normal and\n%tangential prefactors, $C_n=4 G / (1-\\nu)$ and $C_t=8 G / (2-\\nu)$,\n%are specified in terms of the shear modulus $G$ and the Poisson's\n%ratio, $\\nu$, of the material from which the grains are made. We take\n%$G=29$ GPa and $\\nu = 0.2$, typical values for glass. In our simulations\n%half the grains had $R_1=0.105$ mm and the other half had \n%$R_2=0.095$ mm. An extra condition of sliding is also added. When the\n%transverse force exceeds the Coulomb threshold ($\\mu_c f_n$) the\n%contacting grains slide and $f_t = \\mu_c f_n$, where $\\mu_c$ is the\n%coefficient of friction between the spheres (typically $\\mu_c = 0.3$).\n%The normal force acts only in traction, $f_n \\equiv 0$ when $w<0$.\n\n\nOur simulations employ periodic boundary conditions and begin with a\ngas of 10000 non-overlapping spheres located at random positions in a\ncube 4 mm on a side. Generating a mechanically stable packing is not\na trivial task \\cite{torquato}. At the outset, a series of\nstrain-controlled isotropic compressions and expansions are applied\nuntil a volume fraction slightly below the critical density. At this\npoint the system is at zero pressure and zero coordination number. We\nthen compress along the $z$ direction,\n%holding the lateral system size fixed (uniaxial compression test),\nuntil the system equilibrates at a desired vertical stress $\\sigma$\n and a non-zero average coordination number $\\langle Z \\rangle$.\n%During the compression many grains (roughly \n%$20\\%$ of the contacts) are sliding. However, once the system reaches \n%static equilibrium, all the contacts satisfy $f_t<\\mu_c f_n$. \n\nFigure \\ref{pressure-Z-pi}a shows the behavior of the stress as a\nfunction of the volume fraction. We find that the pressure vanishes at\na critical $\\phi_c=0.6284(2)$. Although we cannot rule out a\ndiscontinuity in the pressure at $\\phi_c$--- as we could expect for a\nsystem of hard spheres--- our results indicates that the transition is\ncontinuous and the behavior of the pressure can be fitted to a power\nlaw form\n\\begin{equation}\n\\sigma \\sim (\\phi - \\phi_c)^{\\beta},\n\\label{p}\n\\end{equation}\nwhere $\\beta = 1.6(2)$. Our 3D results contrast with recent\nexperiments of slowly sheared grains in 2D Couette geometries\n\\cite{behringer} where a faster than exponential approach to $\\phi_c$\nwas found, while they agree qualitative with similar continuous\ntransition found in compressed emulsions and foams \\cite{bubbles}.\n\nFigure \\ref{pressure-Z-pi}b shows the behavior of the mean\ncoordination number, $ \\langle Z \\rangle$, as a function of $\\phi$.\nWe find\n\\begin{equation}\n \\langle Z \\rangle - Z_c \\sim (\\phi - \\phi_c)^{\\theta},\n\\label{z}\n\\end{equation}\nwhere $Z_c=4$ is a minimal coordination number, and $\\theta=0.29(5)$\nis a critical exponent. At criticality the system is very loose and\nfragile with a very low coordination number. The value of $Z_c$ can be\nunderstood in term of constraint arguments as discussed in\n\\cite{edwards}; in the rigid ball limit, for a disordered system with\nboth normal and transverse forces, we find $Z_c = D+1 = 4$\n\\cite{edwards}. As we compress the system more contacts are created,\nproviding more constraints so that the forces become overdetermined.\n%The grains start\n%to deform, and the fragile initial RCP structure gives way to a more\n%compa\n\nWe notice that $\\phi_c$ obtained for this system is considerably lower\nthan the best estimated value at RCP \\cite{berryman}, \n$\\phi_{\\mbox{\\scriptsize\nRCP}}=0.6366(4)$ obtained by Finney \n\\cite{finney} using ball bearings. This latter value is obtained by\ncarefully vibrating the system and letting the grains settle into the\nmost compact packing. Numerically, this is achieved by allowing the\ngrains reach the state of mechanical equilibrium interacting only via\nnormal forces. By removing the transverse forces, grains can slide\nfreely and find most compact packings than with transverse forces.\nNumerically we confirm this by equilibrating the system at zero\ntransverse force. The critical packing fraction found in\nthis way is $\\phi_c=0.634(4)$($ \\approx \\phi_{\\mbox{\\scriptsize RCP}}$\nwithin error bars). The stress behaves as in Eq. (\\ref{p}) but with a\ndifferent exponent $\\beta=2.0(2)$ (Fig. \\ref{pressure-Z-pi}a). At the\ncritical volume fraction the average coordination number is now $\nZ_c=6$ [and $\\theta=0.94(5)$, Fig. \\ref{pressure-Z-pi}b], which again\ncan be understood using constraint arguments which would give a\nminimal coordination number equal to 2D for frictionless rigid balls\n\\cite{edwards}.\n\n\n\\begin{figure}\n%\\centerline{ \\vbox{ \\hbox{\\epsfxsize=12.cm \\epsfbox{fig1a.ps} }\n%\\epsfxsize=12.cm \\epsfbox{fig1b.ps} }} \n\\centerline{ \n\\hbox{\n\\epsfxsize=4.cm \\epsfbox{fig2a.ps} \n\\epsfxsize=4.cm \\epsfbox{fig2b.ps} } }\n\\centerline{ \n\\epsfxsize=5.cm \\epsfbox{fig2c.ps} }\n\\narrowtext\n%\\vspace{1.cm}\n\\caption{Distribution of forces for different confining stresses $\\sigma$\nobtained (a) in\nthe numerical simulations of friction balls, and (b) in the carbon paper\nexperiments. The straight solid lines are fittings to\nexponential forms and the dashed lines are fittings to Gaussian forms.\nIn both graphs we shift down the distributions corresponding to the\ntwo larger stresses for clarity.\n(c) Participation number versus external stress for the same system as in \n(a).}\n\\label{distri}\n\\end{figure}\n\\vspace{-.5cm}\nWe conclude that the value $\\phi_c \\approx\n0.6288<\\phi_{\\mbox{\\scriptsize RCP}}$ found with transverse forces\ncorresponds to the RLP limit, experimentally achieved by pouring balls\nin a container but without allowing for further rearrangements \\cite{ibm}.\nExperimentally, stable loose \npackings with $\\phi$ as low as 0.60 have been\nfound \\cite{ibm}. In our simulations, $\\phi_c$ lower than $0.6288$ can\nbe obtained by increasing the strength of the tangential forces. This\nis in agreement with experiments of Scott and Kilgour \\cite{scott2}\nwho found that the maximum packing density of spheres decreases with\nthe surface roughness (friction) of the balls.\n\nWhile previous studies characterized RCP's and RLP's by using radial\ndistribution functions and Voronoi constructions \\cite{finney}, we\ntake a different approach which allow us to compare our results\ndirectly with recent work on force transmissions in granular matter.\n%Dense random packings have been characterized by their radial\n%distribution functions and also by using Voronoi constructions \n%\\cite{finney}. Here we take a different approach and study the \n%changes in the spatial distribution of interparticle\n%forces in the packing as $\\phi_c$ is approached.\nPrevious studies of granular media indicate that, for forces greater\nthan the average value, the distribution of inter-grain contact forces\nis exponential \\cite{chicago,radjai}.\n%, a result verified using scalar\n%models of force transmission in granular materials, such as the\n%``q-model'' \\cite{chicago}, and through numerical simulations of rigid\n%particles packs \\cite{radjai}. \nIn addition, photo-elastic\nvisualization experiments and simulations \\cite{force_chains,chicago}\nshow that contact forces are strongly localized along ``force chains''\nwhich carry most of the applied stress.\n%The continuum limit of \n%a vector version of the q-model,\n%with a particular constitutive relation \n% \\cite{bouchaud} (see\n%\\cite{savage} for alternative formulations)\n%between the stress components, gives rise to hyperbolic equations \n%for stress propagation whose characteristics are believed to describe\n%force chains. \nThe existence of force chains and exponential force distributions are\nthought to be intimately related.\n\nHere we analyze this scenario in the entire range of pressures: from\nthe $\\phi_c$ limit and up. Figure \\ref{distri}a shows the force\ndistribution obtained in the simulations with friction balls. At low\nstress, the distribution is exponential in agreement with previous\nexperiments and models.\n%Our results \n%indicate that this behavior extends to the full range of forces. There \n%is no indication of a power-law distribution for forces smaller than\n%$\\langle f \\rangle$ as suggested by other models \n%\\cite{chicago,radjai}. \nWhen the system is compressed further, we find a gradual transition to\na Gaussian force distribution. We observe a similar transition in our\nsimulations involving frictionless grains under isotropic compression.\nThis suggests that our results are generic, and do not depend,\nqualitatively, on the preparation history or on the existence of\nfriction generated transverse forces between the grains.\n%Of course, the precise location of the transition may depend on the system's \n%preparation. \n%For example, we see in Figs. \\ref{distri} that our simulations \n%show a crossover to Gaussian behavior for \n%stresses that are, in general, larger \n%than those found in the experiments. This may be due to differences \n%in the experimental and numerical set up protocols. [In the experiments \n%Janssen's effect \\cite{pgg} affects the total pressure at the bottom \n%of the bin while in the simulations, where we have periodic boundary\n%conditions and no rigid lateral walls, there is no corresponding effect.]\n%Moreover, we will see below indications that the transition to a Gaussian \n%behavior starts at $\\approx 2.1$MPa, in better agreement with the \n%experimental data.\n\n\nPhysically, we find that the transition from Gaussian to exponential\nforce distribution is driven by the localization of force chains as\nthe applied stress is decreased. In granular materials, with\nparticles of similar size, localization is induced by the disorder of\nthe packing arrangement. To quantify the degree of localization, we\nconsider the participation number $\\Pi$:\n\\begin{equation}\n\\Pi\\equiv\\left(M\\sum_{i=1}^M q_i^2\\right)^{-1} \\: \\: .\n\\label{pi}\n\\end{equation}\nHere $M$ is the number of contacts between the spheres, $ \\langle\nZ\\rangle =2 M / N$ is the average coordination number, and $N$ is the\nnumber of spheres. $q_i\\equiv {f_i}/{\\sum_{j=1}^{M}{f_j}}$, where\n$f_i$ is the magnitude of the total force at every contact. From the\ndefinition (\\ref{pi}), $\\Pi=1$ indicates a limiting state with a\nspatially homogeneous force distribution (${q_i}=1/M$, $\\forall i$).\nOn the other hand, in the limit of complete localization, $\\Pi\\approx\n1/M\\to 0$ and $M\\rightarrow\\infty$.\n\n\nFigure \\ref{distri}c shows our results for $\\Pi$ versus\n $\\sigma$. Clearly, the system is more localized at low stress than at\n high stress. Initially, the growth of $\\Pi$ is logarithmic,\n indicating a smooth delocalization transition. This behavior is seen\n up to $\\sigma\\approx$ 2.1 MPa, after which the participation number\n saturates to a higher value:\n\\begin{equation}\n\\begin{array}{rcll}\n\\Pi(\\sigma) & \\propto & \\log (\\sigma) & \\qquad[ \\mbox{$\\sigma<$2.1\nMPa}]\\\\ \\Pi(\\sigma) & \\approx & 0.62 & \\qquad[ \\mbox{$\\sigma>$2.1\nMPa}]\\\\\n\\end{array}\n\\label{62}\n\\end{equation}\nThis behavior suggests that, near the critical density, the forces are\nlocalized in force chains sparsely distributed in space. As the\napplied stress is increased, the force chains become more dense, and\nare thus distributed more homogeneously.\n\n\nHow might we expect the participation number to depend upon other\nsystem parameters when the forces are transmitted principally by force\nchains? In an idealized situation, the system has $N_{FC}$ force\nchains, each of which has $N_z$ spheres. Each sphere in a force chain\nhas two major load bearing contacts, which loads must be approximately\nequal. In the lateral directions, roughly four weak contacts are\nrequired for stability. These contacts carry a fraction $\\alpha<1$ of\nthe major vertical load. All other contacts have $f_i \\approx 0$.\nUnder these assumptions,\n\n\\begin{equation}\n\\Pi = \\frac{2}{ \\langle Z \\rangle}\\frac{(1+2\\alpha)^2}{(1+2\\alpha^2)}\n\\frac{ N_{FC} N_z}{ N} \\leq \\frac{2}{ \\langle Z \\rangle\n}\\frac{(1+2\\alpha)^2}{(1+2\\alpha^2)} \\:\\:\\:.\n\\label{pifc}\n\\end{equation}\nThe last inequality becomes an equality {\\it iff} all the balls are in\nforce chains. From our simulations at large pressure $\\alpha\\approx\n2/5$, so at $\\langle Z \\rangle \\approx 8$, $\\Pi \\approx 0.62$, which\nimplies that the system has been completely homogenized.\n%that at $\\sigma$ = 100 MPa, $\\Pi = 0.62 = 4/ \\langle Z \\rangle$,\nAlthough Eq. (\\ref{pifc}) is oversimplified, we believe that the\nchange in slope in Fig. \\ref{distri}c is emblematic of the complete\ndisappearance of well-separated chains.\n\n\\begin{figure}\n\\centerline{\n\\hbox{\n\\epsfxsize=5.cm\n%\\epsfxsize=5cm\n\\epsfbox{fig3a.ps}\n\\epsfxsize=5.cm\n%\\epsfxsize=5cm\n\\epsfbox{fig3b.ps}\n}\n}\n%\\vspace{.2cm}\n\\narrowtext\n\\caption{Example of percolating\nforce chains for the same system as in Fig. \\protect\\ref{distri}a: \n(a) near $\\phi_c$ and (b) away from $\\phi_c$ at large\nconfining\nstress. The color code of the chains is according to the total force\nin N carried by the\nchains.}\n\\label{chains}\n\\end{figure}\n\n\n\nThe localization transition can be understood by studying the behavior\nof the forces during the loading of the sample. Clearly, visualizing\nforces in 3D systems is a complicated task. In order to exhibit the\nrigid structure from the system we visually examine all the forces\nlarger than the average force; these carry most of the stress of the\nsystem. The forces smaller than the average are thought to act as an\ninterstitial subset of the system providing stability to the buckling\nof force chains \\cite{force_chains,radjai}. We look for force chains\nby starting from a sphere at the top of the system, and following the\npath of maximum contact force at every grain. We look only for the\npaths which percolate, i.e., stress paths spanning the sample from the\ntop to the bottom. In Fig. \\ref{chains} we show the evolution of the\nforce chains thus obtained for two extreme cases of confining stress.\nWe clearly see localization at low confining stress: the force-bearing\nnetwork is concentrated in a few percolating chains. At this point\nthe grains are weakly deformed but still well connected. We expect a\nbroad force distribution, as found in this and previous studies. As\nwe compress further, new contacts are created and the density of force\nchains increases. This in turn gives rise to a more homogeneous\nspatial distribution of forces, which is consistent with the crossover\nto a narrow Gaussian distribution.\n\n{\\it Experiments}: Some of the predictions of our numerical study can\nbe tested using standard carbon paper experiments \\cite{chicago},\nwhich have been used successfully in the past to study the force\nfluctuations in granular packings. A Plexiglas cylinder, 5 cm\ndiameter and varying height (from 3 cm to 5 cm), is filled with\nspherical glass beads of diameter $0.8\\pm0.05$ mm. At the bottom of\nthe container we place a carbon paper with white glossy paper\nunderneath. We close the system with two pistons and we allow the top\npiston to slide freely in the vertical direction, while the bottom\npiston is held fixed to the cylinder. The system is compressed in the\nvertical direction with an Inktron$^{TM}$ press and the beads at the\nbottom of the cylinder left marks on the glossy paper. We digitize\nthis pattern and calculate the ``darkness'' \\cite{chicago} of every\nmark on the paper. To calibrate the relationship between the marks\nand the force, a known weight is placed on top of a single bead; for\nthe forces of interest in this study (i.e., from $\\approx 0.05$N to 6\nN), there is a roughly linear relation between the darkness of the dot\nand the force on the bead.\n\nWe perform the experiment for different external forces, ranging from\n2000 N to 9000 N, and different cylinder heights. The corresponding\nvertical stress, $\\sigma$, at the bottom of the cylinder ranges\nbetween 100 KPa and 2.3 MPa (as measured from the darkness of the\ndots).\n%The stress at the bottom of the cylinder is smaller than the\n%stress applied at the top due to the screening of the walls \n%(Janssen's\n%effect \\cite{pgg}). \nThe results of four different measurements are shown in\nFig. \\ref{distri}b. For $\\sigma$ smaller than $\\approx 750$ KPa, the\ndistribution of forces, $f$, at the bottom piston decays\nexponentially:\n\\begin{equation}\nP(f) = \\langle f \\rangle ^{-1}\\exp{[- f/\\langle f \\rangle ]}, \\qquad[\n\\mbox{ $ \\sigma<$ 750 KPa}],\n\\end{equation}\nwhere $ \\langle f \\rangle$ is the average force. When the stress is\nincreased above 750 KPa there is a gradual crossover to a Gaussian\nforce distribution as we find in the simulations. For example, at 2.3\nMPa we have\n\\begin{equation}\nP(f) \\propto \\exp{\\left[- k^2 \\left(f-f_o\\right)^2\\right]}, \\qquad[\n\\mbox{$\\sigma=$ 2.3 MPa}].\n\\end{equation}\n%Because the distribution is cut off for negative values of $f$, one\n%has $ \\langle f\\rangle \n%= f_o + \\exp[-k^2 f_o^2]/\\sqrt{\\pi}k[1 + {\\rm erf}(kf_o)]$. We\n%find \nwhere $k f_o \\approx 1$, and therefore $ \\langle f \\rangle \\approx\nf_o$. Similar results have been found in 2D geometries\n\\cite{behringer}.\n\n\n{\\it Discussion: } In summary, using both numerical simulations and\nexperiments, we have studied unconsolidated compressible granular\nmedia in a range of pressures spanning almost four decades. In the\nlimit of weak compression, the stress vanishes continuously as\n$(\\phi-\\phi_{c})^\\beta$, where $\\phi_c$ corresponds to RLP or RCP\naccording to the existence or not of transverse forces between the\ngrains, respectively. At criticality, the coordination number\napproaches a minimal value $Z_c$ (=4 for friction and 6 for\nfrictionless grains) also as a power law. Our result $Z_c=6$ agrees\nwith experimental analysis of Bernal packings for close contacts\nbetween spheres fixed by means of wax \\cite{bernal}, and our own\nanalysis of the Finney packings \\cite{finney} using the actual sphere\ncenter coordinates of 8000 steel balls. However, no similar\nexperimental study exists for RLP which could be able to confirm\n$Z_c=4$. A critical slowing down--- the time to equilibrate the\nsystem increases near $\\phi_c$--- and the emergency of shear rigidity\n(to be discussed elsewhere) is also found at criticality. The\ndistribution of forces is found to decay exponentially. The system is\ndominated by a fragile network of \nrelatively few force chains which span the system.\n\nWhen the stress is increased away from $\\phi_c$ to the point that the\nnumber of contacts has significantly increased from its initial value\n$Z_c$ we find: (1) the distribution of forces crosses over to a\nGaussian (2) the participation number increases, and then abruptly\nsaturates and (3) the density of force chains increases to the point\nwhere it no longer makes sense to describe the system in those terms.\nOur simulations indicate that the crossover is associated with a loss\nof localization and the ensuing homogenization of the force-bearing\nstress paths.\n\n%The system has become homogeneous down to a scale comparable to the\n%grain size.\n%Our results could be interpreted as a crossover \n%between a fragile state at $\\langle Z \\rangle \\approx Z_c$ and low \n%confining pressure,\n%to an elastic state at larger pressures.\n%In this regime the balls are considerably deformed, and an analysis of\n%stress distributions independent of the state of strain of the sample,\n%as assumed in the q-model, is not valid. Thus deviations from the\n%prediction of the q-model are due to the deformability of particles in\n%realistic compressible media.\n%\\cite{socolar2}. \n\n\n%Interestingly, it has been recently conjectured that cohesionless \n%rigid balls at the minimal coordination $Z_c$ are \n%in a ``fragile state'' \\cite{fragile}, \n%i.e.,\n%that they are able to support only certain loads without severe \n%rearrangements. Further,\n%it is conjectured that finite deformability of the grains could restore the \n%elastic \n%response of the system \\cite{savage}. \n% Thus, our results could be interpreted as a crossover \n%between a fragile state at $\\langle Z \\rangle \\approx Z_c$ and low \n%confining pressure,\n%to an elastic state at larger pressures.\n\n\\vspace{-.7cm}\n\n\\begin{references}\n\n\\vspace{-1.8cm}\n\n\\bibitem{bernal} \nJ. D. Bernal, Nature {\\bf 188}, 910 (1960);\n%Proc. Roy. Soc. London, Ser. A {\\bf 280}, 299 (1964);\n{\\it Disorder and Granular Media}, edited by D. Bideau and A. Hansen \n(Elsevier, Amsterdam, 1993).\n\n\\bibitem{finney}\nJ. L. Finney, Proc. Roy. Soc. London, Ser. A {\\bf 319}, 479 (1970).\n\n\\bibitem{berryman}\nJ. G. Berryman, Phys. Rev. A {\\bf 27}, 1053 (1983).\n\n\\bibitem{ibm}\n G. D. Scott, Nature {\\bf 188}, 908 (1960);\nG. Y. Onoda and E. G. Liniger, Phys. Rev. Lett. {\\bf 64}, 2727 (1990);\n\n\\bibitem{torquato}\nM. D. Rintoul and S. Torquato, Phys. Rev. Lett. {\\bf 77}, 4198 (1996).\n\n\\bibitem{nedderman}\n{\\it Statics and Kinematics of Granular Materials}, by R. M.\nNedderman (Cambridge University Press, 1992).\n\n\n\\bibitem{bouchaud}\nJ. P. Bouchaud, {\\it et al.},\n%P. Claudin, M. E. Cates, and J. P. Wittmer, \nin \n{\\it Physics of Dry Granular Media}, H. J. Herrmann, J. P. Hovi,\nand S. Luding (eds) (Kluwer, Dordrecht, 1998).\n\n\n\\bibitem{chicago}\nC.-H. Liu, {\\it et al.},\n% S. R. Nagel, D. A. Schecter, S. N. Coppersmith, S. N. Majumdar,\n%O. Narayan, T. A. Witten, \nScience {\\bf 269}, 513 (1995); \n% S. N. Coppersmith, {\\it et al.},\n%C.-H. Liu, S. N. Majumdar, O. Narayan, T. A. Witten,\n%Phys. Rev. E {\\bf 53}, 4673 (1996);\nD. M. Mueth,\nH. M. Jaeger, and S. R. Nagel, Phys. Rev. E {\\bf 57}, 3164 (1998).\n\n\n\n\\bibitem{radjai}\n%F. Radjai, D. E. Wolf, S. Roux, M. Jean, and J. J. Moreau,\n%Phys. Rev. Lett. (1997).\nF. Radjai, {\\it et al.},\n%M. Jean, and J. J. Moreau, S. Roux,\nPhys. Rev. Lett. {\\bf 77}, 3110 (1997).\n\n\n%\\bibitem{jenkins}\n%P. A. Cundall, J. T. Jenkins, and I. Ishibashi, in {\\it Powder and Grains\n%1989}, (Biarez and Gourv\\`es, eds) (Balkema, Rotterdam, 1989).\n%G. W. Baxter, in {\\it Powders and Grains 97}, R. P. Behringer and \n%J. T. Jenkins (eds)\n%(Balkema, Rotterdam, 1997).\n\n\n\\bibitem{bubbles}\nAnother experimental realization of frictionless balls is the packing of \nbubbles and compressed microemulsions. \n D. J. Durian, Phys. Rev. Lett. {\\bf 75}, 4780 (1995);\n M.-D. Lacasse, {\\it et al.},\n%G. S. Grest, D. Levine, T. G. Mason,\n%and D. A. Weitz, \nPhys. Rev. Lett. {\\bf 76}, 3448 (1996)\n\n\n\\bibitem{behringer}\nD. Howell, R. P. Behringer, and C. Veje, Phys. Rev. Lett. {\\bf 82}, 5241 \n(1999).\n\n\n\\bibitem{cundall}\nP. A. Cundall and O. D. L. Strack, G\\'eotechnique {\\bf 29}, 47 (1979).\n\n\n\\bibitem{johnson}\n{\\it Contact Mechanics}, by\nK. L. Johnson (Cambridge University Press, 1985).\n\n%\\bibitem{makse}\n%H. A. Makse, N. Gland, D. L. Johnson, L. Schwartz, \n%Phys. Rev. Lett. {\\bf 83}, 5070 (1999).\n\n\n\n\n\\bibitem{edwards}\nS. Alexander, Phys. Rep. {\\bf 296}, 65 (1998);\nS. F. Edwards and D. V. Grinev, Phys. Rev. Lett. {\\bf 82}, 5397 (1999);\nA. Tkachenko and T. A. Witten, Phys. Rev. E {\\bf 60},\n687 (1999).\n\n\n\\bibitem{scott2}\nG. D. Scott and D. M. Kilgour, Br. J. Appl. Phys. {\\bf 2}, 863 (1969).\n\n\n\\bibitem{force_chains}\n%P. Dantu, in\n%{\\it Proc. 4th Int. Conf. on Soil Mechanics Foundation Engineering}, p. 144\n% (Butterworths, London, 1957);\nP. Dantu, Ann. Ponts Chauss. {\\bf IV}, 193 (1967).\n%A. Drescher and G. De Josselin De Jong, J. Mech. Phys. Solids {\\bf 20}, 337\n%(1972).\n%\\bibitem{cundall2}\n%P. A. Cundall and O. D. L. Strack, in {\\it Mechanics of Granular Materials:\n%New Models and Constitutive Relations},\n%(J. T. Jenkins and M. Satake, eds) (1983);\n%C. Thornton, and D. J. Barnes, Acta Mechanica {\\bf 64}, 45 (1986).\n\n\n%\\bibitem{pgg}\n%P.-G. de Gennes, Physica A {\\bf 261}, 267 (1998);\n\n\n\n%\\bibitem{socolar}\n%J. E. S. Socolar, Phys. Rev. E {\\bf 57}, 3204 (1998).\n\n\n\n\n\n%\\bibitem{savage}\n%S. B. Savage, in {\\it Powders and Grains 97}, R. P. Behringer and \n%J. T. Jenkins (eds)\n%(Balkema, Rotterdam, 1997).\n\n\n%\\bibitem{socolar2}\n%Other scalar lattice models of deformable springs\n%give rise to Gaussian distribution of forces. See, \n%M. G. Sexton, J. E. S. Socolar, and D. G. Schaeffer, Phys. Rev. E {\\bf 60} 1999\n%(1999).\n\n\n%\\bibitem{fragile}\n%M. E. Cates, {\\it et al}., Phys. Rev. Lett. {\\bf 81}, 1841 (1998).\n\n%\\bibitem{wolf}\n%J. Sch\\\"afer, S. Dippel, and D. E. Wolf, J. Phys. I France {\\bf 6}, 5 (1996).\n\n\n\n\n%\\bibitem{ball}\n%R. C. Ball and D. V. Grinev, (unpublished).\n\n\n\\end{references}\n\n\n%ACKNOWLEDGMENTS.\n%We would like to thank J. St. Germain and B. Andrews for their valuable help\n%in the experiments and visualization, and N. Gland\n%and D. Pissarenko\n%for discussions.\n\n%\\newpage\n\n%FIG. \\ref{pressure-Z-pi}. (a) Confining stress and (b)\n%average coordination number\n%as a function of volume fraction for friction and frictionless balls.\n\n\n%FIG. \\ref{distri}. \n%Distribution of forces for different confining stresses $\\sigma$\n%obtained (a) in\n%the numerical simulations of friction balls, and (b) in the carbon paper\n%experiments. The straight solid lines are fittings to\n%exponential forms and the dashed lines are fittings to Gaussian forms.\n%In both graphs we shift down the distributions corresponding to the\n%two larger stresses for clarity.\n%(c) Participation number versus external stress for the same system as in \n%(a).\n\n%FIG. \\ref{chains}. \n%Example of percolating\n%force chains for the same system as in Fig. \\protect\\ref{distri}a: \n%(a) near $\\phi_c$ and (b) away from $\\phi_c$ at large\n%confining\n%stress. The color code of the chains is according to the total force\n%in N carried by the\n%chains.\n\n\n%\\newpage\n\n\n%\\newpage\n\n\n%\\newpage\n\n\n\n\n\\end{multicols}\n\n\n\n\\end{document}\n\n\n%We study random packings of compressible spherical grains subjected to\n%an external confining stress near and above the densest packing\n%fraction in 3D. The rigid ball limit is described as a continuous\n%transition where the stress of the system vanishes as\n%$(\\phi-\\phi_c)^\\beta$, where $\\phi_c$ is the critical volume density;\n%a transition which is coincident with the appearance of shear\n%rigidity. If the grains interact via normal forces only, the value of\n%$\\phi_c$ corresponds to the random close packing (RCP) fraction. By\n%contrast, if the grains interact by combined normal and friction\n%generated transverse forces, we find random loose packing (RLP) at the\n%critical point. Our results indicate that the transition is driven by\n%localized force chains. Near $\\phi_c$ there is a percolative fragile\n%structure which we characterize by the participation number, the\n%probability distribution of forces, and visualization techniques.\n%Moreover, we find a smooth transition from strong localization of the\n%force distribution along ``force chains'' at low pressures\n%to a delocalized and more\n%homogeneous state as the grains deform with increased confining\n%stress; a crossover evidenced in the change of the probability\n%distribution of forces.\n%%we find that the exponential distribution of intergranular forces\n%%observed by recent authors is valid only at low confining stresses\n%%where the grains are, in effect, rigid. At larger stresses, above\n%%RCP, we find a crossover to a narrower, Gaussian distribution. This\n%%crossover reflects \n%%In real granular materials the grains are compressible, a fact that\n%%strongly influences the distribution of contact forces. \n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%{\\bf old}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Dense random packing of spheres \\cite{bernal,berryman} are useful\n%models for many physical systems such as the arrangements of atoms in\n%simple fluids, amorphous metals, or glasses as well as colloidal\n%suspensions, or biological systems \\cite{torquato}.\n\n%Random packings have been studied experimentally and in molecular\n%dynamics and Monte Carlo computer simulations of hard spheres at a\n%finite temperature. An equilibrium hard sphere system presents a\n%first order liquid-solid phase transition as the volume fraction\n%$\\phi$ (defined as the ratio between the volume occupied by the\n%spheres and the volume of the system) is increased up to the freezing\n%point, above which the disordered state can persist in a metastable\n%branch until $\\phi\\to\\phi_{\\mbox{\\scriptsize RCP}}^-$ \\cite{torquato},\n%where $\\phi_{\\mbox{\\scriptsize RCP}}$ is the density of random close\n%packing (RCP)--- the densest possible packing of random hard spheres.\n\nAnother realization of the problem is related to macroscopic granular\nmaterials. In a typical experiment spherical balls are poured in a\ncontainer which is squeezed and shaken to achieve the best possible\npacking. The best estimation of the volume fraction at RCP was\nobtained by Finney \\cite{finney} for ball bearings:\n$\\phi_{\\mbox{\\scriptsize RCP}}=0.6366(4)$ \\cite{berryman}. Looser\npackings of mechanically stable spheres, with packing fractions\nranging from $\\approx 0.60$ to RCP are much less reproducible and are\ngenerated by rolling spheres into a container without shaking--- this\nlimit is known as the random loose packing (RLP) \\cite{ibm}.\n\nUnlike glasses and amorphous solids, granular materials are zero\ntemperature systems whose interparticle forces are exclusively\nrepulsive: the rigidity of the system arises due to the applied\nexternal stress.\n%Thus, the only way to\n%achieve a mechanically stable packing is by imposing an external\n%pressure. \nIn this paper we study the approach to RCP and RLP for realistic\ncompressible granular materials interacting via non-linear\n(Hertz-Mindlin) normal and friction generated transverse contact\nforces. Since the balls are deformable, the volume fraction of the\nsystem can be increased above the hard sphere limit by the action of\nthe external pressure.\n% and we are interested in the limit \n%$\\phi\\to\\phi_{\\mbox{\\scriptsize RCP}}^+$.\n\n\n% We show that the RCP\n%state is achieved in our system of ``soft'' spheres\n%as a limit where the pressure of\n%the system vanishes as a power-law as $\\phi$ approaches a critical density \n%$\\phi_c=0.634(4)(\\approx\\phi_{\\mbox{\\scriptsize RCP}}) $.\n%This critical value is obtained \n% when the grains are allowed to rearrange, sliding and \n%rotating freely (interacting \n%via normal forces only), mimicking the rearrangements of grains \n%during shaking.\nWe describe the rigid ball limit, as a continuous phase transition\nwhere the order parameter is the stress, $\\sigma$, which vanishes\ncontinuously as $(\\phi-\\phi_c)^\\beta$, where $\\phi_c$ is the critical\nvolume density, and $\\beta$ is the corresponding critical exponent.\nThus, we characterize the hard sphere limit\n%state where deformations are negligible by\nby looking at the limit $\\phi\\to\\phi^+_c$ from above.\n\n\nOf particular importance is the fact that $\\phi_c$ depends on the type\nof interaction between the grains. If the grains interact via normal\nforces only (so that they slide and rotate freely mimicking the\nrearrangements of grains during shaking \\cite{bubbles}), we obtain the\nvalue of $\\phi_c= 0.634(4)(\\approx\\phi_{\\mbox{\\scriptsize RCP}})$\ncorresponding to RCP. By contrast, if the grains interact by combined\nnormal and friction generated transverse forces, we find random loose\npacking (RLP) at the critical point with\n$\\phi_c=0.6284(2)<\\phi_{\\mbox{\\scriptsize RCP}}$. The power-law\nexponents characterizing the approach to $\\phi_c$ are not universal\nand depend on the strength of friction generated shear forces as well.\n\n%When the balls are allowed to interact \n%with friction generated transverse forces we find that the \n%pressure vanishes but at a lower packing fraction \n%$\\phi_c=0.6284(2)<\\phi_{\\mbox{\\scriptsize RCP}}$. \n%We argue that this corresponds to the limit \n%experimentally achieved by\n%pouring balls in a container but without allowing for \n%further rearrangements. \n%The power-law exponents characterizing the approach to $\\phi_c$ are not\n%universal and depend\n%on the strength of friction generated shear forces.\n\nIn both cases we show that the system at the critical density is at\nthe minimal average coordination number, $Z_c$, required for\nmechanical stability. Our results indicate that the transition at the\nRCP or RLP is driven by localized force chains. Near the critical\ndensity there is a percolative fragile structure which we characterize\nby the participation number (which measures localization of force\nchains), the probability distribution of forces, and also by\nvisualization techniques. Part of these results are experimentally\nverified using standard carbon paper experiments to study force\ndistributions in the granular assembly, and we also compare with\nrecent experiments in 2D Couette geometries relevant to the present\nstudy \\cite{behringer}.\n\n%For grains\n%interacting via normal and tangential forces this number is D+1 (D\n%is the dimension), and for frictionless \n%balls it is\n%2D as conjectured recently using constraint counting arguments\n%\\cite{edwards}, and we confirm these results with our numerical\n%simulations, for D=3. \n%Concomitant with the approach to the critical\n%volume fraction we find a\n%change in the probability distribution and in the degree of spatial\n%correlation of the interparticle forces: \n%localized force chains with an exponential probability\n%distribution--- as observed in recent studies \n%\\cite{force_chains,chicago}---\n%are found at low confining stress\n%giving way to a considerably less localized and homogeneous\n%arrangement of the forces with a Gaussian probability distribution at\n%higher stress away from the critical density. \n\n\n%The densest possible acking of equal sized spheres is the ordered\n%close packing (fcc) with a volume fraction $\\phi=\\pi/18^2\\approx 0.7405$ in \n%three dimensions.\n\n%Specifically, we consider the force distribution \n%when the packing is subjected to uniaxial strains which can be large \n%enough to\n%deform the individual beads and significantly change the average \n%coordination \n%number. For low confining stresses (i.e., in the rigid grain limit) we \n%confirm previous results; both experiment and simulation indicate that \n%the force distribution is exponential. \n%%For grains interacting via a combination \n%%of normal and tangential forces, constraint counting arguments predict\n%%$\\langle Z \\rangle = D+1$, where $D$ is the spatial dimension \n%%\\cite{edwards}. \n%However, when the system is compressed at large \n%confining stresses we find a transition to a Gaussian force distribution. \n%At this point the grains are considerable\n%deformed and the system is like \n% an amorphous elastic solid \\cite{edwards} with a correspondingly\n%more homogeneous force distribution. \n\n\n\n\n\n\n"
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{
"name": "cond-mat0002102.extracted_bib",
"string": "\\bibitem{bernal} \nJ. D. Bernal, Nature {\\bf 188}, 910 (1960);\n%Proc. Roy. Soc. London, Ser. A {\\bf 280}, 299 (1964);\n{\\it Disorder and Granular Media}, edited by D. Bideau and A. Hansen \n(Elsevier, Amsterdam, 1993).\n\n\n\\bibitem{finney}\nJ. L. Finney, Proc. Roy. Soc. London, Ser. A {\\bf 319}, 479 (1970).\n\n\n\\bibitem{berryman}\nJ. G. Berryman, Phys. Rev. A {\\bf 27}, 1053 (1983).\n\n\n\\bibitem{ibm}\n G. D. Scott, Nature {\\bf 188}, 908 (1960);\nG. Y. Onoda and E. G. Liniger, Phys. Rev. Lett. {\\bf 64}, 2727 (1990);\n\n\n\\bibitem{torquato}\nM. D. Rintoul and S. Torquato, Phys. Rev. Lett. {\\bf 77}, 4198 (1996).\n\n\n\\bibitem{nedderman}\n{\\it Statics and Kinematics of Granular Materials}, by R. M.\nNedderman (Cambridge University Press, 1992).\n\n\n\n\\bibitem{bouchaud}\nJ. P. Bouchaud, {\\it et al.},\n%P. Claudin, M. E. Cates, and J. P. Wittmer, \nin \n{\\it Physics of Dry Granular Media}, H. J. Herrmann, J. P. Hovi,\nand S. Luding (eds) (Kluwer, Dordrecht, 1998).\n\n\n\n\\bibitem{chicago}\nC.-H. Liu, {\\it et al.},\n% S. R. Nagel, D. A. Schecter, S. N. Coppersmith, S. N. Majumdar,\n%O. Narayan, T. A. Witten, \nScience {\\bf 269}, 513 (1995); \n% S. N. Coppersmith, {\\it et al.},\n%C.-H. Liu, S. N. Majumdar, O. Narayan, T. A. Witten,\n%Phys. Rev. E {\\bf 53}, 4673 (1996);\nD. M. Mueth,\nH. M. Jaeger, and S. R. Nagel, Phys. Rev. E {\\bf 57}, 3164 (1998).\n\n\n\n\n\\bibitem{radjai}\n%F. Radjai, D. E. Wolf, S. Roux, M. Jean, and J. J. Moreau,\n%Phys. Rev. Lett. (1997).\nF. Radjai, {\\it et al.},\n%M. Jean, and J. J. Moreau, S. Roux,\nPhys. Rev. Lett. {\\bf 77}, 3110 (1997).\n\n\n%\n\\bibitem{jenkins}\n%P. A. Cundall, J. T. Jenkins, and I. Ishibashi, in {\\it Powder and Grains\n%1989}, (Biarez and Gourv\\`es, eds) (Balkema, Rotterdam, 1989).\n%G. W. Baxter, in {\\it Powders and Grains 97}, R. P. Behringer and \n%J. T. Jenkins (eds)\n%(Balkema, Rotterdam, 1997).\n\n\n\n\\bibitem{bubbles}\nAnother experimental realization of frictionless balls is the packing of \nbubbles and compressed microemulsions. \n D. J. Durian, Phys. Rev. Lett. {\\bf 75}, 4780 (1995);\n M.-D. Lacasse, {\\it et al.},\n%G. S. Grest, D. Levine, T. G. Mason,\n%and D. A. Weitz, \nPhys. Rev. Lett. {\\bf 76}, 3448 (1996)\n\n\n\n\\bibitem{behringer}\nD. Howell, R. P. Behringer, and C. Veje, Phys. Rev. Lett. {\\bf 82}, 5241 \n(1999).\n\n\n\n\\bibitem{cundall}\nP. A. Cundall and O. D. L. Strack, G\\'eotechnique {\\bf 29}, 47 (1979).\n\n\n\n\\bibitem{johnson}\n{\\it Contact Mechanics}, by\nK. L. Johnson (Cambridge University Press, 1985).\n\n%\n\\bibitem{makse}\n%H. A. Makse, N. Gland, D. L. Johnson, L. Schwartz, \n%Phys. Rev. Lett. {\\bf 83}, 5070 (1999).\n\n\n\n\n\n\\bibitem{edwards}\nS. Alexander, Phys. Rep. {\\bf 296}, 65 (1998);\nS. F. Edwards and D. V. Grinev, Phys. Rev. Lett. {\\bf 82}, 5397 (1999);\nA. Tkachenko and T. A. Witten, Phys. Rev. E {\\bf 60},\n687 (1999).\n\n\n\n\\bibitem{scott2}\nG. D. Scott and D. M. Kilgour, Br. J. Appl. Phys. {\\bf 2}, 863 (1969).\n\n\n\n\\bibitem{force_chains}\n%P. Dantu, in\n%{\\it Proc. 4th Int. Conf. on Soil Mechanics Foundation Engineering}, p. 144\n% (Butterworths, London, 1957);\nP. Dantu, Ann. Ponts Chauss. {\\bf IV}, 193 (1967).\n%A. Drescher and G. De Josselin De Jong, J. Mech. Phys. Solids {\\bf 20}, 337\n%(1972).\n%\n\\bibitem{cundall2}\n%P. A. Cundall and O. D. L. Strack, in {\\it Mechanics of Granular Materials:\n%New Models and Constitutive Relations},\n%(J. T. Jenkins and M. Satake, eds) (1983);\n%C. Thornton, and D. J. Barnes, Acta Mechanica {\\bf 64}, 45 (1986).\n\n\n%\n\\bibitem{pgg}\n%P.-G. de Gennes, Physica A {\\bf 261}, 267 (1998);\n\n\n\n%\n\\bibitem{socolar}\n%J. E. S. Socolar, Phys. Rev. E {\\bf 57}, 3204 (1998).\n\n\n\n\n\n%\n\\bibitem{savage}\n%S. B. Savage, in {\\it Powders and Grains 97}, R. P. Behringer and \n%J. T. Jenkins (eds)\n%(Balkema, Rotterdam, 1997).\n\n\n%\n\\bibitem{socolar2}\n%Other scalar lattice models of deformable springs\n%give rise to Gaussian distribution of forces. See, \n%M. G. Sexton, J. E. S. Socolar, and D. G. Schaeffer, Phys. Rev. E {\\bf 60} 1999\n%(1999).\n\n\n%\n\\bibitem{fragile}\n%M. E. Cates, {\\it et al}., Phys. Rev. Lett. {\\bf 81}, 1841 (1998).\n\n%\n\\bibitem{wolf}\n%J. Sch\\\"afer, S. Dippel, and D. E. Wolf, J. Phys. I France {\\bf 6}, 5 (1996).\n\n\n\n\n%\n\\bibitem{ball}\n%R. C. Ball and D. V. Grinev, (unpublished).\n\n\n"
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cond-mat0002103
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Pulse driven switching in one-dimensional nonlinear photonic band gap materials: a numerical study
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[
{
"author": "E.~Lidorikis \\cite{lido} and C.~M.~Soukoulis"
}
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[
{
"name": "lidorikis_main_rev2.tex",
"string": "%lidorikis_main.tex\n%\\documentstyle[twocolumn,aps,psfig]{revtex}\n\\documentstyle[preprint,aps,psfig]{revtex}\n%\\documentstyle[prb,aps]{revtex}\n%\\documentclass[12pt]{article}\n%c for double space, comment out the \\tightlines\n%\\tightenlines\n%\\hoffset=-1.65in\n%\\textwidth=39pc\n%\\math-with-secnums\n\\begin{document}\n%\\draft\n%\\preprint{Ames Laboratory-U.S. DOE Preprint }\n\\title{\nPulse driven switching in one-dimensional nonlinear photonic band gap \nmaterials:\na numerical study \n}\n\\author{\nE.~Lidorikis \\cite{lido} and C.~M.~Soukoulis\n}\n\\address{\nAmes Laboratory-U.S. DOE and Department of Physics and Astronomy, \nIowa State University, Ames, Iowa 50011 \\\\\n}\n\\author{\\parbox[t]{5.5in}{\\small\nWe examine numerically the time-dependent properties of nonlinear\nbistable multilayer structures for constant wave illumination. \nWe find that our system exhibits both steady-state and \n self-pulsing solutions. In the steady\nstate regime, we examine the dynamics of driving the system between\ndifferent transmission states by injecting pulses, and we find the\noptimal pulse parameters. We repeat this work for the case of a linear\nperiodic system with a nonlinear impurity layer. \n\\\\ \\\\\nPACS numbers: 42.65.Pc, 41.20.Jb, 42.65.Sf, 42.70.Qs }}\n\\maketitle\n\\normalsize \n%\\tightenlines\n%\\newpage\n\n\\section{INTRODUCTION}\n\nNonlinear dielectric materials \\cite{newell_book} \nexhibiting a bistable response\nto intense radiation are key elements for an all-optical digital\ntechnology. For certain input optical powers\nthere may exist two distinct transmission branches forming\na hysteresis loop, which incorporates a history dependence \nin the system's response. Exciting applications involve \n optical switches, logic gates, set-reset fast memory elements etc \n\\cite{gibbs_book}. \nMuch interest has been given lately to periodic nonlinear structures\n\\cite{winful},\nin which because of the distributed feedback mechanism, the nonlinear\neffect is greatly enhanced. In the low intensity limit, these\nstructures are just Bragg reflectors characterized by high transmission\nbands separated by photonic band gaps \\cite{soukoulis}. \nFor high intensities and \nfrequencies inside the transmission band, bistability results from the \nmodulation of transmission by an intensity-dependent phase shift. \nFor frequencies inside the gap bistability originates from gap soliton\nformation \\cite{chen}, which can lead to much lower switching thresholds \n\\cite{desterke2}.\n\n\nThe response of nonlinear periodic structures illuminated by a \nconstant wave (CW) with a frequency inside the photonic gap is\ngenerally separated into three regimes: i) steady state response via\nstationary gap soliton formation, ii) self-pulsing via excitation \nof solitary waves, and iii) chaotic. Much theoretical work has\nbeen done for systems with a weak sinusoidal \nrefractive index modulation and uniform nonlinearity \n\\cite{winful,mills,desterke,wabnitz,john}, or deep modulation \nmultilayered systems \\cite{chen,dowling,tran}, \nas well as experimental \\cite{sankey,eggleton}. \nOne case of interest\nis when the system is illuminated by a CW bias and switching between\ndifferent transmission states is achieved by means of external pulses. \nSuch switching has already been \ndemonstrated experimentally for various kinds of nonlinearities \n\\cite{sankey,tarng,cada2}, \nbut to our knowledge, a detailed study\nof the dynamics, the optimal pulse parameters and the stability \nunder phase variations during injection, has yet to be performed.\n\nIn this paper we use the Finite-Difference-Time-Domain (FDTD) \\cite{taflove}\n method to study the time-dependent properties of CW propagation\nin multilayer structures with a Kerr type nonlinearity. We find our \nresults generally in accord to those obtained for systems with weak \nlinear index modulation \\cite{desterke}, which were solved with approximate \nmethods. \nWe next examine the dynamics of driving the system from one transmission \nstate to \nthe other by injecting a pulse, and try to find the optimal pulse \nparameters for this switching. We also test how these parameters \nchange for a different initial phase or frequency of the \npulse.\nFinally, we will repeat all work for the case of a linear multilayer\nstructure with a nonlinear impurity layer.\n\n\\section{FORMULATION} \n\nElectromagnetic wave propagation in dielectric media is governed by\nMaxwell's equations\n\\begin{eqnarray}\n\\mu \\frac{\\partial \\vec{H}}{\\partial t} = \n -\\vec{\\nabla} \\times \\vec{E} & \\hspace{1.5cm} & \n\\frac{\\partial \\vec{D}}{\\partial t} = \n\\vec{\\nabla} \\times \\vec{H}\n\\end{eqnarray}\nAssuming here a Kerr type saturable nonlinearity and an isotropic medium, \nthe electric flux density\n$\\vec{D}$ is related to the electric field $\\vec{E}$ by\n\\begin{eqnarray}\n\\vec{D}=\\epsilon_0 \\vec{E} +\\vec{P}_L +\\vec{P}_{NL}=\\epsilon_0\n\\left( \\epsilon_r +\\frac{\\alpha \n\\vert \\vec{E} \\vert^2}{1+\\gamma \\vert \\vec{E} \\vert^2}\\right) \\vec{E}\n\\end{eqnarray}\nwhere $\\gamma \\geq 0$.\n$\\vec{P}_L$ and $\\vec{P}_{NL}$ are the induced linear and nonlinear \nelectric polarizations\nrespectively. Here we will assume zero linear dispersion and so a \nfrequency independent $\\epsilon_r$.\nInverting this to obtain $\\vec{E}$ from $\\vec{D}$ involves the \nsolution of a cubic equation in $\\vert \\vec{E} \\vert$. For $\\alpha \\geq 0$\nthere is always only one real root, so there is no ambiguity. For \n$\\alpha < 0$ this is true only for $\\gamma > 0$. In our study we\nwill use $\\alpha =-1$ and $\\gamma = (\\epsilon_r-1)^{-1} $ so that \nfor $\\vert \\vec{E} \\vert \\rightarrow \\infty$, \n$\\vec{D}\\rightarrow\\epsilon_0\\vec{E}$.\n\nThe structure we are considering consists of a periodic array of 21\nnonlinear dielectric layers in vacuum, each 20 nm wide with \n$\\epsilon_r=3.5$, separated by a lattice constant $a=$200 nm. The linear, \nor low intensity,\ntransmission coefficient as a function of frequency is shown in \nFig.~1a. In the numerical setting each unit cell is divided into 256 grids, \nhalf of them defining the highly refractive nonlinear layer. For the midgap\nfrequency, this corresponds to about 316 grids per wavelength in the\nvacuum area and 1520 grids per effective wavelength in the nonlinear \ndielectric, where of course the length scale is different in the two regions.\n Stability considerations only require more than 20 grids\nper wavelength \\cite{taflove}. Varying the number of the grids used we found \nour results\nto be completely converged. On the two sides of the system we\napply absorbing boundary conditions \\cite{taflove}. \n\nWe first study the structure's response to an incoming constant\nplane wave of frequency\nclose to the gap edge $\\omega a/2\\pi c=0.407$. For each value of the \namplitude, we wait until the system reaches a steady state and then \ncalculate the corresponding transmission and reflection coefficients.\nIf no steady state is achievable, we approximate them by averaging\nthe energy transmitted and reflected over a certain period of time, \nalways checking that energy conservation is satisfied. Then the\nincident amplitude is increased to its next value, which is done \nadiabatically over a time period of 20 wave cycles, and measurements\nare repeated. This procedure continues until a desired maximum \nvalue is reached, and then start decreasing the amplitude, repeating \nbackwards the same routine. The form of the incident CW is\n\\begin{eqnarray}\nE_{\\tiny{CW}}(t)=\\left(A_{\\tiny{CW}}+dA_{\\tiny{CW}}\n\\frac{min \\{ (t-t_0),20T \\} }{20T}\\right)e^{i\\omega t}\n\\end{eqnarray}\nwhere $t_0$ is the time when the amplitude change started, $A_{\\tiny{CW}}$ is \nthe last amplitude value considered and $dA_{\\tiny{CW}}$ the amplitude \nincrement. \nOne wave cycle $T$ involves about 2000 time steps.\n\n\\section{RESPONSE TO A CW BIAS}\n\nThe amplitude of the CW is varied from zero to a maximum of 0.7 \nwith about 40 measurements in between. Results are shown in Fig.~1b,\nalong with the corresponding one from a time-independent approximation.\nThe agreement between the two methods is exact for small intensities,\nhowever, after a certain input the output waves are not constant any\nmore but pulsative. This is in accord with the results\nobtained with the slowly varying envelope approximation for systems\nwith a weak refractive index modulation \\cite{desterke}. It is interesting\n that the averaged output power is still in agreement with\nthe time-independent results, something not mentioned in earlier work.\nFor higher input values, the solution will again reach a steady state just\nbefore going to the second nonlinear jump, after which it will \nagain become pulsative. This time though, the averaged transmitted power \nis quantitatively different from the one predicted from time-independent\ncalculations. \n\nThe nonlinear transmission jump originates from the excitation of a \nstationary gap soliton when the incident intensity \nexceeds a certain threshold value. Due to the nonlinear change of the \ndielectric constant, the photonic gap is shifted locally in the area \nunderneath the soliton, which becomes effectively \ntransparent, resembling a quantum well with the soliton being\nits bound state solution \\cite{lidorikis1}.\nThe incident radiation coupled to that soliton \ntunnels through the structure and large transmission is achieved. \nWe obtain a maximum switching time of the order of 100 $T_r$, or \na frequency of 360 GHz, where $T_r=2L/c$ is roundtrip time in vacuum.\nThe second transmission jump is related to the excitation of\ntwo gap solitons, which however, are not stable and so transmission\nis pulsative. The Fourier transform of the output shows that,\nafter the second transmission jump, the system pulsates at a frequency\n$\\omega a/2\\pi c=0.407\\pm n \\times 0.024$, \nexactly three times the one of the first pulsating solution \n $\\omega a/2\\pi c=0.407\\pm n \\times 0.008$, where $n$ is an integer. \nFor much higher input values the response eventually becomes chaotic.\nA more detailed description of the switching process as well as the\nsoliton generation dynamics can be found in \\cite{desterke}.\n\n\n\\section{PULSE DRIVEN SWITCHING}\n\nWe next turn to the basic objective of this work. We assume a \nspecific constant input amplitude $\\vert A_{\\tiny{CW}}\\vert=0.185$ \ncorresponding to the middle of \nthe first bistable \nloop. Depending on the system's history, it can be either in\nthe low transmission state I, shown in Fig.~1c, or in the high \ntransmission state II, shown in Fig.~1d, which are both steady states. \nWe want to study the dynamics of a pulse\ninjected into a system like that. More specifically, if it will\ndrive the system to switch from one state to the other, how the \nfields change in the structure during switching, for which\npulse parameters this will happen and if these parameters change\nfor small phase and frequency fluctuations. We assume Gaussian envelope\npulses\n\\begin{eqnarray}\nE_{\\tiny{P}}(0,t)=A_{\\tiny{P}}e^{-(t-t_0-5t_w)^2/t_w^2}\ne^{i\\omega t}\n\\end{eqnarray}\nwhere $A_{\\tiny{P}}$ is the pulse amplitude, $t_0$ the time when injection \nstarts,\nand $W=2t_wc$ is the pulse's full width at $1/e$ of maximum amplitude.\nThe beginning of time $t$ is the same as for the CW, so there is\nno phase difference between them. \nAfter injection we wait until the system reaches a steady state again\nand then measure the transmission and reflection coefficients to \ndetermine the final state. During this time we save the field \nvalues inside the structure every few time steps, as well as the\ntransmitted and reflected waves. This procedure is repeated for \nvarious values of $A_{\\tiny{P}}$ and $t_w$, for both possible\ninitial states. Our results are summarized in Fig.~2. White areas \nindicate the pulse parameters for which the intended switch was\nsuccessful while black are for which it failed. In Fig.~2a, or\nthe ``Switch'' graph, the intended switching scheme is for the same\npulse to be able to drive the system from state I to state II\nand {\\em vise versa}. Fig.~2b, or ``Switch All Up'', is for a pulse\n able to drive the system from I to II, but fails to do \nthe opposite, ie. the final state is always II independently from\nwhich the initial state was. Similarly, Fig.~2c or ``Switch All Down'',\nis for the pulse whose final state is always I, and Fig.~2d or\n''No Switch'', for the pulse that does not induce any switch for \nany initial state.\n\nWe find a rich structure on these parameter planes. Note also that\nthere is a specific cyclic order as one crosses the curves moving to\nhigher pulse energies: $\\rightarrow$ d $\\rightarrow$ b $\\rightarrow$ \na $\\rightarrow$ c $\\rightarrow$ etc. This indicates that there must\nbe some kind of energy requirements for each desired switching scheme.\nAfter analyzing the curves it was found that only the first one\nin the ``Switch'' graph could be assigned to a simple constant energy\ncurve $\\mathcal{E}$$\\sim W \\vert A_p\\vert^2$. Since any switching involves\nthe creation or destraction of a stationary soliton, then this\nshould be its energy. In order to put some numbers, if we would assume\na nonlinearity $\\vert \\alpha\\vert=10^{-9}$ cm$^2$/W, then we would need\na CW of energy $\\simeq 34$ MW/cm$^2$ and a pulse of width \n$W/c$ of a few tenths of femtoseconds and energy $\\mathcal{E}$$\\simeq \n2.5 \\mu$J/cm$^2$. These energies may seem large, but they\ncan be sufficiently lowered by increasing the number of layers \nand using an incident frequency closer the the gap edge.\n\nIn order to find more about how the switching occurs, we plotted\nin Fig.~3 the effective transparent areas and the output fields\nas a function of time, for the first three curves of Fig.~2a.\nAs expected, for the pulse from the first curve, the energy for the\nsoliton excitation is just right, and the output fields are small\ncompared to the input. For the other curves however, there is an\nexcess of energy. The system has to radiate this energy away before\na stable gap soliton can be created. It is interesting to note that \nthis energy goes only in the transmitted wave, not the reflected,\nand it consists of a series of pulses \\cite{eggleton}. For the second curve \nin Fig.~2a \nthere is one pulse, for the third there are two etc. The width\nand frequency of the pulses are independent of the incident pulse,\nthey are the known pulsating solutions we found in the CW case. So the \nsystem temporally goes into a pulsating state to radiate away \nthe energy excess before settling down into a stable state. If this\nenergy excess is approximately equal to an integer number of\npulses (the solitary waves from the unstable solutions), then we will have\na successful switch, otherwise it will fail. A similar behavior is\nfound in the system's response during switch down \nfor the first three curves in Fig.~2a, using\nexactly the same pulses as before for the switch up. So the same pulse is\ncapable of switching the system up, and if reused, switching the system\nback down. Using the numbers assumed before for the nonlinearity $\\alpha$, the pulses\nused in Fig.~3 are \n(a) $W/c=14$ fs, $\\mathcal{E}$=2.5 $\\mu$J/cm$^2$,\n(b) $W/c=28$ fs, $\\mathcal{E}$=12 $\\mu$J/cm$^2$,\n(c) $W/c=42$ fs, $\\mathcal{E}$=32 $\\mu$J/cm$^2$.\n\nUp to now, the injected pulse has been treated only as an amplitude\nmodulation of the CW source, ie. they had the same exactly frequency \nand there was no phase difference\nbetween them. The naturally rising question is how an initial\nrandom phase between the CW and the pulse, or a slightly different \nfrequency affect our results. We repeated the simulations for\nvarious values of an initial phase difference, first\nkeeping them with the same frequency. We find that although the results \nshow qualitatively the\nsame stripped structure as in Fig.~2, there are quantitative differences. \nThe main result is that there\nis not a set of pulse parameters that would perform the desired\nswitching successfully for any initial phase difference. \nThus the pulse can not be incoherent with the CW, ie. generated at different \nsources, if a controlled and reproducible switching mechanism is desired,\nbut rather it should be introduced as an amplitude modulation of the CW. \nHowever, if this phase could be controlled, then the switching operation\nwould be controlled, and a single pulse would be able to perform all\ndifferent operations.\n\n\nThe picture does not change if we use pulses\nof slightly different frequency from the source. We used various pulses\nwith frequencies both higher and lower than the CW, and we found a \nsensitive, rather chaotic, dependence on the initial phase at \ninjection time. The origin of this complex response, if it is an \nartifact of the simple Kerr-type nonlinearity model that we used, \nand if it should appear for other kinds of nonlinearities, \n is not yet clear to as.\nMore work is also needed on how these results would change if one\nused a different $\\vert A_{\\tiny{CW}}\\vert$ not in the middle of the\nbistable loop, a wider or narrower bistable loop etc., but these\nwould go more into the scope of engineering.\n\n\\section{LINEAR LATTICE WITH A NONLINEAR IMPURITY LAYER}\n\nBesides increasing the number of layers to achieve lower switching\nthresholds, one can use a periodic array of linear layers $\\epsilon\n=\\epsilon_0\\epsilon_r$ with a nonlinear impurity layer \n\\cite{radic,hattori,wang,lidorikis2}\n$\\epsilon=\\epsilon_0(\\epsilon_r^{\\prime}+\\alpha \n\\vert \\vec{E} \\vert^2)$ where $\\epsilon_r \\neq \\epsilon_r^{\\prime}$\nand we will use $\\alpha=+1$ and $\\gamma=0$.\nThis system is effectively a Fabry-Perot cavity\nwith the impurity (cavity) mode inside the photonic gap, as shown in\nFig.~4a. The bistable response originates from its nonlinear \nmodulation with light intensity. The deeper this mode is in the\ngap, the stronger the linear dispersion for frequencies close to it.\nBecause of the high $Q$ of the mode, we can use frequencies extremely\nclose to it achieving very low switching thresholds \\cite{lidorikis2}.\n Here however we only want\nto study the switching mechanism, so we will use a shallow impurity mode.\n\nThe bistable input-output diagram,\nthe output fields during switching and the field distributions\nin the two transmission branches are also shown in Fig.~4. We observe \na smaller relaxation time and of course the absence of pulsating solutions.\n The parameters\nused are $\\epsilon_r^{\\prime}=1$ and $\\omega a/2 \\pi c=0.407$ which\ncorresponds to a frequency between the mode and the gap edge. \nWe want to test if a pulse can drive this system in switching \nbetween the two different transmission states, and again test our results\nagainst phase and frequency perturbations. The two states shown in Fig.~4\nare for an input CW amplitude of $\\vert A_{\\tiny{CW}}\\vert=0.16$.\nThe results for coherent, pulse and CW, are shown in Fig.5. We see that\nany desired form of switching can still be achieved, but the \nparameter plane graphs no more bare any simple explanations as the \nones obtained for the nonlinear superlattice.\nRepeating the simulations for incoherent beams and different\nfrequencies we obtain the same exactly results as before. Only\nphase-locked beams can produce controlled and reproducible\nswitching.\n\n\\section{CONCLUSIONS}\n\nWe have studied the time-dependent switching properties \nof nonlinear dielectric multilayer systems for frequencies\ninside the photonic band gap of the corresponding linear structure. \nThe system's response is characterized by both stable and self-pulsing\nsolutions. We examined the dynamics of driving the system between\ndifferent transmission states by pulse injection, and found correlations\nbetween the pulse, the stationary gap soliton and the unstable\nsolitary waves. A small dependence on the phase difference\nbetween the pulse and the CW is also found, requiring coherent\nbeams for fully controlled and reproducible\nswitching. Similar results are also found for the case of a\nlinear periodic structure with a nonlinear impurity.\n\n\n\\acknowledgements \nAmes Laboratory is operated for the U. S. Department of Energy by Iowa \nState University under contract No. W-7405-ENG-82. This work was supported\nby the Director of Energy Research office of Basic Energy Science and \nAdvanced Energy Projects, the Army Research office, and a PENED grand.\n\n\n\n\\begin{references}\n\n\\bibitem[*]{lido} Present address: Concurrent Computing Laboratory\nfor Materials Simulation (CCLMS) and Department of Physics and Astronomy,\nLouisiana State University, Baton Rouge, LA 70803.\n\n\\bibitem{newell_book} A.~C.~Newell and J.~V.~Moloney, {\\em Nonlinear \nOptics} (Addison-Wesley, Redwood CA, 1992).\n\n\\bibitem{gibbs_book} H.~M.~Gibbs, {\\em Optical Bistability: Controlling\nLight with Light} (Academic, Orlando FL, 1985).\n\n\\bibitem{winful} H.~G.~Winful, J.~H.~Marburger, and E.~Garmire,\nAppl. Phys. Lett {\\bf 35}, 379 (1979); H.~G.~Winful, and \nG.~D.~Cooperman, Appl. Phys. Lett {\\bf 40}, 298 (1982).\n\n\\bibitem{soukoulis} {\\em Photonic Band Gaps and Localization}, edited\nby C.~M.~Soukoulis (Plenum, New York, 1993); {\\em Photonic Band Gap\nMaterials}, edited by C.~M.~Soukoulis (Klumer, Dordrecht, 1996).\n\n\\bibitem{chen} Wei Chen and D.~L.~Mills, Phys. Rev. B {\\bf 36},\n6269 (1987); Phys. Rev. Lett. {\\bf 58}, 160 (1987).\n\n\\bibitem{desterke2} C.~Martijn de Sterke and J.~E.~Sipe, in \n{\\em Progress in Optics}, edited by E.~Wolf (Elsevier, Amsterdam, 1994),\nVol. 33.\n\n\\bibitem{mills} D.~L.~Mills and S.~E.~Trullinger, Phys. Rev. B {\\bf 36},\n947 (1987).\n\n\\bibitem{desterke} C.~Martijn de Sterke and J.~E.~Sipe, Phys. Rev. A {\\bf 42},\n2858 (1990).\n\n\\bibitem{wabnitz} A.~B.~Aceves, C.~De~Angelis and S.~Wabnitz, Opt. Lett. \n{\\bf 17}, 1566 (1992).\n\n\\bibitem{john} Sajeev John and Ne\\c{s}et Ak\\\"{o}zbek, \nPhys. Rev. Lett. {\\bf 71}, 1168 (1993); Ne\\c{s}et Ak\\\"{o}zbek and\nSajeev John, Phys. Rev. E {\\bf 57}, 2287 (1998).\n\n\\bibitem{dowling} Michael Scalora, Jonathan P.~Dowling, Charles M.~Bowden,\nand Mark J.~Bloemer, Phys. Rev. Lett. {\\bf 73}, 1368 (1994).\n\n\\bibitem{tran} P.~Tran, Opt. Lett. {\\bf 21}, 1138 (1996).\n\n\\bibitem{sankey} N.~D.~Sankey, D.~F.~Prelewitz, and T.~G.~Brown,\nAppl. Phys. Lett. {\\bf 60}, 1427 (1992).\n\n\n\\bibitem{eggleton} B.~J.~Eggleton, C.~Martijn de Sterke, R.~E.~Slusher,\nJ. Opt. Soc. Am. B {\\bf 14}, 2980 (1997).\n\n\\bibitem{tarng} S.~S.~Tarng, K.~Tai, J.~L.~Jewell, H.~M.~Gibbs, A.~C.~Gossard,\nS.~L.~McCall, A.~Passner, T.~N.~C.~Venkatesan, W.~Wiegmann,\nApp. Phys. Lett. {\\bf 40}, 205 (1982).\n\n\\bibitem{cada2} J.~He, M.~Cada, M.-A.~Dupertuis, D.~Martin,\nF.~Morier-Genoud, C.~Rolland, A.~J.~SpringThorpe, Appl. Phys. Lett. {\\bf 63},\n866 (1993).\n\n\\bibitem{taflove} Allen Taflove, {\\em Computational Electrodynamics: The \nFinite-Difference Time-Domain Method} (Artech House, Boston, 1995).\n\n\\bibitem{lidorikis1} Elefterios Lidorikis, Qiming Li, and Costas M. Soukoulis,\nPhys. Rev. B {\\bf 54}, 10249 (1996).\n\n\\bibitem{radic} Stojan Radic, Nicholas George, and Govind P.~Agrawal,\nJ. Opt. Soc. Am. B {\\bf 12}, 671 (1995).\n\n\\bibitem{hattori} Toshiaki Hattori, Noriaki Tsurumachi, and Hiroki Nakatsuka,\nJ. Opt. Soc. Am. B {\\bf 14}, 348 (1997).\n\n\\bibitem{wang} Rongzhou Wang, Jinming Dong, and D.~Y.~Xing,\nPhys. Rev. E {\\bf 55}, 6301 (1997).\n\n\\bibitem{lidorikis2} E.~Lidorikis, K.~Busch, Qiming Li, C.~T.~Chan, \nand Costas M. Soukoulis,\nPhys. Rev. B {\\bf 56}, 15090 (1997).\n \n\\end{references}\n\n\\begin{figure}\n\\psfig{figure=lidorikis_fig1_rev2.eps,width=16cm,height=16cm,angle=270}\n\\caption{(a) The linear transmission diagram. The\nsmall arrow indicates the frequency we used in the nonlinear study.\n(b) The nonlinear response: closed/(open) circles correspond\nto steady states when increasing/(decreasing) the intensity and\ncrosses/(open squares) correspond to self-pulsing states when \nincreasing/(decreasing) the intensity.\n(c) Intensity configuration for low transmission state.\n(d) Intensity configuration for high transmission state. }\n%\\label{fjkdf}\n\\end{figure}\n\n%\\begin{figure}\n%\\psfig{figure=lidorikis_fig2.eps,width=16cm,height=16cm,angle=270}\n%\\caption{(a) Switch-up and (b) switch-down dynamics. (1) Effective\n%transparent area {\\em vs} time and (2) output fields. \n%Solid and dotted lines correspond to transmitted and reflected\n%waves respectively.}\n%\\label{fjkdf}\n%\\end{figure}\n\n%\\begin{figure}\n%\\psfig{figure=lidorikis_fig3.eps,width=16cm,height=16cm,angle=270}\n%\\caption{(a) First and (b) second self-pulsing solutions.\n%(1) Effective transparent area, (2) output fields; solid/(dotted)\n%for transmitted/(reflected) waves, (3) Fourier transform of\n%output waves.}\n%\\label{fjkdf}\n%\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=lidorikis_fig2_rev2.eps,width=16cm,height=16cm,angle=0}\n\\caption{The four different switching schemes: \n(a) Final state opposite of initial,\n(b) final state always high transmission state,\n(c) final state always low transmission state,\n(d) no change of state. White areas indicate successful \noperation while black indicate failure.\n}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=lidorikis_fig3_rev2.eps,width=16cm,height=16cm,angle=270}\n\\caption{Switch-up dynamics for (a) First, (b) second and \n(c) third white curves in the ``Switch'' graph of Fig.~2.\n(1) Effectively transparent area and (2) input and output waves.}\n%\\label{fjkdf}\n\\end{figure}\n\n%\\begin{figure}\n%\\psfig{figure=lidorikis_fig4_rev.eps,width=16cm,height=16cm,angle=270}\n%\\caption{Switch-down dynamics for the first three curves of the\n%``Switch'' graph of Fig.~4. Labeling is the same as Fig.~5.}\n%\\label{fjkdf}\n%\\end{figure}\n\n%\\begin{figure}\n%\\psfig{figure=lidorikis_fig5_rev.eps,width=16cm,height=16cm,angle=0}\n%\\caption{The ``Switch'' area for four different values\n%of phase difference between CW and pulse. No overlap of these\n%curves is found.}\n%\\label{fjkdf}\n%\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=lidorikis_fig4_rev2.eps,width=16cm,height=16cm,angle=270}\n\\caption{Linear lattice with a nonlinear impurity layer: \n(a) Linear transmission diagrams, (b) nonlinear\nresponse; solid/(open) circles for increasing/(decreasing)\nCW intensity, (c) output waves during switch-up and \n(d) switch-down, (e) intensity configuration for the low transmission\nstate and (f) high transmission state.}\n%\\label{fjkdf}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=lidorikis_fig5_rev2.eps,width=16cm,height=16cm,angle=0}\n\\caption{The four switching schemes described in Fig.~2 for the\ncase of a linear lattice with a nonlinear impurity layer.\nNo simple curved structure is found here. }\n%\\label{fjkdf}\n\\end{figure}\n\n\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"
}
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[
{
"name": "cond-mat0002103.extracted_bib",
"string": "\\bibitem[*]{lido} Present address: Concurrent Computing Laboratory\nfor Materials Simulation (CCLMS) and Department of Physics and Astronomy,\nLouisiana State University, Baton Rouge, LA 70803.\n\n\n\\bibitem{newell_book} A.~C.~Newell and J.~V.~Moloney, {\\em Nonlinear \nOptics} (Addison-Wesley, Redwood CA, 1992).\n\n\n\\bibitem{gibbs_book} H.~M.~Gibbs, {\\em Optical Bistability: Controlling\nLight with Light} (Academic, Orlando FL, 1985).\n\n\n\\bibitem{winful} H.~G.~Winful, J.~H.~Marburger, and E.~Garmire,\nAppl. Phys. Lett {\\bf 35}, 379 (1979); H.~G.~Winful, and \nG.~D.~Cooperman, Appl. Phys. Lett {\\bf 40}, 298 (1982).\n\n\n\\bibitem{soukoulis} {\\em Photonic Band Gaps and Localization}, edited\nby C.~M.~Soukoulis (Plenum, New York, 1993); {\\em Photonic Band Gap\nMaterials}, edited by C.~M.~Soukoulis (Klumer, Dordrecht, 1996).\n\n\n\\bibitem{chen} Wei Chen and D.~L.~Mills, Phys. Rev. B {\\bf 36},\n6269 (1987); Phys. Rev. Lett. {\\bf 58}, 160 (1987).\n\n\n\\bibitem{desterke2} C.~Martijn de Sterke and J.~E.~Sipe, in \n{\\em Progress in Optics}, edited by E.~Wolf (Elsevier, Amsterdam, 1994),\nVol. 33.\n\n\n\\bibitem{mills} D.~L.~Mills and S.~E.~Trullinger, Phys. Rev. B {\\bf 36},\n947 (1987).\n\n\n\\bibitem{desterke} C.~Martijn de Sterke and J.~E.~Sipe, Phys. Rev. A {\\bf 42},\n2858 (1990).\n\n\n\\bibitem{wabnitz} A.~B.~Aceves, C.~De~Angelis and S.~Wabnitz, Opt. Lett. \n{\\bf 17}, 1566 (1992).\n\n\n\\bibitem{john} Sajeev John and Ne\\c{s}et Ak\\\"{o}zbek, \nPhys. Rev. Lett. {\\bf 71}, 1168 (1993); Ne\\c{s}et Ak\\\"{o}zbek and\nSajeev John, Phys. Rev. E {\\bf 57}, 2287 (1998).\n\n\n\\bibitem{dowling} Michael Scalora, Jonathan P.~Dowling, Charles M.~Bowden,\nand Mark J.~Bloemer, Phys. Rev. Lett. {\\bf 73}, 1368 (1994).\n\n\n\\bibitem{tran} P.~Tran, Opt. Lett. {\\bf 21}, 1138 (1996).\n\n\n\\bibitem{sankey} N.~D.~Sankey, D.~F.~Prelewitz, and T.~G.~Brown,\nAppl. Phys. Lett. {\\bf 60}, 1427 (1992).\n\n\n\n\\bibitem{eggleton} B.~J.~Eggleton, C.~Martijn de Sterke, R.~E.~Slusher,\nJ. Opt. Soc. Am. B {\\bf 14}, 2980 (1997).\n\n\n\\bibitem{tarng} S.~S.~Tarng, K.~Tai, J.~L.~Jewell, H.~M.~Gibbs, A.~C.~Gossard,\nS.~L.~McCall, A.~Passner, T.~N.~C.~Venkatesan, W.~Wiegmann,\nApp. Phys. Lett. {\\bf 40}, 205 (1982).\n\n\n\\bibitem{cada2} J.~He, M.~Cada, M.-A.~Dupertuis, D.~Martin,\nF.~Morier-Genoud, C.~Rolland, A.~J.~SpringThorpe, Appl. Phys. Lett. {\\bf 63},\n866 (1993).\n\n\n\\bibitem{taflove} Allen Taflove, {\\em Computational Electrodynamics: The \nFinite-Difference Time-Domain Method} (Artech House, Boston, 1995).\n\n\n\\bibitem{lidorikis1} Elefterios Lidorikis, Qiming Li, and Costas M. Soukoulis,\nPhys. Rev. B {\\bf 54}, 10249 (1996).\n\n\n\\bibitem{radic} Stojan Radic, Nicholas George, and Govind P.~Agrawal,\nJ. Opt. Soc. Am. B {\\bf 12}, 671 (1995).\n\n\n\\bibitem{hattori} Toshiaki Hattori, Noriaki Tsurumachi, and Hiroki Nakatsuka,\nJ. Opt. Soc. Am. B {\\bf 14}, 348 (1997).\n\n\n\\bibitem{wang} Rongzhou Wang, Jinming Dong, and D.~Y.~Xing,\nPhys. Rev. E {\\bf 55}, 6301 (1997).\n\n\n\\bibitem{lidorikis2} E.~Lidorikis, K.~Busch, Qiming Li, C.~T.~Chan, \nand Costas M. Soukoulis,\nPhys. Rev. B {\\bf 56}, 15090 (1997).\n \n"
}
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cond-mat0002105
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Soft Phonon Anomalies in the Relaxor Ferroelectric Pb(Zn$_{1/3}$Nb$_{2/3}$)$_{0.92}$Ti$_{0.08}$O$_3$
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[
{
"author": "P.\\ M.\\ Gehring$^{(1)}$"
},
{
"author": "S.\\ -E.\\ Park$^{(2)}$"
},
{
"author": "G.\\ Shirane$^{(3)}$"
}
] |
Neutron inelastic scattering measurements of the polar TO phonon mode dispersion in the cubic relaxor Pb(Zn$_{1/3}$Nb$_{2/3}$)$_{0.92}$Ti$_{0.08}$O$_3$ at 500~K reveal anomalous behavior in which the optic branch appears to drop precipitously into the acoustic branch at a finite value of the momentum transfer $q = 0.2$~\AA$^{-1}$ measured from the zone center. We speculate this behavior is the result of nanometer-sized polar regions in the crystal.
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[
{
"name": "prl.tex",
"string": "% VERSION 1/28/2000\n% \\documentstyle[preprint,prl,aps]{revtex}\n\\documentstyle[aps,prl,twocolumn,epsf]{revtex}\n\\begin{document}\n\\input{psfig}\n\\draft\n%\\preprint{To be submitted to Physical Review Letters}\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname \n@twocolumnfalse\\endcsname \n\n\\title{Soft Phonon Anomalies in the Relaxor Ferroelectric\n Pb(Zn$_{1/3}$Nb$_{2/3}$)$_{0.92}$Ti$_{0.08}$O$_3$}\n\n\\author{ P.\\ M.\\ Gehring$^{(1)}$, S.\\ -E.\\ Park$^{(2)}$, G.\\ Shirane$^{(3)}$ }\n\n\\address{ $^{(1)}$NIST Center for Neutron Research, National Institute\nof Standards and Technology, Gaithersburg, Maryland 20899 }\n\n\\address{ $^{(2)}$Materials Research Laboratory, The Pennsylvania\nState University, University Park, Pennsylvania 16802 }\n\n\\address{ $^{(3)}$Physics Department, Brookhaven National Laboratory,\nUpton, New York 11973 }\n\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\n Neutron inelastic scattering measurements of the polar TO phonon\n mode dispersion in the cubic relaxor\n Pb(Zn$_{1/3}$Nb$_{2/3}$)$_{0.92}$Ti$_{0.08}$O$_3$ at 500~K reveal\n anomalous behavior in which the optic branch appears to drop\n precipitously into the acoustic branch at a finite value of the\n momentum transfer $q = 0.2$~\\AA$^{-1}$ measured from the zone\n center. We speculate this behavior is the result of nanometer-sized\n polar regions in the crystal.\n\\end{abstract}\n\n\\pacs{PACS numbers: 77.84.Dy, 63.20.Dj, 77.80.Bh, 64.70.Kb }\n\n]\n\nThe discovery by Kuwata {\\it et al.} in 1982 that it was possible to\nproduce single crystals of the relaxor-ferroelectric material\nPb(Zn$_{1/3}$Nb$_{2/3}$)$_{1-x}$Ti$_x$O$_3$ represented an important\nachievement in the field of ferroelectrics \\cite{Kuwata}. Because the\nparent compounds Pb(Zn$_{1/3}$Nb$_{2/3}$)O$_3$ (PZN) and PbTiO$_3$\n(PT) form a solid solution, it was possible to tune the stoichiometry\nof the material to lie near the morphotropic phase boundary (MPB) that\nseparates the rhombohedral and tetragonal regions of the phase diagram\n\\cite{Kuwata,Park1,Park2}. Such MPB compositions in\nPb(Zr$_{1-x}$Ti$_x$)O$_3$ (PZT), the material of choice for the\nfabrication of high-performance electromechanical actuators, exhibit\nexceptional piezoelectric properties, and have generated much\nscientific study \\cite{Noheda}. However, in contrast to PZN-$x$PT,\nall attempts to date to grow large single crystals of PZT near the MPB\nhave failed, and this has impeded progress in fully characterizing the\nPZT system.\n\nThe dielectric and piezoelectric properties of single crystals of both\nPZN-$x$PT and PMN-$x$PT (M = Mg) have since been examined by Park {\\it\n et al.} who measured the strain as a function of applied electric\nfield \\cite{Park1,Park2}. These materials were found to exhibit\nremarkably large piezoelectric coefficients $d_{33} > 2500$~pC/N and\nstrain levels $S \\sim$ 1.7\\% for rhombohedral crystals oriented along\nthe pseudo-cubic [001] direction. This level of strain represents an\norder of magnitude increase over that presently achievable by\nconventional piezoelectric and electrostrictive ceramics including\nPZT. That these ultrahigh strain levels can be achieved with nearly\nno dielectric loss ($< 1$\\%) due to hysteresis suggests both PMN-$x$PT\nand PZN-$x$PT hold promise in establishing the next generation of\nsolid state transducers \\cite{Service}. A very recent theoretical\nadvance in our understanding of these materials occurred when it was\nshown using first principles calculations that the {\\it intrinsic}\npiezoelectric coefficient $e_{33}$ of MPB PMN-40\\%PT was dramatically\nenhanced relative to that for PZT by a factor of 2.7 \\cite{Bellaiche}.\nMotivated by these experimental and theoretical results, we have\nstudied the dynamics of the soft polar optic phonon mode in a high\nquality single crystal of PZN-8\\%PT, for which the measured value of\n$d_{33}$ is a maximum, using neutron inelastic scattering methods.\n\nIn prototypical ferroelectric systems such as PbTiO$_3$ it is well\nknown that the condensation or softening of a zone-center transverse\noptic (TO) phonon is responsible for the transformation from a cubic\nparaelectric phase to a tetragonal ferroelectric phase. This is\nreadily seen in neutron inelastic scattering measurements made at\nseveral temperatures above the Curie temperature. In the top panel of\nFig.~1 we show the dispersion of the lowest-energy TO branch in\nPbTiO$_3$ where at 20~K above $T_c$ the zone center ($\\zeta = 0$)\nenergy has fallen to 3~meV \\cite{Shirane}.\n\nIn relaxor compounds, however, there is a built-in disorder that\nproduces a diffuse phase transition in which the dielectric\npermittivity $\\epsilon$ exhibits a broad maximum as a function of\ntemperature at $T_{max}$. In the case of PMN and PZN, both of which\nhave the simple $ABO_3$ perovskite structure, the disorder results\nfrom the $B$-site being occupied by ions of differing valence (either\nMg$^{2+}$ or Zn$^{2+}$, and Nb$^{5+}$). This breaks the translational\nsymmetry of the crystal. Despite years of intensive research, the\nphysics of the observed diffuse phase transition is still not well\nunderstood \\cite{Westphal,Colla,Blinc}. Moreover, it is interesting\nto note that no definitive evidence for a soft mode has been found in\nthese systems. The bottom panel of Fig.~1 shows neutron scattering\ndata taken by Naberezhnov {\\it et al.} on PMN \\cite{Naberezhnov}\nexactly analogous to that shown in the top panel for PbTiO$_3$, except\nthat the temperature is $\\sim 570$~K higher than $T_{max}$.\n\nA seminal model for the disorder inherent to relaxors was first\nproposed by Burns and Dacol in 1983 \\cite{Burns}. Using measurements\nof the optic index of refraction on both ceramic samples of\n(Pb$_{1-3x/2}$La$_x$)(Zr$_y$Ti$_{1-y}$)O$_3$ (PLZT) and single\ncrystals of PMN and PZN \\cite{Burns}, they demonstrated that a\nrandomly-oriented local polarization $P_d$ develops at a well-defined\ntemperature $T_d$, frequently referred to as the Burns temperature,\nseveral hundred degrees above the apparent transition temperature\n$T_{max}$. Subsequent studies have provided additional evidence of\nthe existence of $T_d$ \\cite{Mathan,Bokov,Zhao}. The spatial extent\nof these locally polarized regions was conjectured to be $\\sim$\nseveral unit cells, and has given rise to the term ``polar\nmicro-regions,'' or PMR \\cite{Tsurumi}. For PZN-8\\%PT, the formation\nof PMR occurs at $T_d \\sim$ 700~K, well above the cubic-to-tetragonal\nphase transition at $T_c \\sim$ 450~K. We find striking anomalies in\nthe TO phonon branch (the same branch that goes soft at the zone\ncenter at $T_c$ in PbTiO$_3$) that we speculate are directly caused by\nthese PMR.\n\n% \n%==============================Fig.1==================================\n\\vspace{0.15in} \n\\noindent \n\\parbox[b]{3.4in}{ \\psfig{file=Fig1xv.ps,width=2.85in} {Fig.~1. \\small\n Top - Dispersion of the lowest energy TO mode and the TA mode in\n PbTiO$_3$, measured just above $T_c$ (from Ref.\\ \\cite{Shirane}).\n Bottom - Dispersion curves of the equivalent modes in PMN measured\n far above $T_g$ (from Ref.\\ \\cite{Naberezhnov}). }}\n\\vspace{0.05in}\n%=====================================================================\n% \n\nAll of the neutron scattering experiments were performed on the BT2\nand BT9 triple-axis spectrometers located at the NIST Center for\nNeutron Research. The (002) reflection of highly-oriented pyrolytic\ngraphite (HOPG) was used to monochromate and analyze the incident and\nscattered neutron beams. An HOPG transmission filter was used to\neliminate higher-order neutron wavelengths. The majority of our data\nwere taken holding the final neutron energy $E_f$ fixed at 14.7~meV\n($\\lambda_f = 2.36$~\\AA) while varying the incident neutron energy\n$E_i$, and using horizontal beam collimations\n60$'$-40$'$-S-40$'$-40$'$. The single crystal of PZN-8\\% PT used in\nthis study weighs 2.8 grams and was grown using the high-temperature\nflux technique described elsewhere \\cite{Park2}. The crystal was\nmounted onto an aluminum holder and oriented with the either the cubic\n[$\\bar{1}$10] or [001] axis vertical. It was then placed inside a\nvacuum furnace capable of reaching temperatures up to 670~K.\n\n%Inelastic measurements were made holding either the final neutron\n%energy $E_f$ fixed at 14.7~meV ($\\lambda_f = 2.36$~\\AA) while varying\n%the incident neutron energy $E_i$, or by holding the incident energy\n%fixed at 30.5~meV ($\\lambda_i = 1.64$~\\AA) and varying the final\n%energy. Typical horizontal beam collimations used were\n%60$'$-40$'$-40$'$-40$'$ and 40$'$-48$'$-48$'$-80$'$. \n\n% \n%==============================Fig.2==================================\n\\vspace{0.15in} \n\\noindent \n\\parbox[b]{3.4in}{ \\psfig{file=Fig2xv.ps,width=2.95in} {Fig.~2. \\small\n Solid dots represent positions of peak scattered neutron intensity\n taken from constant-$\\vec{Q}$ and constant-E scans at 500~K along\n both [110] and [001] symmetry directions. Vertical (horizontal)\n bars represent phonon FWHM linewidths in $\\hbar \\omega$ ($q$).\n Solid lines are guides to the eye indicating the TA and TO phonon\n dispersions. }}\n\\vspace{0.05in}\n%=====================================================================\n% \n\nTwo types of scans were used to collect data. Constant energy scans\nwere performed by keeping the energy transfer $\\hbar \\omega = \\Delta E\n= E_f - E_i$ fixed while varying the momentum transfer $\\vec{Q}$.\nConstant-$\\vec{Q}$ scans were performed by holding the momentum\ntransfer $\\vec{Q} = \\vec{k_f} - \\vec{k_i}$ ($k = 2\\pi/\\lambda$) fixed\nwhile varying the energy transfer $\\Delta E$. Using these scans, the\ndispersions of both the transverse acoustic (TA) and the lowest-energy\ntransverse optic (TO) phonon modes were mapped out at a temperature of\n500~K (still in the cubic phase, but well below the Burns temperature\nof $\\sim$ 700~K). In Fig.~2 we plot where the peak in the scattered\nneutron intensity occurs as a function of $\\hbar \\omega$ and\n$\\vec{q}$, where $\\vec{q} = \\vec{Q} - \\vec{G}$ is the momentum\ntransfer measured relative to the $\\vec{G} = (2,2,0)$ and $(4,0,0)$\nBragg reflections along the symmetry directions [001] and [110],\nrespectively. The horizontal scales of the left and right halves of\nthe figure have been adjusted so that each corresponds to the same $q$\n(\\AA$^{-1}$) per unit length. The sizes of the vertical and\nhorizontal bars represent the phonon FWHM (full width at half maximum)\nlinewidths in $\\hbar \\omega$ (meV) and $q$ (\\AA$^{-1}$), respectively,\nand were derived from Gaussian least-squares fits to the\nconstant-$\\vec{Q}$ and constant-$E$ scans. The lowest energy data\npoints trace out the TA phonon branch along [110] and [001]. Solid\nlines have been drawn through these points as a guide to the eye, and\nare nearly identical to that shown for PMN in Fig.~1.\n\nBy far the most striking feature in Fig.~2 is the unexpected collapse\nof the TO mode near the zone center where the polar optic branch\nappears to drop precipitously, like a waterfall, into the acoustic\nbranch. This anomalous behavior, shown by the shaded regions in\nFig.~2, stands in stark contrast to that of PMN at high temperature\nwhere the same phonon branch intercepts the $\\hbar \\omega$-axis at a\nfinite energy (see bottom panel of Fig.~1). The strange drop in the\nTO phonon energy occurs for $q \\sim 0.13$ r.l.u. measured along [001],\nand for $q \\sim 0.08$ r.l.u. measured along [110] (1 r.l.u. = $2\\pi/a$\n= 1.54~\\AA$^{-1}$). It is quite intriguing to note that these\n$q$-values are both approximately equal to 0.2 \\AA$^{-1}$.\n\nTo clarify the nature of this unusual observation, we show an extended\nconstant-$E$ scan taken at $\\Delta E = 6$~meV in Fig.~3 along with a\nconstant-$\\vec{Q}$ scan in the insert. Both scans were taken at the\nsame temperature of 500~K, near the (2,2,0) Bragg peak, and along the\n[001] direction. The small horizontal bar shown under the left peak\nof the constant-$E$ scan represents the instrumental FWHM\n$q$-resolution, and is clearly far smaller than the instrinsic peak\nlinewidth. We see immediately that the constant-$\\vec{Q}$ scan shows\nno evidence of any well-defined phonon peak, most likely because the\nphonons near the zone center are overdamped. However, the\nconstant-$E$ scan indicates the presence of a ridge of scattering\nintensity at $\\zeta = q = 0.13$~r.l.u., or about 0.2~\\AA$^{-1}$, that\nsits atop the scattering associated with the overdamped phonons. Thus\nthe sharp drop in TO branch that appears to take place in Fig.~2 does\nnot correspond to a real dispersion curve as such. Rather, it simply\nindicates a region of ($\\hbar \\omega, q$)-space in which the phonon\nscattering cross section is enhanced. The origin of this enhancement\nis unknown, however we speculate that it is a direct result of the PMR\ndescribed by Burns and Dacol \\cite{Burns}. If the length scale\nassociated with this enhancement is of order $2\\pi/q$, this\ncorresponds to $\\sim 31$~\\AA, or about 7 to 8 unit cells, consistent\nwith Burns and Dacol's conjecture.\n\n% \n%==============================Fig.3==================================\n\\vspace{0.15in} \n\\noindent \n\\parbox[b]{3.4in}{ \\psfig{file=Fig3xv.ps,width=3.15in} {Fig.~3. \\small\n Single constant-E scan measured along [001] at 6~meV at 500~K near\n the (2,2,0) Bragg peak. Solid line is a fit to a double Gaussian\n function of $\\zeta$. The inset shows no peak in the scattered\n intensity measured along the energy axis. The arrow indicates the\n position of the constant-E scan. }}\n\\vspace{0.05in}\n%=====================================================================\n% \n\nLimited data were also taken as a function of temperature to determine\nthe effect on this anomalous ridge of scattering. In Fig.~4 we show\ntwo constant-$E$ scans, both measured at an energy transfer $\\Delta E\n= 5$~meV along the [010] direction, with one taken at 450~K, and the\nother at 600~K. The solid and dashed lines are fits to simple\nGaussian functions of $q$. As is clearly seen, the ridge of\nscattering shifts to smaller $q$, i.e. {\\it towards} the zone center,\nwith increasing temperature. These data strongly suggest a picture,\nshown schematically in the inset to Fig.~4, in which the ridge of\nscattering evolves into the expected classic TO phonon branch behavior\nat higher temperature. A single data point, obtained briefly at 670~K\nto avoid damaging the crystal, is plotted in the inset to Fig.~4, and\ntentatively corroborates this picture.\n\nWe have discovered an anomalous enhancement of the polar TO phonon\nscattering cross section that occurs at a special value of $q =\n0.2$~\\AA$^{-1}$, independent of whether we measure along the [001] or\n[110] direction. We believe this to be direct microscopic evidence of\nthe PMR proposed by Burns and Dacol \\cite{Burns}. The presence of\nsuch small polarized regions of the crystal above $T_c$ should\neffectively prevent the propagation of long-wavelength ($q \\rightarrow\n0$) soft mode phonons. A similar conclusion was reached by Tsurumi\n{\\it et al.} based on dielectric measurements of PMN \\cite{Tsurumi}.\nThe observation that the phonon scattering cross section is enhanced\n0.2~\\AA$^{-1}$ from the zone center gives a measure of the size of the\nPMR consistent with the estimates of Burns and Dacol. If true, then\nthis unusual behavior should be observed in other related relaxor\nsystems. Indeed, tentative evidence for this has already been\nobserved at room temperature in neutron scattering measurements on PMN\n\\cite{Gehring}. This enhancement should also be reflected in x-ray\ndiffuse scattering intensities (\\cite{Vakhrushev,You}), although it\nmay be masked by the superposition of strong acoustic modes.\n\n% \n%==============================Fig.4==================================\n\\vspace{0.15in} \n\\noindent \n\\parbox[b]{3.4in}{ \\psfig{file=Fig4xv.ps,width=2.95in} {Fig.~4. \\small\n Two constant-E scans measured along [010] at 5~meV at different\n temperatures. The peak shifts towards the zone center with\n increasing temperature. The inset suggests schematically how the\n TO branch dispersion recovers at higher temperatures. }}\n\\vspace{0.05in}\n%=====================================================================\n% \n\nOur picture is not yet complete. Whereas Fig.~3 demonstrates that\nthese anomalies appear as ridges on top of a broad overdamped cross\nsection, the complete nature of this cross section can only be\nrevealed by an extensive contour map of the Brillouin zone, for which\nwe lack sufficient data. Another important aspect which requires\nfurther study is exactly how the ``waterfall\" evolves, at much higher\ntemperatures, into the standard optic mode dispersion as shown in\nFig.~1 for PMN. We have not yet carried out this experiment because\nof the concern of possible crystal deterioration at these high\ntemperatures under vacuum \\cite{Park2}. We intend to do so only after\nall other key experiments have been completed \\cite{X}.\n\nOur current picture suggests that the TO phonon dispersion should\nchange if one alters the state of the PMR. It is known that a macro\nferroelectric phase can be created in these relaxor crystals by\ncooling the crystal in a field, or by application, at room\ntemperature, of a sufficiently strong field. We are now planning\nneutron inelastic measurements on such a crystal, as well as on PZN.\n\nWe thank S.\\ Vakhrushev, S.\\ Wakimoto, as well as D.\\ E.\\ Cox, L.\\ E.\\ \nCross, R.\\ Guo, B.\\ Noheda, N.\\ Takesue, and G.\\ Yong for stimulating\ndiscussions. Financial support by the U.\\ S.\\ Dept.\\ of Energy under\ncontract No.\\ DE-AC02-98CH10886, by ONR under project MURI\n(N00014-96-1-1173), and under resource for piezoelectric single\ncrystals (N00014-98-1-0527) is acknowledged. We acknowledge the\nsupport of NIST, U.\\ S.\\ Dept.\\ of Commerce, in providing the neutron\nfacilities used in this work.\n\n\n\n%\\newpage\n\n%\\begin{figure}\n%\\centerline{\\psfig{figure=PRLFIG1.eps,width=110mm}}\n%\\caption{ Top panel - dispersion of the lowest TO mode and the TA mode\n%in PbTiO$_3$ taken 20~K above $T_c$ from Ref.\\ \\cite{Shirane}. Bottom\n%panel - dispersion curves for the equivalent modes in PMN at very high\n%temperature, 570~K above $T_g$, taken from Fig.~1 of Ref.\\\n%\\cite{Naberezhnov}. }\n%\\end{figure}\n\n%\\begin{figure}\n%\\centerline{\\psfig{figure=PRLFIG2.eps,width=110mm}}\n%\\caption{Data taken from constant-$\\vec{Q}$ and constant-E scans on\n%PZN-8PT at 500~K along both [110] and [001] symmetry directions.\n%Solid dots represent positions of peak scattered neutron intensity.\n%Vertical (horizontal) bars represent the phonon FWHM linewidths in\n%$\\hbar \\omega$ ($q$). The solid lines are guides to the eye\n%indicating the TA phonon branch dispersion.}\n%\\end{figure}\n\n%\\begin{figure}\n%\\centerline{\\psfig{figure=PRLFIG3.eps,width=110mm}}\n%\\caption{Single constant-E scan measured along [001] at 6~meV at 500~K\n%taken near the (2,2,0) Bragg peak. Solid line is a fit to a double\n%Gaussian function of $\\zeta$. The inset shows the absence of any peak\n%in the scattered intensity when measured along the energy axis. The\n%arrow indicates the position of the constant-E scan.}\n%\\end{figure}\n\n%\\begin{figure}\n%\\centerline{\\psfig{figure=PRLFIG4.eps,width=110mm}}\n%\\caption{Two constant-E scans measured along [010] at 5~mev at\n%different temperatures. Note the shift of the peak towards the zone\n%center with increasing temperature. The inset suggests schematically\n%how the TO branch dispersion recovers at higher temperatures. Solid\n%dot represents actual data point at 670~K. }\n%\\end{figure}\n\n%\\newpage\n\n\\begin{references}\n\n\\bibitem{Kuwata} J.\\ Kuwata, K.\\ Uchino, and S.\\ Nomura,\n Ferroelectrics {\\bf 37}, 579 (1981); {\\it ibid}. Jpn.\\ J.\\ Appl.\\ \n Phys.\\ {\\bf 21}, 1298 (1982).\n\n\\bibitem{Park1} S.-E.\\ Park and T.\\ R.\\ Shrout, J.\\ Appl.\\ Phys.\\ {\\bf\n 82}, 1804 (1997).\n\n\\bibitem{Park2} S.\\ -E.\\ Park, M.\\ L.\\ Mulvihill, G.\\ Risch, and T.\\ \n R.\\ Shrout, Jpn.\\ J.\\ Appl.\\ Phys.\\ 1 {\\bf 36}, 1154 (1997).\n\n\\bibitem{Noheda} See, for example, B.\\ Noheda {\\it et al.}, Appl.\\ \n Phys.\\ Lett.\\ {\\bf 74}, 2059 (1999).\n\n% D.\\ E.\\ Cox, G.\\ Shirane, J.\\ A.\\ Gonzalo, L.\\ E.\\ Cross, and S.\\ \n% -E.\\ Park,\n\n\\bibitem{Service} R.\\ F.\\ Service, Science {\\bf 275}, 1878 (1997).\n\n\\bibitem{Bellaiche} L.\\ Bellaiche, J.\\ Padilla, and D.\\ Vanderbilt,\n Phys.\\ Rev.\\ B {\\bf 59}, 1834 (1999); L.\\ Bellaiche and D.\\ \n Vanderbilt, Phys.\\ Rev.\\ Lett.\\ {\\bf 83}, 1347 (1999).\n\n\\bibitem{Shirane} G.\\ Shirane, J.\\ D.\\ Axe, J.\\ Harada, and J.\\ P.\\ \n Remeika, Phys.\\ Rev.\\ Lett.\\ {\\bf 2}, 155 (1970).\n\n\\bibitem{Westphal} V. Westphal, W.\\ Kleeman, and M.\\ D.\\ Glinchuk,\n Phys.\\ Rev.\\ Lett.\\ {\\bf 68}, 847 (1992).\n\n\\bibitem{Colla} E.\\ V.\\ Colla, E.\\ Yu.\\ Koroleva, N.\\ M.\\ Okuneva, and\n S.\\ B.\\ Vakhrushev, Phys.\\ Rev.\\ Lett.\\ {\\bf 74}, 1681 (1995).\n\n\\bibitem{Blinc} R.\\ Blinc {\\it et al.}, Phys.\\ Rev.\\ Lett. {\\bf 83},\n 424 (1999).\n\n\\bibitem{Naberezhnov} A.\\ Naberezhnov, S.\\ Vakhrushev, B.\\ Dorner, and\n H.\\ Moudden, Eur.\\ Phys.\\ J.\\ B {\\bf 11}, 13 (1999).\n\n\\bibitem{Burns} G.\\ Burns and F.\\ H.\\ Dacol, Phys.\\ Rev.\\ B {\\bf 28},\n 2527 (1983); {\\it ibid}. Sol.\\ Stat.\\ Comm.\\ {\\bf 48}, 853, (1983).\n\n\\bibitem{Tsurumi} T.\\ Tsurumi, K.\\ Soejima, T.\\ Kamiya, and M.\\ \n Daimon, Jpn.\\ J.\\ Appl.\\ Phys.\\ Part 1 {\\bf 33}, 1959 (1994).\n\n\\bibitem{Mathan} N.\\ de Mathan {\\it et al.}, J.\\ Phys.\\ Condens.\\ \n Matter {\\bf 3}, 8159 (1991).\n%E.\\ Husson, C.\\ Calvarin, J.\\ Gavarri,\n% A.\\ Hewat, and A.\\ Morell, \n\n\\bibitem{Bokov} A.\\ A.\\ Bokov, Ferroelectrics {\\bf 131}, 49 (1992).\n\n\\bibitem{Zhao} J.\\ Zhao, A.\\ E.\\ Glazounov, Q.\\ M.\\ Zhang, and B.\\ \n Toby, Appl.\\ Phys.\\ Lett.\\ {\\bf 72}, 1048 (1998).\n\n\\bibitem{Gehring} P.\\ M.\\ Gehring, S.\\ Vakhrushev, and G.\\ Shirane,\n to be published.\n\n\\bibitem{Vakhrushev} S.\\ Vakhrushev, A.\\ Naberezhnov, S.\\ K.\\ Sinha,\n Y.\\ -P.\\ Feng, and T.\\ Egami, J.\\ Phys.\\ Chem.\\ Solids {\\bf 57},\n 1517 (1996).\n\n\\bibitem{You} H.\\ You and Q.\\ M.\\ Zhang, Phys.\\ Rev.\\ Lett.\\ {\\bf 79},\n 3950 (1997).\n\n\\bibitem{X} The PMN data below $q = 0.1$ r.l.u. at high temperature\n were interpreted in terms of a mode coupling model by Naberezhnov\n {\\it et al.} \\cite{Naberezhnov}. Our current data do not cover the\n same $\\hbar \\omega ,q$-range for the PZN-8\\%PT sample.\n\n\\end{references}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002105.extracted_bib",
"string": "\\bibitem{Kuwata} J.\\ Kuwata, K.\\ Uchino, and S.\\ Nomura,\n Ferroelectrics {\\bf 37}, 579 (1981); {\\it ibid}. Jpn.\\ J.\\ Appl.\\ \n Phys.\\ {\\bf 21}, 1298 (1982).\n\n\n\\bibitem{Park1} S.-E.\\ Park and T.\\ R.\\ Shrout, J.\\ Appl.\\ Phys.\\ {\\bf\n 82}, 1804 (1997).\n\n\n\\bibitem{Park2} S.\\ -E.\\ Park, M.\\ L.\\ Mulvihill, G.\\ Risch, and T.\\ \n R.\\ Shrout, Jpn.\\ J.\\ Appl.\\ Phys.\\ 1 {\\bf 36}, 1154 (1997).\n\n\n\\bibitem{Noheda} See, for example, B.\\ Noheda {\\it et al.}, Appl.\\ \n Phys.\\ Lett.\\ {\\bf 74}, 2059 (1999).\n\n% D.\\ E.\\ Cox, G.\\ Shirane, J.\\ A.\\ Gonzalo, L.\\ E.\\ Cross, and S.\\ \n% -E.\\ Park,\n\n\n\\bibitem{Service} R.\\ F.\\ Service, Science {\\bf 275}, 1878 (1997).\n\n\n\\bibitem{Bellaiche} L.\\ Bellaiche, J.\\ Padilla, and D.\\ Vanderbilt,\n Phys.\\ Rev.\\ B {\\bf 59}, 1834 (1999); L.\\ Bellaiche and D.\\ \n Vanderbilt, Phys.\\ Rev.\\ Lett.\\ {\\bf 83}, 1347 (1999).\n\n\n\\bibitem{Shirane} G.\\ Shirane, J.\\ D.\\ Axe, J.\\ Harada, and J.\\ P.\\ \n Remeika, Phys.\\ Rev.\\ Lett.\\ {\\bf 2}, 155 (1970).\n\n\n\\bibitem{Westphal} V. Westphal, W.\\ Kleeman, and M.\\ D.\\ Glinchuk,\n Phys.\\ Rev.\\ Lett.\\ {\\bf 68}, 847 (1992).\n\n\n\\bibitem{Colla} E.\\ V.\\ Colla, E.\\ Yu.\\ Koroleva, N.\\ M.\\ Okuneva, and\n S.\\ B.\\ Vakhrushev, Phys.\\ Rev.\\ Lett.\\ {\\bf 74}, 1681 (1995).\n\n\n\\bibitem{Blinc} R.\\ Blinc {\\it et al.}, Phys.\\ Rev.\\ Lett. {\\bf 83},\n 424 (1999).\n\n\n\\bibitem{Naberezhnov} A.\\ Naberezhnov, S.\\ Vakhrushev, B.\\ Dorner, and\n H.\\ Moudden, Eur.\\ Phys.\\ J.\\ B {\\bf 11}, 13 (1999).\n\n\n\\bibitem{Burns} G.\\ Burns and F.\\ H.\\ Dacol, Phys.\\ Rev.\\ B {\\bf 28},\n 2527 (1983); {\\it ibid}. Sol.\\ Stat.\\ Comm.\\ {\\bf 48}, 853, (1983).\n\n\n\\bibitem{Tsurumi} T.\\ Tsurumi, K.\\ Soejima, T.\\ Kamiya, and M.\\ \n Daimon, Jpn.\\ J.\\ Appl.\\ Phys.\\ Part 1 {\\bf 33}, 1959 (1994).\n\n\n\\bibitem{Mathan} N.\\ de Mathan {\\it et al.}, J.\\ Phys.\\ Condens.\\ \n Matter {\\bf 3}, 8159 (1991).\n%E.\\ Husson, C.\\ Calvarin, J.\\ Gavarri,\n% A.\\ Hewat, and A.\\ Morell, \n\n\n\\bibitem{Bokov} A.\\ A.\\ Bokov, Ferroelectrics {\\bf 131}, 49 (1992).\n\n\n\\bibitem{Zhao} J.\\ Zhao, A.\\ E.\\ Glazounov, Q.\\ M.\\ Zhang, and B.\\ \n Toby, Appl.\\ Phys.\\ Lett.\\ {\\bf 72}, 1048 (1998).\n\n\n\\bibitem{Gehring} P.\\ M.\\ Gehring, S.\\ Vakhrushev, and G.\\ Shirane,\n to be published.\n\n\n\\bibitem{Vakhrushev} S.\\ Vakhrushev, A.\\ Naberezhnov, S.\\ K.\\ Sinha,\n Y.\\ -P.\\ Feng, and T.\\ Egami, J.\\ Phys.\\ Chem.\\ Solids {\\bf 57},\n 1517 (1996).\n\n\n\\bibitem{You} H.\\ You and Q.\\ M.\\ Zhang, Phys.\\ Rev.\\ Lett.\\ {\\bf 79},\n 3950 (1997).\n\n\n\\bibitem{X} The PMN data below $q = 0.1$ r.l.u. at high temperature\n were interpreted in terms of a mode coupling model by Naberezhnov\n {\\it et al.} \\cite{Naberezhnov}. Our current data do not cover the\n same $\\hbar \\omega ,q$-range for the PZN-8\\%PT sample.\n\n"
}
] |
cond-mat0002106
|
Quantum Criticality at the Metal Insulator Transition
|
[
{
"author": "{D. Schmeltzer}"
},
{
"author": "Physics Department"
},
{
"author": "The City College"
},
{
"author": "CUNY"
},
{
"author": "Convent Ave. at 138 ST"
},
{
"author": "New York"
},
{
"author": "NY 10031"
},
{
"author": "USA"
}
] |
We introduce a new method to analysis the many-body problem with disorder. The method is an extension of the real space renormalization group based on the operator product expansion. We consider the problem in the presence of interaction, large elastic mean free path, and finite temperatures. As a result scaling is stopped either by temperature or the length scale set by the diverging many-body length scale (superconductivity). Due to disorder a superconducting instability might take place at $T_{SC}\rightarrow 0$ giving rise to a metallic phase or $T>T_{SC}$. For repulsive interactions at $T\rightarrow 0$ we flow towards the localized phase which is analized within the diffusive Finkelstein theory. For finite temperatures with strong repulsive backward interactions and non-spherical Fermi surfaces characterized by $|\frac{d\ln N(b)}{\ln b}|\ll 1$ one finds a fixed point $(D^*,\Gamma^*_2)$ in the plane $(D,\Gamma_2^{(s)})$. ($D\propto(K_F\ell)^{-1}$ is the disorder coupling constant, $\Gamma_2^{(s)}$ is the particle-hole triplet interaction, $b$ is the length scale and $N(b)$ is the number of channels.) For weak disorder, $D<D^*$, one obtains a metallic behavior with the resistance $\rho(D,\Gamma_2^{(s)},T)=\rho(D,\Gamma_2^{(s)},T)\simeq \rho^*f(\frac{D-D^*}{D^*}\frac{1}{T^{z\nu_1}})$ ($\rho^*=\rho(D^*,\Gamma_2^*,1)$, $z=1$, and $\nu_1>1$) in good agreement with the experiments.\\
|
[
{
"name": "cond-mat0002106.tex",
"string": "\\documentstyle[preprint,prl,aps,epsf]{revtex}\n\\begin{document}\n\\draft\n\n\\title{\nQuantum Criticality at the Metal Insulator Transition\n}\n\n\\author{\n{\\bf D. Schmeltzer}\\\\\nPhysics Department, The City College, CUNY\\\\\nConvent Ave. at 138 ST, New York, NY 10031, USA}\n\n%\\date{Received \\today}\n\\maketitle\n\\begin{abstract}\nWe introduce a new method to analysis the many-body problem with disorder.\nThe method is an extension of the real space renormalization group based\non the operator product expansion. We consider the problem in the presence\nof interaction, large elastic mean free path, and finite temperatures. As\na result scaling is stopped either by temperature or the length scale set by\nthe diverging many-body length scale (superconductivity). Due to disorder\na superconducting instability might take place at $T_{SC}\\rightarrow 0$\ngiving rise to a metallic phase or $T>T_{SC}$. For repulsive interactions\nat $T\\rightarrow 0$ we flow towards the localized phase which is analized\nwithin the diffusive Finkelstein theory. For finite temperatures with strong\nrepulsive backward interactions and non-spherical Fermi surfaces characterized\nby $|\\frac{d\\ln N(b)}{\\ln b}|\\ll 1$ one finds a fixed point $(D^*,\\Gamma^*_2)$\nin the plane $(D,\\Gamma_2^{(s)})$. ($D\\propto(K_F\\ell)^{-1}$ is the\ndisorder coupling constant, $\\Gamma_2^{(s)}$ is the particle-hole triplet\ninteraction, $b$ is the length scale and $N(b)$ is the number of channels.)\nFor weak disorder, $D<D^*$, one obtains a metallic behavior with the resistance\n$\\rho(D,\\Gamma_2^{(s)},T)=\\rho(D,\\Gamma_2^{(s)},T)\\simeq\n\\rho^*f(\\frac{D-D^*}{D^*}\\frac{1}{T^{z\\nu_1}})$\n($\\rho^*=\\rho(D^*,\\Gamma_2^*,1)$, $z=1$, and $\\nu_1>1$) in good agreement\nwith the experiments.\\\\\n\\end{abstract}\n \n\\section{Introduction}\n\\label{sec-1}\n\nThe Metal-Insulator (M-I) transition has been understood within the seminal\npaper \\cite{01} in 1979. Focusing on noninteracting electrons the authors\ndemonstrated that in two dimension (2D) even weak disorder is sufficient to\nlocalize the electrons at $T=0$. Few years later \\cite{02} it has been\nrealized by Finkelstein that the particle-hole interaction in the triplet\nchannel might enhance the conductivity. However a detailed analysis\nrevealed that at long scale the interaction term diverges making difficult\nto determine what will happen at long scales. Recently a remarkable\nexperiment \\cite{03} has been performed on a 2D electron gas in zero\nmagnetic field strongly points towards a M-I transition in two dimensions.\nThe characteristic of this experiment performed on a 2DES silicon (\n$n_s\\sim 10^{11} cm^{-2}$) the mean free path ``$\\ell$\" is large, the\nelectron-electron interaction was $\\sim 5 mev$, while the Fermi energy is\nonly $0.6 mev$. The lowest temperature in the experiment was $0.2 K$.\nThese experimental condition might suggest that the non-linear sigma model\nintroduced in ref.\\cite{02} might not be applicable since it ignores the\ninteraction effects at length scales shorter than the mean free path.\nSince the mean free path is large quantum effects in the momentum range\n$2\\pi/\\ell\\leq \\mid q\\mid \\leq \\Lambda$ ($\\Lambda^{-1}\\sim a \\sim$\nparticle separation) might be important for weak disorder,\n$\\ell\\longrightarrow\\infty$. This suggests that a phase transition due to a\ncollective many body interaction might occur before the diffusive limit is\nreached. One might have a phase transition from a superconductor to\ninsulator \\cite{04},\nWigner crystal \\cite{05,06}, or quantum Hall-insulator transition \\cite{07}.\nIn one dimension it is known that attractive interaction or ferromagnetic\nspin fluctuations can suppress the $2k_F$ backscattering leading to a\ndelocalization transition \\cite{08}. We investigate the problem in the\npresence of interaction and large mean free paths.\nIn order to clarify the situation in 2D we propose to use the Renormalization\nGroup (RG) analysis. Motivated by the fact that the mean free path\n``$\\ell$\" can be large with respect to the particle separation\n$a\\sim\\Lambda^{-1}$ (standard transport theories start at the scale\n``$\\ell$\" and investigate only processes at larger scales governed by\ndiffusion) we investigate at finite temperatures the competition between\nlocalization and interaction. The competition between multiple scattering\n(due to disorder) and the interactions is investigated within a RG theory.\nThe method used here is different from the procedure used in ref.\\cite{02}.\nIn ref.\\cite{02} one emphasizes the disorder by replacing the multiple\nelastic scattering by a diffusion theory and in the second step the\ninteractions are treated perturbatively. We consider a situation where the\nelastic mean free path is much larger than any microscopic length.\nTherefore we might have a situation that before entering the diffusive\nregion we have to stop scaling. This can happen if the thermal wave length\nis shorter than the elastic mean free path or that the Cooper channel\ndiverges giving rise to superconductivity.\nIn the quantum region the single particle\nexcitations are well-defined and the Fermi surface is parametrized in\nterms of $N_o=\\frac{\\pi k_F}{\\Lambda}$ channels. When the cutoff $\\Lambda$\nis reduced $\\Lambda \\longrightarrow \\Lambda/b$, one finds that the\ninteractions scale like $\\Gamma\\longrightarrow \\Gamma b^{1-d}$ and the\nnumber of channels, increases like $N=N_ob$ \\cite{09}. The disorder scales\nlike $D\\longrightarrow D b^{2-d}$. Due to the fact that the number of\nchannels increase under scaling, we find that the interaction is marginal\nand the disorder is relevant. The quantum region gives rise to a set of\nscaling equations for the interaction term $\\Gamma$: \n$\\Gamma_2^{(c)}$--particle-hole singlet,\n$\\Gamma_2^{(s)}$--particle-hole triplet, $\\Gamma_3^{(s)}$--particle-particle\nsinglet and disorder $D$ ($d_3^{(s)}$--the Cooperon). Our results show that\ndue to disorder $\\Gamma_3^{(s)}$ might becomes negative resulting in a\nsuperconducting instability at $T\\longrightarrow 0$. This might give\nrise to an Insulator-Superconductor transition similar to what one has\nfor superconducting films where a phase transition is\nexpected \\cite{04}. In the absence of an instability the standard method\nat length scale $b>b_{Dif}$, $b_{Dif}=\\frac{\\Lambda}{2\\pi/\\ell}$ is the\ndiffusion theory developed by Finkelstein. Here we consider the situation\nwhere the system is in the clean limit such that the microscopic mean free\npath $\\ell_o=\\ell(b=1)$ is large. Due to interaction we obtain that the\nmean free path $\\ell(b>1)$ increases, $\\ell(b)>\\ell_o$.\\\\\n\nIn this paper we will\nwork at finite temperatures such the the thermal wavelength is shorter than\nthe mean free path $\\ell$. We introduce a thermal length scale\n$b_T=\\frac{v_F\\Lambda}{T}$ and consider the situation where $b_{Dif}>b_T$.\nSince we have to stop the scaling scaling at $b=b_T$ we are allowed to ignore\nthe diffusive region. In the recent transport experiment $E_F/T\\sim 5$ and\n$K_F\\ell \\gg 5$, therefore the condition $b_{Dif}>b_T$ is realized. The\npresence of the cutoff $b_T$ prevent the number of channels to scale to\ninfinity, instead we have $N_o<N(b)\\le N(b_T)=\\bar{N}=\\frac{E_F}{T}$. We\nsolve the model under the condition $b_{Dif}>b_T$ and find that the physics\nis controlled by the disorder ``$D$\" and the particle-hole triplet\n$\\Gamma_2^{(s)}$. We find that when the number of channels does not scale\n(This might be the case at finite temperature or for non-spherical Fermi\nsurfaces, which obeys $N(b)\\simeq \\; Const.$), a fixed point in the plane\n$\\Gamma_2^{(s)}$ and $D$ is obtained. This fixed point separates a metallic\nphase from a localized one. The metallic phase is caused by the fact that\nthe particle-hole triplet flows to a stable fixed point causing a shift in the\ncritical dimension from $d=2$ to $d<2$. The presence of the stable fixed\npoint in the triplet channel causes power law behavior of the spin-spin\ncorrelations. The resistivity is expected to obey the scaling behavior:\n$\\rho(D,\\Gamma_2^{(s)},T)=\n\\rho(D(b),\\Gamma_2^{(s)}(b),Tb^z) ; \\; \\; z\\simeq 1$ where\n$\\Gamma_2^{(s)}(b)=\\Gamma_2^*+(\\Gamma_2^{(s)}-\\Gamma_2^*) b^{-1/\\nu_2}$ and\n$D(b)=D^*+(D-D^*)b^{1/\\nu_1}$. Choosing $Tb^z=T_o$ we obtain:\n$\\rho(D,\\Gamma_2^{(s)},T)\\simeq \\rho(D^*,\\Gamma_2^*,T_o)+\n{\\textstyle const.}(\\frac{D-D^*}{D^*})(\\frac{T_o}{T})^{1/z\\nu_1}$. In\nagreement with the experimental results given in ref.\\cite{01} the resistivity\nincreases for $D>D^*$ and decreases for $D<D^*$.\nIn the literature alternative theories have been proposed already:\nref.\\cite{14} (phenomenological), ref.\\cite{10} (within the Finkelstein\ntheory), as well\nas models which focus on the insulating side ref.\\cite{15,06}.\\\\\n\nThe plan of this paper is: We introduce in Chapter \\ref{sec-2} our microscopic\nmodel. We consider a two dimensional gas in the presence of a screened two-body\npotential and a static random potential. We follow a standard method for\ntreating\ndisorder. We use the ``replica\" method and perform the statistical average\nover the disorder. In the second step we parametrize the Fermi Surface (FS)\nin terms of $N$ channels. Using this parametrization we identify in Appendix\n\\ref{app-1} all the possible interaction and disorder terms. We find that the\ninteraction and disorder is best described in terms of chiral currents\ncarrying indices of charge, spin, replica, and channel.\nIn Chapter \\ref{sec-3} the method of the Renormalization Group (RG) based on\nthe Operator Product Expansion (OPE) is introduced. We compute the OPE\nrules for the different interaction terms, particle-hole (p-h) singlet,\np-h triplet, particle-particle (p-p) and the Cooperon (the effective\ninteraction induced by the disorder).\nChapter \\ref{sec-4} is devoted to the derivation of the RG equations based\non the OPE results obtained in Chapter \\ref{sec-3}.\nIn Chapter \\ref{sec-5} we consider the scaling equations in the quantum limit.\nChapter \\ref{sec-6} is devoted to the possible superconducting instability\nwhich might occur in the quantum region.\nIn Chapter \\ref{sec-7} we investigate the scaling equations at finite\ntemperatures. Here we observe that the physics is determined\nby the effective number of channels $\\bar{N}$.\nIn Chapter \\ref{sec-8} we solve the RG equations and compute the resistivity.\nChapter \\ref{sec-9} is limited to discussions and conclusions.\\\\\n\n\\section{The Microscopic Model}\n\\label{sec-2}\n\nWe introduce the screened two-body potential and perform a statistical average\nover the disorder using the replica method. We parametrize the FS in terms\nof $N$ Fermions. Using these Fermions we replace the interaction terms and\nthe Cooperon by chiral currents.\nThe starting point of our investigation is the averaged disorder \\cite{11}\npartition function, $\\bar{Z^{\\alpha}}, \\; \\alpha=1,...,\\alpha\\rightarrow 0$,\n \n\\begin{equation}\n\\label{eq-001}\n \\bar{Z^{\\alpha}}=\\int D[\\bar{\\psi},\\psi]e^{-S}, \\; \\; \\; \\; \\;\n \\alpha=1,..., \\; \\alpha\\rightarrow 0\n\\end{equation}\n\n\\begin{equation}\n\\label{eq-002}\n S_o=\\int d^dx \\int dt \\{\\sum_{\\sigma}\\sum_{\\alpha}[\n\\bar{\\psi}_{\\sigma,\\alpha} \\partial_t \\psi_{\\sigma,\\alpha}-\n\\bar{\\psi}_{\\sigma,\\alpha}(\\frac{\\nabla^2}{2m}+E_F)\\psi_{\\sigma,\\alpha}]\\}\n\\end{equation}\n \n\\begin{equation}\n\\label{eq-003}\n S_{int}=\\int d^dx\\int d^dy\\int dt \\sum_{\\sigma,\\sigma^{\\prime}}\\sum_{\\alpha}\n\\{\\bar{\\psi}_{\\sigma,\\alpha}(x)\\bar{\\psi}_{\\sigma^{\\prime},\\alpha}(y) v(x-y)\n\\psi_{\\sigma^{\\prime},\\alpha}(y)\\psi_{\\sigma,\\alpha}(x) \\}\n\\end{equation}\n \n\\begin{equation}\n\\label{eq-004}\n S_D=-\\int dt_1 \\int dt_2 \\int d^dx\\int d^dy \\sum_{\\sigma,\\sigma^{\\prime}}\n \\sum_{\\alpha,\\beta} \\{\\overline{V(x)V(y)}\n \\bar{\\psi}_{\\sigma,\\alpha}(x,t_1)\\bar{\\psi}_{\\sigma^{\\prime},\\beta}(y,t_2)\n \\psi_{\\sigma^{\\prime},\\beta}(y,t_2)\\psi_{\\sigma,\\alpha}(x,t_1) \\}\n\\end{equation}\n \n\\noindent ``$v(x-y)$\" is the two body screened potential and\n$\\overline{V(x)V(y)}=D\\delta(x-y)$ where $D=\\frac{v_F^2}{K_F\\ell}$ is the\ndisorder parameter controlled by the elastic scattering time $\\tau=\\ell/v_F$.\nNext we parametrize the Fermi surface (FS) in terms of $N$ Fermions or\n$N/2$ pairs of right and left movers ( see ref. \\cite{09} ):\n \n\\begin{equation}\n\\label{eq-005}\n \\psi_{\\sigma,\\alpha}(\\vec{x})=\\sum_{n=1}^{N/2}\n (e^{ik_F\\hat{n}\\cdot\\vec{x}}R_{n,\\sigma,\\alpha}(\\vec{x})+\n e^{-ik_F\\hat{n}\\cdot\\vec{x}}L_{n,\\sigma,\\alpha}(\\vec{x}))\n\\end{equation}\n \n\\noindent $R_{n,\\sigma,\\alpha}(\\vec{x})$ and $L_{n,\\sigma,\\alpha}(\\vec{x})$\nare right and left movers defined by momenta $\\mid q_{\\parallel}\\mid<\\Lambda$,\n$\\mid q_{\\perp}\\mid<\\Lambda$ around each Fermi point $k_F=k_F\\hat{n}$.\nThe Fermi momentum is determined by the renormalized Fermi energy\n$\\bar{E}_F$ which is related to the non-interacting Fermi energy $E_F$ by\nthe relation $\\bar{E}_F=E_F+\\delta\\mu_F$, such that\n$\\bar{E}_F=\\frac{k_F^2}{2m^*}$. The value of $\\delta\\mu_F$ is obtained\nfrom the interaction.\nThe two dimensional Fermions are expressed in terms of the one dimensional\nFermions $\\hat{R}_{n,\\sigma,\\alpha}(x_{\\parallel})$ and\n$\\hat{L}_{n,\\sigma,\\alpha}(x_{\\parallel})$:\n \n\\[\n R_{n,\\sigma,\\alpha}(\\vec{x})=\\hat{R}_{n,\\sigma,\\alpha}(x_{\\parallel})\n Z_n(x_{\\perp}), \\; \\; \\; \\;\\;\n L_{n,\\sigma,\\alpha}(\\vec{x})=\\hat{L}_{n,\\sigma,\\alpha}(x_{\\parallel})\n Z_n(x_{\\perp})\n\\]\n \n\\noindent $Z_n(x_{\\perp})$ is scalar function which ensures the conservation\nof momentum in the transversal direction. The number of channels (Fermions)\nis related to $k_F$ and cutoff $\\Lambda<k_F$, $N_o=\\frac{\\pi k_F}{\\Lambda}$.\nUsing the representation given in Eq.\\ref{eq-005}, we introduce the normal\norder currents $J^R_{n,\\alpha,\\sigma}(Z)$ (right mover) and\n$J^L_{n,\\alpha,\\sigma}(\\bar{Z})$ (left mover) with $Z$ and $\\bar{Z}$\ngiven by $Z=(Z_{\\parallel},Z_{\\perp})$,\n$\\bar{Z}=(\\bar{Z}_{\\parallel},\\bar{Z}_{\\perp})$,\n$Z_{\\parallel}=v_Ft-ix_{\\parallel}$,\n$\\bar{Z}_{\\parallel}=v_Ft+ix_{\\parallel}$,\nand $Z_{\\perp}=\\bar{Z}_{\\perp}=x_{\\perp}$,\n\n\\begin{equation} \n\\label{eq-006}\n J^R_{n,\\alpha,\\sigma}(Z)=:R^{\\dagger}_{n,\\alpha,\\sigma}(Z)\n R_{n,\\alpha,\\sigma}(Z): \\equiv\n R^{\\dagger}_{n,\\alpha,\\sigma}(Z+\\epsilon)R_{n,\\alpha,\\sigma}(Z)-\n \\langle R^{\\dagger}_{n,\\alpha,\\sigma}(Z+\\epsilon)R_{n,\\alpha,\\sigma}(Z)\n \\rangle_o\n\\end{equation}\n\n\\noindent with $\\epsilon=\\varepsilon_x-i\\delta$, $\\epsilon\\rightarrow 0$ and\nthe expectation value:\n\n\\begin{equation}\n\\label{eq-007}\n \\langle R^{\\dagger}_{n,\\alpha,\\sigma_1}(\\vec{x},t_1)R_{m,\\beta,\\sigma_2} \n (\\vec{y},t_2)\\rangle_o \\sim \\delta_{n,m} \\delta_{\\alpha,\\beta}\n \\delta_{\\sigma_1,\\sigma_2} \\delta_{\\Lambda}^{d-1}(x_{\\perp}-y_{\\perp})\n [v_F(t_1-t_2)-i(x_{\\parallel}-y_{\\parallel})]^{-1}\n\\end{equation}\n\n\\noindent Similarly we introduce for the left movers:\n\n\\begin{equation} \n\\label{eq-008}\n J^L_{n,\\alpha,\\sigma}(\\bar{Z})=:L^{\\dagger}_{n,\\alpha,\\sigma}(\\bar{Z})\n L_{n,\\alpha,\\sigma}(\\bar{Z}): \\equiv\n L^{\\dagger}_{n,\\alpha,\\sigma}(\\bar{Z}+\\bar{\\epsilon})\n L_{n,\\alpha,\\sigma}(\\bar{Z})-\\langle\n L^{\\dagger}_{n,\\alpha,\\sigma}(\\bar{Z}+\\bar{\\epsilon}) \n L_{n,\\alpha,\\sigma}(\\bar{Z}) \\rangle_o\n\\end{equation}\n\n\\noindent We write the interaction and the disorder parts in the normal\norder form. From the disorder part we obtain the elastic scattering term\n$\\frac{1}{2\\tau} \\propto D$ (see ref.\\cite{12}). From the disorder part\n(Eq.\\ref{eq-004}) we obtain the normal order form $\\tilde{S}_D$.\n\nFrom the interaction part we find the normal order representation\n$\\tilde{S}_{int}$ plus a shift of the Fermi energy: $\\delta\\mu_{int}\n(J^R_{n,\\alpha,\\sigma}(\\vec{x},t)+J^L_{n,\\alpha,\\sigma}(\\vec{x},t))$.\nWe choose $\\delta\\mu_F$ such that it cancels the interaction shift,\n$\\delta\\mu_F+\\delta\\mu_{int}=0$. As a result $S_0$ becomes:\n\n\\begin{equation}\n\\label{eq-009}\n \\tilde{S}_o=\\sum_{n=1}^{N/2} \\sum_{\\sigma} \\sum_{\\alpha} \\int d^dx\\int dt\n\\{ \\bar{R}_{n,\\alpha,\\sigma} [\\partial_t-v_F\\hat{n}\\cdot\\vec{\\partial}]\nR_{n,\\alpha,\\sigma}+\n \\bar{L}_{n,\\alpha,\\sigma} [\\partial_t+v_F\\hat{n}\\cdot\\vec{\\partial}]\nL_{n,\\alpha,\\sigma}\\}\n\\end{equation}\n\n\\noindent Using the representation given in Eq.\\ref{eq-005} we replace the\ninteraction term and disorder in terms of the currents ( see appendix \n\\ref{app-1} ).\nThe interaction part is decomposed in terms of forward\nscattering $Q_{n,m}^{(F)}(t,\\vec{x},\\vec{y})$ (charge part)\n$H_{n,m}^{(F)}(t,\\vec{x},\\vec{y})$ (spin part),\n$Q_{n,m}^{(B)}(t,\\vec{x},\\vec{y})$ (particle-hole in the singlet channel),\n$H_{n,m}^{(B)}(t,\\vec{x},\\vec{y})$ (particle-hole in the triplet channel),\n$O_{n,m}^{(s)}(t,\\vec{x},\\vec{y})$ (particle-particle in the singlet channel),\n$O_{n,m}^{(t)}(t,\\vec{x},\\vec{y})$ (particle-particle in the triplet channel).\nFrom the screened two-body potential $v(\\mid\\vec{q}\\mid)$ we obtain the\nscattering matrix elements for the different processes,\n$\\Gamma^{(c)}(\\vec{n},\\vec{m})$,$\\Gamma^{(s)}(\\vec{n},\\vec{m})$,\n$\\Gamma_2^{(c)}(\\vec{n},\\vec{m})$,$\\Gamma_2^{(s)}(\\vec{n},\\vec{m})$,\n$\\Gamma_3^{(s)}(\\vec{n},\\vec{m})$,$\\Gamma_3^{(t)}(\\vec{n},\\vec{m})$.\nFor the screened case the matrix elements $\\Gamma(\\vec{n},\\vec{m})$\ndepend only on the angles ``$\\theta$\" on the FS. For example, if $\\kappa$\nis the inverse of the screening length we have\n$ \\Gamma_2^{(s)}(\\vec{n},\\vec{m})=2\\kappa[1+\\frac{2k_F}{\\kappa}\n\\cos\\theta/2]^{-1}, \\; \\; 0\\leq\\theta\\leq\\pi$\n(``$\\theta$\" is the angle between the unit vectors $\\vec{n}$ and $\\vec{m}$).\nThe particle-particle matrix is,\n$ \\Gamma_3^{(s)}(\\vec{n},\\vec{m})=\\frac{\\kappa}{2}[\n(1+\\frac{2k_F}{\\kappa}\\sin\\theta/2)^{-1}+\n(1+\\frac{2k_F}{\\kappa}\\cos\\theta/2)^{-1}],\\;\\; 0\\leq\\theta\\leq\\pi$.\nWe introduce the left and right currents and obtain the representations for\nthe interaction and disorder terms:\n \n\\[\n J^R_{n,\\alpha,\\sigma_1;m,\\beta,\\sigma_2}(\\vec{x},t_1,t_2)=\n :R^{\\dagger}_{n,\\alpha,\\sigma_1}(\\vec{x},t_1)\n R_{m,\\beta,\\sigma_2}(\\vec{x},t_2):\n\\]\n \n\\begin{equation} \n\\label{eq-010}\n J^L_{n,\\alpha,\\sigma_1;m,\\beta,\\sigma_2}(\\vec{x},t_1,t_2)=\n :L^{\\dagger}_{n,\\alpha,\\sigma_1}(\\vec{x},t_1)\n L_{m,\\beta,\\sigma_2}(\\vec{x},t_2):\n\\end{equation}\n \n\\noindent For the interaction term we have $t_1=t_2$ and\n$\\alpha=\\beta$. We obtain that the interaction part for a screened two-body\npotential takes the form:\n \n\\[\n \\tilde{S}_{int}=\\frac{\\Lambda^{1-d}}{2N_o}\\sum_n\\sum_m\\int d^dx\n \\int dt\\sum_{\\alpha}\n \\{\\Gamma^{(c)}(\\vec{n},\\vec{m})Q^{(F)}_{n,m;\\alpha}(\\vec{x},t)-\n \\Gamma^{(s)}(\\vec{n},\\vec{m})H^{(F)}_{n,m;\\alpha}(\\vec{x},t)\n\\]\n \n\\begin{equation}\n\\label{eq-011}\n +\\Gamma^{(c)}_2(\\vec{n},\\vec{m})Q^{(B)}_{n,m;\\alpha}(\\vec{x},t)-\n \\Gamma^{(s)}_2(\\vec{n},\\vec{m})H^{(B)}_{n,m;\\alpha}(\\vec{x},t)+\n \\Gamma^{(s)}_3(\\vec{n},\\vec{m})O^{(s)}_{n,m;\\alpha}(\\vec{x},t)+\n \\Gamma^{(t)}_3(\\vec{n},\\vec{m})O^{(t)}_{n,m;\\alpha}(\\vec{x},t)\\}\n\\end{equation}\n\n\\noindent In Eq.\\ref{eq-011} we have to restrict\n$\\Gamma^{(s)}_3(\\vec{n},\\vec{m})$ and $\\Gamma^{(t)}_3(\\vec{n},\\vec{m})$\nto $\\vec{n}\\not=\\vec{m}$ in order to avoid double counting. If we ignore\nthe angle dependence of $\\Gamma^{(t)}_3$ we have $\\Gamma^{(t)}_3\\simeq 0$.\nFor $\\vec{n}=\\vec{m}$ we have the relation $\\Gamma^{(s)}_3(\\vec{n},\\vec{n})\n=\\frac{1}{2}\\Gamma^{(s)}_2(\\vec{n},\\vec{n})$. Based on dimensional analysis\nwe obtain that the interaction term has the dimension of $\\Lambda^{1-d}$.\nDue to the fact that the interaction is defined at the scale $\\Lambda<k_F$\nwe have the relation\n$k_F^{1-d}\\Gamma(k_F)=\\Lambda^{1-d}(\\frac{k_F}{\\Lambda})^{d-1}\n\\Gamma(\\Lambda)\\propto\\frac{\\Lambda^{1-d}}{N_o}\\Gamma(\\Lambda)$\nwhere $N_o=\\pi(\\frac{k_F}{\\Lambda})^{d-1}$ is the number of channels.\nThe operators $Q^{(F)}_{n,m;\\alpha}$, $H^{(F)}_{n,m;\\alpha}$,\n$Q^{(B)}_{n,m;\\alpha}$, $H^{(B)}_{n,m;\\alpha}$, $O^{(s)}_{n,m;\\alpha}$, and\n$O^{(t)}_{n,m;\\alpha}$ are given by:\n \n\\[\n Q^{(F)}_{n,m;\\alpha}(\\vec{x},t)=J^R_{n,\\alpha}(\\vec{x},t)\n J^R_{m,\\alpha}(\\vec{x},t)+J^L_{n,\\alpha}(\\vec{x},t)J^L_{m,\\alpha}(\\vec{x},t),\n\\]\n\\[\n H^{(F)}_{n,m;\\alpha}(\\vec{x},t)=\\vec{J}^R_{n,\\alpha}(\\vec{x},t)\\cdot\n \\vec{J}^R_{m,\\alpha}(\\vec{x},t)+\\vec{J}^L_{n,\\alpha}(\\vec{x},t)\\cdot\n \\vec{J}^L_{m,\\alpha}(\\vec{x},t),\n\\]\n\\[\n Q^{(B)}_{n,m;\\alpha}(\\vec{x},t)=J^R_{n,\\alpha}(\\vec{x},t)\n J^L_{m,\\alpha}(\\vec{x},t)+J^L_{n,\\alpha}(\\vec{x},t)J^R_{m,\\alpha}(\\vec{x},t),\n\\]\n\\[\n H^{(B)}_{n,m;\\alpha}(\\vec{x},t)=\\vec{J}^R_{n,\\alpha}(\\vec{x},t)\\cdot\n \\vec{J}^L_{m,\\alpha}(\\vec{x},t)+\\vec{J}^L_{n,\\alpha}(\\vec{x},t)\\cdot\n \\vec{J}^R_{m,\\alpha}(\\vec{x},t),\n\\]\n\\[\n O^{(s)}_{n,m;\\alpha}(\\vec{x},t)=O^{(\\parallel)}_{n,m;\\alpha}(\\vec{x},t)-\n O^{(\\perp)}_{n,m;\\alpha}(\\vec{x},t), \\; \\;\\;\n O^{(t)}_{n,m;\\alpha}(\\vec{x},t)=O^{(\\parallel)}_{n,m;\\alpha}(\\vec{x},t)+\n O^{(\\perp)}_{n,m;\\alpha}(\\vec{x},t),\n\\]\n\\[\n O^{(\\perp)}_{n,m;\\alpha}(\\vec{x},t)=\\sum_{\\sigma}\n :R^{\\dagger}_{n,\\alpha,\\sigma}(\\vec{x},t)R_{m,\\alpha,-\\sigma}(\\vec{x},t):\n :L^{\\dagger}_{n,\\alpha,-\\sigma}(\\vec{x},t)L_{m,\\alpha,\\sigma}(\\vec{x},t):\n\\]\n\\[\n +:L^{\\dagger}_{n,\\alpha,\\sigma}(\\vec{x},t)L_{m,\\alpha,-\\sigma}(\\vec{x},t):\n :R^{\\dagger}_{n,\\alpha,-\\sigma}(\\vec{x},t)R_{m,\\alpha,\\sigma}(\\vec{x},t):,\n\\]\n\\[\n O^{(\\parallel)}_{n,m;\\alpha}(\\vec{x},t)=\\sum_{\\sigma}\n :R^{\\dagger}_{n,\\alpha,\\sigma}(\\vec{x},t)R_{m,\\alpha,\\sigma}(\\vec{x},t):\n :L^{\\dagger}_{n,\\alpha,\\sigma}(\\vec{x},t)L_{m,\\alpha,\\sigma}(\\vec{x},t):\n\\]\n\\begin{equation}\n\\label{eq-012}\n +:L^{\\dagger}_{n,\\alpha,\\sigma}(\\vec{x},t)L_{m,\\alpha,\\sigma}(\\vec{x},t):\n :R^{\\dagger}_{n,\\alpha,\\sigma}(\\vec{x},t)R_{m,\\alpha,\\sigma}(\\vec{x},t):,\n\\end{equation}\n \n\\noindent where\n \n\\begin{equation}\n\\label{eq-013}\n J^R_{n,\\alpha}(\\vec{x},t)=\\sum_{\\sigma}\n :R^{\\dagger}_{n,\\alpha,\\sigma}(\\vec{x},t)\n R_{n,\\alpha,\\sigma}(\\vec{x},t):,\\;\\;\\;\\;\\;\n \\vec{J}^R_{n,\\alpha}(\\vec{x},t)=\\frac{1}{2}\n :R^{\\dagger}_{n,\\alpha,\\sigma_1}(\\vec{x},t)\n \\vec{\\sigma}_{\\sigma_1,\\sigma_2} R_{n,\\alpha,\\sigma_2}(\\vec{x},t):\n\\end{equation}\n\n\\noindent with similar expressions for the left movers. $``\\Lambda\"<k_F$\nis the cutoff of the theory and we find that the naive dimension of the\ninteraction field is $\\Lambda^{1-d}\\;\\;(d=2)$. This follows from the fact\nthat Eq.\\ref{eq-009} is invariant under the scaling $\\Lambda\\longrightarrow\n\\Lambda/b$, $x=x^{\\prime} b$, $t=t^{\\prime} b$,\n$R_n(\\vec{x},t)=b^{-d/2}R_n(\\vec{x}^{\\prime},t^{\\prime})$,\n$L_n(\\vec{x},t)=b^{-d/2}L_n(\\vec{x}^{\\prime},t^{\\prime})$, and $N(b)=bN_o$.\nFollowing the same procedure as for the interaction we express the disorder\npart using again the respective part:\n \n\\[\n \\tilde{S}_D=-\\frac{\\Lambda^{2-d}}{N_o}\\sum_n\\sum_m\\sum_{\\alpha,\\beta}\n \\int dt_1 \\int dt_2 \\int d^dx \\{\n d^{(d)}_2\\rho_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2)-\n d^{(c)}_2 q_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2)\n\\]\n\\begin{equation} \n\\label{eq-014}\n -d^{(s)}_2 h^{(B)}_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2)+\n d^{(s)}_3 c^{(s)}_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2)+\n d^{(t)}_3 c^{(t)}_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2)\n\\end{equation}\n \n\\noindent The operators in Eq.\\ref{eq-014} are in complete analogy with the\nones in Eq.\\ref{eq-011}, except\nthat they are at different times and have double replica index:\n \n\\[\n \\rho_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2) \\longleftrightarrow\n Q^{(F)}_{n,m,\\alpha}(\\vec{x},t);\n\\]\n\\[\n q_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2) \\longleftrightarrow\n Q^{(B)}_{n,m,\\alpha}(\\vec{x},t);\n\\]\n\\[\n h^{(B)}_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2) \\longleftrightarrow\n H^{(B)}_{n,m,\\alpha}(\\vec{x},t);\n\\]\n\\[\n C^{(s)}_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2) \\longleftrightarrow\n O^{(s)}_{n,m,\\alpha}(\\vec{x},t);\n\\]\n\\[\n C^{(t)}_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2) \\longleftrightarrow\n O^{(t)}_{n,m,\\alpha}(\\vec{x},t)\n\\]\n \n\\noindent The corresponding constants in Eq.\\ref{eq-014} have the initial\nvalues:\n$d_3^{(s)}=d_2^{(c)}=\\frac{1}{2}d_2^{(s)}\\equiv D$, $d_3^{(t)}=0$.\nWe will find that only the Cooperon term,\n$ d^{(s)}_3 c^{(s)}_{n,m,\\alpha,\\beta}(\\vec{x};t_1,t_2)$ is important.\nFor the rest part of this paper we will ignore the rest of the\nterms and consider only the Cooperon part.\\\\\n\n\\section{The Renormalization Group Method}\n\\label{sec-3}\n\nIn the first part of this chapter we will introduce the RG method based on\nthe OPE. This method is needed in order to analyze the possible phase\ndiagram of our problem. The real space method based on the Operator\nProduct Expansion (OPE) introduced in ref.\\cite{13} is in particular\nadvantageous. In order to explain how this works we express the action\nin Eq.\\ref{eq-011} by a formal expression $S\\sim\\sum\\Gamma_iA_i$\nwhere $A_i$ are the operators and $\\Gamma_i$ are the coupling constants. Using\nthe fact that the time ordered product of the single particle operator is\ngiven by,\n\\[\n R_{n,\\alpha,\\sigma}(\\vec{x},t_1)\n R^{\\dagger}_{m,\\beta,\\sigma_1}(\\vec{y},t_2)\\sim\n \\frac{1}{2\\pi}\\delta_{n,m}\\delta_{\\alpha,\\beta}\\delta_{\\sigma,\\sigma_1}\n \\delta_{\\Lambda}^{d-1}(x_{\\perp}-y_{\\perp}) \\theta(t_1-t_2)\n\\]\n\\begin{equation} \n\\label{eq-015}\n [v_F(t_1-t_2)-i(x_{\\parallel}-y_{\\parallel})]^{-1}\n\\end{equation}\n\n\\noindent where $x_{\\parallel}=\\hat{n}\\cdot\\vec{x}$,\n$x_{\\perp}=\\vec{x}-\\hat{n}\\cdot\\vec{x}$. We find for any two\noperators given in Eq.\\ref{eq-011} the OPE:\n \n\\begin{equation} \n\\label{eq-016}\n A_i(\\vec{x},t_1)A_j(\\vec{x}+a,t_2)\\sim\\sum_K\n \\frac{C^K_{ij}F_K(\\mid t_1-t_2\\mid)\n A_K(\\vec{x},\\frac{t_1+t_2}{2})}{[a^2+v_F^2(t_1-t_2)^2]^{x_i+x_j-x_K}}\n\\end{equation}\n \n\\noindent with $C^K_{ij}$ the structure constant and\n$F_K(\\mid t_1-t_2\\mid)\\sim 1$.\nAs a result the product of any number of operators can be reduced to a sum\nof operators. This implies that once the cutoff $\\Lambda$ is reduced to\n$\\Lambda/b$, one can obtain the scaling equations for coupling constants\n$\\Gamma_i$. For $\\Gamma_i$ with the scaling dimension\n$\\Gamma_i\\longrightarrow\\Gamma_ib^{(x_i-d)}$, one obtains:\n \n\\begin{equation} \n\\label{eq-017}\n \\frac{d\\Gamma_K}{d\\ln b}=-(d-x_K)\\Gamma_K-\\frac{1}{2}\n \\sum_{i,j}\\tilde{C}_{i,j}^K\\Gamma_i\\Gamma_j+\n \\frac{1}{3!}\\sum_{i,j}\\sum_{p,q} \\tilde{C}_{i,j}^p\\tilde{C}_{p,q}^K\n \\Gamma_i\\Gamma_j\\Gamma_q\n\\end{equation}\n \n\\noindent where the $\\tilde{C}_{i,j}^K$ are proportional to the structure\nconstants $C_{i,j}^{K}$. In order to be able to complete the RG equation\ngiven in Eq.\\ref{eq-017} we have to compute the operator product expansion\nof the operators which appear in Eqs.\\ref{eq-011} and \\ref{eq-014}.\nThe second part of this chapter will be devoted to the calculation of the\nOPE for the interaction and disorder operators. Using current algebra of\nthe chiral currents given in ref.\\cite{16} we will establish the OPE rules\nfor our problem. The calculation is based on the Wick theorem which replaces\nthe time order product by the normal ordered form plus all the possible ways\nof contracting pairs of Fermion fields. This calculation is standard and\nlengthy therefore we will present only the results.\nWe start with the results for the p-p singlet:\n\n\\[\n O^{(s)}_{n,m,\\alpha}(\\vec{x},t) O^{(s)}_{k,l,\\beta}(\\vec{x}+\\vec{a},t+\\tau)=\n \\frac{1}{(2\\pi)^2}(\\frac{\\Lambda}{2\\pi})^{d-1} \\delta_{\\alpha,\\beta}\n [a^2+(v_F\\tau)^2]^{-1} \\{ O^{(s)}_{k,m,\\alpha}(\\vec{x},t)\\delta_{n,l}\n\\]\n\\begin{equation} \n\\label{eq-018}\n + O^{(s)}_{n,l,\\alpha}(\\vec{x},t)\\delta_{k,m}\n -\\delta_{n,l}\\delta_{k,m}\n (Q^{(B)}_{n,m,\\alpha}(\\vec{x},t)+Q^{(B)}_{m,n,\\alpha}(\\vec{x},t))\\}\n +[``c\" number].\n\\end{equation}\n\n\\noindent From Eq.\\ref{eq-018} we learn that the OPE generates p-p and p-h\nsinglets.\\\\\n\nFor the p-h in the triplet channel no new terms are generated:\n\n\\[\n H^{(B)}_{n,m,\\alpha}(\\vec{x},t) H^{(B)}_{k,l,\\beta}(\\vec{x}+\\vec{a},t+\\tau)=\n \\frac{1}{(2\\pi)^2}(\\frac{\\Lambda}{2\\pi})^{d-1} \\delta_{\\alpha,\\beta}\n [a^2+(v_F\\tau)^2]^{-1}\n\\]\n\\begin{equation}\n\\label{eq-019}\n \\{ -2H^{(B)}_{n,m,\\alpha}(\\vec{x},t) [\n \\delta_{n,l}\\delta_{k,m}+\\delta_{n,k}\\delta_{l,m}]\\} + [``c\" number]\n\\end{equation}\n\n\\noindent The p-h singlet generates only a ``c\" number:\n\n\\begin{equation}\n\\label{eq-020}\n Q^{(B)}_{n,m,\\alpha}(\\vec{x},t) Q^{(B)}_{k,l,\\beta}(\\vec{x}+\\vec{a},t+\\tau)=\n [``c\" number]\n\\end{equation}\n\n\\noindent The OPE for the Cooperon do not generate new terms:\n\n\\[\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)\n C^{(s)}_{k,l;\\alpha^{\\prime}\\beta^{\\prime}}\n (\\vec{x}+\\vec{a};t_1+\\tau_1,t_2+\\tau_2)=\n \\frac{1}{(2\\pi)^2}(\\frac{\\Lambda}{2\\pi})^{d-1}\n [\\frac{1/2}{(v_F\\tau_1-ia)(v_F\\tau_2+ia)}\n\\]\n\\[\n +\\frac{1/2}{(v_F\\tau_1+ia)(v_F\\tau_2-ia)}]\n \\{ 2\\delta_{\\alpha,\\beta^{\\prime}}\\delta_{\\alpha^{\\prime},\\beta}\n [\\delta_{m,k}C^{(s)}_{n,l;\\alpha,\\beta}(\\vec{x};t_1,t_2)+\n \\delta_{n,l}C^{(s)}_{k,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)]\n\\]\n\\begin{equation} \n\\label{eq-021}\n -\\frac{1}{2}\\delta_{\\alpha,\\alpha^{\\prime}}\\delta_{\\beta,\\beta^{\\prime}}\n \\delta_{m,k}\\delta_{n,l} [ C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)\n +C^{(s)}_{m,n;\\alpha,\\beta}(\\vec{x};t_1,t_2)] \\} +[``c\" number]\n\\end{equation}\n\n\\noindent The OPE between the p-p and p-h triplet generates the p-p operator\nand the p-h singlet:\n\n\\[\n O^{(s)}_{n,m;\\alpha}(\\vec{x},t)H^{(B)}_{k,l;\\beta}(\\vec{x}+\\vec{a},t+\\tau)=\n \\frac{1}{(2\\pi)^2}(\\frac{\\Lambda}{2\\pi})^{d-1}\\delta_{\\alpha,\\beta}\n [a+(v_F\\tau)^2]^{-1}\\{ \\frac{3}{4}\\delta_{l,k}\\delta_{n,l}\\delta_{m,k}\n Q^{(B)}_{n,m})\\vec{x},t)]\\}\n\\]\n\\begin{equation}\n\\label{eq-022}\n -\\frac{\\delta_{l,k}}{8}[\\delta_{n,l}\n O^{(s)}_{l,m;\\alpha}(\\vec{x},t)+\\delta_{m,l}O^{(s)}_{n,l;\\alpha}(\\vec{x},t)]\n \\}+[``c\" number]\n\\end{equation}\n\n\\noindent The OPE between the p-p term and the p-h singlet generates a ``c\"\nnumber:\n\n\\begin{equation} \n\\label{eq-023}\n O^{(B)}_{n,m;\\alpha}(\\vec{x},t)Q^{(B)}_{k,l;\\beta}(\\vec{x}+\\vec{a},t+\\tau)\n =[``c\" number]\n\\end{equation}\n\n\\noindent The product for the product p-h triplet and p-h singlet gives a\n``c\" number:\n\n\\begin{equation} \n\\label{eq-024}\n H^{(B)}_{n,m;\\alpha}(\\vec{x},t)Q^{(B)}_{k,l;\\beta}(\\vec{x}+\\vec{a},t+\\tau)=\n [``c\" number]\n\\end{equation}\n\n\\noindent In the remaining part we present the OPE between the Cooperon and\nthe interaction operators. For the p-p case we generate the Cooperon and\np-p operator:\n\n\\[\n O^{(s)}_{n,m;\\gamma}(\\vec{x},t)\n C^{(s)}_{k,l;\\alpha,\\beta}(\\vec{x}+\\vec{a},t_1+\\tau_1,t_2+\\tau_2)=\n \\frac{1}{(2\\pi)^2}(\\frac{\\Lambda}{2\\pi})^{d-1}\\{\n [\\frac{1/2}{(v_F\\tau_1-ia)(v_F\\tau_2+ia)}+\n\\]\n\\[\n \\frac{1/2}{(v_F\\tau_1+ia)(v_F\\tau_2-ia)}] \\delta_{\\gamma,\\alpha}\n \\delta_{\\gamma,\\beta} [O^{(s)}_{k,m;\\gamma}(\\vec{x},t)\\delta_{n,l}+\n O^{(s)}_{n,l;\\gamma}(\\vec{x},t)\\delta_{k,m}]\n +\\frac{1/2}{(v_F\\tau_1)^2+a^2}\\delta_{k,m}\\delta_{n,l}\\delta_{n,m}\n (\\delta_{\\gamma,\\alpha}+\\delta_{\\gamma,\\beta})\n\\]\n\\begin{equation}\n\\label{eq-025}\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x},t,t+\\tau_2)\n +\\frac{1/2}{(v_F\\tau_2)^2+a^2}\\delta_{k,m}\\delta_{n,l}\\delta_{n,m}\n (\\delta_{\\gamma,\\alpha}+\\delta_{\\gamma,\\beta})\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x},t+\\tau_1,t)\\}\n +[``c\" number]\n\\end{equation}\n\n\\noindent For the p-h triplet one obtains the p-p and Cooperon terms:\n\n\\[\n H^{(B)}_{n,m;\\gamma}(\\vec{x},t)\n C^{(s)}_{k,l;\\alpha,\\beta}(\\vec{x}+\\vec{a},t_1+\\tau_1,t_2+\\tau_2)=\n \\frac{1}{(2\\pi)^2}(\\frac{\\Lambda}{2\\pi})^{d-1}\\{\n -\\frac{1}{2}[\\frac{1}{a^2+(v_F\\tau_1)^2}\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x},t,t+\\tau_2)\n\\]\n\\[\n +\\frac{1}{a^2+(v_F\\tau_2)^2}C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x},t+\\tau_1,t)]\n [\\delta_{k,m}\\delta_{n,l}\\delta_{n,m}(\\delta_{\\gamma,\\alpha}+\n \\delta_{\\gamma,\\beta})\\frac{3}{4}]-[\\frac{1/2}{(v_F\\tau_1-ia)(v_F\\tau_2+ia)}\n\\]\n\\begin{equation}\n\\label{eq-026}\n + \\frac{1/2}{(v_F\\tau_1+ia)(v_F\\tau_2-ia)}][\\delta_{m,k}\\delta_{m,l}+\n \\delta_{n,k}\\delta_{n,l}][\\delta_{\\gamma,\\alpha}\\delta_{\\gamma,\\beta}]\n [\\frac{1}{4}O^{(s)}_{n,m;\\gamma}(\\vec{x},t)+\n \\frac{1}{4}O^{(t)}_{n,m;\\gamma}(\\vec{x},t)]\\}+[``c\" number]\n\\end{equation}\n\n\\noindent When we consider the p-h singlet we generate the p-p and Cooperon\nterms.\n\n\\[ \n Q^{(B)}_{n,m;\\gamma}(\\vec{x},t) \n C^{(s)}_{k,l;\\alpha,\\beta}(\\vec{x}+\\vec{a},t_1+\\tau_1,t_2+\\tau_2)= \n \\frac{1}{(2\\pi)^2}(\\frac{\\Lambda}{2\\pi})^{d-1}\\{\n -\\frac{1}{2}[\\frac{1}{a^2+(v_F\\tau_1)^2} (\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x},t,t+\\tau_2) \n\\]\n\\[\n -C^{(t)}_{n,m;\\alpha,\\beta}(\\vec{x},t,t+\\tau_2)]))\n +\\frac{1}{a^2+(v_F\\tau_2)^2} (C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x},t+\\tau_1,t)\n -C^{(t)}_{n,m;\\alpha,\\beta}(\\vec{x},t+\\tau_1,t))]\n\\]\n\\[\n \\frac{1}{2}(\\delta_{n,k}\\delta_{m,l}+\\delta_{m,k}\\delta_{n,l})\n (\\delta{\\gamma,\\alpha}+\\delta_{\\gamma,\\beta})+\n [\\frac{1/2}{(v_F\\tau_1-ia)(v_F\\tau_2+ia)}+\n \\frac{1/2}{(v_F\\tau_1+ia)(v_F\\tau_2-ia)}]\n\\]\n\\begin{equation}\n\\label{eq-027}\n \\delta_{\\gamma,\\alpha}\\delta_{\\gamma,\\beta}\n (\\delta_{m,k}\\delta_{m,l}+\\delta_{n,k}\\delta_{n,l})\n (O^{(s)}_{n,m;\\gamma}(\\vec{x},t)-O^{(t)}_{n,m;\\gamma}(\\vec{x},t))\\}\n +[``c\" number]\n\\end{equation}\n\n\\noindent The Eqs.\\ref{eq-018}-\\ref{eq-027} have been obtained using the\nfree Fermion action given in Eq.\\ref{eq-009}. We will work at a finite\ntemperature, therefore Eqs.\\ref{eq-015} is an approximation of the exact\npropagator\n$\\{\\frac{v_F\\beta}{\\pi}\\sin[\\frac{\\pi}{v_F\\beta}(v_F(t_1-t_2)\n-i(x_{\\parallel}-y_{\\parallel}))]\\}^{-1}$. This means that the ``$t$\"\nrange of integration in Eqs.\\ref{eq-017} and \\ref{eq-027} is restricted to\n$t<\\beta$. \\\\\n\n\\section{Derivation of the RG equations}\n\\label{sec-4}\n\nThis chapter is designated to the computation of the RG equations. This will\nbe done by expanding the partition function $Z$ in terms of the interaction\nand disorder operators. Using the OPE rules derived in\nEqs.\\ref{eq-018}-\\ref{eq-027} will allow to replace the product of operators\nin terms of a sum of operators. When rescaling the minimal distance ``a\" to\n``ba\" will allow to find the scaling equations.\nIt is important to remark that the\nmethod used here is different from the standard method used for problems\nwith disorder. The traditional method \\cite{02} starts from the diffusion\ntheory and includes the interaction terms as a perturbation. Here we start\nfrom the Fermion theory and include simultaneously on equal footing the\neffects of interaction and disorder. In the standard approach the quantum\ndiffusion theory ignores completely the effects of interactions at short\ndistances (distances shorter than the mean free path). We will see that\nconsidering the disorder and interaction on equal footing new terms will\nappear in the RG equations. The scaling equations will contain terms which\nare controlled by the number of channels.\\\\\n\nFollowing the analysis given in section \\ref{sec-2} we have:\n\n\\begin{equation} \n\\label{eq-028}\n \\tilde{S}_o=\\sum_{n=1}^{N/2} \\sum_{\\sigma} \\sum_{\\alpha} \\int d^dx\\int dt\n\\{ \\bar{R}_{n,\\alpha,\\sigma} [\\partial_t-v_F\\hat{n}\\cdot\\vec{\\partial}]\nR_{n,\\alpha,\\sigma}+\n\\bar{L}_{n,\\alpha,\\sigma} [\\partial_t+v_F\\hat{n}\\cdot\\vec{\\partial}]\nL_{n,\\alpha,\\sigma}\\}.\n\\end{equation}\n\n\\noindent The action $\\tilde{S}_o$ determines the partition function $Z_o$,\n\n\\[\n Z_o=\\int {\\it D}[\\bar{\\psi},\\psi] e^{-\\tilde{S}_o}.\n\\]\n\n\\noindent We perturb the partition function $Z_o$ by the interaction\n$\\tilde{S}_{int}$ and disorder $\\tilde{S}_D$:\n\n\\[\n \\tilde{S}_{int}=\\frac{\\Lambda^{1-d}}{2N_o}\\sum_n\\sum_m\\sum_{\\alpha}\n \\int d^dx\\int dt \\{\n \\Gamma^{(c)}(\\vec{n},\\vec{m})Q^{(F)}_{n,m;\\alpha}(\\vec{x},t)\n -\\Gamma^{(s)}(\\vec{n},\\vec{m})H^{(F)}_{n,m;\\alpha}(\\vec{x},t)\n +\\Gamma^{(c)}_2(\\vec{n},\\vec{m})Q^{(B)}_{n,m;\\alpha}(\\vec{x},t)\n\\]\n\\[\n -\\Gamma^{(s)}_2(\\vec{n},\\vec{m})H^{(B)}_{n,m;\\alpha}(\\vec{x},t)\n +(1-\\delta_{n,m})\\Gamma^{(s)}_3(\\vec{n},\\vec{m})\n O^{(s)}_{n,m;\\alpha}(\\vec{x},t)\\}\n\\]\n\\[\n =\\frac{\\Lambda^{1-d}}{2N_o}\\sum_n\\sum_m\\sum_{\\alpha}\n \\int d^dx\\int dt \\{\n \\Gamma^{(c)}(\\vec{n},\\vec{m})Q^{(F)}_{n,m;\\alpha}(\\vec{x},t)\n -\\Gamma^{(s)}(\\vec{n},\\vec{m})H^{(F)}_{n,m;\\alpha}(\\vec{x},t) \n +e^{(c)}_2(\\vec{n},\\vec{m})Q^{(B)}_{n,m;\\alpha}(\\vec{x},t) \n\\] \n\\begin{equation} \n\\label{eq-029}\n -e^{(s)}_2(\\vec{n},\\vec{m})H^{(B)}_{n,m;\\alpha}(\\vec{x},t) \n +e^{(s)}_3(\\vec{n},\\vec{m}) O^{(s)}_{n,m;\\alpha}(\\vec{x},t)\\}\n\\end{equation}\n\n\\noindent In Eq.\\ref{eq-029} we have ignored for simplicity the\nparticle-particle triplet and consider only the particle-particle singlet.\\\\\n\nDue to the relation between the particle-particle and the particle-hole\ntriplets we remove the term $(1-\\delta_{n,m})$ by defining new coupling\nconstants:\n\n\\[\n e^{(c)}_2(\\vec{n},\\vec{m})=\\Gamma^{(c)}_2(\\vec{n},\\vec{m})-\\frac{1}{2}\n \\delta_{n,m}\\Gamma^{(s)}_3(\\vec{n},\\vec{m}),\n\\]\n\\[\n e^{(s)}_2(\\vec{n},\\vec{m})=\\Gamma^{(s)}_2(\\vec{n},\\vec{m})-2\n \\delta_{n,m}\\Gamma^{(s)}_3(\\vec{n},\\vec{m}),\n\\]\n\\begin{equation} \n\\label{eq-030}\n e^{(s)}_3(\\vec{n},\\vec{m})=\\Gamma^{(s)}_3(\\vec{n},\\vec{m}).\n\\end{equation}\n\n\\noindent The results in Eqs.\\ref{eq-030} follows from the operator identity\n\n\\begin{equation}\n\\label{eq-031}\n O^{(s)}_{n,m;\\alpha}(\\vec{x},t)=(1-\\delta_{n,m})\n O^{(s)}_{n,m;\\alpha}(\\vec{x},t)+\\delta_{n,m}[\\frac{1}{2}\n Q^{(B)}_{n,m;\\alpha}(\\vec{x},t)-2H^{(B)}_{n,m;\\alpha}(\\vec{x},t)]\n\\end{equation}\n\n\\noindent Eq.\\ref{eq-031} follows directly from the definitions of the\nparticle-particle singlet for $\\vec{n}=\\vec{m}$ in terms of the currents\n( see Eqs.\\ref{eq-012}-\\ref{eq-013} ). For $\\vec{n}=\\vec{m}$ the\nparticle-hole\ntriplet $\\Gamma^{(s)}_2(\\vec{n},\\vec{n})$ is related to the particle-particle\nsinglet $\\Gamma^{(s)}_3(\\vec{n},\\vec{n})$ ( see Eq.\\ref{eq-a4} )\n\n\\begin{equation}\n\\label{eq-032}\n \\frac{1}{2}\\Gamma^{(s)}_2(\\vec{n},\\vec{n})=\\Gamma^{(s)}_3(\\vec{n},\\vec{n}).\n\\end{equation}\n\n\\noindent For the disorder part we will consider only the dominant\nCooperon term $d^{(s)}_3C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)$\nand we will ignore the effect of forward disorder\n\n\\begin{equation} \n\\label{eq-033}\n \\tilde{S}_D=-\\frac{\\Lambda^{2-d}}{N_o}\\sum_n\\sum_m\\sum_{\\alpha}\\sum_{\\beta}\n \\int d^dx \\int dt_1 \\int dt_2 \\{ d^{(s)}_3\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2) \\}.\n\\end{equation}\n\n\\noindent Following ref.\\cite{13} we compute the partition function $Z$\nof the action $\\tilde{S}_o+\\tilde{S}_{int}+\\tilde{S}_D$ by expanding\nup to the third order in $\\tilde{S}_{int}+\\tilde{S}_D$. Using $Z_o$ we obtain:\n\n\\[\n Z=Z_o\\{1-[\\langle\\tilde{S}_{int}\\rangle_a+\\langle\\tilde{S}_D\\rangle_a\n -\\frac{1}{2}\\langle\\tilde{S}^2_{int}\\rangle_a-\n \\langle\\tilde{S}_{int}\\tilde{S}_D\\rangle_a-\n \\frac{1}{2}\\langle\\tilde{S}^2_D\\rangle_a\n\\]\n\\begin{equation} \n\\label{eq-034}\n +\\frac{1}{3!}\\langle\\tilde{S}^3_{int}\\rangle_a\n +\\frac{1}{3!}\\langle\\tilde{S}^3_D\\rangle_a\n +\\frac{1}{2}\\langle\\tilde{S}^2_{int}\\tilde{S}_D\\rangle_a\n +\\frac{1}{2}\\langle\\tilde{S}_{int}\\tilde{S}^2_D\\rangle_a ]\\}.\n\\end{equation}\n\n\\noindent The meaning of $\\langle\\cdots\\rangle_a$ is to take the expectation\nvalue with respect to $\\tilde{S}_o$ defined in Eq.\\ref{eq-028}. Since\nwe want to perform a RG analysis we will take the expectation value only\nin the interval $(\\Lambda,\\Lambda/b)$, $b\\geq 1$. In real space this means to\nintegrate from the microscopic distance $a$ to $ba$. \\\\\n\nNext we will compute the first term in Eq.\\ref{eq-034}\n\n\\begin{equation} \n\\label{eq-035}\n \\langle\\tilde{S}_{int}\\rangle_{ba}=b^{2-d}\\frac{N_o}{N(b)}\n \\langle\\tilde{S}_{int}\\rangle_a, \\;\\;\\;\\;\\;\n N(b)=N_ob\n\\end{equation}\n\n\\noindent where $\\langle\\cdots\\rangle_{ba}$ represents the expectation\nvalue with respect to Eq.\\ref{eq-028} with the new cutoff \n$\\Lambda/b=2\\pi/ba$.\nThe expectation value of $\\langle\\tilde{S}_D\\rangle_a$ is different from\nEq.\\ref{eq-035}. The difference is due to the two times $t_1$ and $t_2$.\nFor times $\\mid t_1-t_2\\mid\\leq a/v_F$,\n$C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)$ is replaced by the singlet\nparticle-particle interaction.\n\n\\begin{equation} \n\\label{eq-036}\n \\langle\\tilde{S}_D\\rangle_{ba}=b^{3-d}\\frac{N_o}{N(b)}\n \\langle\\tilde{S}_D\\rangle_a\n\\end{equation}\n\\begin{equation} \n\\label{eq-037}\n \\Delta\\langle\\tilde{S}_{int}\\rangle_{ba}=\n -\\frac{2a}{v_F}b^{2-d}\\frac{N_o}{N(b)}\\langle\\tilde{S}_D(t_1=t_2)\\rangle_a.\n\\end{equation}\n\n\\noindent Eq.\\ref{eq-037} represents the contribution from the disorder\nCooperon to the singlet particle-particle term when\n$\\mid t_1-t_2\\mid\\leq a/v_F$.\\\\\n\nIn order to compute the higher order term we have to use the rule of the\noperator product expansion defined in Eqs.\\ref{eq-018}-\\ref{eq-027}, and\nhave to perform the time integration. We introduce the notation\n$\\langle\\cdots\\rangle_{da}$ which stands for the expectation value in the\ndomain ($a$,$ba$)\n\\[\n -\\frac{1}{2}\\langle\\tilde{S}^2_{int}\\rangle_{da}=\\frac{da}{a}\\{\n \\frac{\\Lambda^{1-d}}{2N}\\frac{A^{-1}}{4N}(-1)\\sum_n\\sum_m\\sum_{\\alpha}\n \\int d^dx\\int dt \\{ T(\\vec{n},\\vec{m})Q^{(B)}_{n,m;\\alpha}(\\vec{x},t)\n\\]\n\\begin{equation}\n\\label{eq-038}\n -S(\\vec{n},\\vec{m})H^{(B)}_{n,m;\\alpha}(\\vec{x},t)\n +R(\\vec{n},\\vec{m})O^{(B)}_{n,m;\\alpha}(\\vec{x},t)\\}\n\\end{equation}\n\n\\noindent where $\\frac{da}{a}=\\frac{ba-a}{a}\\sim d\\ln b$. \n$R(\\vec{n},\\vec{m})$, $S(\\vec{n},\\vec{m})$, and $T(\\vec{n},\\vec{m})$ are a\nset of polynomials defined by the rules of the OPE given by\nEqs.\\ref{eq-018}-\\ref{eq-027}.\n\n\\[\n T(\\vec{n},\\vec{m})=-[2(e^{(s)}_3(\\vec{n},\\vec{m}))^2+\\frac{3}{4}\\delta_{n,m}\n e^{(s)}_2(\\vec{n},\\vec{m})e^{(s)}_3(\\vec{n},\\vec{m})],\n\\]\n\\[\n R(\\vec{n},\\vec{m})=2\\sum_{\\vec{l}}e^{(s)}_3(\\vec{n},\\vec{l})\n e^{(s)}_3(\\vec{l},\\vec{m})+\\frac{1}{4}e^{(s)}_3(\\vec{n},\\vec{m})\n e^{(s)}_2(0),\n\\]\n\\begin{equation} \n\\label{eq-039}\n S(\\vec{n},\\vec{m})=4(e^{(s)}_2(\\vec{n},\\vec{m}))^2\n\\end{equation}\n\n\\noindent where $e^{(s)}_2(0)\\equiv e^{(s)}_2(\\vec{n},\\vec{n})$. $A^{-1}$\nis determined by the time integration\n\n\\begin{equation} \n\\label{eq-040}\n A^{-1}=\\frac{I_1(\\hat{\\beta})}{(2\\pi)^{d-1}2\\pi v_F};\n \\;\\;\\;\\;\\;\n \\hat{\\beta}\\equiv\\frac{\\beta}{\\pi a}\n\\end{equation}\n\n\\begin{equation}\n\\label{eq-041}\n I_1(\\hat{\\beta})=\\frac{2}{\\pi}\\int_0^{\\infty}dx \\frac{\\cos x/\\hat{\\beta}}\n {x^2+1}\n\\end{equation}\n\n\\noindent where $\\hat{\\beta}$ is the dimensionless inverse temperature.\nThe function $I_1(\\hat{\\beta})$ originates at $T\\not= 0$.\nIn the limit $\\beta\\gg 1$,\n$I_1(\\hat{\\beta})\\rightarrow 1$. In the limit $\\beta\\sim 1$,\n$I_1(\\hat{\\beta}) \\ll 1$ and the time integration can be neglected.\n\n\\begin{equation} \n\\label{eq-042}\n -\\frac{1}{2}\\langle\\tilde{S}^2_D\\rangle_{da}=\\frac{da}{a}\\{\n \\frac{\\Lambda^{2-d}}{N}\\frac{B^{-1}}{2N}(-1)\\sum_n\\sum_m\\sum_{\\alpha}\n \\sum_{\\beta}\\int d^dx\\int dt_1\\int dt_2 [2(d^{(s)}_3)^2(1-\\frac{1}{8N})N\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)]\\}\n\\end{equation}\n\n\\noindent and\n\n\\[\n -\\langle\\tilde{S}_{int}\\tilde{S}^2_D\\rangle_{da}=\\frac{da}{a}\\{\n -\\frac{\\Lambda^{1-d}}{2N}\\frac{B^{-1}}{N}\\sum_n\\sum_m\\sum_{\\alpha}\n \\int d^dx\\int dt [-d^{(s)}_3\\hat{L}(\\vec{n},\\vec{m})\n O^{(s)}_{n,m;\\alpha}(\\vec{x},t)]\n\\]\n\\begin{equation}\n\\label{eq-043}\n -\\frac{\\Lambda^{2-d}}{N}\\frac{A^{-1}}{2N}\\sum_n\\sum_m\\sum_{\\alpha}\n \\sum_{\\beta}\n \\int d^dx\\int dt_1\\int dt_2 [-d^{(s)}_3\\hat{M}(\\vec{n},\\vec{m}) \n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)]\\}\n\\end{equation}\n\n\\noindent where\n\n\\begin{equation} \n\\label{eq-044}\n B^{-1}=\\frac{I_2(2\\hat{\\beta})}{(2\\pi)^{d-1}2v_F^2}\n\\end{equation}\n\n\\begin{equation} \n\\label{eq-045}\n I_2(\\hat{\\beta})=[I_1(\\hat{\\beta})]^2\n\\end{equation}\n\n\\noindent The term $\\hat{L}(\\vec{n},\\vec{m})$ and $\\hat{M}(\\vec{n},\\vec{m})$\nare given by:\n\n\\begin{equation} \n\\label{eq-046}\n \\hat{L}(\\vec{n},\\vec{m})=2\\sum_{\\vec{l}}e^{(s)}_3(\\vec{l},\\vec{m})\n +\\frac{1}{2}e^{(s)}_2(\\vec{n},\\vec{m})-2e^{(c)}_2(\\vec{n},\\vec{m})\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation} \n\\label{eq-047}\n \\hat{M}(\\vec{n},\\vec{m})=\\frac{3}{2}e^{(s)}_2(\\vec{n},\\vec{m})\n +2\\gamma^{(s)}_3(\\vec{n},\\vec{m})\\delta_{n,m}-2e^{(c)}_2(\\vec{n},\\vec{m})\n\\end{equation} \n\n\\noindent The set of Eqs.\\ref{eq-043}-\\ref{eq-047} concludes the RG \ncalculation to second order.\\\\\n\nThe presence of the elastic mean free path introduces a cutoff in the time\ndomain and allows us to apply the method of OPE to higher order. To third\norder in the interaction parameters $e^{(s)}_2$, $e^{(c)}_2$, $e^{(s)}_3$,\nand disorder $d^{(s)}_3$ we obtain:\n\n\\[\n \\langle\\tilde{S}^2_{int}\\tilde{S}_D\\rangle_{da}=\\frac{da}{a}\\{\n \\frac{\\Lambda^{2-d}}{N}\\frac{A^{-1}}{2N^2}\\sum_n\\sum_m\\sum_{\\alpha}\n \\sum_{\\beta} \\int d^dx\\int dt_1\\int dt_2 [ \\frac{A^{-1}}{2}\n \\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})} G_1(\\vec{n},\\vec{m})\n\\]\n\\[\n +\n B^{-1}\\frac{J_2(\\hat{2\\beta})}{I_2(2\\hat{\\beta})}\\hat{\\ell}\n G_2(\\vec{n},\\vec{m})]\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)\\}\n +\\frac{da}{a}\\{\\frac{\\Lambda^{1-d}}{2N}\\frac{A^{-1}B^{-1}}{N^2}\n \\sum_n\\sum_m\\sum_{\\alpha}\\int d^dx\\int dt\n\\]\n\\[\n [\\frac{J_2(\\hat{\\beta})}{2I_2(\\hat{\\beta})}K_3(\\vec{n},\\vec{m})\n +\\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})} K_2(\\vec{n},\\vec{m})\n +\\frac{B}{A}\\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})} K_3(\\vec{n},\\vec{m})]\n O^{(s)}_{n,m;\\alpha}(\\vec{x},t)\\}\n\\]\n\\begin{equation}\n\\label{eq-048}\n +\\frac{da}{a}\\{\\frac{\\Lambda^{1-d}}{2N}\\frac{A^{-1}B^{-1}}{N^2}\n \\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}\n \\sum_n\\sum_m\\sum_{\\alpha}\\int d^dx\\int dt\n F(\\vec{n},\\vec{m})Q^{(B)}_{n,m;\\alpha}(\\vec{x},t)\\}\n\\end{equation}\n\n\\noindent In Eq.\\ref{eq-048} the time integration introduces:\n\n\\begin{equation} \n\\label{eq-049}\n J_1(\\hat{\\beta})\\simeq I_1(\\hat{\\beta}),\n\t\\;\\;\\;\\;\\;\n J_2(\\hat{\\beta})\\simeq I_2(\\hat{\\beta}).\n\\end{equation}\n\n\\noindent The integral in Eq.\\ref{eq-049} depends explicitly on the\ndimensionless $\\hat{\\beta}$. At the scale $b=1$ we have\n$\\hat{\\beta}(b=1)=\\hat{\\beta}\\gg 1$ and for $b=b_T=\\beta/a$ we have\n$\\hat{\\beta}(b)=1$ and have to stop scaling. By Using the OPE\nrules we generate the polynomials $G_1$, $G_2$, $K_1$, $K_2$, $K_3$, and $F$.\nThese polynomials are obtained from the microscopic couplings and the OPE results\nobtained at second order ( the polynomials $R$, $S$, $T$, $L$, and $M$). \n\n\\begin{equation} \n\\label{eq-050}\n G_1(\\vec{n},\\vec{m})=-d^{(s)}_3[\\frac{3}{2}S(\\vec{n},\\vec{m})\n +2R(0)\\delta_{n,m} -T(\\vec{n},\\vec{m})],\n\\end{equation}\n\n\\begin{equation} \n\\label{eq-051}\n G_2(\\vec{n},\\vec{m})=-d^{(s)}_3[2\\hat{M}(0)e^{(s)}_3(0)\\delta_{n,m}\n +\\frac{3}{2}\\hat{M}(\\vec{n},\\vec{m})e^{(s)}_2(\\vec{n},\\vec{m})-2\n \\hat{M}(\\vec{n},\\vec{m})e^{(c)}_2(\\vec{n},\\vec{m})],\n\\end{equation}\n\n\\begin{equation} \n\\label{eq-052}\n K_1(\\vec{n},\\vec{m})=-d^{(s)}_3[2\\sum_{\\vec{l}}R(\\vec{l},\\vec{m})+\n \\frac{1}{2}S(\\vec{n},\\vec{m})-T(\\vec{n},\\vec{m})],\n\\end{equation}\n\n\\begin{equation} \n\\label{eq-053}\n K_2(\\vec{n},\\vec{m})=-d^{(s)}_3[2\\sum_{\\vec{l}}\\vec{L}(\\vec{n},\\vec{l})\n e^{(s)}_3(\\vec{l},\\vec{m})+\\frac{1}{4}\\vec{L}(\\vec{n},\\vec{m})\n e^{(s)}_2(0)],\n\\end{equation}\n\n\\begin{equation} \n\\label{eq-054}\n K_3(\\vec{n},\\vec{m})=-d^{(s)}_3[2\\sum_{\\vec{l}}\\vec{M}(\\vec{n},\\vec{l}) \n e^{(s)}_3(\\vec{l},\\vec{m})+\\frac{1}{2}\\vec{M}(\\vec{n},\\vec{m}) \n e^{(s)}_2(\\vec{n},\\vec{m})+2\\vec{M}(\\vec{n},\\vec{m})\n e^{(c)}_2(\\vec{n},\\vec{m})], \n\\end{equation}\n\n\\begin{equation} \n\\label{eq-055}\n F(\\vec{n},\\vec{m})=-d^{(s)}_3[2\\vec{L}(\\vec{n},\\vec{m})\n e^{(s)}_3(\\vec{n},\\vec{m})-\\frac{3}{4}\\delta_{n,m}\n e^{(s)}_2(0)\\hat{L}(\\vec{n},\\vec{m})].\n\\end{equation}\n\n\\noindent Next we compute:\n\n\\[\n \\langle\\tilde{S}_{int}\\tilde{S}^2_D\\rangle_{da}=\\frac{da}{a}\\{\n \\frac{\\Lambda^{2-d}}{N}\\frac{A^{-1}B^{-1}}{N^2}\\sum_n\\sum_m\\sum_{\\alpha}\n \\sum_{\\beta} \\int d^dx\\int dt_1\\int dt_2 [\n \\frac{J_1(\\hat{\\beta})}{2I_1(\\hat{\\beta})}(d^{(s)}_3)^2\n \\hat{M}(\\vec{n},\\vec{m})(4N-1)\n\\]\n\\begin{equation}\n\\label{eq-056}\n +\\frac{J_2(2\\hat{\\beta})}{I_2(2\\hat{\\beta})}(d^{(s)}_3)^2\n (2\\hat{L}(0)\\delta_{n,m}+4\\sum_{\\hat{l}}\\hat{M}(\\vec{l},\\vec{m})\n -\\hat{M}(\\vec{n},\\vec{m}))]\n C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)\\}.\n\\end{equation}\n\n\\noindent The OPE in Eq.\\ref{eq-056} determines the behavior of the \nCooperon as a function of the polynomials\n$\\hat{M}$ (see Eq.\\ref{eq-047}) and Cooperon coupling $d^{(s)}_3$.\n\n\\[\n \\frac{1}{3!} \\langle\\tilde{S}^3_{int}\\rangle_{da}=\\frac{da}{a}\\{\n \\frac{\\Lambda^{1-d}}{2N}\\frac{(A^{-1})^2}{3!4N^2}\n \\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})} \n \\sum_n\\sum_m\\sum_{\\alpha}\\int d^dx\\int dt\n\\]\n\\begin{equation} \n\\label{eq-057}\n [W(\\vec{n},\\vec{m})Q^{(B)}_{n,m;\\alpha}(\\vec{x},t)-\n V(\\vec{n},\\vec{m})H^{(B)}_{n,m;\\alpha}(\\vec{x},t)+\n U(\\vec{n},\\vec{m})O^{(s)}_{n,m;\\alpha}(\\vec{x},t)]\\}\n\\end{equation}\n\n\\noindent where the functions $U$, $V$, and $W$ are defined in terms of the\nmicroscopic couplings and the second order functions $R$ and $S$ defined\nin Eq.\\ref{eq-039}.\n\n\\[\n W(\\vec{n},\\vec{m})=-[2R(\\vec{n},\\vec{m})e^{(s)}_3(\\vec{n},\\vec{m})\n +\\frac{3}{4}\\delta_{n,m}(R(0)e^{(s)}_2(0)+S(0)e^{(s)}_3(0))]\n\\]\n\\[\n V(\\vec{n},\\vec{m})=4S(\\vec{n},\\vec{m})e^{(s)}_2(\\vec{n},\\vec{m})\n\\]\n\\begin{equation} \n\\label{eq-058}\n U(\\vec{n},\\vec{m})=2\\sum_{\\vec{l}}R(\\vec{n},\\vec{l})\n e^{(s)}_3(\\vec{l},\\vec{m})+\\frac{1}{4}(\n R(\\vec{n},\\vec{m})e^{(s)}_2(0)+S(0)e^{(s)}_3(\\vec{n},\\vec{m}))\n\\end{equation}\n\n\\noindent and\n\n\\[\n \\frac{1}{3!} \\langle\\tilde{S}^3_D\\rangle_{da}=\\frac{da}{a}\\{\n -\\frac{\\Lambda^{2-d}}{N}(B^{-1})^2\\frac{J_2(2\\hat{\\beta})}{I_2(2\\hat{\\beta})}\n \\frac{16}{3!}\\sum_n\\sum_m\\sum_{\\alpha}\\sum_{\\beta}\\int d^dx\\int\n dt_1\\int dt_2\n\\]\n\\begin{equation} \n\\label{eq-059}\n (d^{(s)}_3)^3(1-\\frac{1}{4N})^3C^{(s)}_{n,m;\\alpha,\\beta}(\\vec{x};t_1,t_2)\\}\n\\end{equation}\n\n\\noindent Using the results given in Eqs.\\ref{eq-029}-\\ref{eq-059} we will\nobtain the RG equations.\\\\\n\n\\section{The RG equations in the Quantum limit}\n\\label{sec-5}\n\nThe quantum region is defined by $\\Lambda/b_T<|q|<\\Lambda$ where\n$b_T=\\frac{v_F\\Lambda}{T}$. In principle it is possible that before the\nscale $b_T$ has been reached, one of the coupling constants has reached\nvalues of order one. If this happens at a scale $b_o<b_T$ we have to stop at\n$b_o$ and for the interval $\\frac{\\Lambda}{b_T}\\le |q|<\\frac{\\Lambda}{b_o}$ we\nhave a different theory. If the Cooperon coupling constant\n$\\frac{\\hat{t}}{N}\\propto(k_F\\ell)^{-1}$ reaches values of order one at\n$b_o<b_T$ we must crossover to the Finkelstein diffusion theory. From the\nother hand if one of the two-body interactions reaches large values we have to\nconstruct a new theory. If the two-body interaction which grows under scaling\nis the Cooper coupling constant we have to construct a theory based on a\nsuperconductivity with disorder. We will consider here the situation where\nthe effects of interactions are such that the value of $b_o\\equiv b_{Dif}$\nobeys $b_{Dif}>b_T$ or $b_o\\equiv b_{SC}$, $b_{SC}<b_T$ ($b_{SC}$ is the\nlength scale where the Cooper coupling constant diverges.). Therefore we\nwill ignore the diffusive region.\\\\\n\nWe introduce the following rescaled coupling constants:\n\n\\[\n d^{(s)}_3=\\hat{t}B; \\;\\;\\;\\;\\;\n e^{(c)}_2(\\vec{n},\\vec{m})=\\hat{e}^{(c)}_2(\\vec{n},\\vec{m})A;\n\\]\n\\[\n \\Gamma^{(c)}_2(\\vec{n},\\vec{m})=\\hat{\\gamma}^{(c)}_2(\\vec{n},\\vec{m})A;\n \\;\\;\\;\\;\\;\n e^{(s)}_2(\\vec{n},\\vec{m})=\\hat{e}^{(s)}_2(\\vec{n},\\vec{m})A;\n\\]\n\\[\n \\Gamma^{(s)}_2(\\vec{n},\\vec{m})=\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})A; \n \\;\\;\\;\\;\\;\n e^{(s)}_3(\\vec{n},\\vec{m})=\\hat{e}^{(s)}_3(\\vec{n},\\vec{m})A;\n\\]\n\\begin{equation}\n\\label{eq-060}\n \\Gamma^{(s)}_3(\\vec{n},\\vec{m})=\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})A.\n\\end{equation}\n\n\\noindent where the constants $A$ and $B$ are defined in Eqs.\\ref{eq-040} and\n\\ref{eq-044}. In the quantum regime the number of channels obeys\n$N_0\\rightarrow N(b)=\\pi(\\frac{k_F}{\\Lambda/b})=N_0b$.\nDue to the fact that when the cutoff $\\Lambda$ is reduced to $\\Lambda/b$\nthe number of channels scales like $N(b)=N_0b$, it follows that the naive\nscaling dimension of the interaction and disorder will be\n\n\\[\n \\frac{\\gamma}{N}\\stackrel{\\Lambda/b}{\\longrightarrow}\n \\frac{\\gamma}{N}b^{2-d} \n \\;\\;\\;\\;\n and\n \\;\\;\\;\\;\n \\frac{\\hat{t}}{N}\\stackrel{\\Lambda/b}{\\longrightarrow}\n \\frac{\\hat{t}}{N}b^{3-d}.\n\\]\n\n\\noindent We observe that the interaction becomes marginal while the\ndisorder is relevant. In the opposite situation where the number of\nchannels does not scale, we have: $\\gamma\\rightarrow\\gamma b^{1-d}$\nand $\\hat{t}\\rightarrow\\hat{t}b^{2-d}$.\\\\\n\nFor the disorder Cooperon coupling constant $\\hat{t}$ we\nhave the scaling equation:\n\n\\[\n \\frac{d\\hat{t}}{d\\ln b}=\\hat{t}[1-\\frac{1}{N}(\\frac{3}{4}\n \\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})-\\hat{\\gamma}^{(c)}_2(\\vec{n},\\vec{m})\n +\\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}\\delta_{n,m}\n [\\hat{\\gamma}^{(s)}_3]^2_{n,m})]\n\\]\n\\begin{equation}\n\\label{eq-061}\n +2\\hat{t}^2[1-\\frac{1}{N}\\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}\n (\\frac{3}{2}\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})-2\n \\hat{\\gamma}^{(c)}_2(\\vec{n},\\vec{m}))-\\frac{1}{N}\n \\frac{J_2(\\hat{\\beta})}{I_2(\\hat{\\beta})}(\n 3\\langle\\hat{\\gamma}^{(s)}_2\\rangle\n -2\\langle\\hat{\\gamma}^{(c)}_2\\rangle)].\n\\end{equation}\n\n\\noindent In Eq.\\ref{eq-061} we use the notation:\n\n\\begin{equation}\n\\label{eq-062}\n [\\hat{\\gamma}^{(s)}_3]^2_{n,m} \\equiv \\frac{1}{N} \\sum_{\\vec{l}}\n \\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{l})\\hat{\\gamma}^{(s)}_3(\\vec{l},\\vec{m})\n\\end{equation}\n\n\\begin{equation} \n\\label{eq-063}\n \\langle\\hat{\\gamma}^{(s)}_2\\rangle=\\frac{1}{N}\\sum_{\\vec{l}}\n \\gamma^{(s)}_2(\\vec{l},\\vec{n}),\n \\;\\;\\;\\;\\;\n \\langle\\hat{\\gamma}^{(c)}_2\\rangle=\\frac{1}{N}\\sum_{\\vec{l}}\n \\gamma^{(c)}_2(\\vec{l},\\vec{n}).\n\\end{equation}\n\n\\noindent From Eq.\\ref{eq-061} we see that we can have a M-I transition in\ntwo dimensions when the p-p interaction $\\gamma^{(s)}_3$ and the p-h\n$\\gamma^{(s)}_2$ increases such that the linear term in ``$\\hat{t}$\"\nbecomes negative (see Eq.\\ref{eq-061}). We observe in Eq.\\ref{eq-061}\nthat the\neffect of the p-h singlet $\\gamma^{(c)}_2$ is opposite to the p-h triplet\n$\\gamma^{(s)}_2$. $\\gamma^{(c)}_2$ enhances the localization while\n$\\gamma^{(s)}_2$ drives the system metallic. This is consistent with the known\nfact that a ``Hartree\" term ($\\gamma^{(c)}_2$) favors localization while the\n``Fock\" exchange term ($\\gamma^{(s)}_2$) drives the system metallic. From\ndimensional analysis it follows that Eq.\\ref{eq-061} must be linear in\n$\\hat{t}$. In addition we have that the number of channels obey the scaling\nlaw, $N=N(b)=N_ob$.\\\\\n\nThe scaling equation for the particle-hole singlet is:\n\n\\[\n \\frac{d\\hat{\\gamma}^{(c)}_2(\\vec{n},\\vec{m})}{d \\ln b}=\\frac{1}{N}\\{\n (\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m}))^2+\n \\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})} [\n \\hat{t}\\langle\\hat{\\gamma}^{(s)}_3\\rangle(\n \\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})(1-3\\delta_{n,m})+\n \\frac{3}{2}\\delta_{n,m}\\hat{\\gamma}^{(s)}_2(0))\n\\]\n\\begin{equation} \n\\label{eq-064}\n -\\frac{1}{24}[\\hat{\\gamma}^{(s)}_3]^2_{n,m}(\n \\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})(4-3\\delta_{n,m})\n +\\frac{3}{2}\\delta_{n,m}\\hat{\\gamma}^{(s)}_3(0))]\\}\n +\\frac{1}{2}\\delta_{n,m}\n \\frac{d\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})}{d \\ln b}\n\\end{equation}\n\n\\noindent and the particle-hole triplet \n$ \\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})$ is given by:\n\n\\begin{equation} \n\\label{eq-065}\n \\frac{d\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})}{d \\ln b}=\\frac{1}{N}\\{\n -(\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m}))^2+\\frac{1}{6N}\n \\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}\n (\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m}))^3\\})+\n 2\\delta_{n,m}\\frac{d\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})}{d \\ln b}\n\\end{equation}\n\n\\noindent From Eqs.\\ref{eq-064} and \\ref{eq-065} we see that the\nparticle-particle channel affects the particle-hole singlet. In addition for\n$\\vec{n}=\\vec{m}$ the particle-particle channel \n$\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{n})$ is identicle to the particle-hole\ntriplet $\\frac{1}{2}\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{n})$,\n$\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{n})=\n\\frac{1}{2}\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{n})$.\\\\\n\nThe particle-particle singlet term obeys the scaling equation:\n\n\\[\n \\frac{d\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})}{d \\ln b}=\n -\\frac{1}{2}[\\hat{\\gamma}^{(s)}_3]^2_{n,m}\n +\\hat{t}\\langle\\hat{\\gamma}^{(s)}_3\\rangle+\\frac{1}{3!}\n \\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}[\\hat{\\gamma}^{(s)}_3]^3_{n,m}\n\\]\n\\[\n -2\\frac{J_2(\\hat{\\beta})}{I_2(\\hat{\\beta})}\\hat{t}\n \\langle(\\hat{\\gamma}^{(s)}_3)^2\\rangle\n -4\\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}\\hat{t}\n \\langle(\\hat{\\gamma}^{(s)}_3)^2\\rangle\n +8\\frac{J_2(\\hat{\\beta})}{I_2(\\hat{\\beta})}\\hat{t}^2\n \\langle(\\hat{\\gamma}^{(s)}_3)\\rangle\n +8\\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}\\hat{t}^2\n \\langle(\\hat{\\gamma}^{(s)}_3)\\rangle\n\\]\n\\[\n +\\frac{1}{N}\\{[\\hat{t}+4\\hat{t}^2(\n \\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}+\\frac{A}{B}\n \\frac{J_2(\\hat{\\beta})}{I_2(\\hat{\\beta})})]\n [\\frac{1}{2}\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})\n -2\\hat{\\gamma}^{(c)}_2(\\vec{n},\\vec{m})]\\}\n +\\frac{1}{N} \\{\\frac{1}{8}\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})\n \\hat{\\gamma}^{(s)}_3(0)-\\frac{1}{16}\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})\n \\hat{\\gamma}^{(s)}_2(0)\n\\]\n\\begin{equation} \n\\label{eq-066}\n -\\frac{1}{3!}\\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}\\delta_{n,m}\n [\\hat{\\gamma}^{(s)}_3]^2_{n,m}\\hat{\\gamma}^{(s)}_3(0)\n +\\frac{1}{2!3!}\\frac{J_1(\\hat{\\beta})}{I_1(\\hat{\\beta})}\n [\\hat{\\gamma}^{(s)}_3]^2_{n,m}\\hat{\\gamma}^{(s)}_2(0)\n -\\frac{J_2(2\\hat{\\beta})}{I_2(2\\hat{\\beta})}\n \\langle(\\hat{\\gamma}^{(s)}_3)\\rangle\n (\\frac{1}{2}\\hat{\\gamma}^{(s)}_2(0)-\\hat{\\gamma}^{(s)}_3(0)\\delta_{n,m})\\}\n\\end{equation}\n\n\\noindent where\n\n\\[\n [\\hat{\\gamma}^{(s)}_3]^2_{n,m}=\\frac{1}{N}\\sum_{\\vec{l}}\n \\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{l})\\hat{\\gamma}^{(s)}_3(\\vec{l},\\vec{m})\n\\]\n\\[\n \\langle(\\hat{\\gamma}^{(s)}_3)^2\\rangle=\\frac{1}{N}\\sum_{\\vec{l}}\n (\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{l}))^2\n\\]\n\\[ \n \\langle(\\hat{\\gamma}^{(s)}_3)\\rangle=\\frac{1}{N}\\sum_{\\vec{l}} \n \\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{l})\n\\] \n\\begin{equation} \n\\label{eq-067}\n [\\hat{\\gamma}^{(s)}_3]^3_{n,m}=\\frac{1}{N^2}\\sum_{\\vec{l}}\n \\sum_{\\vec{l}^{\\prime}}\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{l})\n \\hat{\\gamma}^{(s)}_3(\\vec{l},\\vec{l}^{\\prime})\n \\hat{\\gamma}^{(s)}_3(\\vec{l}^{\\prime},\\vec{m}).\n\\end{equation}\n\n\\noindent The scaling relation for the forward part are trivial:\n\n\\begin{equation} \n\\label{eq-068}\n \\frac{d\\Gamma^{(c)}}{d\\ln b}= \\frac{d\\Gamma^{(s)}}{d\\ln b}=0\n\\end{equation}\n\n\\noindent The set of Eqs.\\ref{eq-064}-\\ref{eq-066} show that in the limit of\n$N\\rightarrow\\infty$ the interaction is controlled only by the\nparticle-particle singlet $\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})$. In\naddition we observe that the disorder renormalizes the \n$\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})$. We observe that the scaling equation\nfor $\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})$ can be negative at $b=1$.\nThe origin of the negative\nterm is given by Eq.\\ref{eq-037}, where it has been shown that at short times\nthe Cooperon behaves like a Cooper p-p singlet. As a result the initial\nvalues of the particle-particle singlet\n$\\hat{\\gamma}_3^{(s)}(\\vec{n},\\vec{m}; b=1)$ are replaced by\n$\\hat{\\gamma}_3^{(s)}(\\vec{n},\\vec{m}; b=1)-2v_F(\\frac{B}{A})\\hat{t}$.\nIn Eqs.\\ref{eq-061},\n\\ref{eq-064}, \\ref{eq-065}, and \\ref{eq-066} the scaling of the number of the\nchannels is stopped when diffusive region is reached. At finite temperature we\nstop scaling at the scale $b=b_T=E_F/T$. This will fix the\nnumber of channels to $\\bar{N}\\equiv N_T=E_F/T$ (see ref.\\cite{09}).\nIt might be possible that in two dimensions the decoherency introduced by the\ntemperature might be stronger than $T$. This might be the case if we have in\nmind dephasing effects in two dimensions which can define an effective\ntemperature $T_{eff}(T)>T$ replacing $\\bar{N}$ by $E_F/T_{eff}$.\\\\\n\n\\section{The conducting phase due to the superconducting instability in the\nquantum region}\n\\label{sec-6}\n\nIn the low temperature limit we can ignore all the many body effect except\nthe particle-particle singlet $\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})$. The\nreason being the $1/N$ factor which appears in Eqs.\\ref{eq-064} and\n\\ref{eq-065} and is missing for the particle-particle singlet in\nEq.\\ref{eq-066}. The growth of the number of\nchannels $N(b)$ is determined by the topology of the Fermi surface. In\nparticular this is the case for spherical Fermi surface where $N(b)=N_ob$.\nFor $T\\not= 0$ we obtain $N(b=b_T)=\\frac{E_F}{T}$. (In chapter\n\\ref{sec-8} we will consider non-spherical Fermi surface with repulsive\ninteraction which might lead to a Ferromagnetic instability.)\\\\\n\nDue to the fact that the $1/N$ factor is only absent for the particle-particle\nsinglet, we will investigate the\nproblem in the parameter space $(\\hat{\\gamma}^{(s)}_3,\\hat{t})$\nusing the angular momentum representation:\n\n\\[\n \\gamma^{(s)}_3(r)\\equiv\\gamma_r=\\int_0^{\\pi}\\frac{d\\theta}{\\pi}\n \\gamma^{(s)}_3(\\theta)\\cos(r\\theta),\n \\;\\;\\;\\;\\;\n r=0,2,4,\\cdots.\n\\]\n\n\\noindent For the singlet case $r=0$ we have $\\gamma_{r=0}=\\gamma_o$:\n\n\\begin{equation}\n\\label{eq-069}\n \\gamma_o\\simeq\\frac{1}{N}\\sum_{\\vec{l}}\n \\hat{\\gamma}^{(s)}_3(\\vec{l},\\vec{n})\n\\end{equation}\n\n\\noindent From Eq.\\ref{eq-066} we obtain to leading order in $1/N$\nthe following equation for particle-particle singlet:\n\n\\begin{equation}\n\\label{eq-070}\n \\frac{d\\gamma_o}{d\\ln b}=-\\frac{1}{2}\\gamma_o^2\n +\\gamma_o\\hat{t}-6\\gamma_o^2\\hat{t}+16\\gamma_o\\hat{t}^2\n +\\frac{1}{3!}\\gamma_o^3,\n\\end{equation}\n\n\\noindent with\n\n\\[\n \\gamma_o(b=1)\\rightarrow\\gamma_o(b=1)-2\\tilde{v_F}\\hat{t}\n\\]\n\n\\noindent In Eq.\\ref{eq-070} we have used $I_1\\sim I_2\\sim J_1\\sim 1$ and\n$\\tilde{v_F}=v_FB/A$ where the constants $A$ and $B$ have been defined\nin Eqs.\\ref{eq-040} and \\ref{eq-044}.\\\\\n\n\\noindent We investigate Eq.\\ref{eq-070} in the limit of weak disorder\n$\\hat{t}\\rightarrow 0$. We find that even for positive value of $\\gamma_o$\nthe effect of disorder is to drive $\\gamma_o(b)$ to negative values.\nThe reason for this is the fact that the negative linear term in $\\hat{t}$ can\ncause an initial negative value for $\\gamma_o(b=1)$. As a result the term\n$-\\frac{1}{2}\\gamma_o^2$ (for negative value of $\\gamma_o$, $\\gamma_o(b=1)<0$)\nmight drive the particle-particle interaction towards a superconducting\ninstability. This behavior can be seen in the following way. In the\nlimit of $\\hat{t}\\rightarrow 0$ we keep in Eq.\\ref{eq-070} only the\ntwo first order terms and obtain the solution for $\\gamma_o(b)$:\n\n\\begin{equation}\n\\label{eq-071}\n \\gamma_o(b=e^{\\ell})=\\gamma_o(b=1)\\; exp(\\int_0^{\\ell}\\hat{t}(x)\\;dx)\n[1+\\frac{1}{2}\\gamma_o(b=1)\\int_0^{\\ell}dy\\;exp(\\int_0^{y}\\hat{t}(x)\\;dx)]^{-1} \n\\end{equation}\n\n\\noindent For $\\gamma_o(b=1)<0$, $\\gamma_o(b)$ diverges at a length scale\n$b=b_{SC}\\equiv\\frac{v_F\\Lambda}{T_{SC}}$ where $T_{SC}$ represents\nthe superconducting instability temperature,\n\n\\[\n T_{SC}=v_F\\Lambda(1+\\frac{2\\hat{t}}{|\\gamma_o(b=1)|})^{-\\frac{1}{t}}\n \\stackrel{\\hat{t}\\rightarrow 0}{\\longrightarrow}\n v_F\\Lambda\\; exp(-\\frac{2}{|\\gamma_o(b=1)|}).\n\\]\n\n\\noindent Next we consider the RG equation for the Cooperon (see\nEq.\\ref{eq-061} with\n$J_1(\\hat{\\beta})\\sim I_1(\\hat{\\beta})\\sim I_2(\\hat{\\beta})\\sim 1$).\nFrom Eq.\\ref{eq-061} we observe that in the limit of\nvanishing interactions the Cooperon\ncoupling constant scales like $\\frac{\\hat{t}(b)}{N(b)}\\sim\n\\frac{\\hat{t}(b=1)}{N_o}[1-2\\hat{t}(b=1)\\log b]^{-1}$ and diverges at\n$b\\equiv b_{Loc} \\equiv \\frac{v_F\\Lambda}{T_{Loc}}$\n($b_{Loc}\\ge b_{Dif}$, $T_{Dif}\\ge T_{Loc}$),\n$T_{Loc}\\simeq v_F\\Lambda\\;exp[-\\frac{1}{2\\hat{t}(b=1)}]$.\nIn order to understand the physics of the system we have to compare\nthe physical temperature $T$ with the other two, $T_{SC}$ and $T_{Loc}$.\nWe have to consider separately the cases:\na) $T<T_{SC}<T_{Loc}$; b) $T<T_{Loc}<T_{SC}$; c) $T_{SC}<T_{Loc}<T$;\nd) $T_{SC}<T<T_{Loc}$; e) $T_{Loc}<T<T_{SC}$; f) $T_{Loc}<T_{SC}<T$.\\\\\n\n\\noindent a) $T<T_{SC}<T_{Loc}$\\\\\nThis is the localized case where the mean free path ``$\\ell$\" is the shortest\nlength scale in the problem. This case will not be analyzed here. Most of the\nwork in the past has been concentrated towards this case, in particular the\nFinkelstein theory which has investigated the interactions within the diffusion theory.\\\\\n\n\\noindent b) $T<T_{Loc}<T_{SC}$\\\\\nHere the shortest length scale is the Cooper coherence length. Physically\none can describe this region by a system of disorder bosons (the bosons\ndescribe the pairs). The critical theory might correspond to a disorder X-Y\nmodel.\\\\\n\n\\noindent c) $T_{SC}<T_{Loc}<T$\\\\\nThis is a region where interactions are not important. The physics is controlled\nby classical hopping transport.\\\\\n\n\\noindent d) $T_{SC}<T<T_{Loc}$\\\\\nAs in case a) here the system is localized. This case will not be considered\nhere. (See the Finkelstein theory.)\\\\\n\n\\noindent e) $T_{Loc}<T<T_{SC}$\\\\\nAgain a bosonic X-Y theory with disorder is applicable here as in case b).\\\\\n\n\\noindent f) $T_{Loc}<T_{SC}<T$\\\\\nIn this region we will have transport controlled by pair breaking.\\\\\n\nIn the rest part of this section we will investigate the RG equation for the\nnegative particle-particle singlet $\\hat{\\gamma}_3^{(s)}(\\vec{n},\\vec{m})$\nand the Cooperon coupling constant $\\hat{t}$. In agreement with\nEq.\\ref{eq-069} we introduce the angular momentum representation for the\nCooper and Cooperon channels:\n$\\gamma_o=\\frac{1}{N}\\sum_{\\vec{\\l}}\\hat{\\gamma}_3^{(s)}(\\vec{l},\\vec{n})$,\n$t_o=\\frac{1}{N}\\sum_{\\vec{\\l}}\\hat{t}(\\vec{l},\\vec{n})$. We obtain from\nEq.\\ref{eq-061} and Eq.\\ref{eq-070} the following RG equations:\n\n\\[\n \\frac{d\\lambda}{d\\ln b}=\\frac{1}{2}\\lambda^2+\\lambda t_o,\n \\;\\;\\;\\;\\;\n \\lambda\\equiv -\\gamma_o\n\\]\n\\[\n \\frac{dt_o}{d\\ln b}=t_o[1-(\\frac{\\lambda}{N})^2]+2t_o^2\n\\]\n\\begin{equation}\n\\label{eq-072}\n \\rho(b)\\propto\\frac{t_o(b)}{N(b)}\\equiv\\bar{t}_o(b),\n \\;\\;\\;\\;\\;\n N(b)=N_ob\n\\end{equation}\n\n\\noindent $\\rho(b)$ is the resistance with $b$ restricted to\n$1<b\\le\\frac{v_F\\Lambda}{T}$. From Eq.\\ref{eq-072} we observe that in the \nlimit\n$b\\rightarrow\\infty$ ($T\\rightarrow 0$) the parameter $\\lambda$ diverges.\nIn particular we observe that the ratio\n$\\frac{\\lambda(b)}{N(b)}\\stackrel{b\\rightarrow\\infty}{\\longrightarrow}\\infty$.\nAs a result the RG equation behaves like\n$\\frac{dt_o}{d\\ln b}=-t_o(\\frac{\\lambda}{N})^2$. Due to the large value of\n$(\\frac{\\lambda}{N})^2$ it follows that\n$t_o(b)\\stackrel{b\\rightarrow\\infty}{\\longrightarrow}0$. As a result we obtain\na superconducting ground state. At finite temperature we consider the case\n$b_T<b_{SC}<b_{Loc}$. We substitute the solution of $\\lambda(b)$ into\n$t_o(b)$ and obtain\n\n\\begin{equation}\n\\label{eq-073}\n \\bar{t}_o(b_T)=\\frac{t_o}{N_o}\n exp\\{-\\int_0^{\\log b_T}(\\frac{\\lambda(x)}{N(x)})^2 dx\\}\n \\sim \\frac{t_o}{N_o}\n exp\\{-\\frac{(4/N_o^2)T_{SC}}{|T-T_{SC}|}\\},\n \\;\\;\\;\n T>T_{SC}\n\\end{equation}\n\n\\noindent $N_o\\simeq\\frac{\\pi k_F}{\\Lambda}\\sim 1$ and $T_{SC}$ is given by\nEq.\\ref{eq-071}. As a result we obtain that the resistance obeys\n$\\rho(T)\\stackrel{T\\rightarrow T_{SC}}{\\longrightarrow}0$, \n$ \\rho(T)\\sim\\;Const.\\; exp\\{-\\frac{(4/N_o^2)T_{SC}}{|T-T_{SC}|}\\}$.\nTo conclude this section ($\\gamma_o<0$) we remark that the transport\ndata \\cite{03} show some similarity with the one reported for disorder\nbosons in ref.\\cite{04}. This might suggest that the correct starting point\nmight be a disordered bosonic system instead of a diffusion theory \\cite{02}.\\\\\n\n\\section{The RG equation at finite temperature}\n\\label{sec-7}\n\nAt a temperature $T$ the scaling is restricted to\n$\\frac{\\Lambda}{b_T}<|q|<\\Lambda$ where $b_T=\\frac{v_F\\Lambda}{T}$.\nIn this interval the number of channels is restricted to\n$\\bar{N}=N(b_T)=\\frac{E_F}{T}$,\nwith $N(b)$ obeying the condition $N_o<N(b)\\le\\bar{N}$. We replace in\nEqs.\\ref{eq-061}-\\ref{eq-068} $J_1(\\hat{\\beta})\\sim J_2(\\hat{\\beta})\\sim\nI_1(\\hat{\\beta})\\sim I_2(\\hat{\\beta})\\sim 1$ and find a simplified form\n\n\n\\[\n \\frac{d\\hat{t}}{d\\ln b}=\\epsilon(b)\\hat{t}-\\frac{\\hat{t}}{N}(\\frac{3}{4}\n \\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})-\\hat{\\gamma}^{(c)}_2(\\vec{n},\\vec{m})\n +\\delta_{n,m}[\\hat{\\gamma}^{(s)}_3]^2_{n,m})\n\\]\n\\begin{equation}\n\\label{eq-074}\n +2\\hat{t}^2[1-\\frac{1}{N}(\\frac{3}{2}\n \\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})-2\\hat{\\gamma}^{(c)}_2(\\vec{n},\\vec{m}))\n -\\frac{1}{N}(3\\langle\\hat{\\gamma}^{(s)}_2\\rangle\n -2\\langle\\hat{\\gamma}^{(c)}_2\\rangle)]\n\\end{equation}\n\n\\noindent The parameter $\\epsilon(b)$ controls the crossover at finite\ntemperatures. $\\epsilon(b)$ is given by, $\\epsilon(b)=1$ for $b<b_T$ and\n$\\epsilon(b)\\simeq 0$ for $b>b_T$. Eq.\\ref{eq-074} replaces the scaling\nEq.\\ref{eq-061} for the disorder\ncoupling constant $\\hat{t}$. In Eq.\\ref{eq-074} we observe that the\ninteraction\nhas produced a shift in the critical dimensionality. The disorder parameter\n$\\hat{t}$ has accumulated a finite anomalous dimension,\n$\\frac{1}{N}(\\frac{3}{4}\\hat{\\gamma}_2^{(s)}\\cdots)$, which will control the\nM-I transition. (In the limit $T\\rightarrow 0$, $N\\rightarrow\\infty$\ncausing this term to disappear.)\\\\\n\nAt finite temperatures the scaling Eqs.\\ref{eq-064} and \\ref{eq-065} for\nthe interactions $\\hat{\\gamma}^{(s)}_2$ and $\\hat{\\gamma}^{(c)}_2$ are\nthe same except that linear terms of the form\n$[\\epsilon(b)-1]\\hat{\\gamma}^{(s)}_2$ and\n$[\\epsilon(b)-1]\\hat{\\gamma}^{(c)}_2$ are added to the Eqs.\\ref{eq-064}\nand \\ref{eq-065}, respectively.\nFor the particle-particle singlet $\\hat{\\gamma}^{(s)}_3$ we have\n\n\\[\n \\frac{d\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})}{d\\ln b}=\n [\\epsilon(b)-1]\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})-\\frac{1}{2}\n [\\hat{\\gamma}^{(s)}_3]^2_{n,m}+\\hat{t}\\langle\\hat{\\gamma}^{(s)}_3\\rangle\n +\\frac{1}{3!}[\\hat{\\gamma}^{(s)}_3]^3_{n,m}-6\n \\hat{t}\\langle(\\hat{\\gamma}^{(s)}_3)^2\\rangle+16\n \\hat{t}\\langle\\hat{\\gamma}^{(s)}_3\\rangle\n\\]\n\\[\n +\\frac{1}{N}\\{(\\hat{t}+8\\hat{t}^2)(\\frac{1}{2}\n \\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{m})-2\\hat{\\gamma}^{(c)}_2(\\vec{n},\\vec{m}))\n \\}+\\frac{1}{N}\\{\\frac{1}{8}\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})\n \\hat{\\gamma}^{(s)}_3(0)-\\frac{1}{16}\\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})\n \\hat{\\gamma}^{(s)}_2(0)\n\\]\n\\begin{equation}\n\\label{eq-075}\n -\\frac{1}{3!}\\delta_{n,m}[\\hat{\\gamma}^{(s)}_3]^2_{n,m}\n \\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{m})+\\frac{1}{2!3!}\n [\\hat{\\gamma}^{(s)}_3]^2_{n,m}\\hat{\\gamma}^{(s)}_2(0)\n -\\langle\\hat{\\gamma}^{(s)}_3\\rangle(\\frac{1}{2}\\hat{\\gamma}^{(s)}_2(0)\n -\\hat{\\gamma}^{(s)}_3(0)\\delta_{n,m})\\}\n\\end{equation}\n\nIn Eq.\\ref{eq-075} we use the same definitions as given in Eq.\\ref{eq-067}.\nEq.\\ref{eq-075} must be supplemented by the condition\n$\\frac{1}{2}\\hat{\\gamma}^{(s)}_2(\\vec{n},\\vec{n})=\n \\hat{\\gamma}^{(s)}_3(\\vec{n},\\vec{n})$ plus Eqs.\\ref{eq-064} and\n\\ref{eq-065}.\\\\\n\n\\section{The scaling equations for the resistivity at finite temperatures\nand strong repulsive interactions}\n\\label{sec-8}\n\nWe restrict ourselves to finite temperatures or/and cases where the scaling\nof the number of channels is different from $N(b)=N_ob$ (spherical Fermi\nsurface). For flat Fermi surface the number of the channels does not scale.\nWe have $N(b)\\sim N(b=1)\\sim N_o$. At finite temperature for spherical Fermi\nsurface the number of channels is finite and is restricted by the temperature\n$N_o<N(b)<N(b_T)\\sim \\frac{E_F}{T}$. Since the coupling constants depend on\nthe number of channels (finite), we will normalize the coupling constant by\n$N$, the number of channels\n\n\\begin{equation}\n\\label{eq-076}\n \\bar{\\gamma}_2^{(c)}\\equiv\\frac{\\hat{\\gamma}_2^{(c)}}{N},\n \\;\\;\\;\n \\bar{\\gamma}_2^{(s)}\\equiv\\frac{\\hat{\\gamma}_2^{(s)}}{N},\n \\;\\;\\; \n \\bar{\\gamma}_3^{(s)}\\equiv\\frac{\\hat{\\gamma}_3^{(s)}}{N},\n \\;\\;\\;\n \\bar{t}\\equiv\\frac{\\hat{t}}{N}.\n\\end{equation}\n\n\\noindent As a result the new RG equations are given in terms of the original\nEqs. \\ref{eq-075}, \\ref{eq-065}, and \\ref{eq-066}:\n\n\\[\n \\frac{d\\bar{\\gamma}_2^{(c)}}{d\\ln b}=\\frac{1}{N}\n (\\frac{d\\hat{\\gamma}_2^{(c)}}{d\\ln b})-\\epsilon_T\\bar{\\gamma}_2^{(c)};\n\\]\n\\[\n \\frac{d\\bar{\\gamma}_2^{(s)}}{d\\ln b}=\\frac{1}{N} \n (\\frac{d\\hat{\\gamma}_2^{(s)}}{d\\ln b})-\\epsilon_T\\bar{\\gamma}_2^{(s)}; \n\\]\n\\begin{equation} \n\\label{eq-077}\n \\frac{d\\bar{\\gamma}_3^{(s)}}{d\\ln b}=\\frac{1}{N} \n (\\frac{d\\hat{\\gamma}_3^{(s)}}{d\\ln b})-\\epsilon_T\\bar{\\gamma}_3^{(s)}\n\\end{equation}\n\n\\noindent The parameter $\\epsilon_T$ depends on the topology of the Fermi\nsurface and temperature\n\n\\begin{equation} \n\\label{eq-078}\n \\epsilon_T\\equiv|\\frac{d\\ln N(b)}{d \\ln b}|,\n \\;\\;\\;\n N_o\\le N(b)\\le N(b_T).\n\\end{equation}\n\n\\noindent The parameter $\\epsilon_T$ takes values of $0\\le\\epsilon_T\\le 1$. The\nvalue of $\\epsilon_T=1$ is obtained for spherical Fermi surface $N(b)=N_ob$ and\n$\\epsilon_T=0$ is obtained for flat Fermi surface or high temperatures,\n$N(b)\\sim\\bar{N}\\sim\\frac{E_F}{T}$.\\\\\n\nHere we consider a special case of repulsive interactions such that the\nparticle-particle singlet and particle-hole triplet are strong in the\nbackward direction. This\nmeans that the most relevant interactions are those with $\\vec{n}=\\vec{m}$.\nIn order to be specific we will consider a special model for which the terms\n$\\hat{\\gamma}_2^{(c)}(\\vec{n},\\vec{m})$,\n$\\hat{\\gamma}_2^{(s)}(\\vec{n},\\vec{m})$, and\n$\\hat{\\gamma}_3^{(s)}(\\vec{n},\\vec{m})$ are zero for $\\vec{n}\\not=\\vec{m}$.\nWe keep only terms with $\\vec{n}=\\vec{m}$ and introduce the definition:\n\n\\[\n \\hat{\\gamma}_2^{(c)}\\equiv\\hat{\\gamma}_2^{(c)}(\\vec{n},\\vec{n}),\n \\;\\;\\;\n \\hat{\\gamma}_2^{(s)}\\equiv\\hat{\\gamma}_2^{(s)}(\\vec{n},\\vec{n})\n =2\\hat{\\gamma}_3^{(s)}(\\vec{n},\\vec{n})\n\\]\n\\begin{equation} \n\\label{eq-079}\n \\hat{\\gamma}_2^{(c)}(\\vec{n}\\not=\\vec{m})\\simeq\n \\hat{\\gamma}_2^{(s)}(\\vec{n}\\not=\\vec{m})\\simeq\n \\hat{\\gamma}_3^{(s)}(\\vec{n}\\not=\\vec{m})\\simeq 0\n\\end{equation}\n\n\\noindent Using Eqs. \\ref{eq-076}, \\ref{eq-077}, and \\ref{eq-078} we obtain:\n\n\\begin{equation}\n\\label{eq-080}\n \\frac{\\bar{\\gamma}_2^{(c)}}{d\\ln b}=\n -\\bar{\\gamma}_2^{(c)}(\\epsilon_T+\\hat{t})\n +\\frac{1}{2}\\hat{t}\\bar{\\gamma}_2^{(s)}\n -\\frac{3}{4}(\\bar{\\gamma}_2^{(s)})^2(\\hat{t}-\\frac{1}{4})\n\\end{equation}\n\n\\begin{equation}\n\\label{eq-081}\n \\frac{\\bar{\\gamma}_2^{(s)}}{d\\ln b}=\n \\bar{\\gamma}_2^{(s)}(2\\hat{t}+8\\hat{t}^2-\\epsilon_T)\n -(\\bar{\\gamma}_2^{(s)})^2(\\frac{5}{4}+3\\hat{t})\n +\\frac{5}{24}(\\bar{\\gamma}_2^{(s)})^3\n -4\\bar{\\gamma}_2^{(c)}(\\hat{t}+8\\hat{t}^2)\n\\end{equation}\n\n\\begin{equation} \n\\label{eq-082}\n \\frac{d\\hat{t}}{d\\ln b}=\n \\hat{t}[1-\\frac{3}{4}\\bar{\\gamma}_2^{(s)}+\\bar{\\gamma}_2^{(c)}\n -\\frac{1}{4}(\\bar{\\gamma}_2^{(s)})^2]\n +2\\hat{t}^2[1-\\frac{3}{2}\\bar{\\gamma}_2^{(s)}+2\\bar{\\gamma}_2^{(c)}]\n\\end{equation}\n\n\\begin{equation} \n\\label{eq-083}\n \\bar{t}=\\frac{\\hat{t}}{N},\n \\;\\;\\;\n N=N(b),\n \\;\\;\\;\n 1\\le b\\le b_T=\\frac{E_F}{T}\n\\end{equation}\n\n\\noindent From Eq.\\ref{eq-080} we conclude that the particle-hole singlet\n$\\bar{\\gamma}_c$ is irrelevant. Therefore we will take $\\bar{\\gamma}_c=0$\nand ignore Eq.\\ref{eq-080}. We will solve the RG equation in the space of\n$\\bar{\\gamma}_2^{(s)}$ and $\\hat{t}$ (Eqs. \\ref{eq-081} and \\ref{eq-082}).\nIn the parameter space $(\\bar{\\gamma}_2^{(s)},\\hat{t})$ we find a non-trivial\nfixed point. In the limit $\\epsilon_T\\rightarrow 0$ we find\n$(\\bar{\\gamma}_2^{(s)})^*=\\frac{8}{5}(\\hat{t})^*\\simeq\\frac{4}{25}$.\\\\\n\n\nWe linearize the equations around this fixed point and find:\n$\\hat{t}(b)=\\hat{t}^*+(\\hat{t}-\\hat{t}^*)b^{1/\\nu_1}$,\n$\\nu_1\\simeq 1+\\frac{2}{25}$ and\n$\\bar{\\gamma}_2^{(s)}(b)=\\bar{\\gamma}_2^*+(\\bar{\\gamma}_2^{(s)}-\n\\bar{\\gamma}_2^*)b^{-1/\\nu_2}$. \nThese equations show that for $\\hat{t}<\\hat{t}^*$\nthe disorder decreases and in the same time $\\bar{\\gamma}_2^{(s)}$ flows to\n$\\bar{\\gamma}_2^*$. For large value of disorder we obtain that $\\hat{t}$\nincrease and $\\bar{\\gamma}_2^{(s)}$ flows to $\\bar{\\gamma}_2^*$.\nExperimentally the presence of the stable fixed point $\\bar{\\gamma}_2^*$\nmight be identified by a power law behavior in the spin-spin\ncorrelation. This is similar to what one has in one dimension and might\ncorresponds to a spin-liquid phase. For the transport properties, we\nbelieve that our predictions are in a qualitative agreement with the\nexperiments \\cite{01}, we find for the resistivity\n$\\rho(\\bar{t},\\bar{\\gamma}_2^{(s)},T)$ at a finite temperature and the\ndynamical exponent $z\\simeq 1$: \n$\\rho(\\bar{t},\\bar{\\gamma}_2^{(s)},T)=\n\\rho(\\bar{t}(b),\\bar{\\gamma}_2^{(s)}(b),Tb^z)=\n\\rho(\\bar{t}^*+(\\bar{t}-\\bar{t}^*)b^{1/\\nu_1}$, $\\bar{\\gamma}_2^*+\n(\\bar{\\gamma}_2^{(s)}-\\bar{\\gamma}_2^*) b^{-1/\\nu_2},Tb^z)$.\nWe introduce $Tb^z\\simeq T_o\\Rightarrow b\\simeq(\\frac{T_o}{T})^{1/z}$ and use\nthe definitions $\\bar{t}\\equiv\\frac{\\hat{t}}{\\bar{N}}$.\nAs a result we find:\n\n\\begin{equation} \n\\label{eq-084}\n \\rho(\\bar{t},\\bar{\\gamma}_2^{(s)},T)\\simeq\n \\rho(\\bar{t}^*,\\bar{\\gamma}_2^*,T_o)+{\\textstyle const.}(\n \\frac{\\bar{t}-\\bar{t}^*}{\\bar{t}^*})(\\frac{T_o}{T})^{1/z\\nu_1}\n\\end{equation}\n\n\\noindent Eq.\\ref{eq-084} shows that for $\\bar{t}<\\bar{t}^*$ the resistivity\n$\\rho$ decreases as we lower the temperature and increases when\n$\\bar{t}>\\bar{t}^*$. In order to make contact with the experiments we replace:\n$\\bar{t}\\propto(k_F\\ell)^{-1}$, $k_F\\propto n_c^{1/2}$,\n$\\bar{t}^*\\propto (n_c^*)^{-1/2}$ ($n_c^*$ is the critical density) and identify$\\frac{\\bar{t}-\\bar{t}^*}{\\bar{t}^*}\\propto\\frac{n_c^*-n_c}{n_c^*}\n\\equiv\\delta$. As a result we find:\n$\\rho(n_c,\\bar{\\gamma}_2^{(s)},T)\\simeq\\rho^*f(\\frac{\\delta}{T^{1/z\\nu_1}})$,\n$\\rho^*\\equiv\\rho(n_c^*,\\gamma_2^*,T_o)$ which is the result observed in\nref.\\cite{03}. We hope that more accurate experiments will confirm the\nexistence of the suggested fixed point.\\\\\n\n\\section{Conclusion}\n\\label{sec-9}\n\nA new method for studying many-body systems and disorder has been introduced.\nThe method is based on the extension of the OPE to two dimensional systems.\nUsing a real space version of RG we have derived a set of RG equations for\ndisorder and interaction. We have constructed an alternative theory to the\none constructed by Finkelstein \\cite{02}. The basic assumption in\nref.\\cite{02} is that the elastic mean free path is the shortest length in\nthe problem. As a result the multiple elastic scatterings are replaced by\na diffusion theory (the non-linear $\\sigma$-model) and the interactions are\nconsidered as a perturbation of the diffusion theory. The method used here is\nbased on a RG analysis which studies the competitions between the\nmultiple elastic scattering and the interaction. We identified the following\nregions:\\\\\n1) The multiple elastic scattering is the shortest length scale and diverges\nfirst. For this case we agree with the results given in ref.\\cite{02} and\ndo not have anything to add.\\\\\n2) The particle-particle singlet is negative and a superconducting\ninstability occurs for $T\\le T_{SC}$ where $T_{SC}>T_{Loc}$. As a result\none has to treat first the interaction within an effective\nGinzburg-Landau theory. We reproduced a bosonic model (X-Y) which is\nperturbed by disorder.\\\\\n3) The interactions are positive and the particle-hole is dominant in the\nbackward direction. At finite temperature and non-spherical Fermi\nsurfaces which obey $|\\frac{d\\ln N(b)}{d\\ln b}|\\ll 1$ one obtains a\nnon-trivial fixed point in the plane $(\\bar{\\gamma}_2^{(s)},\\hat{t})$\nwhich separates the conducting from the insulating phase. This fixed\npoint is characterized by a stable fixed point in the $\\bar{\\gamma}_2^{(s)}$\ndirection. No divergence in the particle-hole triplet occurs expect\nthe infinite correlation length for the spin-spin ferromagnetic correlations\nwhen $\\bar{\\gamma}_2^{(s)}\\rightarrow \\bar{\\gamma}_2^*$.\\\\\n\n\\begin{center}\n {\\bf ACKNOWLEDGMENTS\\\\}\n\\end{center} \n\nD. Schmeltzer would like to thank professor A.M. Finkelstein for his valuable\ncomment concerning the differences between his theory and the one presented\nhere.\\\\\n\n\\newpage\n\n\\appendix\n\\section{}\n\\label{app-1}\n\nThe Fermion field $\\psi_{\\sigma,\\alpha}(\\vec{x})$ is decomposed into\n$N$ Fermions, $\\psi_{\\sigma,\\alpha}(\\vec{x})=\\sum_{\\vec{\\omega}=1}^N\ne^{ik_F\\vec{\\omega}\\vec{x}}\\psi_{\\vec{\\omega},\\sigma,\\alpha}$.\nUsing this representation we\nobtain from Eq.\\ref{eq-003} the result:\n\n\\[\n S_{int}\\simeq\\int d^dx\\int dt\\sum_{\\sigma,\\sigma^{\\prime}}\\sum_{\\alpha}\n \\sum_{\\vec{\\omega}_1}\\sum_{\\vec{\\omega}_2}\\sum_{\\vec{\\omega}_3}\n \\sum_{\\vec{\\omega}_4}\n \\delta_{\\vec{\\omega}_1+\\vec{\\omega}_2=\\vec{\\omega}_3+\\vec{\\omega}_4}\n\\]\n\\begin{equation}\n\\label{eq-a1}\n v(\\vec{\\omega}_1,\\vec{\\omega}_2,\\vec{\\omega}_3,\\vec{\\omega}_4)\n \\psi^{\\dagger}_{\\vec{\\omega}_1,\\sigma,\\alpha}(\\vec{x})\n \\psi^{\\dagger}_{\\vec{\\omega}_2,\\sigma^{\\prime},\\alpha}(\\vec{x})\n \\psi_{\\vec{\\omega}_3,\\sigma^{\\prime},\\alpha}(\\vec{x})\n \\psi_{\\vec{\\omega}_4,\\sigma,\\alpha}(\\vec{x})\n\\end{equation}\n\n\\noindent\nwhere $v(\\vec{\\omega}_1,\\vec{\\omega}_2,\\vec{\\omega}_3,\\vec{\\omega}_4)$\nrepresents the projection of the screened two-body potential on the Fermi\nsurface. The presence of the Kroneker-delta function imposes the condition\n$\\vec{\\omega}_1+\\vec{\\omega}_2=\\vec{\\omega}_3+\\vec{\\omega}_4$.\nAs a result we separate the interaction term into three processes:\n1) direct, 2) exchange, and 3) Cooperon channel:\n\n\\begin{enumerate}\n \\item The direct process is realized when $\\vec{\\omega}_1=\\vec{\\omega}_4$,\n $\\vec{\\omega}_2=\\vec{\\omega}_3$.\n \\item The exchange process:\n $\\vec{\\omega}_1=\\vec{\\omega}_3\\equiv\\vec{\\omega}$,\n $\\vec{\\omega}_2=\\vec{\\omega}_4\\equiv\\vec{\\omega}^{\\prime}$\n \\item The Cooperon channel:\n $\\vec{\\omega}\\equiv\\vec{\\omega}_1=-\\vec{\\omega}_2$,\n $\\vec{\\omega}^{\\prime}\\equiv\\vec{\\omega}_3=-\\vec{\\omega}_4$\n\\end{enumerate}\n\n\\noindent As a result Eq.\\ref{eq-a1} becomes\n\n\\[\n S_{int}\\simeq\\int d^dx\\int dt\\sum_{\\sigma,\\sigma^{\\prime}}\\sum_{\\alpha}\n \\sum_{\\vec{\\omega}}\\sum_{\\vec{\\omega}^{\\prime}}\\{\n v(0)\\psi^{\\dagger}_{\\vec{\\omega},\\sigma,\\alpha}(\\vec{x})\n \\psi_{\\vec{\\omega},\\sigma,\\alpha}(\\vec{x})\n \\psi^{\\dagger}_{\\vec{\\omega}^{\\prime},\\sigma^{\\prime},\\alpha}(\\vec{x})\n \\psi_{\\vec{\\omega}^{\\prime},\\sigma^{\\prime},\\alpha}(\\vec{x})\n\\]\n\\[\n -v(\\vec{\\omega},\\vec{\\omega}^{\\prime},\\vec{\\omega},\\vec{\\omega}^{\\prime})\n \\psi^{\\dagger}_{\\vec{\\omega},\\sigma,\\alpha}(\\vec{x})\n \\psi_{\\vec{\\omega},\\sigma^{\\prime},\\alpha}(\\vec{x})\n \\psi^{\\dagger}_{\\vec{\\omega}^{\\prime},\\sigma^{\\prime},\\alpha}(\\vec{x})\n \\psi_{\\vec{\\omega}^{\\prime},\\sigma,\\alpha}(\\vec{x})\n\\]\n\\begin{equation}\n\\label{eq-a2}\n +v(\\vec{\\omega},-\\vec{\\omega},\\vec{\\omega}^{\\prime},-\\vec{\\omega}^{\\prime})\n \\psi^{\\dagger}_{\\vec{\\omega},\\sigma,\\alpha}(\\vec{x}) \n \\psi_{-\\vec{\\omega},\\sigma^{\\prime},\\alpha}(\\vec{x}) \n \\psi^{\\dagger}_{\\vec{\\omega}^{\\prime},\\sigma^{\\prime},\\alpha}(\\vec{x}) \n \\psi_{-\\vec{\\omega}^{\\prime},\\sigma,\\alpha}(\\vec{x})\\}\n\\end{equation}\n\n\\noindent In Eq.\\ref{eq-a2} we observe that the Cooperon channel is identical\nto the exchange one if we substitute in the exchange term\n$\\vec{\\omega}^{\\prime}=-\\vec{\\omega}$. This means that we have to take into\nconsideration this identity in order to avoid double counting.\\\\\n\nWe replace the ``$N$\" fermions by $N/2$ pairs of chiral fermions\n(see Eq.\\ref{eq-005}).\nIn the second step we replace Eq.\\ref{eq-a2} by the current representation:\n\n\\[\n S_{int}\\simeq\\int d^dx\\int dt\\sum_{\\alpha}\\sum_{\\vec{n},\\vec{m}}\\{\n (v(0)-\\frac{1}{2}v(\\vec{n},\\vec{m}))\n (J^R_{n,\\alpha}(\\vec{x})J^R_{m,\\alpha}(\\vec{x}) \n +J^L_{n,\\alpha}(\\vec{x})J^L_{m,\\alpha}(\\vec{x}))\n\\]\n\\[\n -2v(\\vec{n},\\vec{m})\n (J^R_{n,\\alpha}(\\vec{x})J^R_{m,\\alpha}(\\vec{x}) \n +J^L_{n,\\alpha}(\\vec{x})J^L_{m,\\alpha}(\\vec{x}))\n +(v(0)-\\frac{1}{2}v(\\vec{n},\\vec{m}+\\pi))\n (J^R_{n,\\alpha}(\\vec{x})J^L_{m,\\alpha}(\\vec{x}) \n\\]\n\\[\n +J^L_{n,\\alpha}(\\vec{x})J^R_{m,\\alpha}(\\vec{x})) \n -2v(\\vec{n},\\vec{m}+\\pi) \n (J^R_{n,\\alpha}(\\vec{x})J^L_{m,\\alpha}(\\vec{x}) \n +J^L_{n,\\alpha}(\\vec{x})J^R_{m,\\alpha}(\\vec{x})) \n\\]\n\\[\n +(1-\\delta_{n,m})[v(\\vec{n},\\vec{m})\\sum_{\\sigma=\\uparrow,\\downarrow}\n (J^R_{n,\\sigma,\\alpha;m,\\sigma,\\alpha}(\\vec{x})\n J^L_{n,-\\sigma,\\alpha;m,-\\sigma,\\alpha}(\\vec{x})\n +J^L_{n,\\sigma,\\alpha;m,\\sigma,\\alpha}(\\vec{x})\n J^R_{n,-\\sigma,\\alpha;m,-\\sigma,\\alpha}(\\vec{x}))\n\\]\n\\begin{equation} \n\\label{eq-a3}\n -v(\\vec{n},\\vec{m}+\\pi)\\sum_{\\sigma=\\uparrow,\\downarrow}\n (J^R_{n,\\sigma,\\alpha;m,-\\sigma,\\alpha}(\\vec{x})\n J^L_{n,-\\sigma,\\alpha;m,\\sigma,\\alpha}(\\vec{x})\n +J^L_{n,\\sigma,\\alpha;m,-\\sigma,\\alpha}(\\vec{x}) \n J^R_{n,-\\sigma,\\alpha;m,-\\sigma,\\alpha}(\\vec{x}))]\\}\n\\end{equation}\n\n\\noindent We introduce the following definitions:\n\n\\[\n \\tilde{\\Gamma}^{(c)}(\\vec{n},\\vec{m})\\equiv v(0)-\\frac{1}{2}\n v(\\vec{n},\\vec{m}),\n \\;\\;\\;\\;\\;\n \\tilde{\\Gamma}^{(s)}(\\vec{n},\\vec{m})\\equiv 2v(\\vec{n},\\vec{m}),\n\\]\n\\[\n \\Gamma^{(c)}_2(\\vec{n},\\vec{m})\\equiv v(0)-\\frac{1}{2}\n v(\\vec{n},\\vec{m}+\\pi),\n \\;\\;\\;\\;\\;\n \\Gamma^{(s)}_2(\\vec{n},\\vec{m})\\equiv 2v(\\vec{n},\\vec{m}+\\pi),\n\\]\n\\begin{equation} \n\\label{eq-a4}\n \\Gamma^{(s)}_3(\\vec{n},\\vec{m})\\equiv\\frac{1}{2}\n (v(\\vec{n},\\vec{m})+v(\\vec{n},\\vec{m}+\\pi)),\n \\;\\;\\;\\;\\;\n \\Gamma^{(t)}_3(\\vec{n},\\vec{m})\\equiv\\frac{1}{2} \n (v(\\vec{n},\\vec{m})-v(\\vec{n},\\vec{m}+\\pi)).\n\\end{equation}\n\n\\noindent Using the definitions of the interaction operators given in\nEqs.\\ref{012} and \\ref{eq-013} we obtain the result given in\nEq.\\ref{eq-011}.\\\\\n\n\\begin{references}\n\\bibitem {01}\n E. Abrahams, P.W. Anderson, D.C. Licciardelo, and T.V. Ramakrishnan,\n Phys. Rev. Lett. {\\bf 42}, 673 (1979).\n\\bibitem {02}\n A.M. Finkelstein, Z. Phys. {\\bf B56}, 189 (1984);\n C. Castelani, et al, Phys. Rev {\\bf B30}, 527 (1984).\n\\bibitem {03}\n S.V. Kravchenko, D. Simonian, M.P. Sarachik, W. Mason, and J.E. Furneou,\n Phys. Rev. Lett. {\\bf 77}, 4938 (1996);\n D. Popovic, A.B. Fowler, and S. Washburn, Phys. Rev. Lett. {\\bf 79},\n 1543 (1997).\n\\bibitem {04}\n A.M. Goldman and Nina Markovic, Phys. Today, {\\bf 39} Nov. (1998).\n\\bibitem {05}\n Y. Hanein, U. Meirav, D. Shahar, C.C. Lin, D.C. Tsui, and H. Shtrikman,\n Phys. Rev. Lett. {\\bf 80}, 1288 (1998).\n\\bibitem {06}\n S. Chakraverty, S. Kivelson, C. Nayak, and K. Voelker, Phys. Rev.\n {\\bf B58}, R559 (1998).\n\\bibitem {07}\n M. Hilke, D. Shahar, S.H. Strong, D.C. Tsui, Y.H. Xie, and D. Monroe,\n Nature, {\\bf 395}, 675 (1998).\n\\bibitem {08}\n T. Giamarchi and H.J. Schultz, Phys. Rev. {\\bf B37}, 325 (1988).\n\\bibitem {09}\n D. Schmeltzer, R. Berkovits, M. Kaveh, and E. Kogan, Letter to the editor,\n Cond. Matt. {\\bf 10} L651 (1998).\n\\bibitem {10}\n C. Castellani, C.Dicastro, and P.A. Lee, Phys. Rev. {\\bf B57}, R9381\n (1998).\n\\bibitem {11}\n D. Belitz and T.R. Kirkpatrick, Phys. Rev. {\\bf B58}, 9710 (1998);\n D. Belitz and T.R. Kirkpatrick, Rev. Mod. Phys. {\\bf 66}, 261 (1994).\n\\bibitem {12}\n S. Hikami, Phys. Rev. {\\bf B24}, 2671 (1981).\n\\bibitem {13}\n J. Cardy, ``Scaling and Renormalization in Statistical Physics\",\n Chapter 5, Cambridge Univ. Press (1996)\n\\bibitem {14}\n V. Dobrosavljevic, E. Abrahams, E. Miranda, and S. Chakraverty,\n Phys. Rev. Lett. {\\bf 79}, 455 (1997).\n\\bibitem {15}\n Qimiao Si and C.M. Varma, Phys. Rev. Lett. {\\bf 81}, 4951 (1998).\n\\bibitem {16}\n P. Di Francesco, P. Mathieu, and D. Senechal, ``Conformal Field Theories\",\n Chapters 6 and 15, Springer-Verlag, New York (1997).\n\\end{references}\n\\end{document}\n"
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"name": "cond-mat0002106.extracted_bib",
"string": "\\bibitem {01}\n E. Abrahams, P.W. Anderson, D.C. Licciardelo, and T.V. Ramakrishnan,\n Phys. Rev. Lett. {\\bf 42}, 673 (1979).\n\n\\bibitem {02}\n A.M. Finkelstein, Z. Phys. {\\bf B56}, 189 (1984);\n C. Castelani, et al, Phys. Rev {\\bf B30}, 527 (1984).\n\n\\bibitem {03}\n S.V. Kravchenko, D. Simonian, M.P. Sarachik, W. Mason, and J.E. Furneou,\n Phys. Rev. Lett. {\\bf 77}, 4938 (1996);\n D. Popovic, A.B. Fowler, and S. Washburn, Phys. Rev. Lett. {\\bf 79},\n 1543 (1997).\n\n\\bibitem {04}\n A.M. Goldman and Nina Markovic, Phys. Today, {\\bf 39} Nov. (1998).\n\n\\bibitem {05}\n Y. Hanein, U. Meirav, D. Shahar, C.C. Lin, D.C. Tsui, and H. Shtrikman,\n Phys. Rev. Lett. {\\bf 80}, 1288 (1998).\n\n\\bibitem {06}\n S. Chakraverty, S. Kivelson, C. Nayak, and K. Voelker, Phys. Rev.\n {\\bf B58}, R559 (1998).\n\n\\bibitem {07}\n M. Hilke, D. Shahar, S.H. Strong, D.C. Tsui, Y.H. Xie, and D. Monroe,\n Nature, {\\bf 395}, 675 (1998).\n\n\\bibitem {08}\n T. Giamarchi and H.J. Schultz, Phys. Rev. {\\bf B37}, 325 (1988).\n\n\\bibitem {09}\n D. Schmeltzer, R. Berkovits, M. Kaveh, and E. Kogan, Letter to the editor,\n Cond. Matt. {\\bf 10} L651 (1998).\n\n\\bibitem {10}\n C. Castellani, C.Dicastro, and P.A. Lee, Phys. Rev. {\\bf B57}, R9381\n (1998).\n\n\\bibitem {11}\n D. Belitz and T.R. Kirkpatrick, Phys. Rev. {\\bf B58}, 9710 (1998);\n D. Belitz and T.R. Kirkpatrick, Rev. Mod. Phys. {\\bf 66}, 261 (1994).\n\n\\bibitem {12}\n S. Hikami, Phys. Rev. {\\bf B24}, 2671 (1981).\n\n\\bibitem {13}\n J. Cardy, ``Scaling and Renormalization in Statistical Physics\",\n Chapter 5, Cambridge Univ. Press (1996)\n\n\\bibitem {14}\n V. Dobrosavljevic, E. Abrahams, E. Miranda, and S. Chakraverty,\n Phys. Rev. Lett. {\\bf 79}, 455 (1997).\n\n\\bibitem {15}\n Qimiao Si and C.M. Varma, Phys. Rev. Lett. {\\bf 81}, 4951 (1998).\n\n\\bibitem {16}\n P. Di Francesco, P. Mathieu, and D. Senechal, ``Conformal Field Theories\",\n Chapters 6 and 15, Springer-Verlag, New York (1997).\n"
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cond-mat0002107
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Development of Magnetism in Strongly Correlated Cerium Systems: Non-Kondo Mechanism for Moment Collapse
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% DON'T CHANGE THIS LINE We present an {ab initio} based method which gives clear insight into the interplay between the hybridization, the coulomb exchange, and the crystal-field interactions, as the degree of 4{f} localization is varied across a series of strongly correlated cerium systems. The results for the ordered magnetic moments, magnetic structure, and ordering temperatures are in excellent agreement with experiment, including the occurence of a {moment collapse} of non-Kondo origin. In contrast, standard {ab initio} density functional calculations fail to predict, even qualitatively, the trend of the unusual magentic properties.
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"string": "%%%%%%%%%%%%%%%%%% file template.tex %%%%%%%%%%%%%%%%%%%%\n% %\n% Copyright (c) Optical Society of America, 1992. %\n% %\n%%%%%%%%%%%%%%%%%%% November 17, 1992 %%%%%%%%%%%%%%%%%%%\n%\n% THIS FILE IS A TEMPLATE TO PRODUCE AN ARTICLE SUBMISSION\n% TO THE OSA JOURNALS, JOSA-A, JOSA-B, and APPLIED OPTICS.\n%\n% THIS TEMPLATE CONTAINS TYPESETTING COMMANDS WHICH BEGIN WITH A\n% BACKSLASH. THESE COMMANDS WILL BE READ BY LATEX, USING THE\n% REVTEX 3.0 STANDARD MACROS. PLEASE FILL IN THE REQUIRED DATA\n% FOR THE MACROS, BUT DO NOT ALTER THE DEFINITIONS.\n%\n% EXAMPLE: IN \\author{Authors' names} , PLEASE FILL IN THE\n% AUTHORS' NAME(S).\n%\n% COMMENTS BEGIN WITH THE PERCENT (%) SYMBOL. AFTER A %, ANY\n% DATA ON THE REST OF A LINE WILL NOT PRINT.\n%\n%\\documentstyle[12pt,aps,amsfonts,floats,epsf]{revtex} % DON'T CHANGE\n\\documentstyle[aps,prl,floats,epsf]{revtex} % DON'T CHANGE\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% \\renewcommand{\\baselinestretch}{2}\n%%%%%%%%%%%%%%%%%%%%%%%%%% above command to stretch the spcaing %%%%\n%\n%\n%\n%\n\\begin{document} % INITIALIZE - DONT CHANGE\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n %%%%%%%% TWO COLUMN FORMAT %%%%%%%%%%%%%\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname\n @twocolumnfalse\\endcsname\n%\\draft\n%\n%\n%\n\\title{Development of Magnetism in \nStrongly Correlated Cerium Systems: Non-Kondo Mechanism for\nMoment Collapse}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\title{Magnetic Properties of Strongly Correlated Cerium Systems: Comparison\n%%% of two {\\it ab initio} based approaches}\n\\author{Eric M. Collins, Nicholas Kioussis, and Say Peng Lim}\n\\address{Department of Physics and Astronomy, California State University \nNorthridge,Northridge, CA 91330-8268}\n\\author{Bernard R. Cooper}\n\\address{Department of Physics, West Virginia University, Morgantown, \nWV 26506-6315}\n\\maketitle\n\\begin{abstract} % DON'T CHANGE THIS LINE\nWe present an {\\it ab initio} based method which gives clear insight into\nthe interplay between the hybridization, the coulomb exchange, and \nthe crystal-field interactions, as the degree of 4{\\it f} localization \nis varied across a series of strongly correlated cerium systems.\nThe results for the ordered magnetic moments, magnetic structure, and ordering\ntemperatures are in excellent agreement with experiment, including\nthe occurence of a {\\it moment collapse} of non-Kondo origin. \nIn contrast, standard {\\it ab initio} density functional calculations\nfail to predict, even qualitatively, the trend of the unusual magentic\nproperties.\n\n\\end{abstract}\n\nPACS: 71.27+a, 71.28+d, 71.10Fd, 75.30Mb, 75.20Hr, 75.10Lp\n\\vspace{+12pt}\n\n]\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n%\n\\newpage\nThe difficulties and interest in treating strongly correlated \nelectron systems, and the consequences of correlation \neffects on magnetic behavior in the transitional 4{\\it f} or \n5{\\it f} localization regime, \nprovide one of the central problems of condensed matter physics.$^{1-3}$\nThe transitional regime behavior is neither atomiclike nor itinerant.\nThis gives rise to an extremely interesting range of phenomena, \nbut also causes very great difficulties in treating the\ntheory of these phenomena adequately, especially in a way providing the\nability to predict the behavior of specific materials.$^{1-3}$\nAn adequate treatment requires treating the interelectronic coulomb \ninteraction, i.e. the correlation effects, as constrained by \nexchange symmetry.$^{4-6}$ In this letter, we demonstrate an approach\nfor treating these difficulties in predicting the interesting and\ncomplex behavior of an important series of cerium compounds.\n\nThe isostructural (rock-salt structure) series of the cerium\nmonopnictides CeX (X = P, As, Sb, Bi) and monochalcogenides (X=S, Se, Te)\nhave become prototype model systems for study,\nbecause of their unusual magnetic properties.$^{7-13}$\nThis series of strongly correlated electron systems offers the opportunity\nto vary systematically, through chemical pressure, the lattice constant\nand the cerium-cerium separation on going down the pnictogen\nor chalcogen column, and hence tailor the degree of 4{\\it f} localization\nfrom the strongly correlated limit in the heavier systems to the weakly\ncorrelated limit in the lighter systems.$^{7-13}$\nThe calculated single-impurity Kondo temperature, T$_K$,\n presented below, is much smaller than the \nmagnetic ordering temperature in these systems, \n and hence this series lies in \n the {\\it magnetic} regime of the Kondo phase diagram.$^{14}$ \nNevertheless, in this work we demonstrate that \nthe sensitivity of the hybridization, \ncoulomb exchange, and crystal-field\ninteractions with the chemical environment gives rise to a variety of\nunusual and interesting magnetic properties across the series, \nin agreement with \nexperiment, {\\it including the occurence \nof a non-Kondo magnetic moment collapse}.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%% OLD %%%%%%%%%%%%%%%%%%%%%%%%%%\n%% The wide variety of ground states is associated with the partial \n%% delocalization of the 4{\\it f} or 5{\\it f}-electrons and the \n%% cooperative hybridization of the correlated {\\it f} electrons \n%% with non-{\\it f}-band electrons.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThis class of cerium systems exhibits large magnetic anisotropy which\nchanges from the $<001>$ direction in the pnictides to the\n$<111>$ direction in the chalcogenides. The low-temperature\nordered magnetic moment increases with increasing lattice constant\nfor the pnictides from 0.80$\\mu_B$ in CeP to 2.1$\\mu_B$ in \nCeSb and CeBi,$^{7-8}$\nwhile it decreases with increasing lattice constant for the chalcogenides\nfrom 0.57$\\mu_B$ in CeS to 0.3$\\mu_B$ in CeTe.$^{7-9}$\nThe {\\it magnetic moment collapse} from CeSb to CeTe, with\nboth systems having about the\nsame lattice constant, is indicative of the sensitivity of the magnetic\n interactions to chemical environment.\nThe experimentally observed low-temperature structure \nin CeBi and CeSb\nis the \n$<001>$ antiferromagnetic \ntype IA ($\\uparrow \\uparrow \\downarrow \\downarrow)$, \nwhereas in CeAs and CeP the structure is the \n$<001>$ antiferromagnetic type I ($\\uparrow \\downarrow$).$^{7,15}$ \nThe ordering temperature increases from 8K in CeP\nto 26K in CeBi for the pnictides, whereas it decreases from\n8.4K in CeS to an unusually low 2.2K in CeTe.$^{7-11}$\nAnother unusual feature of this series of cerium compounds \nis the large {\\it suppression} of the crystal field (CF) splitting\nof the Ce$^{3+}$ free-ion 4$f_{5/2}$ multiplet from\nvalues expected from the behavior of the heavier\nisostructural rare-earth pnictides or chalcogenides.$^{16}$\nThis can be understood$^{17}$ as arising from band-{\\it f} hybridization\neffects.\nIn both the cerium monopnictides and monochalcogenides,\n the CF splitting between the ${\\Gamma_{7}}$ doublet and the\n${\\Gamma_{8}}$ quartet\ndecreases with increasing anion size,\n from 150 K for CeP to 10 K in CeBi and from 130 K for CeS to 30 K for CeTe,\nand it is about the same for the same row in both series,\n a rather surprising result in view of the\nadditional valence electron on the chalcogen ion.$^{18}$\n Neutron scattering experiments have shown$^{19}$ that\nthe ${\\Gamma_{7}}$-doublet\nis the CF ground state in all the cerium pnictides and\nchalcogenides.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%% NEW ADDITION %%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nIn this paper we present material-predictive results from \ntwo {\\it ab initio} based \nmethods to study the change of magnetic properties \n across this series of cerium systems.\nThe first, {\\it ab initio} \nbased, method \ngives clear insight into the role of the three pertinent interactions:\n1) The band-{\\it f} hybridization-induced inter-cerium magnetic \ncoupling; 2) the corresponding effects of band-{\\it f} coulomb exchange; and\n3) the crystal-field interaction. This approach \nallows us also to understand the interplay between \nthese interactions\nas the degree of 4{\\it f} localization is varied across the series.\nThe predictive calculations give results \nfor the magnetic moments, magnetic structure, and ordering \ntemperatures in excellent agreement with experiment. \nThus, this approach allows to understand and predict\n a number of key features \nof observed behavior. \n First, is the very low moment and low ordering \ntemperature of the antiferromagnetism \nobserved in CeTe, \n an incipient heavy Fermion system. (For a review of theory \nand experimental behavior of heavy Fermion systems \nsee references 2,3,20,21.) \nThis {\\it ab initio}-based method, \ndescribed below, predicts the {\\it magnetic moment and \nordering temperature collapse}\nfrom CeSb to CeTe, both systems having about the same lattice\nconstant but CeTe having an additional {\\it p} electron.\nThe origin of the moment collapse is of non-Kondo origin.\nThe earlier work of Sheng and Cooper$^5$ showed that this \nmagnetic ordering reduction is accurately predicted without\nincluding any crystal-field effects. \nAn erroneous statement appears in the recent review \narticle by Santini {\\it et al.}$^{22}$ stating that \ncrystal-field effects played an important role in the \ncalculated results of Sheng and Cooper.$^5$ This is incorrect, \nsince crystal-field effects were {\\it not} included in these calculations.\nWe show in this \npaper that including the crystal-field effects modifies\nthis behavior only quantitatively. \nSecond, our results demonstrate that, while the band-{\\it f} coulomb exchange\nmediated interatomic 4{\\it f}-4{\\it f} interactions dominate \nthe magnetic behavior for the heavier systems, \nwhich are more localized because of the larger Ce-Ce separation, \nthe opposite is true for the lighter, more delocalized systems, \nwhere the hybridization-mediated coupling dominates the\nmagnetic behavior. This reflects the great sensitivity\nof the relative importance of hybridization and coulomb \nexchange effects on magnetic ordering depending on the\ndegree of 4{\\it f} localization.\nThird, we show that \nfor the lighter more delocalized systems the crystal-field\ninteractions are much larger than the inter-cerium interactions\nand hence dominate the magnetic behavior.\nFinally, \n we predict the experimentally observed \nchange of the ground-state magnetic \nstructure from the \n$<001>$ antiferromagnetic\ntype IA ($\\uparrow \\uparrow \\downarrow \\downarrow)$ in CeBi and CeSb\n to the $<001>$ antiferromagnetic type I ($\\uparrow \\downarrow$) \n in CeAs and CeP. \nOn the other hand, the second {\\it ab initio} method, based on \ndensity functional theory within the local density\napproximation (LDA),$^{23,24}$ fails to predict, even qualitatively, \nthe trend of magnetic properties in this series of strongly\ncorrelated electron systems.\n\n\n\nThe first, {\\it ab initio} based, method employs the \ndegenerate Anderson lattice model which incorporates \nexplicitly the hybridization and the coulomb exchange\n interactions on an equal footing$^{4,5}$ \n\n\\small\n\\begin{eqnarray*}\nH &=&\\sum_{k} \\epsilon_{k} c_{k}^{+} c_{k}\n + \\sum_{Rm} \\epsilon_{m} f_{m}^{+}(R) f_{m}(R) \\\\\n & & + \\frac{U}{2} \\sum_{R,m \\ne m'} n_{m}(R) n_{m'}(R) \\\\ \n & & +\\sum_{kmR} [ V_{km} e^{-i{\\bf k \\cdot R}} c_{k}^{+} f_{m}(R)\n + H.C. ] \\\\ \n & & - \\sum_{kk'} \\sum_{mm'R} J_{mm'}({\\bf k,k'})\n e^{-i{\\bf (k-k') \\cdot R}} c_{k}^{+} f_{m}^{+}(R) c_{k'}\n f_{m'}(R).\n\\end{eqnarray*}\n\\vspace{-30pt}\n\\begin{equation}\n\\end{equation}\n\\normalsize\nThe parameters entering the model Hamiltonian, i.e., \nthe band energies $\\epsilon_k$, the {\\it f}-state energy $\\epsilon_m$, \nthe on-site coulomb\nrepulsion U, the hybridization matrix elements, V$_{km}$, and the\nband-{\\it f} coulomb exchange\n$J_{m m'}({\\bf k, k'}) =\n \\left< \\phi_{k}^{*}({\\it r}_{1}) \\psi_{m}^{*}({\\it r}_{2})\n \\left|\\frac{1}{{\\it r}_{12}}\\right|\n \\psi_{m'}({\\it r}_{1})\\phi_{k'}({\\it r}_{2})\\right>$\nare evaluated on a wholly {\\it ab initio} basis \nfrom non-spin polarized full potential linear muffin tin orbital$^{23}$ \n(FPLMTO) calculations.\nHere, ${\\it r}_{12}$ stands for $|{\\it r}_{1}-{\\it r}_{2}|$;\n$\\phi_{k}$ are the non-{\\it f} basis states of the FPLMTO, and\n$\\psi_{m}$ are the localized {\\it f} states.\nBecause the size of both the hybridization and coulomb exchange\nmatrix elements are much smaller ($\\sim$ 0.1 eV) than the intraatomic\n coulomb interaction U (6eV), one can apply perturbation\ntheory and evaluate the anisotropic two-ion 6X6 \ninteraction matrices$^{4,5}$, \n$E_{m_{1}m'_{1}}^{m_{2}m'_{2}}{\\bf (R_{2}-R_{1})}$, \nwhich couple the \n two {\\it f}-ions. \n%%%%%%%%%%%\n% As shown in Figure~\\ref{figa},\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe exchange interactions have three contributions:\n the wholly band-{\\it f} coulomb exchange\nmediated interaction \nproportional to $J_{mm'}^{2}({\\bf k,k'})$, the wholly hybridization-mediated\nexchange interaction proportional to $V_{km}^{4}$,\nand the cross term proportional to $V_{km}^{2} J_{mm'}({\\bf k,k'})$. \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% \\begin{figure}[bp]\n% \\epsfxsize=3.0in\n% \\epsfysize=1.0in\n% \\epsffile{nick.ps}\n% \\caption{Schematic contributions to the two-ion exchange interactions: \n% the pure hybridization exchange proportional to $V^{4}$, the\n% pure coulomb exchange proportional to $J^2 $ and the cross term \n% proportional to $V^{2}J$.}\n% \\label{figa}\n% \\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWith the two-ion interactions having been determined, the low-temperature\nmagnetic moment and the ordering temperature can be determined by use\nof a mean field calculation.$^{4,5,8}$\nWe have previously\n applied this {\\it ab initio} based method to \ninvestigate the effect of hybridization-induced\ncerium-cerium interactions$^{4,17}$ and the combined effect \nof both the hybridization\nand coulomb induced interactions$^{5}$ on the magnetic properties of\nthe heavier cerium pnictides and chalcogenides (CeBi, CeSb, and CeTe).\nHowever, these calculations did not take into account\nthe crystal field interaction and employed a warped muffin-tin LMTO\ncalculation for the parameters entering the model.\nThe excellent agreement found$^{5}$ with experiment for the low-temperature\nmagnetic moment and ordering\ntemperature is \nrelatively unaffected by the CF interaction, \nbecause the CF interaction in the heavier\ncerium systems is smaller than the two-ion exchange interactions.\n\nThe second method employs {\\it ab initio} \nspin polarized electronic structure calculations\nbased on the FPLMTO method$^{23}$ \nusing 1) only spin polarization,\nwith the orbital polarization included\nonly through the spin-orbit coupling, and 2) both the spin and orbital\npolarization polarization.$^{24}$\n In these calculations the 4{\\it f} states are treated as band states.\nThe orbital polarization is\n taken into account\nby means of an eigenvalue shift$^{24}$,\n$\\Delta V_{m} = - E^{3} L_{z} m_{l}$, for the 4{\\it f} atom.\nHere, L$_{z}$ is the z-component of the cerium total orbital moment,\nm$_{l}$ is the magnetic quantum number, and \nE$^{3}$ is the Racah parameter evaluated \nself-consistently at each iteration. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe crystalline field, which was neglected in \nthe previous calculations,$^{4,5}$ is expected \n to affect the magnetic behavior considerably, \nif it is large.\nIt is important to emphasize that\nsince in the first method the 4{\\it f} states are treated as core\nstates, they interact\n only with the spherical component of the effective\none-electron potential. Thus, the interaction of the {\\it atomic-like}\n4f state with the {\\it non-spherical} components of the potential,\ngiving rise\nto the CF splitting, $\\Delta_{CF} = \\epsilon_{\\Gamma _8} -\n\\epsilon_{\\Gamma _7}$, is not\n included in the calculation of the model Hamiltonian \nparameters. \nIn this paper, we generalize the first, {\\it ab initio} based, method\n to include both the interatomic 4{\\it f}-4{\\it f} coupling \nand the crystal-field interactions \n on an equal footing and \nto employ a full potential LMTO evaluation\nof the model Hamiltonian parameters. While the effect of the full potential\non both the hybridization and coulomb exchange interactions\nis small, including the CF interaction will\nbe shown to play a role as important as the \ninteratomic 4{\\it f}-4{\\it f} interactions for\nunderstanding and predicting the {\\it overall trend}\n in the unusual magnetic properties,\nas as one chemically tunes the degree of 4{\\it f} localization across\nthis series of strongly correlated electron systems.\nThe resultant Hamiltonian is$^{4,5}$ \n\n\\small \n\\begin{eqnarray*}\n{\\it H}& =& -\\sum_{i,j}\\sum_{~^{\\mu,\\nu}_{\\epsilon,\\sigma}}\n \\xi^{\\epsilon \\sigma}_{\\mu \\nu}(\\theta_{ij})\n e^{-{\\it i}(\\mu-\\nu+\\epsilon-\\sigma)\\phi_{ij}} \n c^{\\dag}_{\\epsilon}(j)c_{\\sigma}(j)\n c^{\\dag}_{\\mu}(i)c_{\\nu}(i) \\\\\n && + B_{4}\\sum_{i}\\left(O_{4}^{0}(i)+5O_{4}^{4}(i)\\right) , \n\\end{eqnarray*}\n\\vspace{-30pt}\n\\begin{equation}\n\\end{equation}\n\\normalsize\nwhere the ${\\xi}^{\\epsilon \\sigma}_{\\mu \\nu}(\\theta_{ij})$ are the\ntwo-ion 4{\\it f}-4{\\it f} interaction matrices \n rotated to a common crystal-lattice \naxis,\nand the $O_{4}^{0}$ and\n$O_{4}^{4}$ are the Stevens operators equivalents acting on the \nCe$^{3+}$ free-ion 4$f_{5/2}$ multiplet.$^{25}$\nThe CF splitting is $\\Delta_{CF}$ = 360$B_4$;\na positive $B_4$\nvalue gives the $\\Gamma_7$ ground state, which is \n experimentally observed.$^{19}$\nWhile our work in progress is aimed at evaluating the CF splitting \non a wholly {\\it ab initio} basis, in the absence of\n an {\\it ab initio} value of the CF interaction in this\nclass of strongly correlated cerium systems,\nthe $\\Delta_{CF}$ is\nset to the experimental values listed in Table 3.$^{10,19}$\n\nIn Table~\\ref{taba},\nwe present the calculated values of the zero-temperature\ncerium magnetic moment from the FPLMTO electronic structure calculations.\nListed in the table are values both with and without the orbital polarization\ncorrection taken into account. Note, the importance of including the\norbital polarization in these 4f correlated electron systems. \nAs expected, in all\ncases, the orbital polarization is found to be opposite to the spin\npolarization. Comparison of the total energies predicts that\nthe magnetic anisotropy changes from the $<001>$ direction in the pnictides\nto the $<111>$ in the chalcogenides, in agreement with experiment.\nOn the other hand, except perhaps for the lighter chalcogenides (CeS and CeSe),\ncomparison of the {\\it ab initio} and experimental values for the\nmagnetic moment indicates the {\\it failure} of the LDA calculations\nto treat properly the correlation effects of the 4{\\it f} states\n(treated as valence states) within the LDA as the degree of 4{\\it f} \ncorrelations increases in the heavier pnictide systems.\nFurthermore, the {\\it ab inito} calculations fail to predict\nthe large {\\it moment collapse} from CeSb to CeTe, the latter being\ndescribed as an incipient heavy Fermion system.$^{2,3,20,21}$\n\n\\begin{table}[tbp]\n\\caption{\nValues of the calculated and experimental $^{7-11}$ \nlow-temperature ordered magnetic\nmoments for the cerium chalcogenides and pnictides in\nunits of $\\mu_{B}$. Listed are the LMTO values for\nthe spin moment $\\mu_{S}$, the orbital\nmoment $\\mu_{L}$, and total moment $\\mu$, for the\nspin polarized only calculation and for the calculation with\nspin polarization and orbital polarization correction.\n}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n & \\multicolumn{2}{c}{ FP+SP } & &\n \\multicolumn{2}{c}{ FP+SP+OP } & &\n EXPT \\\\\n & $\\mu_{S}$ & $\\mu_{L}$ & $\\mu$ & $\\mu_{S}$ & $\\mu_{L}$\n & $\\mu$ & $\\mu$ \\\\ \\hline\n CeS & -1.00 & 0.91 &-0.09 &-1.24&1.99 & 0.75 & 0.57 \\\\\n CeSe & -1.08 & 1.02 &-0.06 &-1.26&2.07 & 0.81 & 0.57 \\\\\n CeTe & -1.15 & 1.28 & 0.07 &-1.31&2.29 & 0.98 & 0.30 \\\\ \\hline\n CeP & -0.80 & 0.55 &-0.25 &-0.85&1.27 & 0.43 & 0.80 \\\\\n CeAs & -0.84 & 0.64 &-0.20 &-0.85&1.42 & 0.57 & 0.80 \\\\\n CeSb & -0.86 & 0.74 &-0.12 &-0.91&1.61 & 0.70 & 2.06 \\\\\n CeBi & -0.86 & 0.74 &-0.12 &-0.95&1.69 & 0.74 & 2.10 \\\\ \n \\end{tabular}\n \\label{taba}\n \\end{table}\n\nIn Table~\\ref{tabb}, we list the values of the $m$ = $m'$ =1/2 matrix elements\n(characteristic matrix elements of the 6X6 exchange interaction matrix)\nfor the first three nearest-neighbor shells for the light (CeP and CeS)\nand the heavier compounds (CeSb and CeTe). Listed separately in this table\nare the three contributions to the interatomic 4{\\it f}-4{\\it f} \ninteractions arising \n from band-{\\it f} hybridization\n(V$^4$), band-{\\it f} coulomb exchange (J$^2$), and the cross term.\nIt is important to note that while the coulomb exchange mediated interactions\ndominate the magnetic behavior for the heavier, more localized,\n4{\\it f} systems, the opposite is true for the lighter, more delocalized,\n systems where the\nhybridization mediated interactions dominate the magnetic behavior.\nThis change of behavior of the interatomic \n4{\\it f}-4{\\it f} interactions is a result\nof the sensitivity of the hybridization and coulomb exchange to \nthe degree of 4{\\it f} localization. Equally important, is that\nwhile both first and second nearest-neighbor \n 4{\\it f}-4{\\it f} interactions are ferromagnetic\nfor CeSb, there is an {\\it interplay} between ferromagnetic \nfirst nearest-neighbor and antiferromagnetic second nearest-neighbor\ninteractions for CeTe.(These interactions are mediated via scattering\nof conduction electrons). \nThis results in a saturated ordered moment for CeSb and \nin the ordered {\\it magnetic moment collapse}\n for CeTe (see Table III). \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nIn order to determine whether the magnetic moment collapse might be of\nKondo origin, we have evaluated the single-impurity Kondo temperature,$^{26}$ \nk$_B$T$_K$ = De$^{-\\frac{1}{2\\rho(E_F)|J(E_F)|}}$, \nacross the series. Here, D is the bandwidth of the \nconduction electron states, $\\rho(E_F)$ is the density of states of \nthe conduction electrons at the Fermi energy, and $J(E_F)$ is the \nconduction electron-{\\it f} exchange interaction at the Fermi energy, \nwhich has contributions \n both from the coulomb exchange interaction \nin Eq. (2), provided that it is negative, \nand the hybridization-induced exchange interaction \n $|J_{hyb}(E_F)| = \\frac{V^2(E_F)U}{(|E_f - E_F|)(|E_f - E_F| + U)}$, \n where $J_{hyb}(E_F) < 0$.\nWe find that the coulomb exchange interaction in Eq. (2) evaluated \nat the Fermi energy is positive across the entire series and hence\ncannot give rise to the Kondo effect.\nThus, only the hybridization-induced exchange interaction, \n$J_{hyb}(E_F)$, \ncan give rise to the Kondo effect.$^{26}$\nUsing the {\\it ab initio} values of the parameters entering the \nexpression for T$_K$, we find that T$_K \\ll$ T$_{ord}$ across \nthe entire series(T$_K <$ 10$^{-4}$K).\nThe Kondo temperatures in the \nmonopnictide series is smaller than that in the chaclogenides, \ndue to the fact that in the pnictides the Fermi \nenergy lies in the pseudogap, resulting in low $\\rho(E_F)$. \nThese results suggest that \n the moment collapse \nfrom CeSb to CeTe is of non-Kondo origin.\nRather, it results from an interplay of ferromagnetic and \nantiferromagnetic interatomic 4{\\it f}-4{\\it f} \ninteractions which arises purely from differences \nin the underlying electronic structure.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{table}[btp]\n\\caption{\nValues of the $m$ = $m'$ =1/2 matrix elements \n(characteristic matrix elements of the 6X6 interatomic \n4{\\it f}-4{\\it f} interaction matrix\n$E^{m_b m'_b}_{m_a m'_a}({\\bf R_{b}-R_{a}})$, for the first, second, and \nthird nearest-neighbor shells in degrees Kelvin. Listed are the values\nof the hybridization\ninduced ($E_{V^{4}}$), cross terms ($E_{V^{2}J}$), and pure\ncoulomb exchange ($E_{J^{2}}$) contributions.}\n\n\\begin{tabular}{|c||r|r|r||r|r|r|}\n & \\multicolumn{3}{c}{CeP} & \\multicolumn{2}{r}{CeS} & \\\\ \\hline\n & $E_{V^{4}}$ & $E_{V^{2}J}$ & $E_{J^{2}}$\n & $E_{V^{4}}$ & $E_{V^{2}J}$ & $E_{J^{2}}$ \\\\ \\hline\n R = ( $\\frac{1}{2}$ $\\frac{1}{2}$ 0 )\n & 2.23 & 0.64 & 1.53\n & 0.85 & -0.40 & 1.50\n \\\\ \\hline\n R = ( 1 0 0 )\n & 6.39 & 0.27 & 1.65\n & -1.60 & 0.04 & -0.80\n \\\\ \\hline\n R = ( 1 $\\frac{1}{2}$ $\\frac{1}{2}$ )\n & -0.08 & -0.02 & 0.16\n & 0.38 & -0.16 & 0.13 \\\\ \n\\end{tabular}\n\n\\begin{tabular}{|c||r|r|r||r|r|r|} \n & \\multicolumn{3}{c}{CeSb} & \\multicolumn{2}{r}{CeTe} & \\\\ \\hline\n & $E_{V^{4}}$ & $E_{V^{2}J}$ & $E_{J^{2}}$\n & $E_{V^{4}}$ & $E_{V^{2}J}$ & $E_{J^{2}}$ \\\\ \\hline\n R = ( $\\frac{1}{2}$ $\\frac{1}{2}$ 0 )\n & 0.70 & 0.34 & 7.30\n & 0.17 & -0.19 & 2.90\n \\\\ \\hline\n R = ( 1 0 0 )\n & 2.07 & 0.07 & 10.21\n & -0.19 & 0.04 & -1.69\n \\\\ \\hline\n R = ( 1 $\\frac{1}{2}$ $\\frac{1}{2}$ )\n & -0.02 & -0.03 & 0.40\n & 0.04 & -0.06 & -0.01\n \\\\\n\\end{tabular}\n\\label{tabb}\n\\end{table}\n\nListed in Table~\\ref{tabc} \nare the calculated zero-temperature ordered moment and\nordering temperature, T$_N$, from the first, {\\it ab initio} based, method,\nwith and without the CF interaction. It is clear that for the heavier\nsystems (CeBi, CeSb, CeTe) the effect of the CF interaction on the\nmagnetic moments is small, and it is slightly more pronounced on the\nordering temperatures. This is due to the fact that for the more localized\nsystems the CF interaction is smaller than the two-ion interactions.\nThis is the reason that the previous calculations,$^5$ neglecting the CF\ninteraction, gave results in very good agreement with experiment.\nOn the other hand, for the lighter more delocalized systems the CF\ninteractions are much larger than the interatomic \n4{\\it f}-4{\\it f} interactions, \nand hence dominate the magnetic behavior. The overall decrease of the magnetic\nmoments in the presence of the CF interaction in all systems,\narises from the mixing\nof the off-diagonal angular momentum states $|\\pm5/2>$ and $|\\mp3/2>$ states\nfrom the CF interaction with $\\Gamma_7$ ground state.\nOverall, we find that \nthe first, {\\it ab initio} based, approach \nwhich takes into account all three pertinent interactions \n(hybridization, coulomb exchange, and CF\ninteractions) on an equal footing, \nyields results \nfor both the zero-temperature moment and the\nordered temperature (a more stringent test for the theory)\nin excellent agreement with experiment.\n\nA final corroboration of the success of the first {\\it ab initio} based\nmethod is that it predicts the experimentally observed \nchange of the ground-state magnetic \nstructure from the \n$<001>$ antiferromagnetic\ntype IA ($\\uparrow \\uparrow \\downarrow \\downarrow)$ in CeBi and CeSb\n to the $<001>$ antiferromagnetic type I ($\\uparrow \\downarrow$) \n in CeAs and CeP.$^{7,15}$ More specifically, the sign [ferromagnetic (F) or \nantiferromagnetic (AF)] of the \n$|\\pm 5/2>$ matrix elements of the 6X6 exchange matrix determines \nthe {\\it interplanar} interaction between successive (001) Ce planes. \nWe find, that for the heavier compounds (CeBi and CeSb) \nthe $|\\pm 5/2>$ matrix elements of the {\\it coulomb exchange} matrix \nare FM and hence favor the $\\uparrow \\uparrow \\downarrow \\downarrow$ \ntype, while in the lighter systems (CeAs and CeP) the \n$|\\pm 5/2>$ matrix elements of the {\\it hybridization}-induced two-ion matrix \nare AF, and hence they favor the $\\uparrow \\downarrow$ type. \n\n\n\\begin{table}[tbp]\n\\caption{\nCalculated values (from the first {\\it ab initio} based method)\nof the zero-temperature ordered moment $\\mu$ in $\\mu_{B}$, \nand the ordering temperature\n$T_{N}$ in degrees Kelvin, \nwith and without the crystalline field (CF) interaction \nacross the cerium pnictide and chalcogenides series. Also listed \nare the experimental values$^{7-11}$ of $\\mu$ and $T_{N}$, and the CF \nsplitting$^{19}$ between the $\\Gamma_7$ ground state and the $\\Gamma_8$ state \nin degrees Kelvin.}\n\\begin{tabular}{|d|d|c|c|c|c|c|c|d|} \n & $\\Delta_{CF}$ &\n \\multicolumn{2}{r}{ $\\mu_{0}$ }& & &\n \\multicolumn{2}{r}{ $T_{N}$ } & \\\\ \\hline\n & & no CF & CF & exp & & no CF & CF & exp \\\\ \\hline\nCeS & 140 & 1.80 & 0.73 & 0.57 & & 1.0 & 11.0 & 8.4 \\\\\nCeSe & 116 & 1.10 & 0.79 & 0.57 & & 2.5 & 14.0 & 5.7 \\\\\nCeTe & 32 & 0.60 & 0.46 & 0.30 & & 8.0 & 5.0 & 2.2 \\\\ \\hline\nCeP & 150 & 2.10 & 0.73 & 0.81 & & 14 & 11 & 8 \\\\\nCeAs & 137 & 2.10 & 0.74 & 0.85 & & 16 & 13 & 8 \\\\\nCeSb & 37 & 2.10 & 1.80 & 2.06 & & 20 & 18 & 17 \\\\\nCeBi & 8 & 2.10 & 2.10 & 2.10 & & 40 & 40 & 26 \\\\ \n\\end{tabular}\n\\label{tabc}\n\\end{table}\n\nIn conclusion, we have applied two different, {\\it ab inito} based\nand {\\it ab initio} LDA, \nmethods to study the dramatic change of magnetic properties across\na series of strongly correlated electron systems which offer the\nopportunity to chemically tailor the different type of interactions\n(band-{\\it f} hybridization, band-{\\it f} coulomb exchange, \nand CF interactions),\npertinent to the unusual magnetic behavior. \nThe first, {\\it ab initio} based, approach which explicitly takes into account\nthe interplay of the three pertinent interactions, gives results in excellent\nagreement with experiment for all compounds in the series, including\nthe {\\it moment collapse} from CeSb to CeTe and the trend of moments\nand ordering temperatures across the series.\nThe remaining problem of determining on a wholly {\\it ab initio}\nbasis the {\\it suppressed} crystal-field interactions in this\nclass of systems poses a theoretical challenge for future theoretical\nwork.\nOn the other hand, the second, fully {\\it ab initio} LDA, method\ngives good results for the lighter chalcogenide systems, but it entirely\nfails to give, even qualitatively, the trend of the unusual magnetic behavior.\n\n\nThe research at California State University Northridge (CSUN) was\nsupported by the National Science Foundation under Grant No.\nDMR-9531005, by the US Army Grant No. DAAH04-95, and the CSUN Office\nof Research and Sponsored Projects,\nand at West Virginia University by the NSF under Grant No.\nDMR-9120333.\n\n\n\\begin{references}\n\n\\bibitem{kious} \n{\\it Correlation Effects and Materials Properties}, eds. A. Gonis, \nN. Kioussis, and M. Ciftan, (Kluwer Academic/Plenum, New York, 1999).\n\n\\bibitem{hess}\nD. W. Hess, P.S. Riseborough, and J.L. Smith \n{\\it Heavy Fermion Phenomena}, \nEncyclopedia of Applied Physics, edited by G. Trigg,\n(VCH Publishers Inc., New York, 1993), \nVol. 71, p. 435.\n\n\\bibitem{grewe}\nN. Grewe and F. Steglich, {\\it Heavy Fermions}, pp. 343-479,\nin Handbook on the Physics and Chemistry of the Rare Earths, \neds. K. A. Grchneidner Jr. and L. Eyring, (Elsevier Science Publishers,\nAmsterdam, 1991), Vol. 14, pp. 343.\n\n\\bibitem{kio} N. Kioussis, B. R. Cooper, and J. M. Wills, Phys. Rev. B\n{\\bf 44}, 10003 (1991).\n\n\\bibitem{shg} Q. G. Sheng and B. R. Cooper, J. Appl. Phys. {\\bf 69},\n5472 (1991); Phys. Rev. B {\\bf 50}, 965 (1994).\n\n\\bibitem{coop}\nB. R. Cooper, O. Vogt, Q. G. Sheng, and Y. Lin, Phil. Mag. B {\\bf 79},\n683 (1999).\n\n\\bibitem{jrm1} J. Rossat-Mignod, P. Burlet, S. Quezel, J. M. Effantin, D.\nDelac\\^{o}te, H. Bartholin, O. Vogt, and D. Ravot, J. Magn. Magn. Mater.\n{\\bf 31-34}, 398 (1983).\n\n\\bibitem{coo} B. R. Cooper, R. Siemann, D. Yang, P. Thayamballi, and A.\nBanerjea, in {\\it The Handbook of the Physics and Chemistry of the\nActinides}, edited by A. J. Freeman and G. H. Lander (North-Holland,\nAmsterdam, 1985), Vol. 2, Chap. 6, pp. 435-500.\n\n\\bibitem{don} A. D\\\"{o}nni, A. Furrer, P. Fischer, and F. Hulliger, Physica B\n{\\bf 186-188}, 541 (1993).\n\n\\bibitem{hul} F. Hulliger, B. Natterer, and H. R. Ott, J. Magn. Magn. Mater.\n{\\bf 8}, 87 (1978).\n\n\\bibitem{ott} H. R. Ott, J. K. Kjems, and F. Hulliger, Phys. Rev Lett. {\\bf 42},\n1378 (1979).\n\n\\bibitem{jrm2} J. Rossat-Mignod, J. M. Effantin, P. Burlet, T. Chattopadhyay,\nL. P. Regnault, H. Bartholin, C. Vettier, O. Vogt, D. Ravot, and J. C.\nAchart, J. Magn. Magn. Mater. {\\bf 52}, 111 (1985).\n\n\\bibitem{mori} N. M\\^{o}ri, Y. Okayama, H. Takahashi, Y. Haga, and T. Suzuki,\nPhysica B {\\bf 186-188}, 444 (1993).\n\n\\bibitem{donia}\nS. Doniach, Physica B {\\bf 91}, 231 (1977).\n\n\\bibitem{yellow}\nY. Okayama, Y. Ohara, S. Mituda, H. Takahashi, H. Yoshizawa, T. Osakabe,\nM. Kohgi, Y. Haga, T. Suzuki and N. M\\={o}ri, Physica B {\\bf 186-188}\n531 (1993).\n\n\\bibitem{bir} R. J. Birgeneau, E. Bucher, J. P. Maita, L. Passell, and K. C.\nTurberfield, Phys. Rev. B {\\bf 8}, 5345 (1973).\n\n\\bibitem{wills}\nJ. M. Wills and B. R. Cooper, Phys. Rev. B {\\bf 36}, 3809 (1987).\n\n\\bibitem{ott2} H. R. Ott, J. K. Kjems, and F. Hulliger,\nPhys. Rev. Lett. {\\bf 42}, 1378 (1979); F. Hulliger and H.R. Ott,\nJ. Phys. (Paris) Colloq. {\\bf 40}, C5, 128 (1979).\n\n\\bibitem{jrm3} J. Rossat-Mignod, J. M. Effantin, P. Burlet, T. Chattopadhyay,\nL. P. Regnault, H. Bartholin, C. Vettier, O. Vogt, D. Ravot, and\nJ.C. Achart, J. Magn. Magn. Mater. {\\bf 52} 111 (1985).\n\n\\bibitem{palee} \nP. A. Lee, T. M. Rice, J. W. Serene, L. J. Sham, and J. W. Wilkins,\nCondens. Matter Phys. {\\bf 12}, 99 (1986).\n\n\\bibitem{exps}\nFor an experimental review see, G. R. Stewart, Rev. Mod. Phys. \n{\\bf 56}, 755 (1984).\n\n\\bibitem{santi}\nP. Santini, R. Lemanski, and P. Erdos, Advances in Physics, {\\bf 48}, \n537 (1999). \n\n\\bibitem{pri} D. L. Price and B.R. Cooper, Phys. Rev. B {\\bf 39}, 4945 (1989).\n\n\n\\bibitem{bro} \nM. S.S. Brooks and P. J. Kelly, Phys. Rev. Lett. {\\bf 51}, 1708 (1983);\nO. Eriksson, M. S. S. Brooks, and B. Johansson, Phys. Rev. B\n{\\bf 41},\n7311 (1990).\n\n\n\\bibitem{ste} K. R. Lea, M. J. M. Leask, and W. P. Wolf, J. Phys. Chem. Solids\n {\\bf 23}, 1381 (1962).\n\n\\bibitem{Fulde} Peter Fulde, {\\it Electron Correlations \nin Molecules and Solids}, (Springer-Verlag, Berlin, 1991), pp. 277-281.\n\n\\end{references}\n\n\n\n\\end{document}\n\n\n\n% end of file Template.tex\n\n\n\n"
}
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[
{
"name": "cond-mat0002107.extracted_bib",
"string": "\\bibitem{kious} \n{\\it Correlation Effects and Materials Properties}, eds. A. Gonis, \nN. Kioussis, and M. Ciftan, (Kluwer Academic/Plenum, New York, 1999).\n\n\n\\bibitem{hess}\nD. W. Hess, P.S. Riseborough, and J.L. Smith \n{\\it Heavy Fermion Phenomena}, \nEncyclopedia of Applied Physics, edited by G. Trigg,\n(VCH Publishers Inc., New York, 1993), \nVol. 71, p. 435.\n\n\n\\bibitem{grewe}\nN. Grewe and F. Steglich, {\\it Heavy Fermions}, pp. 343-479,\nin Handbook on the Physics and Chemistry of the Rare Earths, \neds. K. A. Grchneidner Jr. and L. Eyring, (Elsevier Science Publishers,\nAmsterdam, 1991), Vol. 14, pp. 343.\n\n\n\\bibitem{kio} N. Kioussis, B. R. Cooper, and J. M. Wills, Phys. Rev. B\n{\\bf 44}, 10003 (1991).\n\n\n\\bibitem{shg} Q. G. Sheng and B. R. Cooper, J. Appl. Phys. {\\bf 69},\n5472 (1991); Phys. Rev. B {\\bf 50}, 965 (1994).\n\n\n\\bibitem{coop}\nB. R. Cooper, O. Vogt, Q. G. Sheng, and Y. Lin, Phil. Mag. B {\\bf 79},\n683 (1999).\n\n\n\\bibitem{jrm1} J. Rossat-Mignod, P. Burlet, S. Quezel, J. M. Effantin, D.\nDelac\\^{o}te, H. Bartholin, O. Vogt, and D. Ravot, J. Magn. Magn. Mater.\n{\\bf 31-34}, 398 (1983).\n\n\n\\bibitem{coo} B. R. Cooper, R. Siemann, D. Yang, P. Thayamballi, and A.\nBanerjea, in {\\it The Handbook of the Physics and Chemistry of the\nActinides}, edited by A. J. Freeman and G. H. Lander (North-Holland,\nAmsterdam, 1985), Vol. 2, Chap. 6, pp. 435-500.\n\n\n\\bibitem{don} A. D\\\"{o}nni, A. Furrer, P. Fischer, and F. Hulliger, Physica B\n{\\bf 186-188}, 541 (1993).\n\n\n\\bibitem{hul} F. Hulliger, B. Natterer, and H. R. Ott, J. Magn. Magn. Mater.\n{\\bf 8}, 87 (1978).\n\n\n\\bibitem{ott} H. R. Ott, J. K. Kjems, and F. Hulliger, Phys. Rev Lett. {\\bf 42},\n1378 (1979).\n\n\n\\bibitem{jrm2} J. Rossat-Mignod, J. M. Effantin, P. Burlet, T. Chattopadhyay,\nL. P. Regnault, H. Bartholin, C. Vettier, O. Vogt, D. Ravot, and J. C.\nAchart, J. Magn. Magn. Mater. {\\bf 52}, 111 (1985).\n\n\n\\bibitem{mori} N. M\\^{o}ri, Y. Okayama, H. Takahashi, Y. Haga, and T. Suzuki,\nPhysica B {\\bf 186-188}, 444 (1993).\n\n\n\\bibitem{donia}\nS. Doniach, Physica B {\\bf 91}, 231 (1977).\n\n\n\\bibitem{yellow}\nY. Okayama, Y. Ohara, S. Mituda, H. Takahashi, H. Yoshizawa, T. Osakabe,\nM. Kohgi, Y. Haga, T. Suzuki and N. M\\={o}ri, Physica B {\\bf 186-188}\n531 (1993).\n\n\n\\bibitem{bir} R. J. Birgeneau, E. Bucher, J. P. Maita, L. Passell, and K. C.\nTurberfield, Phys. Rev. B {\\bf 8}, 5345 (1973).\n\n\n\\bibitem{wills}\nJ. M. Wills and B. R. Cooper, Phys. Rev. B {\\bf 36}, 3809 (1987).\n\n\n\\bibitem{ott2} H. R. Ott, J. K. Kjems, and F. Hulliger,\nPhys. Rev. Lett. {\\bf 42}, 1378 (1979); F. Hulliger and H.R. Ott,\nJ. Phys. (Paris) Colloq. {\\bf 40}, C5, 128 (1979).\n\n\n\\bibitem{jrm3} J. Rossat-Mignod, J. M. Effantin, P. Burlet, T. Chattopadhyay,\nL. P. Regnault, H. Bartholin, C. Vettier, O. Vogt, D. Ravot, and\nJ.C. Achart, J. Magn. Magn. Mater. {\\bf 52} 111 (1985).\n\n\n\\bibitem{palee} \nP. A. Lee, T. M. Rice, J. W. Serene, L. J. Sham, and J. W. Wilkins,\nCondens. Matter Phys. {\\bf 12}, 99 (1986).\n\n\n\\bibitem{exps}\nFor an experimental review see, G. R. Stewart, Rev. Mod. Phys. \n{\\bf 56}, 755 (1984).\n\n\n\\bibitem{santi}\nP. Santini, R. Lemanski, and P. Erdos, Advances in Physics, {\\bf 48}, \n537 (1999). \n\n\n\\bibitem{pri} D. L. Price and B.R. Cooper, Phys. Rev. B {\\bf 39}, 4945 (1989).\n\n\n\n\\bibitem{bro} \nM. S.S. Brooks and P. J. Kelly, Phys. Rev. Lett. {\\bf 51}, 1708 (1983);\nO. Eriksson, M. S. S. Brooks, and B. Johansson, Phys. Rev. B\n{\\bf 41},\n7311 (1990).\n\n\n\n\\bibitem{ste} K. R. Lea, M. J. M. Leask, and W. P. Wolf, J. Phys. Chem. Solids\n {\\bf 23}, 1381 (1962).\n\n\n\\bibitem{Fulde} Peter Fulde, {\\it Electron Correlations \nin Molecules and Solids}, (Springer-Verlag, Berlin, 1991), pp. 277-281.\n\n"
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cond-mat0002108
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Negative Pressure of Anisotropic Compressible Hall States : Implication to Metrology
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"author": "Kenzo Ishikawa and Nobuki Maeda"
}
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Electric resistances, pressure, and compressibility of anisotropic compressible states at higher Landau levels are analyzed. The Hall conductance varies continuously with filling factor and longitudinal resistances have huge anisotropy. These values agree with the recent experimental observations of anisotropic compressible states at the half-filled higher Landau levels. The compressibility and pressure become negative. These results imply formation of strips of the compressible gas which results in an extraordinary stability of the integer quantum Hall effect, that is, the Hall resistance is quantized exactly even when the longitudinal resistance does not vanish.
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{
"name": "strip7.tex",
"string": "\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\documentstyle[prb,aps,preprint,epsf]{revtex}\n\\documentstyle[prb,aps,multicol,epsf]{revtex}\n\\begin{document}\n\\title{Negative Pressure of Anisotropic Compressible Hall States : \nImplication to Metrology} \n\\author{Kenzo Ishikawa and Nobuki Maeda}\n\\address{\nDepartment of Physics, Hokkaido University, \nSapporo 060-0810, Japan}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nElectric resistances, pressure, and compressibility of anisotropic \ncompressible states at higher Landau levels are analyzed. \nThe Hall conductance varies continuously with filling factor \nand longitudinal resistances have huge anisotropy. \nThese values agree with the recent experimental observations \nof anisotropic compressible states at the half-filled higher Landau levels. \nThe compressibility and pressure become negative. \nThese results imply formation of strips of the compressible gas which \nresults in an extraordinary stability of the integer quantum \nHall effect, that is, the Hall resistance is quantized exactly even when the \nlongitudinal resistance does not vanish. \n\\end{abstract}\n\\draft\n\\pacs{PACS numbers: 73.40.Hm, 73.20.Dx}\n\n\\begin{multicols}{2}\n\nRecently highly correlated anisotropic states have been \nobserved around half-filled higher Landau levels of high mobility GaAs/AlGaAs \nhetero-structures.\\cite{a,b,d,e} \nLongitudinal resistance along one direction tends to vanish at low temperature \nbut that of another direction has a large value of order K$\\Omega$. \nThe Hall resistance is approximately proportional to the filling factor. \nSince one longitudinal resistance is finite, the state is compressible. \nIt has been noted also that the current-induced breakdown and collapse of \nthe quantum Hall effect (QHE) at $\\nu=4$ occurs through several steps \nwhich implies that compressible gas in quantum Hall system has unusual \nproperties.\\cite{f,g} \n\nAnisotropic stripe states were predicted to be favored in higher Landau \nlevels.\\cite{h,i} \nUsing the von Neumann lattice basis, the present authors found an \nanisotropic mean field state in the lowest Landau level to have a negative \npressure and negative compressibility.\\cite{j,eisen} \nTheir energies at higher Landau levels were calculated recently by \none of the present authors\\cite{k} and others\\cite{l} in the \nHartree-Fock approximation. \nThey have lower energies than symmetric states, but their physical properties \nhave not been studied well so that it is not clear \nif these states agree with the states found by experiments.\n\nIn the present work, we study the physical properties of anisotropic mean \nfield states around $\\nu=n+1/2$ and $\\nu=n$, where $n$ is an integer. \nWe point out that this mean field state at $\\nu=n+1/2$ has properties \ndescribed above and can explain the experimental observations of anisotropic \nstates. \nThese states have a negative pressure and negative compressibility and \nperiodic density modulation in one direction. \nWe show that as a consequence of negative pressure, \na compressible gas strip is formed in the bulk around $\\nu=n$ and \na current flows in the strip by a new tunneling mechanism, \nactivation from undercurrent. \nThe current is induced in isolated strip with a temperature dependent \nmagnitude of activation type and it causes a small longitudinal resistance \nin the system of quantized Hall resistance. \nThis solves a longstanding puzzle of the integer QHE, namely Hall \nresistance is quantized exactly even if the system has a small finite \nlongitudinal resistance. \nCollapse phenomena\\cite{f} are shown to be understandable also. \n\nThe von Neumann lattice basis\\cite{m,n} is one of the bases for \ndegenerate Landau levels of the two-dimensional continuum space where \ndiscrete coherent states of guiding center variables $(X,Y)$ are used and \nis quite useful in studying QHE because translational \ninvariance in two dimensions is preserved. \nSpatial properties of extended states and interaction effects were studied \nin systematic ways. \nWe are able to express exact relations such as current conservation, \nequal time commutation relations, and Ward-Takahashi identity\n\\cite{o} in equivalent \nmanners as those of local field theory and to connect the Hall conductance \nwith momentum space topological invariant. \nExact quantization of the Hall conductance in quantum Hall regime (QHR) \nin the systems of disorders, interactions, and finite injected current \nhas been proved in this basis. \nWe use this formalism in the present paper as well.\n\nElectrons in the Landau levels are expressed with the creation and \nannihilation operators \n$a_l^\\dagger({\\bf p})$ and $a_l({\\bf p})$ of having Landau level index, \n$l=$0, 1, 2$\\dots$, and momentum, $\\bf p$. \nThe momentum conjugate to von Neumann lattice coordinates is defined in the \nmagnetic Brillouin zone (MBZ), \n$\n\\vert p_i\\vert \\leq{\\pi/a},\\ a=\\sqrt{2\\pi\\hbar/eB}. \n$\nThe many body Hamiltonian $H$ is written in the momentum representation as \n$H=H_0+H_1$, where\n\\begin{eqnarray}\nH_0&=&\\sum_{l=0}^{\\infty}\\int_{MBZ} {d{\\bf p}\\over (2\\pi/a)^2} \nE_l a_l^\\dagger({\\bf p})a_l({\\bf p}),\\label{ham}\\\\\nH_1&=&\\int_{{\\bf k}\\neq0} \nd{\\bf k}\\rho({\\bf k}){V({\\bf k})\\over2}\\rho(-{\\bf k}).\\nonumber\n\\end{eqnarray}\nHere $E_l$ is the Landau level energy $(\\hbar eB/m)(l+1/2)$ and \n$V({\\bf k})=2\\pi q^2/k$ for the Coulomb interaction, and the charge \nneutrality is assumed. \nIn Eq.~(\\ref{ham}), $H_0$ is diagonal but the charge density \n$\\rho({\\bf k})$ is non-diagonal with respect to $l$. \nWe call this basis the energy basis. \nA different basis called the current basis in which charge density becomes \ndiagonal will be used later in computing current correlation functions \nand electric resistances. \n\nIt is worthwhile to clarify the peculiar symmetry of the system described by \nEq.~(\\ref{ham}). \nThe Hamiltonian is invariant under translation in momentum space, \n${\\bf p}\\rightarrow {\\bf p}+{\\bf K}$, where $\\bf K$ is a constant vector, \nwhich is called the $K$-symmetry. \nThis symmetry emerges because the kinetic energy is quenched due to the \nmagnetic field. \nA state which has momentum dependent single particle energy violates \nthe $K$-symmetry.\\cite{ks} \nIn the present paper, we study a mean field solution which violates \n$K_y$-symmetry but preserves $K_x$-symmetry. \nThe one-particle energy has $p_y$ dependence in this state. \n\nThe compressible gas state is characterized by the following form of \nexpectation values in the coordinate space, \n\\begin{equation}\nU^{(l)}({\\bf X}-{\\bf X}')\\delta_{ll'}=\\langle a^\\dagger_{l'}({\\bf X}')a_l\n({\\bf X})\\rangle,\n\\end{equation}\nwhere the expectation values are calculated self-consistently \nin the mean field approximation\\cite{j,k} using $H_1$ and the mean \nfield $U^{(l)}$. \nIn Figs.~1 and 2, the energy per particle, pressure, \nand compressibility are presented with respect to the filling factor \n$\\nu=n+\\nu'$. \nAs seen in these figures, they become negative. \nThe density is uniform in y-direction but is periodic in \nx-direction.\\cite{k} \nThe present anisotropic state could be identified with the stripe structure \ndiscussed in Refs.~\\cite{h} and \\cite{i}. \nWe have checked that bubble states discussed in Ref.~\\cite{h} \nalso have negative pressure and compressibility. \nThese properties may be common in the compressible states of the quantum Hall \nsystem.\\cite{efros} \n\nThe compressible states thus obtained have negative pressure and are \ndifferent from ordinary gas. \nNaively it would be expected that these gas states were unstable. \nHowever thanks to the background charge of dopants, a stable state with \na negative pressure can exist. \nSince the pressure is negative, charge carriers compress itself while \ndopants do no move. \nThen charge neutrality is broken partly and compressible gas states are \nstabilized by Coulomb energy. \nTotal energy becomes minimum with a suitable shape which depends on the \ndensity of compressible gas. \n\nThe bulk compressible gas states are realized around $\\nu=n+1/2$ \n(called region I), where the Coulomb energy is dominant over negative \npressure and a narrow depletion region is formed at the boundary. \nIts width is determined by the balance between the pressure \nand the Coulomb force. \n\nThe low density compressible gas states are realized around $\\nu =n$ \n(called region II). \nIn this case pressure effect is enhanced relatively compared to Coulomb \nenergy and a strip of compressible gas states is formed as \nshown in Fig.~3 (a). \nReal system has disorders by which most electronic states are localized. \nLet us classify three different regions depending on the relative ratio \nbetween localization length $\\xi$ at the Fermi energy \nand the width at potential probe area $L_p$ and the width at Hall probe \narea $L_h$. \nWe assume $L_p < L_h$. \nIn the region II-(i), $\\xi < L_p$ is satisfied and localized states fill \nwhole system. \nIn the region II-(ii), $L_p < \\xi < L_h$ is satisfied and \nthe Hall probe area is filled with localized states but potential probe \narea is filled partly with compressible gas strip. \nFinally in the region II-(iii), $L_h < \\xi$ is satisfied and \nwhole area are filled partly with compressible gas states. \nIn each case if localization length is longer than the width of the system, \nthen these localized states are regarded as extended states \nwhich behave like compressible gas states with a negative pressure. \nIn the regions II-(ii) and II-(iii), the strip contributes to electric \nconductance if current flows through the strip. \nHowever the strip is unconnected with source drain area. \nHow does the current flow through the strip? \nThis problem has not been studied before. \nIn these regions, extended states below Fermi energy \ncarry non-dissipative current, which we call the undercurrent. \nSee Fig.~3 (b). \nWe show later that the undercurrent actually induces the dragged current. \n\nFirst we calculate the electric conductance of the bulk compressible states \nin the region I. \nIt is convenient to use current basis for computing current correlation\nfunctions. \nField operators $a_l$ and propagator $S_{ll'}$ are transformed \nfrom the energy basis to the current basis as,\n\\begin{eqnarray}\n\\tilde a_l({\\bf p})&=&\\sum_{l'}U_{ll'}({\\bf p})a_{l'}({\\bf p}),\\\\\n\\tilde S_{ll'}(p)&=&\\sum_{l_1 l_2}\nU_{ll_1}({\\bf p})S_{l_1l_2}(p)U^\\dagger_{l_2l'}({\\bf p}),\n\\nonumber\n\\end{eqnarray}\nwhere $U({\\bf p})=e^{-i p_x\\xi}e^{-ip_y\\eta}$ and \n$(\\xi,\\eta)$ are relative coordinates defined by $(x-X,y-Y)$. \nIn the current basis, the equal time commutation relation between the \ncharge density and the field operators are given by,\n\\begin{equation}\n[\\rho({\\bf k}),\\tilde a_l({\\bf p})]=-\\tilde a_l({\\bf p})\n\\delta^{(2)}({\\bf p}-{\\bf k}). \n\\end{equation}\nHence vertex part is given by a derivative of inverse of the propagator,\n$\\tilde \\Gamma_\\mu(p,p)=\\partial_\\mu\\tilde S^{-1}(p)$, \nknown as Ward-Takahashi identity.\\cite{o} \nThe Hall conductance is the slope of the current-current correlation \nfunction at the origin and is given by the topologically invariant \nexpression of the propagator in the current basis as\\cite{m,n}\n\\begin{equation}\n\\sigma_{xy}={e^2\\over h}{1\\over24\\pi^2}\\int{\\rm tr}({\\tilde S}(p)\nd{\\tilde S}^{-1}(p))^3. \n\\label{toppo}\n\\end{equation}\nThis shows that $\\sigma_{xy}$ is quantized exactly in QHR \nwhere the Fermi energy is located in the localized state region. \nNow Fermi energy is in the compressible state band region and $\\sigma_{xy}$ \nis not quantized. \nFor the anisotropic states, the inverse propagator is given by \n$S^{-1}(p)_{ll'}=\\{p_0-(E_l+\\epsilon_l(p_y))\\}\\delta_{ll'}$ where \n$\\epsilon_l(p_y)$ is the one-particle energy.\\cite{j,k} \n$S(p)$ has no topological singurality and its winding number vanishes. \nHence the topological property of the propagator in the current basis, \n$\\tilde S(p)$, is determined solely by the unitary operator \n$U({\\bf p})$, \nand the Hall conductance is written as,\n\\begin{eqnarray} \n\\sigma_{xy}&=&{e^2\\over h}{1\\over4\\pi^2}\\int dp\\epsilon^{ij}\n{\\rm tr}\\left[S(p)U^\\dagger({\\bf p})\\partial_i U({\\bf p})\nU^\\dagger({\\bf p})\\partial_j U({\\bf p})\\right]\\nonumber\\\\\n&=&{e^2\\over h}(n+\\nu').\n\\label{topo}\n\\end{eqnarray}\nTo obtain the final result in the above equation, \nwe assumed that the Landau levels are filled completely up to $n$ th level \nand $(n+1)$ th level is filled partially with filling factor $\\nu'$. \nThe Hall conductance is proportional to the total filling factor. \n\nThe longitudinal conductance in x-direction, $\\sigma_{xx}$, \nvanishes since there is no empty state in this direction. \nIf a momentum is added in x-direction, one particle should be lifted to \na higher Landau level. \nThere needs a finite energy and $\\sigma_{xx}$ vanishes. \nThe longitudinal conductance in y-direction, $\\sigma_{yy}$, \ndoes not vanish. \nOne particle energy has a dependence on only $p_y$, \nand the system is regarded as one dimensional. \nOne dimensional conductance is given by Buttiker-Landauer formula\n\\cite{p}. \nWe have thus, \n\\begin{equation}\n\\sigma_{yy}={e^2/h},\\ \\sigma_{xx}=0.\n\\label{sig}\n\\end{equation}\nThe Hall conductance, Eq.~(\\ref{topo}), and the longitudinal conductances, \nEq.~(\\ref{sig}), agree with the experimental observations of anisotropic \nstates around $\\nu=n+1/2\\ (n\\geq 1)$. \n\nNext we study the low density region, region II. \nIn the first region, II-(i), whole area is filled with localized states, \nhence from the formula Eq.~(\\ref{toppo}), the Hall conductance is \nquantized exactly. \nThe longitudinal resistances vanish. \nWe have \n\\begin{equation}\n\\sigma_{xy}=(e^2/h) n,\\ \\sigma_{xx}=\\sigma_{yy}=0. \n\\end{equation}\nThis corresponds to standard QHR. \n\nIn the regions II-(ii) and II-(iii), a compressible strip bridges one \nedge to the other edge. \nTunneling combined with an interaction causes the dragged current \nin the strip. \nThe conductance due to the tunneling mechanism can be calculated by \na current-current correlation function shown in Fig.~4. \nThe two-loop diagram is the lowest order contribution. \nThe dragged current flows in the compressible strip at potential \nprobe area in the region II-(ii), and at Hall probe area in the \nregion II-(iii). \n\nIn the region II-(ii), the Hall probe area is filled with only \nlocalized states. \nHence the Hall conductance is quantized exactly. \nThe potential probe area has a finite longitudinal resistance due to \nan electric current in the strip. \nThe electric current which flows in the strip makes the strip area to \nhave a finite temperature. \nWe have thus the exactly quantized Hall resistance and a small longitudinal \nresistance in this region as \n\\begin{equation} \nR_{xy}^{-1}=(e^2/h)n,\\ R_{xx}^{-1}={(e^2/h)}\\varepsilon.\n\\end{equation}\nThe small parameter $\\varepsilon$ is proportional to the activation form \n$\\exp[-\\beta(\\Delta+mv^2/2)]$, where $\\Delta$ is the energy gap \nbetween the Fermi energy and the lower Landau level, $\\beta$ is the \ninverse temperature at strip area, and $v$ is the average velocity of the \nundercurrent states. \nThe additional term $mv^2/2$ to $\\Delta$ comes from the Galilean boost. \nThis region has not been taken into account in the metrology of \nQHE.\\cite{q} \n\nIn the region II-(iii), whole area is filled with compressible states. \nHence from Eq.~(\\ref{topo}), the Hall conductance is given by unquantized \nvalue and the longitudinal resistance becomes finite. \nIn this case we have \n\\begin{equation} \nR_{xy}^{-1}={(e^2/h)}(n+\\varepsilon'),\n\\ R_{xx}^{-1}={(e^2/h)}\\varepsilon.\n\\end{equation}\nThat is, QHE is collapsed.\\cite{f} \n$\\varepsilon'$ has the same temperature dependence as $\\varepsilon$. \n\nThe localization length and mobility edge depend on injected current in the \nreal system. \nIn a small current system, localization lengths are small in a \nmobility gap and corresponds to QHR (region II-(i)). \nIn an intermediate current system, they become larger and the strip is \nformed in potential probe area first (region II-(ii)) \nand in Hall probe area second (region II-(iii)). \nQHE is collapsed in this region. \nIn a larger current system, localization lengths become even larger, \nand whole system is filled with extended states. \nQHE is broken down in this region. \nThis is consistent with Kawaji et al.'s recent experiments and proposal.\n\\cite{f} \n\nIn summary, we have shown that the anisotropic mean field states have \nunquantized Hall conductance, huge anisotropic longitudinal resistance, \nnegative pressure, and negative compressibility. \nThese electric properties are consistent with the recent experiments of \nthe anisotropic compressible Hall state and of collapse phenomena. \nNegative pressure of these states does not lead to instability but instead \nleads to a formation of a narrow strip of compressible gas states if its \ndensity is low and formation of the bulk compressible gas with the \ndepletion region if its density is around the half-filling. \nConsequently in the system of low density compressible gas states, \nthe Hall resistance is kept in the exactly quantized value even though the \nlongitudinal resistance is finite. \nThis longstanding puzzle was solved from \nthe unusual property of compressible Hall gas, namely negative pressure. \nHence it plays important roles in the metrology of the QHE. \n\nAuthors would like to thank Y. Hosotani and T. Ochiai for \nuseful discussions. \nOne of the present authors (K. I.) also thanks S. Kawaji and \nB. I. Shklovskii for fruitful discussions. \nThis work was partially supported by the special \nGrant-in-Aid for Promotion of Education and Science in Hokkaido University \nprovided by the Ministry of Education, Science, Sports, and Culture, the \nGrant-in-Aid for Scientific Research on Priority area (Physics of CP \nviolation) (Grant No. 12014201), and the Grant-in-aid for \nInternational Science Research (Joint Research 10044043) from the Ministry \nof Education, Science, Sports and Culture, Japan.\n\n\\begin{references}\n\\bibitem{a}\nM. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, \nand K. W. West, Phys. Rev. Lett. {\\bf 82}, 394 (1999). \n\\bibitem{b}\nR. R. Du, D. C. Tsui, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and \nK. W. West, Solid State Commun. {\\bf 109}, 389 (1999).\n\\bibitem{d}\nW. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, \nK. W. Baldwin, and K. W. West, Phys. Rev. Lett. {\\bf 83}, 820 (1999). \n\\bibitem{e}\nM. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, \nPhys. Rev. Lett. {\\bf 83}, 824 (1999). \n\\bibitem{f}\nS. Kawaji, J. Suzuki, T. Shimada, H. Iizuka, T. Kuga, and T. Okamoto, \nJ. Phys. Soc. Jpn. {\\bf 67}, 1110 (1998); S. Kawaji, H. Iizuka, T. Kuga, \nand T. Okamoto, Physica {\\bf B256-258}, 56 (1998). \n\\bibitem{g}\nJ. Weis, Y. Y. Wei, and K.v. Klizing, Physica {\\bf B256-258}, 1 (1998); \nI. I. Kaya, G. Nachtwei, K. v. Klizing, and K. Eberl, \n{\\it ibid.} {\\bf 256-258}, 8 (1998).\n\\bibitem{h}\nA. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys. Rev. Lett. \n{\\bf 76}, 499 (1996); M. M. Fogler, A. A. Koulakov, and \nB. I. Shklovskii, Phys. Rev. B {\\bf 54}, 1853 (1999). \n\\bibitem{i}\nR. Moessner and J. T. Chalker, Phys. Rev. B {\\bf 54}, 5006 (1996).\n\\bibitem{j}\nK. Ishikawa, N. Maeda, and T. Ochiai, Phys. Rev. Lett. {\\bf 82}, \n4292 (1999).\n\\bibitem{eisen}\nActually, the negative compressibility was observed by \nJ. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. \n{\\bf 68}, 674 (1992). \n\\bibitem{k}\nN. Maeda, Phys. Rev. B {\\bf 61}, 4766 (2000). \n\\bibitem{l}\nT. Jungwirth, A. H. MacDonald, L. Smrcka, and S. M. Girvin, \nPhys. Rev. B {\\bf 60}, 15574 (1999); \nT. Stanescu, I. Martin, and P. Phillips, Phys. Rev. Lett. {\\bf 84}, 1288 \n(2000); \n\\bibitem{m}\nN. Imai, K. Ishikawa, T, Matsuyama, and I. Tanaka, Phys. Rev. B {\\bf 42}, \n10610 (1990); \nK. Ishikawa, Prog. Theor. Phys. Suppl. {\\bf 107}, 167 (1992); \nK. Ishikawa, N. Maeda, K. Tadaki, Phys. Rev. B {\\bf 54}, 17819 (1996). \n\\bibitem{n}\nK. Ishikawa, N. Maeda, T. Ochiai, and H. Suzuki, Physica (Amsterdam) \n{\\bf 4E}, 37 (1999); Phys. Rev. B {\\bf 58}, 1088 (1998); {\\it ibid}. \n{\\bf 58}, R13391 (1998).\n\\bibitem{o}\nJ. C. Ward, Phys. Rev. {\\bf 78}, 1824 (1950); \nY. Takahashi, Nuovo Cimento {\\bf 6}, 370 (1957)\n\\bibitem{ks}\nThe K-symmetry was originally discussed in the context of the composite \nFermion theory. \nIn the composite Fermion theory, the concept of compressibility is \ncontroversial, see \nV. Pasquier and F. D. M. Haldane, Nucl. Phys. {\\bf B516}, 719 (1998); \nB. I. Halperin and A. Stern, Phys. Rev. Lett. {\\bf 80}, 5457 (1998); \nG. Murthy and R. Shankar, {\\it ibid}, {\\bf 80}, 5458 (1998). \n\\bibitem{efros}\nA. L. Efros, Solid State Commun. {\\bf 65}, 1281 (1988). \n\\bibitem{p}\nM. Buttiker, Phys. Rev. B {\\bf 38}, 9375 (1988); \nR. Landauer, IBM J. Res. Dev. {\\bf 1}, 223 (1957).\n\\bibitem{q}\nK. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. {\\bf 45}, 494 \n(1980); S. Kawaji and J. Wakabayashi, in \n{\\it Physics in High Magnetic Fields}, \nedited by S. Chikazumi and N. Miura (Springer-Verlag, Berlin, 1981); \nF. Delahaye, Metrologia, {\\bf 26}, 63 (1989). \n\\end{references}\n%\\end{multicols}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\epsfxsize=2.5in\\epsffile{feg2.eps}\nFig~1. Energy per particle (dashed lines) in the \nunit of $q^2/a$ and pressure (solid lines) in the unit of \n$q^2/a^3$ for $\\nu=n+\\nu'$, $n=$0, 1, 2, and 3. \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\epsfxsize=2.4in\\epsffile{fcg2.eps}\nFig~2. Compressibility times $\\nu'^2$ in the unit of $a^3/q^2$ \nfor $\\nu=n+\\nu'$, $n=$0, 1, 2, and 3. \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\epsfxsize=2.4in\\epsffile{Stripenergy.eps}\nFig~3. (a) Schematical view of a Hall bar in the region II-(ii). \nElectric current is injected from S to D. \n(b) Sketch of the extended states carrying the undercurrent $J_0$ \nand the compressible state carrying the dragged current $J_1$ at \nthe strip area in the energy and the x-position. \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\epsfxsize=1.8in\\epsffile{Stripcurrent.eps}\nFig~4. Feynman diagram for the current-current correlation function. \n$J_0$ is the undercurrent carried by the extended states and \n$J_1$ is the dragged current carried by the compressible states in the \nstrip area. The dashed line stands for an interaction effect. \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\end{multicols}\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002108.extracted_bib",
"string": "\\bibitem{a}\nM. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, \nand K. W. West, Phys. Rev. Lett. {\\bf 82}, 394 (1999). \n\n\\bibitem{b}\nR. R. Du, D. C. Tsui, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and \nK. W. West, Solid State Commun. {\\bf 109}, 389 (1999).\n\n\\bibitem{d}\nW. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, \nK. W. Baldwin, and K. W. West, Phys. Rev. Lett. {\\bf 83}, 820 (1999). \n\n\\bibitem{e}\nM. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, \nPhys. Rev. Lett. {\\bf 83}, 824 (1999). \n\n\\bibitem{f}\nS. Kawaji, J. Suzuki, T. Shimada, H. Iizuka, T. Kuga, and T. Okamoto, \nJ. Phys. Soc. Jpn. {\\bf 67}, 1110 (1998); S. Kawaji, H. Iizuka, T. Kuga, \nand T. Okamoto, Physica {\\bf B256-258}, 56 (1998). \n\n\\bibitem{g}\nJ. Weis, Y. Y. Wei, and K.v. Klizing, Physica {\\bf B256-258}, 1 (1998); \nI. I. Kaya, G. Nachtwei, K. v. Klizing, and K. Eberl, \n{\\it ibid.} {\\bf 256-258}, 8 (1998).\n\n\\bibitem{h}\nA. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys. Rev. Lett. \n{\\bf 76}, 499 (1996); M. M. Fogler, A. A. Koulakov, and \nB. I. Shklovskii, Phys. Rev. B {\\bf 54}, 1853 (1999). \n\n\\bibitem{i}\nR. Moessner and J. T. Chalker, Phys. Rev. B {\\bf 54}, 5006 (1996).\n\n\\bibitem{j}\nK. Ishikawa, N. Maeda, and T. Ochiai, Phys. Rev. Lett. {\\bf 82}, \n4292 (1999).\n\n\\bibitem{eisen}\nActually, the negative compressibility was observed by \nJ. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. \n{\\bf 68}, 674 (1992). \n\n\\bibitem{k}\nN. Maeda, Phys. Rev. B {\\bf 61}, 4766 (2000). \n\n\\bibitem{l}\nT. Jungwirth, A. H. MacDonald, L. Smrcka, and S. M. Girvin, \nPhys. Rev. B {\\bf 60}, 15574 (1999); \nT. Stanescu, I. Martin, and P. Phillips, Phys. Rev. Lett. {\\bf 84}, 1288 \n(2000); \n\n\\bibitem{m}\nN. Imai, K. Ishikawa, T, Matsuyama, and I. Tanaka, Phys. Rev. B {\\bf 42}, \n10610 (1990); \nK. Ishikawa, Prog. Theor. Phys. Suppl. {\\bf 107}, 167 (1992); \nK. Ishikawa, N. Maeda, K. Tadaki, Phys. Rev. B {\\bf 54}, 17819 (1996). \n\n\\bibitem{n}\nK. Ishikawa, N. Maeda, T. Ochiai, and H. Suzuki, Physica (Amsterdam) \n{\\bf 4E}, 37 (1999); Phys. Rev. B {\\bf 58}, 1088 (1998); {\\it ibid}. \n{\\bf 58}, R13391 (1998).\n\n\\bibitem{o}\nJ. C. Ward, Phys. Rev. {\\bf 78}, 1824 (1950); \nY. Takahashi, Nuovo Cimento {\\bf 6}, 370 (1957)\n\n\\bibitem{ks}\nThe K-symmetry was originally discussed in the context of the composite \nFermion theory. \nIn the composite Fermion theory, the concept of compressibility is \ncontroversial, see \nV. Pasquier and F. D. M. Haldane, Nucl. Phys. {\\bf B516}, 719 (1998); \nB. I. Halperin and A. Stern, Phys. Rev. Lett. {\\bf 80}, 5457 (1998); \nG. Murthy and R. Shankar, {\\it ibid}, {\\bf 80}, 5458 (1998). \n\n\\bibitem{efros}\nA. L. Efros, Solid State Commun. {\\bf 65}, 1281 (1988). \n\n\\bibitem{p}\nM. Buttiker, Phys. Rev. B {\\bf 38}, 9375 (1988); \nR. Landauer, IBM J. Res. Dev. {\\bf 1}, 223 (1957).\n\n\\bibitem{q}\nK. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. {\\bf 45}, 494 \n(1980); S. Kawaji and J. Wakabayashi, in \n{\\it Physics in High Magnetic Fields}, \nedited by S. Chikazumi and N. Miura (Springer-Verlag, Berlin, 1981); \nF. Delahaye, Metrologia, {\\bf 26}, 63 (1989). \n"
}
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cond-mat0002109
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No Quasi-long-range Order in Strongly Disordered Vortex Glasses: a Rigorous Proof
|
[
{
"author": "D.E. Feldman"
}
] |
The paper contains a rigorous proof of the absence of quasi-long-range order in the random-field $O(N)$ model for strong disorder in the space of an arbitrary dimensionality. This result implies that quasi-long-range order inherent to the Bragg glass phase of the vortex system in disordered superconductors is absent as the disorder or external magnetic field is strong.
|
[
{
"name": "cond-mat0002109.tex",
"string": "\n\n\\documentstyle[aps]{revtex} \\def\\be{\\begin{equation}} \\def\\ee{\\end{equation}}\n\\def\\bea{\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}} \\begin{document} \\title {No\nQuasi-long-range Order in Strongly Disordered Vortex Glasses: a Rigorous Proof}\n\n\\author {D.E. Feldman}\n\n\\address {Department of Condensed Matter Physics, Weizmann Institute of Science,\n76100 Rehovot, Israel\\\\ and Landau Institute for Theoretical Physics, 142432\nChernogolovka, Moscow region, Russia} \\maketitle\n\n\\begin{abstract} The paper contains a rigorous proof of the absence of\nquasi-long-range order in the random-field $O(N)$ model for strong disorder in\nthe space of an arbitrary dimensionality. This result implies that\nquasi-long-range order inherent to the Bragg glass phase of the vortex system in\ndisordered superconductors is absent as the disorder or external magnetic field\nis strong. \\end{abstract} \n\n%\\PACS{74.60.Ge, 75.10.Nr, 05.20.-y, 05.50.+q}\n\nThe nature of the vortex phases of disordered superconductors is a subject of\nactive current investigations. A plausible picture includes three phases\n\\cite{ph}: vortex liquid (VL), vortex glass (VG) and Bragg glass (BG). In all\nthose phases an Abrikosov lattice is absent \\cite{Larkin} and only short-range\norder (SRO) is expected in VG and VL. A higher degree of ordering is predicted\nin BG. It is argued that in this state the vortex array is\nquasi-long-range-ordered \\cite{qlro1,qlro2}. Thus, Bragg peaks can be observed\nin BG \\cite{Cubitt,yaron} as if the system had an Abrikosov lattice. In the\nother phases Bragg peaks are not found \\cite{Cubitt}. The phase transitions\nfrom BG to VG and VL are presumably associated with topological defects\n\\cite{ph}.\n\nThe above picture is supported by variational and renormalization group\ncalculations for the random-field XY model \\cite{qlro1,qlro2,var,var1} which is\nthe simplest model of the vortex array in disordered superconductors. Besides,\nthis model is useful for our understanding of many other disordered systems\n\\cite{qlro2}. Hence, the detailed knowledge of its properties is important.\nUnfortunately, the only rigorous result about the random-field XY model is the\nabsence of long-range order (LRO) \\cite{rig}. The present paper contains a new\nrigorous result: QLRO is absent as the disorder is sufficiently strong. It is\ninteresting that our proof is quite simple in contrast to the very nontrivial\ndemonstration of the absence of LRO for arbitrarily weak disorder \\cite{rig}.\n\nIt is expected \\cite{ph} that the vortex system of the disordered superconductor\nhas no ordering for strong disorder due to appearance of the dislocations. The\nXY model is a convenient polygon for understanding of the role of the\ntopological defects. Since the vortex glass phase of disordered superconductors\ncorresponds to strong disorder, our rigorous result supports the conjecture that\nthe vortex glass phase has no QLRO.\n\nThe simplest model of the vortex array in disordered superconductors has the\nfollowing Hamiltonian \\cite{qlro1,qlro2}\n\n\\be \\label{1} H=\\int d^3r [K(\\nabla u({\\bf r}))^2 + h\\cos(2\\pi u({\\bf\nr})/a-\\theta({\\bf r}))], \\ee where $u$ is the vortex displacement, $a$ the\nconstant of the Abrikosov lattice in the absence of disorder, $\\theta$ the\nrandom phase. The one-component displacement field $u({\\bf r})$ describes\nanisotropic superconductors. The generalization for the isotropic case is\nstraightforward. The ordering can be characterized in terms of the form-factor\n\n\\be \\label{ff} G(r)=\\langle\\cos2\\pi (u({\\bf 0})-u({\\bf r}))/a\\rangle, \\ee where\nthe angular brackets denote the thermal and disorder average. This correlation\nfunction contains information about neutron scattering. LRO corresponds to a\nfinite large-distance asymptotics $G(r\\rightarrow\\infty)\\rightarrow{\\rm\nconstant}$, QLRO is described by the power law $G(r)\\sim r^{-\\eta}$ and SRO\ncorresponds to the exponential decay of the correlation function $G(r)$ at large\n$r$.\n\nWe demonstrate that as the random-field amplitude $h$ Eq. (\\ref{1}) is large\nthe system possesses only SRO. Since $h$ depends on the strength of the\ndisorder in the sample and the external magnetic field \\cite{qlro2} we see that\nSRO corresponds to the situation at which either the disorder or magnetic field\nis strong. We consider not only the XY model (\\ref{1}) but also the more\ngeneral random-field $O(N)$ model. Its Hamiltonian has the following structure\n\n\\be \\label{2} H=-J\\sum_{\\langle ij\\rangle} {\\bf S}_i{\\bf S}_j -H\\sum_i {\\bf\nS}_i{\\bf n}_i, \\ee where ${\\bf S}_i$ are the $p$-component unit spin vectors on\nthe simple cubic lattice in the D-dimensional space, ${\\bf n}_i$ is the random\nunit vector describing orientation of the random field at site $i$, the angular\nbrackets denote the summation over the nearest neighbors on the lattice. The XY\nHamiltonian (\\ref{1}) corresponds to $p=2$. In this case the relation between\n(\\ref{1}) and (\\ref{2}) is given by the formulae $S_x=\\cos 2\\pi u/a, S_y=\\sin\n2\\pi u/a$, where $S_{x,y}$ are the spin components.\n\nThe idea of our proof is based on the fact that the orientation of any spin\n${\\bf S}_i$ depends mostly on the random fields at the nearest sites as the\nrandom-field amplitude $H$ is sufficiently strong. We shall see that the\nknowledge of the random fields in the region of size $Nb$ with the center in\nsite $i$, where $b$ is the lattice constant, allows us to determine the\norientation of the spin ${\\bf S}_i$ with the accuracy $\\exp(-{\\rm constant}N)$.\nThus, the orientations of any two distant spins depend on the realizations of\nthe random field in two non-intersecting regions up to exponentially small\ncorrections. The values of the random field in these regions are uncorrelated.\nHence, the correlations of the distant spins are exponentially small.\n\nBelow we consider the case of the zero temperature. Hence, the system is in the\nground state. We assume that the amplitude of the random field\n\n\\be \\label{3} H=2DJ(1+\\sqrt{2}+\\delta), \\delta>0. \\ee Let us estimate the angle\nbetween an arbitrary spin ${\\bf S}_i$ and the local random field ${\\bf\nh}_i=H{\\bf n}_i$. The Weiss field ${\\bf H}_W={\\bf h}_i + {\\bf H}_J$ acting on\nthe spin includes the random field ${\\bf h}_i$ and the exchange contribution\n${\\bf H}_J=J\\sum_j {\\bf S}_j$, where ${\\bf S}_j$ are the nearest neighbors of\nthe spin ${\\bf S}_i$. The latter $H_J\\le 2DJ$ since the number of the nearest\nneighbors is $2D$, where $D$ is the spatial dimension. Hence, the minimal\npossible modulus of the Weiss field is $(H-H_J)\\ge (H-2DJ)$. Any spin is\noriented along the local Weiss field ${\\bf H}_W$. Let us consider the triangle\ntwo sides of which are ${\\bf h}_i$ and ${\\bf H}_J$, and the third side is\nparallel to ${\\bf S}_i$. The laws of sinuses allow us to show that the maximal\npossible angle between ${\\bf h}_i$ and ${\\bf S}_i$ is $\\phi_1=\\arcsin(2DJ/H)$.\n\nWe shall now determine the orientation of the spin ${\\bf S}_i$ iteratively. Let\nthe zero approximation ${\\bf S}^0_i$ be oriented along the random field ${\\bf\nh}_i$. Let the first approximation ${\\bf S}^1_i$ be oriented along the Weiss\nfield ${\\bf H}_W^0$, calculated with the zero approximation for the neighboring\nspins: ${\\bf H}_W^0={\\bf h}_i+J\\sum_j {\\bf n}_j$, where $\\sum_j$ denotes the\nsummation over the nearest neighbors. The second approximation ${\\bf S}^2_i$ is\ndetermined with the Weiss field in the first approximation, etc. Any\napproximation ${\\bf S}^k_i$ depends on the random fields only at a finite set of\nthe lattice sites. The distance between any such site and site $i$ is no more\nthan $kb$, where $b$ is the lattice constant.\n\nIn any approximation the Weiss field $H_W^k\\ge H-2DJ$. Let ${\\bf s}_i^k$ be the\ndifference between the $k$th and $(k-1)$th approximations for the spin ${\\bf\nS}_i$. Then $|{\\bf H}_W^k-{\\bf H}_W^{(k-1)}|\\le 2DJm^k$, where $m^k$ denotes\nthe maximal value of $s_l^k$. Hence, we find with the laws of sinuses that the\nangle between ${\\bf S}_i^k$ and ${\\bf S}_i^{(k-1)}$ is less than\n$\\phi_k=\\arcsin(2DJm^k/(H-2DJ))$. Since $s^{(k+1)}_i\\le 2\\sin(\\phi_k/2)$, one\nobtains the following estimation:\n\n\\bea \\label{4} m^{(k+1)}\\le \\sqrt{2\\{1-\\sqrt{1-[2DJm^k/(H-2DJ)]^2}\\}}=\n\\sqrt{2[2DJm^k/(H-2DJ)]^2/[1+\\sqrt{1-[2DJm^k/(H-2DJ)]^2}]}\\le & & \\nonumber \\\\\n2\\sqrt{2} D J m^k / (H-2DJ) = m^k/[1+\\delta/\\sqrt{2}]\\le\nm^1/[1+\\delta/\\sqrt{2}]^{k}, & & \\eea where Eq. (\\ref{3}) is used.\n\nNow we are in the position to estimate the correlation function (\\ref{ff}). In\nterms of the $O(N)$ model it is given by the expression\n\n\\be \\label{5} G(r)=\\langle{\\bf S}({\\bf 0}){\\bf S}({\\bf r})\\rangle. \\ee Let\n$N=[r/2b]-1$, where the square brackets denote the integer part. We decompose\nthe values of the spins in the following way ${\\bf S}({\\bf x})={\\bf S}^N({\\bf\nx})+\\sum_{k>N} {\\bf s}^k({\\bf x})$. The $N$th approximations ${\\bf S}^N({\\bf\nx})$ and ${\\bf S}^N({\\bf 0})$ depend on the orientations of the random fields in\ndifferent regions which have no intersection. Hence, the correlation function\n$\\langle {\\bf S}^N({\\bf 0}){\\bf S}^N({\\bf r})\\rangle$ is the product of the\naverages of the two multipliers ${\\bf S}^N$ and thus equals to zero due to the\nisotropy of the distribution of the random field. All other contributions to\nEq. (\\ref{5}) can be estimated with Eq. (\\ref{4}) and are exponentially small\nas the functions of $N$. This proves that the correlation function $G(r)$ has\nan exponentially small asymptotics at large $r$. Thus, for strong disorder\n(\\ref{3}) both LRO and QLRO are absent.\n\nA challenging question concerns the presence of QLRO in the weakly disordered\nrandom-field systems. Another interesting related problem is the question about\nQLRO in the random-anisotropy $O(N)$ model \\cite{F}. Unfortunately, our\napproach cannot be directly generalized for this problem since the zeroth-order\napproximation ${\\bf S}^0_i$ is not unique in the random-anisotropy model: any\nspin has two preferable orientations.\n\nIn conclusion, we have proved that for strong disorder the random-field $O(N)$\nmodel has no QLRO. The renormalization group calculations \\cite{F} suggest that\nat $N>2$ QLRO is absent for arbitrarily weak disorder. On the other hand, in\nthe random-field $O(2)$ model without dislocations the renormalization group\n\\cite{qlro2} predicts QLRO. Our result shows that in the system with the\ntopological defects QLRO is absent at least as the disorder is strong. This\nprediction is relevant for the vortex glass state of the disordered\nsuperconductors.\n\n\\begin{references} \\bibitem{ph} T. Giamarchi and P. Le Doussal, in {\\it \"Spin\nGlasses and Random Fields\"}, ed. A.P. Young, World Scientific (Singapore)\n1998, p. 321. \\bibitem{Larkin} A.I. Larkin, Zh. Eksp. Teor. Fiz. {\\bf\n58}, 1466 (1970) [Sov. Phys. JETP {\\bf 31}, 784 (1970)]. \\bibitem{qlro1} S.E.\nKorshunov, Phys. Rev. B {\\bf 48}, 3969 (1993). \\bibitem{qlro2} T. Giamarchi\nand P. Le Doussal, Phys. Rev. Lett. {\\bf 72}, 1530 (1994); Phys. Rev. B\n{\\bf 52}, 1242 (1995). \\bibitem{Cubitt} R. Cubitt, E.M. Forgan, G. Yang,\nS.L. Lee, D. McK. Paul, H.A. Mook, M. Yethiraj, P.H. Kes, T.W. Li, A.A.\nMenovsky, Z. Tarnawski, and K. Mortensen, Nature {\\bf 365}, 407 (1993).\n\\bibitem{yaron} U. Yaron, P.L. Gamel, D.A. Huse, R.N. Kleiman, C.S.\nOglesby, E. Bucher, B. Batlogg, D.J. Bishop, K. Mortensen, K. Clausen, C.A.\nBolle, and F. De La Cruz, Phys. Rev. Lett. {\\bf 73}, 2748 (1994).\n\\bibitem{var} J. Kierfeld and T. Nattermann, and T. Hwa, Phys. Rev. B {\\bf\n55}, 626 (1997). \\bibitem{var1} D. Carpentier, P. Le Doussal, and T.\nGiamarchi, Europhys. Lett. {\\bf 35}, 379 (1996). \\bibitem{rig} M. Aizeman\nand J. Wher, Phys. Rev. Lett. {\\bf 62}, 2503 (1989); Comm. Math. Phys.\n{\\bf 150}, 489 (1990). \\bibitem{F} D.E. Feldman, Phys. Rev. B {\\bf 61}, 382\n(2000). \\end{references} \\end{document}\n\n"
}
] |
[
{
"name": "cond-mat0002109.extracted_bib",
"string": "\\bibitem{ph} T. Giamarchi and P. Le Doussal, in {\\it \"Spin\nGlasses and Random Fields\"}, ed. A.P. Young, World Scientific (Singapore)\n1998, p. 321. \n\\bibitem{Larkin} A.I. Larkin, Zh. Eksp. Teor. Fiz. {\\bf\n58}, 1466 (1970) [Sov. Phys. JETP {\\bf 31}, 784 (1970)]. \n\\bibitem{qlro1} S.E.\nKorshunov, Phys. Rev. B {\\bf 48}, 3969 (1993). \n\\bibitem{qlro2} T. Giamarchi\nand P. Le Doussal, Phys. Rev. Lett. {\\bf 72}, 1530 (1994); Phys. Rev. B\n{\\bf 52}, 1242 (1995). \n\\bibitem{Cubitt} R. Cubitt, E.M. Forgan, G. Yang,\nS.L. Lee, D. McK. Paul, H.A. Mook, M. Yethiraj, P.H. Kes, T.W. Li, A.A.\nMenovsky, Z. Tarnawski, and K. Mortensen, Nature {\\bf 365}, 407 (1993).\n\n\\bibitem{yaron} U. Yaron, P.L. Gamel, D.A. Huse, R.N. Kleiman, C.S.\nOglesby, E. Bucher, B. Batlogg, D.J. Bishop, K. Mortensen, K. Clausen, C.A.\nBolle, and F. De La Cruz, Phys. Rev. Lett. {\\bf 73}, 2748 (1994).\n\n\\bibitem{var} J. Kierfeld and T. Nattermann, and T. Hwa, Phys. Rev. B {\\bf\n55}, 626 (1997). \n\\bibitem{var1} D. Carpentier, P. Le Doussal, and T.\nGiamarchi, Europhys. Lett. {\\bf 35}, 379 (1996). \n\\bibitem{rig} M. Aizeman\nand J. Wher, Phys. Rev. Lett. {\\bf 62}, 2503 (1989); Comm. Math. Phys.\n{\\bf 150}, 489 (1990). \n\\bibitem{F} D.E. Feldman, Phys. Rev. B {\\bf 61}, 382\n(2000). "
}
] |
cond-mat0002110
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Spin quantization axis dependent magnetic properties and x-ray magnetic circular dichroism of FePt and CoPt
|
[
{
"author": "I.~Galanakis"
},
{
"author": "M.~Alouani"
},
{
"author": "and H.~Dreyss\\'e"
}
] |
We have performed a theoretical study of the magnetic circular dichroism in the x-ray absorption spectra (XMCD) of the equiatomic CoPt and FePt ordered alloys as a function of the spin quantization axis. We found that the magnetization axis is along the [001] direction and the magneto-crystalline anisotropy energy (MCA) for the FePt compound is twice as large as that of the CoPt compound in agreement with experiment. The band structure and the total density of states confirm that all electronic states contribute to the MCA, and not just the states at the vicinity of the Fermi level. The orbital magnetic moments decrease with respect to the angle between the [001] axis and the spin quantization axis, and are much larger for the CoPt compound. We show that the orbital moment anisotropy is reflected in the XMCD signal.
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[
{
"name": "xpt.tex",
"string": "%===================================================================\n\\documentstyle[prb,aps]{revtex}\n\n\\input{epsf}\n\n\\begin{document}\n\\draft\n\\title{Spin quantization axis dependent magnetic properties \nand x-ray magnetic circular dichroism of \nFePt and CoPt}\n\\author{I.~Galanakis, M.~Alouani, and H.~Dreyss\\'e}\n\\address{IPCMS-GEMME, 23, rue du Loess, F-67037\nStrasbourg Cedex, France}\n\\maketitle\n\n\\begin{abstract}\nWe have performed a theoretical study of the magnetic\ncircular dichroism in the x-ray absorption spectra (XMCD) of the\nequiatomic CoPt and FePt ordered alloys as a function of\nthe spin quantization axis. We found that the magnetization axis is\nalong the [001] direction \nand the magneto-crystalline anisotropy energy (MCA) for the FePt compound\nis twice as large as that of the CoPt compound in agreement with \nexperiment. The band structure and the total\ndensity of states confirm that all electronic states contribute to\nthe MCA, and not just the states at the vicinity of the Fermi level.\nThe orbital magnetic moments decrease with respect to the angle\nbetween the [001] axis and the spin quantization axis, and are\nmuch larger \nfor the CoPt compound. We show that the orbital moment anisotropy \nis reflected in the XMCD signal.\n\\end{abstract}\n\n\\pacs{71.55.Ak, 78.70.Dm, 75.50.Cc}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Introduction}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe disordered equiatomic binary alloys of the XY (X= Fe, Co -- Y= Pd, Pt) type \ncrystallize in the fcc structure \nand the magnetization is along the [111]\naxis.\\cite{razee} \nAt low temperatures these alloys \ntend to order in the $L1_0$ layered-ordered\nstructure and in this case the spontaneous magnetization tends to align\n perpendicular to the layer stacking explaining the behavior\nof CoPt films during the magnetic annealing.\\cite{razee2} \nThe strong perpendicular magnetic anisotropy (PMA) is due to the\nhighly anisotropic $L1_0$ structure and it makes them very\nattractive for magnetic recording devices.\\cite{coffey}\n The latter structure can be also obtained by\nmolecular beam epitaxy (MBE) of alternating layers of pure X and Y atoms\ndue to the substrate induced constraints. The first observation of\nthe $L1_{0}$ long-range order for a [001] CoPt film grown by MBE\nwas made by Harp {\\em et al.}\\cite{harp} in 1993 and for a [001]\nFePt film by Cebollada {\\em et al.}\\cite{cebollada} Lately other techniques \nhave been also employed to develop films presenting \nPMA. \n CoPt films were grown by sputtering by Visokay {\\it et\nal.}\\cite{visokay} and by evaporation by Lin and Gorman,\\cite{lin} and\nFePt films were grown by various sputtering\ntechniques.\\cite{watanabe,lairson,watanabe2,mitani,sato}\n\n\nThe magneto-crystalline anisotropy energy (MCA) can be probed\nby many techniques such as torque or ferromagnetic resonance\nmeasurements. Both these methods describe\nthe MCA in terms of phenomenological\n anisotropy constants. It has been demonstrated by Weller \n{\\em et al.}\\cite{weller} that x-ray\n magnetic circular dichroism (XMCD) is also a suitable technique \nfor probing the MCA,\n\\textit{via} the determination of the anisotropy of the orbital\nmagnetic moment on a specific shell and site. The x-ray\nabsorption spectroscopy (XAS) using polarized radiation\n probes element specific magnetic properties of alloys \nby applying the XMCD \n sum rules to the experimental\nspectra.\\cite{thole,carra,VanL98} However for \nitinerant systems, in particular to low symmetry systems,\nthe use of the sum rules is debated because they are derived from an \natomic \ntheory.\\cite{carra,chen,wu}\nLately angle-dependent XMCD experiments have been used to provide \na deeper understanding for the relation between MCA and the orbital \nmagnetic moments.\\cite{grange2}\n\nThe x-ray absorption for CoPt multi-layers has been already\nstudied experimentally by Nakajima {\\em et al.},\\cite{naka} Koide\n{\\it et al.},\\cite{koide} R\\\"uegg {\\it et al.},\\cite{ruegg} and\nSch\\\"ultz {\\it et al.}\\cite{schultz} \nNakajima {\\em et\nal.}\\cite{naka} revealed a strong enhancement of the cobalt orbital\nmoment when PMA was present, Koide {\\it et al.}\\cite{koide}\nshowed that with decreasing cobalt\nthickness the easy axis rotates from in-plane to out-of-plane, and\nR\\\"uegg {\\it et al.}\\cite{ruegg} that platinum polarization increases also with decreasing cobalt\nthickness. Hlil {\\it et al.}\\cite{hlil} showed by x-ray absorption\nspectroscopy that modifications of platinum edges in different compounds\nare correlated to the change in the number of holes. \n\nSeveral {\\em ab-initio}\ncalculations have already been performed to investigate the\nXMCD.\\cite{Ebert,Meb1,Brouder} The $L_2$- and $L_3$-edges involving\nelectronic excitations of 2$p$-core electrons towards $d$-valence\nstates have primarily attracted much attention due to dependence\nof the dichroic spectra on the exchange-splitting and the\nspin-orbit coupling of both initial core and final valence states.\nFor 5$d$ elements dissolved in 3$d$ transition metals, the\nspin-orbit coupling of the initial 2$p$-core states is large and\nthe resulting magnetic moment is small, while the opposite is true\nfor the 3$d$ elements. This can lead to a pronounced dichroic\nspectra as seen by Sch\\\"utz in the case of 5$d$ elements dissolved\nin iron.\\cite{schultz} \n\nIn this work we study the correlation between the quantization axis dependent\nXMCD and the magnetic properties of both ordered alloys \nFePt and CoPt. Our method is based on an \nall-electron relativistic and spin-polarized full-potential\nmuffin-tin orbital method (LMTO)\\cite{WILLS,LMTO} in conjunction\nwith both the von Barth and Hedin parameterization to the local\ndensity approximation (LSDA)\\cite{barth} and the generalized\ngradient approximation (GGA)\\cite{perdew} to the exchange\ncorrelation potential. \nThe implementation of the calculation of the XMCD spectra \nhas been presented in a previous work.\\cite{Meb1}\nIn section 2 we present the details of the \ncalculations, while in sections 3\nand 4 we discuss our MCA and the magnetic spin and orbital\nmoments, respectively. In section 5 we present \nour calculated XMCD as a function of the spin quantization axis, \nand in section 6 we discuss the interpretation of the MCA using\nthe band structure and the total density of states anisotropy.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Computational Details}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n To compute the electronic properties of CoPt and FePt we used\nthe experimental lattice constants ($a$=3.806\\AA \\ and\n$c$/$a$=0.968 for CoPt\\cite{grange2}; $a$= 3.861\\AA \\ and\n$c$/$a$=0.981 for FePt\\cite{villars}) and a unit cell containing\none atom of cobalt(iron) and one of platinum. The $L1_0$ structure can be\nseen as a system of alternating cobalt(iron) and platinum layers along the\n[001] direction. The MCA and the XMCD are computed with respect to \nthe angle $\\gamma$ between the [001] axis and the spin\nquantization axis on the (010) plane. So $\\gamma$=0$^o$\ncorresponds to the [001] axis and $\\gamma$=90$^o$ to the [100] axis.\n\nMCA can be computed directly using {\\em ab-initio} methods; it is\ndefined as the difference between the total energy for two\ndifferent spin quantization axis. The spin-orbit coupling contribution\nto MCA is implicitly included in our {\\em ab-initio} calculations,\nand we do not take into account the many-body\ninteractions of the spin-magnetic moments\\cite{janssen} since their\ncontribution to the MCA is negligible.\\cite{bruno2} The number of {\\bf\nk} points for performing the Brillouin zone (BZ) integration\ndepends strongly on the interplay between the contributions to the MCA\nfrom the Fermi surface and the remaining band structure contribution to the total\nenergy.\\cite{Sol} When the former contribution to the MCA is\nimportant, a large number of {\\bf k}-points is needed to describe\naccurately the Fermi surface. For the two studied systems we found\nthat 6750 {\\bf k}-points in the BZ are enough to converge the MCA within 0.1 meV.\n\nTo perform the integrals over the BZ we use a Gaussian broadening\nmethod which convolutes each discrete eigenvalue with a Gaussian\nfunction of width 0.1 eV. This method is known to lead to a fast\nand stable convergence of the spin and charge densities compared\nto the standard tetrahedron method. To develop the potential\ninside the MT spheres we calculated a\n basis set of lattice harmonics including functions up to\n $\\ell=8$, while for the FFT we\n used a real space grid of 16$\\times$16$\\times$20.\n We used a double set of basis functions, one set to describe the valence\n states and one for the unoccupied states.\n For the valence electrons we used a basis set containing\n3$\\times s$, 3$\\times p$ and 2$\\times d$ wave functions, and for\nthe unoccupied states 2$\\times s$, 2$\\times p$ and 2$\\times d$\nwave functions.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Magneto-crystalline Anisotropy}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nCoPt and FePt films are known to present a strong uniaxial MCA,\nbecause of the high anisotropic $L1_0$ inter-metallic phase. \nExperimentally the magnetization axis is found to be along the [001]\ndirection.\\cite{eurin,farrow} In a first step we performed\ncalculations with 250 {\\bf k}-points in the BZ. This\nnumber of {\\bf k}-points is large enough to produce accurate total\nenergy when Gaussian smearing is used for the\nintegration in the Brillouin Zone but not enough accurate to\ncompute the MCA. In figure\\ \\ref{fig12} we present\ncalculations with respect to the angle $\\gamma$ between the [001]\naxis and the spin quantization axis on the (010) plane for the\nCoPt compound within the LSDA. We observe that total energy value increases \nwith the angle $\\gamma$. So the ground state corresponds to $\\gamma$=0$^o$,\n{\\em i.e.} the [001] axis. The same behavior occurs for the FePt\ncompound. Because 250\n{\\bf k}-points are not enough to produce an accurate value for the\nMCA, we present in figure \\ref{fig2} the convergence of the MCA with the number of {\\bf\nk}-points for both\ncompounds within LSDA. The MCA is the difference in the total\nenergy between the in-plane axis [100] and [110] and the easy axis\n[001]. We have found that the total energy difference between\nthe [100] and the [110] directions is negligible compared to the difference \nbetween the in-plane axis and the [001] axis.\n Our values are converged up to 6750 {\\bf k}-points and are\ngiven per unit cell (one atom of X and one of platinum).\n The MCA converges to 2.2 eV for CoPt and 3.9 eV for FePt.\nThese behavior confirms the assumption that the system is\nisotropic inside the plane. The GGA MCA calculations\nconverged to 1.9 meV and 4.1 meV for CoPt and FePt,\nrespectively. The GGA results seem to be in good agreement \nwith the LSDA results.\n\n\nDaalderop {\\em et al.}\\cite{Daal} performed calculations for MCA\nin CoPt and FePt by means of an LMTO method in the atomic sphere\napproximation (ASA) within LSDA\\cite{LMTO} using the \n force theorem.\\cite{macintosh} The easy magnetization axis found \nby this latter calculation is the [001] for both systems in agreement\nwith our LSDA and GGA results. Their MCA value is 2 meV \nfor CoPt and is 3.5 meV for FePt. \nSolovyev {\\em et al.}\\cite{Sol} used the same structure within a \nreal-space Green's\nfunction technique framework to find also that the magnetization\nis along the [001] axis. Including the spin-orbit\ninteraction for all the atoms, they found a CoPt MCA value of 2.3\nmeV and a FePt value of 3.4 meV. Our computed value agrees also\nwith the value of 1.5 meV for CoPt and 2.8 meV for FePt, obtained\nby Sakuma\\cite{Sak} using LMTO method in the atomic sphere\napproximation (ASA) in conjunction with the force\ntheorem.\\cite{macintosh} The drawback of the LMTO-ASA method is that it \naccounts only for the spherical part of the potential and \nignores the interstitial region.\nFurthermore, the force theorem does not account directly for the\nexchange-correlation contribution to the MCA. Finally, Oppeneer used the\naugmented spherical waves method (ASW) in the atomic \nsphere approximation and found a MCA value of\n2.8 meV for FePt and 1.0 meV for CoPt,\\cite{oppen} which are the\nsmallest among all the {\\em ab-initio} calculations.\n\n Grange {\\em et al.},\\cite{grange2} using a \ntorque measurement for a MBE deposited CoPt film on a MgO(001)\nsubstrate, obtained a MAE of 1.0 meV. An early measurement of \na monocristal of CoPt, by Eurin and Pauleve, produced a value \nof 1.3 meV.\\cite{eurin} The large \nvalue of Eurin and Pauleve is due to the fact that their\nsample was completely ordered. For FePt the first experiment of\nIvanov {\\em et al.}\\cite{ivanov} produced an anisotropy value\nof 1.2 meV and showed that for a thin film the shape\nanisotropy would be one order of magnitude smaller compared to\nthe MCA. Farrow {\\em et al.}\\cite{farrow} and Thiele {\\em et\nal.}\\cite{thiele} found for MBE deposited FePt films on a MgO(001)\nsubstrate an anisotropy value of 1.8 meV. These films were highly\nordered (more than 95\\% of the atoms were in the correct site).\nAll experiments have been carried out at room temperature, which\nexplain at some extend \nthe difference between the calculated and experimental MCA values\n(the MCA decreases with temperature\\cite{eurin}).\nIt is worth mentioning that the experimental MCA for FePt is \nmuch larger than that of CoPt in agreement with our calculations.\nFor thick films volume shape\nanisotropy (VSA) contributes also to the MAE and it favors always an\nin-plane magnetization axis. We can estimate the VSA as\n$-2\\pi M_V^2$ in c.g.s. units, where $M_V$ is the mean magnetization\ndensity,\\cite{bruno2} and obtain a value of -0.1 meV for FePt\nand -0.06 meV for CoPt. These values are one order of magnitude\nsmaller than the MCA values in agreement with the speculation\nof Ivanov {\\em et\nal.}\\cite{ivanov} \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Magnetic moments}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Density of States}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn figure\\ \\ref{fig3} we present the cobalt-projected partial\ndensity of states for three spin quantization axis corresponding\nto angles $\\gamma$= 0$^o$, 45$^o$ and 90$^o$ calculated with 6750\n{\\bf k}-points. The 3-$d$ states dominate the electronic structure\nof cobalt. The spin-up band is practically totally occupied while the\nspin down band is almost half-occupied. The general form of the cobalt DOS,\nas well as the iron DOS, does not seem to change appreciably with the \nangle $\\gamma$. \nThe platinum projected density of states show similar behavior for \nboth FePt and CoPt compounds. \n Bulk platinum is paramagnetic and the small changes in the\nDOS come from the polarization of the 5$d$ electrons via\nhybridization with the 3$d$ electrons of cobalt (iron). It is worth\nmentioning that the DOS is calculated inside each muffin-tin and the\ninterstitial region is not taken into account. To minimize the\ncontribution of the interstitial region we use almost touching \n muffin-tin spheres. In addition, we have find that remaining \nregion has a negligible spin polarization.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Spin Magnetic Moments}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe spin magnetic moments are isotropic with respect to the\nspin quantization axis as expected. They were calculated by attributing\nall the charge inside each muffin-tin sphere to the atom located in that\nsphere. As outlined above we have found that the interstitial contribution \nto the spin magnetic moment is one order of\nmagnitude smaller than that of the platinum site. In CoPt\nwe found a cobalt spin magnetic moment of 1.79 $\\mu_B$ and a platinum\nmoment of 0.36 $\\mu_B$ within the LSDA. The GGA cobalt spin \nmagnetic moment of 1.83 $\\mu_B$ is slightly larger than the value \nwithin LSDA. This is due to a more\natomic like description of the atoms in a solid within the GGA compared to \nthe LSDA.\nAlthough the GGA underestimates the hybridization between cobalt and platinum\n$d$ valence electrons compared to the LSDA, the larger cobalt spin moment\nleads to a slightly larger GGA platinum spin moment of 0.37 $\\mu_B$. Our\ncalculated values are in good agreement with the experimental values\nof Grange {\\em et al.} (1.75$\\mu_B$ for cobalt and\n0.35$\\mu_B$ for platinum).\\cite{grange2}\nPrevious experiments by van Laar on a\npowder sample gave a value of 1.7 $\\mu_B$ for the cobalt atom and 0.25\nfor the platinum atom.\\cite{laar} The spin magnetic moments have been\npreviously calculated by Solovyev {\\em et al.}\\cite{Sol} (1.72 for cobalt and\n0.37 for platinum), by Daalderop {\\em et al}\\cite{Daal} (1.86 for cobalt), by\nSakuma\\cite{Sak} (1.91 for cobalt and 0.38 for platinum), and finally by\nKootte {\\em et al.}\\cite{kootte} by means of a localized spherical wave\nmethod (1.69 for cobalt and 0.37 for platinum). All previous\ncalculations are in good agreement with our full-potential results.\n\nAs expected the iron spin moments are much larger than the cobalt ones.\nThe LSDA produced a\nvalue of 2.87 $\\mu_B$ while the GGA produced a slightly \nlarger value, 2.96 $\\mu_B$. The hybridization between the iron 3$d$\nstates and the platinum 5$d$ states is less intense than in the case of\nCoPt resulting in a smaller platinum moment for FePt. The LSDA platinum spin moment\nis 0.33 $\\mu_B$ (compared to 0.36 $\\mu_B$ in CoPt) and the GGA platinum\nspin moment is 0.34 $\\mu_B$ (compared to 0.37 $\\mu_B$ in CoPt). The spin\nmagnetic moments have been previously calculated by Solovyev et\nal\\cite{Sol} (2.77 for iron and 0.35 for platinum), by Daalderop et\nal\\cite{Daal} (2.91 for iron), by Sakuma\\cite{Sak} (2.93 for iron and\n0.33 for platinum), and finally by Osterloch {\\em et al.}\\cite{osterloch} (2.92\nfor iron and 0.38 for platinum). Here again all\nprevious calculations are in good agreement with our\nfull-potential results. All the methods produced a smaller induced spin moment\nfor platinum atom in FePt than for CoPt, verifying that the hybridization\neffect in CoPt is much stronger than in FePt.\n\n\\subsection{Orbital Magnetic Moments}\n\nContrary to spin moments, the orbital moments are anisotropic. \nFigure \\ref{fig4} presents the behavior of the orbital moments as\n a function of the angle $\\gamma$ between the [001] direction and \nthe spin quantization axis within LSDA for both CoPt and FePt compounds \n(the lines are guide to the eye). The orbital\nmoments decrease with respect to the angle $\\gamma$ but the values for the\n[100] axis does not follow this general trend. All four lines seem to have the\nsame behavior, but it is interesting to notice that for the CoPt and\nmagnetization along the [100] axis cobalt orbital moment is smaller\nthan the platinum one (for the values see Tables \\ref{table1} and\n\\ref{table2}). The cobalt moments decrease faster than platinum moments in CoPt.\nFor the iron and platinum atoms in FePt the two lines are practically\nparallel. The platinum orbital moments in FePt are smaller than for the CoPt\nwith a factor that varies from 73\\% for $\\gamma$=0$^o$ to 67\\%\nfor $\\gamma$=90$^o$. The ratio of iron and platinum orbital moments\nin FePt varies from 1.56 for $\\gamma$=0$^o$ down to 1.35 for for\n$\\gamma$=90$^o$, which is considerably smaller than the ratio\nof cobalt and platinum orbital moments in CoPt. We notice here\nthat in our calculations we can estimate only the projection of\nthe total orbital moment on the spin quantization axis and we have\nno information concerning the real value of the total magnetic\nmoment or its direction in space. It seems that for the magnetization\nalong the [100] direction, the direction of the orbital\nmagnetic moment undergoes a discontinuous jump, resulting in a\nlarge projection on the spin quantization axis with respect \nto $\\gamma$ at the vicinity of 90$^o$. This behavior is \nalso reproduced by the GGA.\n\nIn Table \\ref{table1} we present the values \nof the orbital moments of iron and cobalt within both the LSDA and\nGGA as a function of the angle $\\gamma$. \nThe GGA values seem to be smaller than the LSDA ones but\nfollow exactly the same trends. The orbital moment anisotropy is\nmore important in the case of cobalt. The LSDA cobalt orbital moment changes\nby 0.048 $\\mu_B$ and the GGA moment by 0.027 $\\mu_B$ as we pass from\nthe easy axis [001] to the hard axis [100]. The LSDA iron moment\nchanges by 0.002 $\\mu_B$ and the GGA moment is the same for the two\nhigh symmetry directions. In the case of cobalt the difference\nbetween values calculated within the two functionals, LSDA and\nGGA, becomes smaller when the angle increases and for \n75$^o$ it changes sign.\nIn the\ncase of iron this difference decreases only slightly with the angle but since the\ndifference is considerably smaller than in CoPt (less than\n0.004), we conclude that the orbital moment in the LSDA and the GGA are\nroughly the same.\n\nIn Table \\ref{table2} we present the values for the platinum orbital\nmoment within both functionals. We see that the GGA produces larger\nmoments than the LSDA for platinum in FePt contrary to CoPt. The platinum moments are\nin general smaller than the moments of the 3$d$ ferromagnets, \nand the difference between\nthe values calculated within LSDA and GGA are small.\nThe absolute values for platinum are comparable to \ncobalt(iron) orbital moments even though the spin moments\non platinum are one order of magnitude smaller than for cobalt(iron). The\nlarge orbital moments for platinum are due to a much larger spin-orbit coupling\nfor the $d$ electrons of the platinum compared to the 3$d$ ferromagnets.\n\nThe orbital moments of FePt and CoPt have been previously calculated by Daalderop and\ncollaborators\\cite{Daal} and by Solovyev and\ncollaborators\\cite{Sol} for the [001] direction using the LSDA. \nThe orbital moment of the cobalt site was found to be 0.12 $\\mu_B$ \nby Daalderop and 0.09 $\\mu_B$ by Solovyev. The\nvalue of Daalderop is closer to our LSDA value of 0.11\n$\\mu_B$. For iron site Daalderop found a value of 0.08 $\\mu_B$ and Solovyev 0.07\n$\\mu_B$ in good agreement with our LSDA value. The platinum orbital moment \nhas been calculated by Solovyev. \n He found a value of 0.06 $\\mu_B$ for platinum in\nCoPt and 0.044 $\\mu_B$ for platinum in FePt, close to our values, 0.06 $\\mu_B$ and\n0.05 $\\mu_B$, respectively.\n\nOn the other hand, experimental data are available for CoPt by Grange;\\cite{grange2}\nobtained by applying the sum rules to the experimental XMCD spectra. The sum rules\ngive the moments per hole in the $d$-band. To compare experiment with\ntheory we calculated the number of $d$-holes by integrating the\n$d$ projected density of states inside each muffin-tin sphere. We\nfound 2.63 $d$ and 2.48 $d$ holes for cobalt and \nplatinum, respectively. The cobalt orbital moment varies from 0.26 $\\mu_B$ \nfor $\\gamma$=10$^o$ down to 0.11 $\\mu_B$ for $\\gamma$=60$^o$. The measured values are also \navailable for two other angles: \n 0.24 $\\mu_B$ for\n$\\gamma$=30$^o$ and 0.17 $\\mu_B$ for $\\gamma$=45$^o$. Our theory\nreproduces qualitatively the experimental trends but underestimates the absolute values \nby more than\n50\\% (see Tables \\ref{table1} and \\ref{table2} for all the values). \nThe calculated values show a less sharp decrease\nwith the angle than the experimental ones. For platinum the experimental data are\navailable only for two angles 10$^o$ and 60$^o$. For \n$\\gamma$=10$^o$ the orbital moment is 0.09 $\\mu_B$ and for\n$\\gamma$=60$^o$ it is 0.06 $\\mu_B$. For the platinum site \nthe calculated values are in much better agreement\nthan for the cobalt site but we must keep in mind that the sum rules have\ntheir origin in an atomic theory and their use for 5$d$ itinerant\nelectrons like in platinum is still debated.\nThe discrepancy between\nthe theory and the experiment comes mainly from the \napproximation to the exchange and correlation. Both\nthe LSDA and the GGA approximations to the density functional theory are\nknown to underestimate the orbital moment values, because the orbital\nmoment is a property directly associated with the current in the\nsolid and a static image is not sufficient. But until now a DFT\nformalism like the current and spin density functional\ntheory (CS-DFT),\\cite{vignale} which can treat at the same footing\nthe Kohn-Sham and the Maxwell equations is too heavy to implement in\na full-potential {\\em ab-initio method}. The other \n problem is that there is no form of the exchange\nenergy of a homogeneous electronic gas in a magnetic\nfield known and this is the main quantity entering the CS-DFT\nformalism. Brooks has also developed an {\\it ad hoc} correction to the \nHamiltonian to account for the\norbital polarization but this correction originates from\nan atomic theory and its application to itinerant systems is \nnot satisfactory.\\cite{brooks}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{XMCD}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nXMCD spectroscopy became popular after the development of the sum\nrules that enable the extraction of reliable information on the\nmicro-magnetism directly from the experimental\nspectra.\\cite{thole,carra} The great advantage of XMCD is that we can\nprobe each atom and orbital in the system so to obtain information \non the local magnetic properties. Lately angle-dependent XMCD\nexperiments allowed the determination \nof magnetic properties for different spin quantization axis. \nAll experimental spectra for\nCoPt have been obtained by Grange {\\it et al},\\cite{grange2} and all\ncalculated spectra presented in this section were obtained using the LSDA.\nThe GGA produced the same results and are not presented. \n\nFigure \\ref{fig5} presents the XMCD spectra for cobalt and iron\natoms for the [100] and [001] magnetization axis. \nWe convoluted our theoretical spectra using a Lorentzian\nwidth of 0.9 eV and a Gaussian width of 0.4 eV as proposed by\nEbert,\\cite{Ebert} in the case of iron to account for the core hole\neffect and the experimental resolution, respectively. The energy\ndifference between the $L_3$ and $L_2$ peaks is given by the\nspin-orbit splitting of the $p_{\\frac{1}{2}}$ and\n$p_{\\frac{3}{2}}$ core states. It is larger in the case of cobalt,\n14.8 eV, than for iron, 12.5 eV. The intensities of the peaks are comparable\nfor both atoms, but the cobalt peak-intensities are larger for the [001]\naxis contrary to the iron spectrum. The most important feature of\nthese spectra is the integrated $L_3$/$L_2$ branching ratio as it\nenters the sum rules. It is larger in the case of cobalt site. This is\nexpected since the sum rules predict that a larger $L_3$/$L_2$ ratio \nis equivalent to a larger orbital moment. For both atoms the integrated $L_3$/$L_2$\nbranching ratio is larger for the [001] axis. But the ratio\nanisotropy is larger for the cobalt site (1.32 for {\\bf M}$\\parallel$[001] and\n1.17 for {\\bf M}$\\parallel$[100]) than for the iron site (1.15 for {\\bf\nM}$\\parallel$[001] and 1.14 for {\\bf M}$\\parallel$[100])\nreflecting the larger orbital moment anisotropy of cobalt compared to\niron (see Table \\ref{table1}). Especially for iron both orbital moment\nand integrated $L_3$/$L_2$ branching ratio are practically the\nsame for both magnetization axis. As these changes in the XMCD\nsignal are related to the change in the orbital magnetic moment\nbetween the two magnetization directions. The XMCD anisotropy\nshould be roughly proportional to the underlying MCA but no relation\nexist that connects these two anisotropies. However\nthey are connected indirectly through the orbital moment \nanisotropy.\\cite{bruno} \n\nIn figure \\ref{fig9} we have plotted the platinum XMCD spectra for two\nangles $\\gamma$=10$^o$ and 60$^o$ for both compounds. The life\ntime of the core-hole in platinum is smaller than for cobalt so the\nbroadening used to account for its life time should be larger. We\nused both a Lorentzian (1 eV) and a Gaussian (1 eV) to represent\nthis life time\n and a Gaussian of 1 eV width for the experimental resolution. As in the\ncase of cobalt and iron the platinum XMCD spectra change with the angle and\ndepend on the surrounding neighbors. The peak intensities are larger in\nCoPt. Also the difference in the intensity due to the anisotropy \nhas different sign in the two\ncompounds. The intensities of FePt spectra for $\\gamma$=10$^o$ are much larger\nthan for $\\gamma$=60$^o$ contrary to the CoPt behavior. As is the\ncase for cobalt in CoPt, the platinum site integrated $L_3$/$L_2$ branching ratio\nshows larger anisotropy than for platinum in FePt. In CoPt it is 1.49\nfor $\\gamma$=10$^o$ and 1.19 for $\\gamma$=60$^o$, while for FePt\nit is 1.20 for $\\gamma$=10$^o$ and 1.14 for $\\gamma$=60$^o$. Here again\nthe XMCD follows the anisotropy of the\norbital magnetic moment in these compounds. The energy difference\nbetween the two peaks is 1727 eV for both compounds. The 2$p$\nelectrons of platinum are deep in energy and are little influenced\nby the local environment, so that their spin-orbit splitting does not\ndepend on the neighboring atoms of platinum.\n\n We expect a better\nagreement between the theoretical and the experimental XMCD spectra \nfor the platinum site than for the cobalt site,\nbecause the core hole is deeper\nand would effect less the final states of the photo-excited\nelectron. In figure \\ref{fig6}, we have plotted the\nabsorption and the XMCD spectra of cobalt for $\\gamma$=0$^o$. We have\nscaled our spectra in a way that the experimental and theoretical\n$L_3$ peaks in the absorption spectra have the same intensity. The\nenergy difference between the $L_3$ and $L_2$ peaks is in good\nagreement with experiment. But the intensity of the $L_2$ peak is\nlarger than the corresponding experimental peak. The high\nintensity of the calculated $L_2$ edge makes the theoretical XMCD\nintegrated $L_3$/$L_2$ branching ratio of 1.32 much smaller than\nthe experimental ratio of 1.72. This is because the LSDA\nfails to represent the physics of the core hole photo-excited\nelectron recombination. In the case of 3$d$ ferromagnets the\ncore hole is shallow and influences the final states seen by \nthe photo-excited electron. A formalism that can treat this\nelectron-hole interaction has been proposed by Schwitalla and Ebert,\\cite{schwit}\nbut it failed to improve the $L_3$/$L_2$ branching ratio\nof XAS of the late transition metals.\nBenedict and Shirley\\cite{benedict} have\nalso developed a scheme to treat this phenomena but its\napplication is limited only to crystalline insulators.\n\nIn figure \\ref{fig7} we have plotted experimental and\ntheoretical total absorptions for the platinum atom in CoPt. For both\n$L_2$ and $L_3$ edges the theory gives a sharp peak which does not\nexist in experiment. As expected the $L_3$ peak is much more intense\nthan the $L_2$. In contrast to what is obtained for cobalt, the\nresults for the platinum XMCD (see figure \\ref{fig8}) show better\nagreement with experiment, due to the fact that the core hole\neffect is less intense (core hole much deeper\n compared to cobalt).\n The\nexperimental and theoretical $L_2$ and $L_3$ edges are separated\nby a spin-orbit splitting of the $2p$ core states of 1709 and\n1727eV respectively. The width of both $L_2$ and $L_3$ edges is\ncomparable to experiment, but the calculated $L_2$ edge is much\nlarger. This produces a calculated integrated branching ratio of\n1.49 which is much smaller than the experimental ratio of 2.66.\nHere again the theory is underestimating the branching ratio.\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Band Structure and Density of States Anisotropy}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nIn figure\\ \\ref{fig10} we present the band structure along the\n[001] and [100] axis in the reciprocal space for different angles\n$\\gamma$ for the CoPt compound within the LSDA. We know that it is\nessentially the area around the Fermi level that changes \nwith respect to the spin quantization axis.\nFor this reason we have enlarged a region of $\\pm$1 eV\n around the Fermi level. In the first panel we plot the\nrelativistic band structure for $\\gamma$=0$^o$. In the second and\nthird panel we have plotted the relativistic band structure for\n$\\gamma$=45$^o$ and $\\gamma$=90$^o$. We remark\nthat as the angle increases there are bands that approach the Fermi\nlevel and cross it. However this information concerns just two\nhigh symmetry directions. \nFor this reason we limited ourselves to the changes in the total DOS. In figure\n\\ref{fig11} we notice that just below the Fermi level the \n DOS for the hard axis is lower than for\nthe easy axis which seems to favor the \nhard axis. This means that the anisotropy does not originate from the \nchanges ate the vicinity of the Fermi level but from that of the whole\nDOS. \nIt is\ndifficult to investigate this phenomena by inspection of the changes \nat the vicinity of the Fermi surface and to \nexplain the sign of the MCA . \nOur results confirm the work of Daalderop and\ncollaborators\\cite{daal2} that argued that not only states in the\nvicinity of the Fermi surface contribute to the MAE, as originally\nthought,\\cite{wang} but states far away make an equally important\ncontribution. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Conclusion}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe have performed a theoretical {\\em ab-initio} study of the\nmagnetic properties of the ordered CoPt and FePt fct alloys\nsystems. The calculated easy axis is the [001] for both compounds\nin agreement with other calculations and with experiments on films\nwhich found a strong perpendicular magnetization axis. The density of\nstates is found to change very little with the direction of the spin \nquantization axis,\n and hence the magnetic moments are isotropic with respect to the\nmagnetization axis. Contrary to the spin moments, the orbital magnetic\nmoments decrease with the angle $\\gamma$ up to 75$^o$\n($\\gamma$ is the angle between the spin quantization\naxis and the [001] axis). \n\nThe calculated x-ray magnetic circular dichroism (XMCD)\nfor all the atoms reflect the behavior of the orbital moments.\nEspecially platinum resolved spectra present large differences between the\ntwo compounds. Cobalt XMCD spectra are in agreement with\nexperiment but as usual the $L_3$/$L_2$ ratio is underestimated by\nthe theory. 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T. Thole,\n Phys. Rev. B {\\bf 43}, 13~401 (1991).\n\n \\bibitem{WILLS}\n J. M. Wills and B. R. Cooper,\n Phys. Rev. B {\\bf 36}, 3809 (1987);\n M. Alouani and J. M. Wills, Phys. Rev. B {\\bf 54}, 2480 (1996).\n\n\\bibitem{LMTO}\n O. K. Andersen,\n Phys. Rev. B {\\bf 12}, 3060 (1975).\n\n\\bibitem{barth} U. von Barth and L. Hedin, J. Phys. C: Solid State Phys. {\\bf 5}, 1629 (1972).\n\n\n\\bibitem{perdew}\n J. P. Perdew and Y. Wang, Phys. Rev. B {\\bf 33}, 8800 (1986);\n {\\em ibid.} {\\bf 45}, 13244 (1992).\n\n\\bibitem{hoh}\n P. Hohenberg and W. Kohn, Phys. Rev. B {\\bf 136}, 864 (1964);\n W. Kohn and L. J. Sham, Phys. Rev. A {\\bf 140}, 1133 (1965).\n\n\\bibitem{villars}\nP. Villars and L. D. Calvet: {\\em Pearson's Handbook of\nCrystallographic Data for Intermetallic Phases}, (Metals Park,\nOH: American Society for Metals,\n 1985).\n\n\\bibitem{janssen}\nH. J. F. Jansen, Phys. Rev. B {\\bf 38}, 8022 (1988).\n\n\\bibitem{bruno2}\nP. Bruno, {\\em Physical Origins and Theoretical Models of Magnetic\nAnisotropy}, Ferienkurse des Forschungszentrum J\\\"ulich, J\\\"ulich\n(1993).\n\n\\bibitem{Sol}\nI. V. Solovyev, P. H. Dederichs, and I. Mertig,\n Phys. Rev. B {\\bf 52}, 13 419 (1995).\n\\bibitem{eurin} P. Eurin and J. Pauleve, IEEE Trans. Mag. {\\bf 5}, 216 (1969).\n\n \\bibitem{farrow}\nR. F. C. Farrow, D. Weller, R. F. Marks, M. F. Toney, A.\nCebollada, and G. R. Harp, J. Appl. Phys. {\\bf 79}, 5967 (1996).\n\n\\bibitem{Daal}\n G. H. O. Daalderop, P. J. Kelly, and M. F. H. Shuurmans, Phys. Rev. B\n{\\bf 44}, 12 054 (1991).\n\n\\bibitem{macintosh}\n A. R. Macintosh and O. K. Andersen, {\\it Electrons\nat the Fermi Surface} (Cambridge/Cambridge Univ. Press, 1980).\n\n\n\\bibitem{Sak}\n A. Sakuma, J. Phys. Soc. Jap. {\\bf 63}, 3053 (1994).\n\n\n\\bibitem{oppen}\nP. Oppeneer, J. Magn. Magn. Mat. {\\bf 188}, 275 (1998).\n\n\\bibitem{ivanov}\nO. A. Ivanov, L. V. Solina, V. A. Demshina, and L. M. Magat, Fiz.\nMetal. Metaloved. {\\bf 35}, 92 (1973).\n\n\n\n\\bibitem{thiele}\nJ. -U. Thiele, L. Folks, M. F. Toney, and D. K. Weller, J. Appl.\nPhys. {\\bf 84}, 5686 (1998).\n\n\n\\bibitem{laar}\nB. van Laar, J. Physique {\\bf 25}, 600 (1964).\n\n\\bibitem{kootte}\nA. Kootte, C. Haas, and R. A. de Groot, J. Phys.: Cond. Matt. {\\bf\n3}, 1133 (1991).\n\n\\bibitem{osterloch}\n I. Osterloch, P. M. Oppeneer, J. Sticht and J. K\\\"ubler,\n J. Phys.: Condens. Matter {\\bf 6}, 285 (1994).\n\n\n\\bibitem{vignale}\nG. Vignale and M. Resolt, Phys. Rev. B {\\bf 37},\n10~685 (1998).\n\n\\bibitem{brooks}\n M. S. S. Brooks, Physica B {\\bf 130}, 6 (1985).\n\n\\bibitem{bruno}\nP. Bruno, Phys. Rev. B \\textbf{39}, 865 (1989); \nG. van der Laan, J. Phys. Cond. Matt. \\textbf{10}, 3239 (1998).\n\n\\bibitem{schwit}\n J. Schwitalla and H. Ebert, Phys. Rev. Lett. {\\bf 80}, 4586 (1998).\n\n\\bibitem{benedict}\n L X. Benedict and E. L. Shirley, Phys. Rev. B {\\bf 59}, 5441 (1999).\n\n\\bibitem{daal2}\nG. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmand, Phys.\nRev. Lett. {\\bf 71}, 2165 (1993).\n\n\\bibitem{wang}\nD. S. Wang, R. Wu, and A. J. Freeman, Phys. Rev. Lett. {\\bf 70},\n869 (1993); D. S. Wang, R. Wu, and A. J. Freeman, Phys. Rev. Lett.\n{\\bf 71}, 2166 (1993).\n\n\\end{thebibliography}\n\n\\begin{table}\n\\caption{Calculated LSDA and GGA cobalt(iron) orbital magnetic moments\nwith respect to the angle $\\gamma$ between the [001] axis and the spin\nquantization axis in the (010) plane. For both atoms the orbital moments\ndecrease with the angle but values for $\\gamma$=90$^o$ do not\nfollow this trend. Both LSDA and GGA produce the same trends. The moments\nat the cobalt site \nare larger than at the iron site. The experimental orbital moments for the cobalt site\nare taken from the Ref. \\protect\\onlinecite{grange2}. The theory \nunderestimates the experimental orbital moments by more than 50\\% but\nreproduces the correct trends. } \\label{table1}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n$\\gamma$ & 0$^o$ & 10$^o$ & 30$^o$ & 45$^o$ & 60$^o$ & 75$^o$ & 90$^o$ \\\\\n\\hline \\hline Fe-LSDA & 0.072 & 0.070 &0.062 &0.051 &0.036 &0.019 &0.070\n\\\\ \\hline\nFe-GGA &0.068 &0.067 &0.059 &0.048 &0.034 &0.018 &0.068 \\\\ \\hline \\hline\nCo-LSDA & 0.109 &0.107& 0.095& 0.076 &0.052& 0.022& 0.061 \\\\ \\hline\nCo-GGA &0.088& 0.087 &0.077 &0.063 &0.044 &0.023& 0.061 \\\\ \\hline\nCo-Exp & & 0.26 & 0.24 & 0.17 & 0.11 & & \n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Calculated LSDA and GGA orbital moments of platinum in FePt and CoPt \nwith respect to the angle $\\gamma$ between the [001] axis and the spin\nquantization axis in the (010) plane.\n For the FePt the GGA produces larger values than LSDA\ncontrary to CoPt. The moments decrease with the angle but increase sharply \nat $\\gamma$=90$^o$. The moments are significantly larger for\nthe CoPt compound. Experimental results for platinum in CoPt are taken\nfrom the Ref. \\protect\\onlinecite{grange2}. The agreement is better\nthan in the case of cobalt.} \\label{table2}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n$\\gamma$ & 0$^o$ & 10$^o$ & 30$^o$ & 45$^o$ & 60$^o$ & 75$^o$ & 90$^o$ \\\\\n\\hline \\hline FePt-LSDA &0.046 &0.045& 0.039& 0.032& 0.023& 0.012 &0.052\n\\\\ \\hline\nFePt-GGA& 0.049& 0.048 &0.043 &0.035 &0.025 &0.013 &0.058\\\\ \\hline \\hline\nCoPt-LSDA &0.063& 0.061& 0.054 &0.044& 0.030& 0.014& 0.078 \\\\ \\hline\nCoPt-GGA &0.061& 0.060& 0.053 &0.043 &0.031 & 0.016& 0.072 \\\\ \\hline\nCoPt-Exp & & 0.09 & & & 0.06 & & \n\\end{tabular}\n\\end{table}\n\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 1\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios1.eps}}\n\\end{minipage}\n\\caption{ \\label{fig12} Total energy for the CoPt system within\nLSDA as a function of the angle $\\gamma$ between the [001] axis\nand the spin quantization axis on the (010) plane. The fundamental\nstate corresponds to $\\gamma$=0$^o$, that means to the [001] axis.\nThe total energy increases as a function of $\\gamma$.}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 2\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios2.eps}}\n\\end{minipage}\n\\caption{ \\label{fig2}\n%\nConvergence of the magneto crystalline anisotropy in meV with \nrespect to the number of\n{\\bf k}-points. The calculation is performed for two in-plane axis. For CoPt\nthe line for the [100] axis converges to 2.2 meV while the line for\nthe [110] axis converges to 2.17 meV. For FePt the two lines differ by \nless than 0.01 meV. The behavior of\nMCA verifies the assumption that the $L1_0$ structure is isotropic\nin the (001) plane.}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 3\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios3.eps}}\n\\end{minipage}\n\\caption{ \\label{fig3} Cobalt projected partial density of states with\nrespect to the angle $\\gamma$ between the [001] axis and \nthe spin quantization axis in the (010) plane. There are no major changes with $\\gamma$.\nThis behavior reflect the isotropic character of the\nspin moments.\n%\n}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 4\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios4.eps}}\n\\end{minipage}\n\\caption{ \\label{fig4}\n%\nCalculated orbital moments for both compounds within LSDA as a function of the angle \n$\\gamma$ between the [001] direction and the spin quantization axis \n(the lines are guides for the eye). All\nfour atoms show the same behavior. The orbital moments decrease with\nthe angle $\\gamma$ until about 75$^o$, then they increase. \nThe moments for the CoPt compound are\nlarger than for these of FePt. \n}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 5\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios5.eps}}\n\\end{minipage}\n\\caption{ \\label{fig5}\n%\n XMCD spectra for the cobalt and iron sites in the two XPt compounds\n and for magnetization along the two high symmetry axis ([001] and \n[100]). The iron and cobalt XMCD show different behavior.\n The picks intensities for the cobalt spectrum lay\n higher for {\\bf M}$\\parallel$[001] and the\n $L_3$/$L_2$ integrated branching ratio is larger for this axis.\n Contrary, the\n picks intensity for the iron site are larger for {\\bf M}$\\parallel$[100]\n and the $L_3$/$L_2$ integrated branching ratio is about constant.}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 6\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios6.eps}}\n\\end{minipage}\n\\caption{ \\label{fig9}\n%\nXMCD spectra for platinum for two values of the angle $\\gamma$ \nbetween the [001] direction and the spin quantization axis in both\ncompounds. The differences in the XMCD are important\nand the $L_3$/$L_2$ integrated branching ratio changes\nconsiderably with the angle. The intensities for the CoPt compound are\nlarger reflecting the bigger orbital moments in this compound. The\nspin-orbit splitting of the core $p_{\\frac{1}{2}}$ and\n$p_{\\frac{3}{2}}$ states is 1727eV in both compounds. }\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 7\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios7.eps}}\n\\end{minipage}\n\\caption{ \\label{fig6}\n%\nTheoretical and experimental absorption and XMCD spectra for the cobalt site \nin CoPt \nfor $\\gamma$=0$^o$, where $\\gamma$ the angle between the [001] \ndirection and the spin quantization axis. The theory overestimates the\n absorption at the $L_2$ edge\nand underestimates the $L_3$/$L_2$ integrated branching ratio.\n}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 8\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios8.eps}}\n\\end{minipage}\n\\caption{ \\label{fig7}\n%\nCalculated and experimental absorption spectra at the platinum \nsite and for $\\gamma$=10$^o$ in\nCoPt, where $\\gamma$ is the angle between the [001]\ndirection and the spin quantization axis.\n The theory produces a peak at the threshold of both\n$L_{2,3}$ edges. Both experimental and theoretical spectra have\nthe same shape for the two edges but $L_3$ edge is much more\nintense. Most of the structures in the experimental spectra are washed out \ndue to a strong broadening effect.}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 9\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios9.eps}}\n\\end{minipage}\n\\caption{ \\label{fig8}\n%\nCalculated and experimental platinum XMCD spectra at $L_2$ and $L_3$ \nedges for $\\gamma$=10$^0$ in\nthe CoPt compound ($\\gamma$ is the angle between the [001]\ndirection and the spin quantization axis).\n The agreement between the theory and the experiment is\nbetter than in the case of cobalt because the core hole is deeper.\n }\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 1O\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios10.eps}}\n\\end{minipage}\n\\caption{ \\label{fig10}\n%\nCoPt band structure around the Fermi level along two\n high symmetry directions of the Brillouin zone as a function\nof the angle $\\gamma$ between the [001] direction and the spin quantization axis.}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURE 11\n%\n\\begin{figure}\n\\begin{minipage}{6.0in}\n\\epsfxsize=5.0in \\epsfysize=4.0in \\centerline{\\epsfbox{ios11.eps}}\n\\end{minipage}\n\\caption{ \\label{fig11}\n%\nTotal density of states for CoPt at the vicinity of the \nFermi level, for the spin quantization axis along the [001] and [100] \nmagnetization axis. The states just below the Fermi level suggest that \nthe hard axis is favored which led as to conclude that the MCA is due to\nthe changes in all the DOS.}\n\\end{figure}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002110.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{razee}\n S. S. A. Razee, J. B. Staunton, and F. J. Pinski, Phys. Rev. B {\\ bf 56}, 8082\n (1996); S. S. A. Razee, J. B.\n Staunton, F. J. Pinski, B. Ginatempo, and E. Bruno J. Appl.\n Phys. {\\bf 83}, 7097 (1998).\n\n\\bibitem{razee2}\nS. S. A. Razee, J. B. Staunton, B. Ginatempo, F. J.\n Pinski, and E. Bruno, Phys. Rev. Lett. {\\bf 82},\n 5369 (1999).\n\n\\bibitem{coffey}\nK. R. Coffey, M. A. Parker, and J. K. Howard, IEEE Trans. Magn.\n{\\bf 31}, 2737 (1995); S. H. Charap, P. -L. Lu, and Y. He, IEEE\nTrans. Magn. {\\bf 33}, 978 (1997); N. Li and B. M. Lairson, IEEE\nTrans. Magn. {\\bf 35}, 1077 (1999).\n\n\\bibitem{harp} G. R. Harp, D. Weller, T. A. Rabedeau, R. F. C. Farrow, and R. F. Marks,\nMat. Res. Soc. Symp. Proc. {\\bf 313}, 493 (1993).\n\n\\bibitem{cebollada}\nA. Cebollada, D. Weller, J. Sticht, G. R. Harp, R. F. C. Farrow,\nR. F. Marks, R. Savoy, and J. C. Scott, Phys. Rev. B {\\bf 50},\n3419 (1994).\n\n\n\\bibitem{visokay}\n M. R. Visokay and R. Sinclair, Appl. Phys. Letters {\\bf 66}, 1692 (1995).\n\n\\bibitem{lin}\n C. -J. Lin and G. L. Gorman, Appl. Phys. Lett. {\\bf 61}, 1600 (1992).\n\n\\bibitem{watanabe}\nM. Watanabe and M. Homma, Jpn. J. Appl. Phys. {\\bf 35}, L1264\n(1996).\n\n\\bibitem{lairson}\nB. M. Lairson, M. R. Visokay, R. Sinclair, and B. M. Clemens,\nAppl. Phys. Lett. {\\bf 62}, 639 (1993).\n\n\n\\bibitem{watanabe2}\nM. Watanabe, K. Takanashi, and . Fujimori, J. Magn. Magn. Mat.\n{\\bf 113}, 110 (1991).\n\n\\bibitem{mitani}\nS. Mitani, K. Takanashi, M. Sano, H. Fujimori, A. Osawa, and H.\nNakajima, J. Magn. Magn. Mat. {\\bf 148}, 163 (1995).\n\n\\bibitem{sato}\nK. Sato, H. Hongu, H. Ikekame, Y. Tosaka, M. Watanabe, K.\nTakanashi, and H. Fujimori, Jpn. J. Appl. Phys. {\\bf 32}, 989\n(1993).\n\n\\bibitem{weller} D. Weller, J. St\\\"{o}hr, R. Nakajima, A. Carl, M. G.\nSamant, C. Chappert, R. M\\'{e}gy, P. Beauvillain, P. Veillet, and\nG. A. Held,\n Phys. Rev. Lett. \\textbf{75}, 3752 (1995).\n\n\\bibitem{thole} B. T. Thole, P. Carra, F. Sette, and G. van der Laan,\nPhys. Rev. Lett. \\textbf{68}, 1943 (1992).\n\n\\bibitem{carra} P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys. Rev.\n B {\\bf 70}, 694 (1993).\n\n\\bibitem{VanL98}\nG. van der Laan,\n Phys. Rev. B {\\bf 57}, 112 (1998).\n\n\\bibitem{chen} C. T. Chen, Y. U. Idzerda, H.- J. Lin, N. V. Smith , G. Meigs,\nE. Chaban, G. H. Ho, E. Pellegrin, and F. Sette, Phys. Rev.\nLett. {\\bf 75}, 152 (1995).\n\n\\bibitem{wu} R. Wu, D. Wang, and A. J. Freeman, Phys. Rev. lett. {\\bf 71}, 3581\n(1993); R. Wu and A. J. Freeman, {\\it ibid} \\textbf{73}, 1994\n(1994).\n\n\\bibitem{grange2}\nW. Grange, thesis, Universit\\'e Louis Pasteur, Strasbourg, 1998.\n\n\\bibitem{naka}\n N. Nakajima, T. Koide, T. Shidara, H. Miyauchi, H. Fukutani, A. Fujimori, K. Iio , T. Katayama,\nM. N\\'yvlt, and Y. Suzuki, Phys. Rev. Lett. {\\bf 81}, 5229\n(1998).\n\n\\bibitem{koide}\n T. Koide, N. Nakajima, T. Shidara, H. Miyauchi, H. Kawabe, H. Fukutani, A. Fujimori,\nK. Iio, T. Katayama, and Y. Suzuki, J. Appl. Phys. {\\bf 79},\n5708 (1996).\n\n\\bibitem{ruegg}\n S. R\\\"uegg, G. Sch\\\"utz, P. Fischer, R. Wienke, W. P. Zeper, and H. Ebert,\nJ. Appl. Phys. {\\bf 69}, 5655 (1991).\n\n\\bibitem{schultz}\n G. Sch\\\"ultz {\\em et al.}, Z. Phys. B {\\bf 75}, 495 (1989).\n\n\\bibitem{hlil}\nE. K. Hlil, R. Baudoing-Savois, B. Moraweck, and A. J. Renouprez,\nJ. Phys. Chem. {\\bf 100}, 3102 (1996).\n\n\\bibitem{Ebert}\nH. Ebert, Rep. Prog. Phys. {\\bf 59}, 1665 (1996).\n\n\\bibitem{Meb1}\n M. Alouani, J. M. Wills, and J. W. Wilkins,\n Phys. Rev. B {\\bf 57}, 9502 (1998).\n\n\\bibitem{galanakis}\nI. Galanakis, S. Ostanin, M. Alouani,\nH. Dreyss\\'e, and J. M. Wills,\nPhys. Rev. B {\\bf 61}, 599 (2000).\n\n\\bibitem{Brouder}\n C. Brouder and M. Hikam,\n Phys. Rev. B {\\bf 43}, 3809 (1991);\nC. Brouder, M. Alouani, and K. H.\nBennamann, Phys. Rev.\n B {\\bf 54}, 7334 (1996);\n G. Y. Guo,\n Phys. Rev. B {\\bf 57}, 10~295 (1998);\nA. Ankudinov and J. J. Rehr, Phys. Rev. B {\\bf 56}, 1712 (1997);\nA. Ankudinov, B. Ravel, J. J. Rehr, and S. Conradson, Phys. Rev. B\n{\\bf 58}, 7565 (1998); \n G. van der Laan and B. T. Thole,\n Phys. Rev. B {\\bf 43}, 13~401 (1991).\n\n \\bibitem{WILLS}\n J. M. Wills and B. R. Cooper,\n Phys. Rev. B {\\bf 36}, 3809 (1987);\n M. Alouani and J. M. Wills, Phys. Rev. B {\\bf 54}, 2480 (1996).\n\n\\bibitem{LMTO}\n O. K. Andersen,\n Phys. Rev. B {\\bf 12}, 3060 (1975).\n\n\\bibitem{barth} U. von Barth and L. Hedin, J. Phys. C: Solid State Phys. {\\bf 5}, 1629 (1972).\n\n\n\\bibitem{perdew}\n J. P. Perdew and Y. Wang, Phys. Rev. B {\\bf 33}, 8800 (1986);\n {\\em ibid.} {\\bf 45}, 13244 (1992).\n\n\\bibitem{hoh}\n P. Hohenberg and W. Kohn, Phys. Rev. B {\\bf 136}, 864 (1964);\n W. Kohn and L. J. Sham, Phys. Rev. A {\\bf 140}, 1133 (1965).\n\n\\bibitem{villars}\nP. Villars and L. D. Calvet: {\\em Pearson's Handbook of\nCrystallographic Data for Intermetallic Phases}, (Metals Park,\nOH: American Society for Metals,\n 1985).\n\n\\bibitem{janssen}\nH. J. F. Jansen, Phys. Rev. B {\\bf 38}, 8022 (1988).\n\n\\bibitem{bruno2}\nP. Bruno, {\\em Physical Origins and Theoretical Models of Magnetic\nAnisotropy}, Ferienkurse des Forschungszentrum J\\\"ulich, J\\\"ulich\n(1993).\n\n\\bibitem{Sol}\nI. V. Solovyev, P. H. Dederichs, and I. Mertig,\n Phys. Rev. B {\\bf 52}, 13 419 (1995).\n\\bibitem{eurin} P. Eurin and J. Pauleve, IEEE Trans. Mag. {\\bf 5}, 216 (1969).\n\n \\bibitem{farrow}\nR. F. C. Farrow, D. Weller, R. F. Marks, M. F. Toney, A.\nCebollada, and G. R. Harp, J. Appl. Phys. {\\bf 79}, 5967 (1996).\n\n\\bibitem{Daal}\n G. H. O. Daalderop, P. J. Kelly, and M. F. H. Shuurmans, Phys. Rev. B\n{\\bf 44}, 12 054 (1991).\n\n\\bibitem{macintosh}\n A. R. Macintosh and O. K. Andersen, {\\it Electrons\nat the Fermi Surface} (Cambridge/Cambridge Univ. Press, 1980).\n\n\n\\bibitem{Sak}\n A. Sakuma, J. Phys. Soc. Jap. {\\bf 63}, 3053 (1994).\n\n\n\\bibitem{oppen}\nP. Oppeneer, J. Magn. Magn. Mat. {\\bf 188}, 275 (1998).\n\n\\bibitem{ivanov}\nO. A. Ivanov, L. V. Solina, V. A. Demshina, and L. M. Magat, Fiz.\nMetal. Metaloved. {\\bf 35}, 92 (1973).\n\n\n\n\\bibitem{thiele}\nJ. -U. Thiele, L. Folks, M. F. Toney, and D. K. Weller, J. Appl.\nPhys. {\\bf 84}, 5686 (1998).\n\n\n\\bibitem{laar}\nB. van Laar, J. Physique {\\bf 25}, 600 (1964).\n\n\\bibitem{kootte}\nA. Kootte, C. Haas, and R. A. de Groot, J. Phys.: Cond. Matt. {\\bf\n3}, 1133 (1991).\n\n\\bibitem{osterloch}\n I. Osterloch, P. M. Oppeneer, J. Sticht and J. K\\\"ubler,\n J. Phys.: Condens. Matter {\\bf 6}, 285 (1994).\n\n\n\\bibitem{vignale}\nG. Vignale and M. Resolt, Phys. Rev. B {\\bf 37},\n10~685 (1998).\n\n\\bibitem{brooks}\n M. S. S. Brooks, Physica B {\\bf 130}, 6 (1985).\n\n\\bibitem{bruno}\nP. Bruno, Phys. Rev. B \\textbf{39}, 865 (1989); \nG. van der Laan, J. Phys. Cond. Matt. \\textbf{10}, 3239 (1998).\n\n\\bibitem{schwit}\n J. Schwitalla and H. Ebert, Phys. Rev. Lett. {\\bf 80}, 4586 (1998).\n\n\\bibitem{benedict}\n L X. Benedict and E. L. Shirley, Phys. Rev. B {\\bf 59}, 5441 (1999).\n\n\\bibitem{daal2}\nG. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmand, Phys.\nRev. Lett. {\\bf 71}, 2165 (1993).\n\n\\bibitem{wang}\nD. S. Wang, R. Wu, and A. J. Freeman, Phys. Rev. Lett. {\\bf 70},\n869 (1993); D. S. Wang, R. Wu, and A. J. Freeman, Phys. Rev. Lett.\n{\\bf 71}, 2166 (1993).\n\n\\end{thebibliography}"
}
] |
cond-mat0002111
|
Perpendicular transport properties of YBa$_2$Cu$_3$O$_{7-\delta}$/PrBa$_2$Cu$_3$O$_{7-\delta}$ superlattices.
|
[
{
"author": "{J.C.}{Mart\\'\\i nez}\\thanksref{thank1}"
}
] |
The coupling between the superconducting planes of YBa$_2$Cu$_3$O$_{7-\delta}$ / PrBa$_2$Cu$_3$O$_{7-\delta}$ superlattices has been measured by c-axis transport. We show that only by changing the thickness of the superconducting YBa$_2$Cu$_3$O$_{7-\delta}$ layers, it is possible to switch between quasi-particle and Josephson tunneling. From our data we deduce a low temperature c-axis coherence length of $\xi_c=0.27$\ nm.
|
[
{
"name": "LT22.tex",
"string": "% --------------------------------------------------------------------\n%\n% LTsamplepap.tex\n%\n% A sample manuscript for LT22. You need both phbauth.cls and\n% elsart.cls to typeset this. You also need an eps figure file with\n% the name figure1.eps.\n%\n% --------------------------------------------------------------------\n\n\\documentclass{phbauth}\n\\usepackage{graphicx}\n\n\\begin{document}\n\n\\begin{frontmatter}\n\n% Use lower case letters in the title.\n\\title{Perpendicular transport properties of \n\tYBa$_2$Cu$_3$O$_{7-\\delta}$/PrBa$_2$Cu$_3$O$_{7-\\delta}$ superlattices.}\n\n\\author[address1]{{J.C.}{Mart\\'\\i nez}\\thanksref{thank1}},\n\\author[address1]{{A.}{Schattke}},\n\\author[address1]{{G.}{Jakob}}\n\\author[address1]{{H.}{Adrian}}\n\n\\address[address1]{Johannes Gutenberg - Universit{\\\"a}t Mainz; Institute of \nPhysics; \n55099 Mainz; Germany}\n\n% The corresponding author should be distinguished and his email\n% address and/or fax number must be given. His mailing address has to\n% be complete: the proofs are send to this address around\n% January 1, 2000. The address for sending proofs has to be indicated\n% as \"present address\", if it is different from the address above.\n\\thanks[thank1]{Corresponding author. E-mail: martinez@mail.uni-mainz.de}\n\n\\begin{abstract}\nThe coupling between the superconducting planes of YBa$_2$Cu$_3$O$_{7-\\delta}$ / \nPrBa$_2$Cu$_3$O$_{7-\\delta}$ superlattices has been measured by c-axis \ntransport. We show that only by changing the thickness of the superconducting \nYBa$_2$Cu$_3$O$_{7-\\delta}$ layers, it is possible \nto switch between quasi-particle and Josephson tunneling.\nFrom our data we deduce a low temperature c-axis coherence length of $\\xi_c=0.27$\\ nm.\n\\end{abstract}\n\n\\begin{keyword}\n% write here 3 or 4 keywords separated by semicolons\nJosephson effect; tunneling; YBa$_2$Cu$_3$O$_7$\n\\end{keyword}\n\\end{frontmatter}\n\n%\\section{Introduction}\n\nArtificial YBa$_2$Cu$_3$O$_{7-\\delta}$/PrBa$_2$Cu$_3$O$_{7-\\delta}$ \nsuperlattices (Y123/Pr123) constitute ideal model systems for isolating \ngiven properties of High Temperature Superconductors. In those systems it \nis possible for instance to modify the c-axis tunneling properties simply by \nvarying the periodicities of the Y123 and Pr123 layers.\n\nSeries of 200\\ nm thick Y123/Pr123 superlattices have been prepared by \nhigh-pressure dc-sputtering. The high quality of the samples was checked by \ndetecting up to third order satellite peaks in x-ray $\\theta-2\\theta$ scans. \nLater on series of mesa structures with dimensions between $15\\times 15$ and \n$50 \\times 50$\\ $\\mu$m$^2$ were prepared by ion milling. In this \nwork we present low temperature data on 2:7 (2 layers of Y123 and 7 of Pr123) \nand 8:8 superlattices.\n\nWe measured simultaneously the $U$ vs. $I$ characteristics and the differential \nconductivity $\\sigma(U)$ by means of a standard Lock-In technique. In Fig.\\ \n\\ref{fig1} we show the $\\sigma(U)$ on a $30\\times 30 \\mu$m$^2$ \nmesa done on a 2:7 superlattice at 2.0\\ K. \nNo superconducting current could be detected. However the peak in \n$\\sigma(U)$ corresponds to a $c$-axis superconducting gap. From the peak to peak voltage $U_{pp}$ \npeak, we estimate that each of the $n=$8 to 10 bi-layers constituting this mesa \nhave a $c$-axis gap $\\Delta_c=U_{pp}/4 n=5.0\\pm 0.5$ \\ meV. \nThis value is in excellent agreement with the value of $\\Delta_c$ given in \nthe literature which scatters between 4 and 6 meV for planar junctions \\cite{iguchi}. \n\nIn Fig.\\ \\ref{fig1}, we observe sharper features in $\\sigma(U)$. In order to \nverify a quasi-periodicity, we marked each minimum by a vertical line which is \nassociated to an integer. In Fig.\\ \\ref{fig1} \nwe plot this index as a function of the minima position. A clear zero crossing of the \nlinear fit is obtained by choosing an index n=9 for the lowest index. \nFrom the linear fit we deduce a period of $(11.1 \\pm 0.5)$\\ mV which gives \na periodicity $\\delta U=(1.2\\pm 0.1)$ meV for a single junction. These \nfeatures are reproducible and temperature independent up to $20$\\ K.\n\n\\begin{figure}[ht]\n%h=here, t=top, b=bottom, p=separate figure page\n\\begin{center}\\leavevmode\n\\includegraphics[width=1\\linewidth]{Fig1.eps}\n\\caption{ \nDifferential conductance $\\sigma$ versus Voltage $U$ for a 2:7 superlattice. The \nvertical lines mark minima observed in the spectrum. The right axis corresponds \nto the index associated to each minimum. The linear fit reveals the existence \nof a quasi-periodicity.\n}\\label{fig1}\\end{center}\\end{figure}\n\nSuch a quasi-periodic structure in the density of states has been theoretically \npredicted by Hahn \\cite{hahn}. According to this work, above the superconducting \ngap, additional structures should appear with a periodicity of: \n$\\xi_c/s=1/\\pi^2 \\delta U/\\Delta_c$ where $s$ is the period of the superlattice and \n$\\xi_c$ the $c$-axis coherence length. From our data \nand by taking $s=10.5$\\ nm we deduce a $c$-axis coherence length of $\\xi_c=0.27$ \nnm. If we assume for Y123 an anisotropy of $\\gamma \\approx 5$ \\cite{janossi} we \nwould obtain a in-plane coherence length $\\xi_{ab}=\\gamma\\xi_c\\approx 1.4$\\ nm. \nThis value is close to the generally quoted $\\xi_{ab}=1.5$\\ nm for Y123 \n\\cite{tinkham}.\n\n%\\section{Josephson tunneling in 8:8 superlattices}\n\nIn Fig.\\ \\ref{fig2}, we show the $I$ vs. $U$ characteristics of a $40\\times 40 \\mu$m$^2$ \nmesa done on a 8:8 \nsuperlattice for zero field and for $B=1$\\ T. From the difference between the \ntwo curves we deduce the presence of two distinct low and high current regimes. \nTo investigate the difference between these two regimes we measured the $B$ \ndependence of $\\sigma(U=0)$ at zero bias current and at $115$\\ $\\mu$A (see \ninserts).\n\n\\begin{figure}[h]\n%h=here, t=top, b=bottom, p=separate figure page\n\\begin{center}\\leavevmode\n\\includegraphics[width=1\\linewidth]{Fig2.eps}\n\\caption{ \nDifferential conductance $\\sigma$ versus Voltage $U$ for a 8:8 superlattice. The \ninserts correspond to the field dependence of $\\sigma(U=0)$ at two different \ncurrent bias.\n}\\label{fig2}\\end{center}\\end{figure}\n\nAt zero current $\\sigma(B)$ shows a modulation of $B_\\Phi=0.7$\\ T. By \nconsidering that $B_\\Phi=\\Phi_0/(s+t)b_{eff}$, s=9.4 nm (Y123 thickness) and \nt=9.4 nm (Pr123 thickness), we deduce $b_{eff}=0.152$\\ $\\mu$m. The low current \nregime can therefore be associated to structural shorts \\cite{remark2}. At $115$\\ $\\mu$A we \nobserve a $\\sigma(B)$ modulation of $2.4$\\ mT. This value is very similar to the \n$2.5$\\ mT predicted for a $40\\times 40$\\ $\\mu$m$^2$ mesa. The high current \nregime can therefore be associated to a c-axis Josephson Effect. The absence of a \nsimilar effect in the 2:7 superlattices is probably due to a non-fully developed \nsuperconducting order parameter in the 2 unit cells thick Y123 layers.\n\n%\\begin{ack}\nThis work was supported by the German BMBF (Contract 13N6916) and the E.U. \n(ERBFMBICT972217).\n%\\end{ack}\n\n\\begin{thebibliography}{9}\n\\bibitem{iguchi} I. Iguchi, Z. Wen, Physica C {\\bf 178} 1 (1991); Q.Y. Ying, C. \nHilbert, Appl.Phys.Lett. {\\bf 65} 3005 (1994); M.A. Bari, F.Baudenbacher, J. \nSantiso, E. J. Tarte, J.E. Evets, M.G.Balmire, Physica C, {\\bf 256} 227 (1996); \nK. Nakajima, T. Arai, S.E. Shafranjuk, T. Yamashita, I. Tanaka, H. Kojima, \nPhysica C, {\\bf 293} 292 (1997)\n\\bibitem{hahn} A. Hahn, Physica B, {\\bf 165-166} 1065 (1990)\n\\bibitem{janossi} B. Janossi D. Prost, S. Pekker, L. Fruchter, Physica C {\\bf \n181} 51 (1991)\n\\bibitem{tinkham} Michael Tinkham, {\\it Introduction to Superconductivity, 2nd \ned.} 325 (1996) \n\\bibitem{remark2} Such shorts can mainly occur in superlattices having similar thicknesses for both layers. Superlattices having Y123 thicknesses much smaller than the Pr123 layer show usually no shorts.\n\\end{thebibliography}\n\\end{document}\n\n\n"
}
] |
[
{
"name": "cond-mat0002111.extracted_bib",
"string": "\\begin{thebibliography}{9}\n\\bibitem{iguchi} I. Iguchi, Z. Wen, Physica C {\\bf 178} 1 (1991); Q.Y. Ying, C. \nHilbert, Appl.Phys.Lett. {\\bf 65} 3005 (1994); M.A. Bari, F.Baudenbacher, J. \nSantiso, E. J. Tarte, J.E. Evets, M.G.Balmire, Physica C, {\\bf 256} 227 (1996); \nK. Nakajima, T. Arai, S.E. Shafranjuk, T. Yamashita, I. Tanaka, H. Kojima, \nPhysica C, {\\bf 293} 292 (1997)\n\\bibitem{hahn} A. Hahn, Physica B, {\\bf 165-166} 1065 (1990)\n\\bibitem{janossi} B. Janossi D. Prost, S. Pekker, L. Fruchter, Physica C {\\bf \n181} 51 (1991)\n\\bibitem{tinkham} Michael Tinkham, {\\it Introduction to Superconductivity, 2nd \ned.} 325 (1996) \n\\bibitem{remark2} Such shorts can mainly occur in superlattices having similar thicknesses for both layers. Superlattices having Y123 thicknesses much smaller than the Pr123 layer show usually no shorts.\n\\end{thebibliography}"
}
] |
cond-mat0002112
|
A reconstruction from small-angle neutron scattering measurements of the real space magnetic field distribution in the mixed state of Sr$_{2}$RuO$_{4}$.
|
[
{
"author": "P.~G.~Kealey$^{1,*}$"
},
{
"author": "T.~M.~Riseman$^{1}$"
},
{
"author": "E.~M.~Forgan$^{1}$"
},
{
"author": "L.~M.~Galvin$^{1}$"
},
{
"author": "A.~P.~Mackenzie$^{1}$"
},
{
"author": "S.~L.~Lee$^{2}$"
},
{
"author": "D.~M$^c$K.~Paul$^{3}$"
},
{
"author": "R.~Cubitt$^{4}$"
},
{
"author": "D.~F.~Agterberg$^{5}$"
},
{
"author": "R.~Heeb$^{6}$"
},
{
"author": "Z.~Q.~Mao$^{7}$"
},
{
"author": "Y.~Maeno$^{7}$."
}
] |
We have measured the diffracted neutron scattering intensities from the square magnetic flux lattice in the perovskite superconductor Sr$_2$RuO$_4$, which is thought to exhibit p-wave pairing with a two-component order parameter. The relative intensities of different flux lattice Bragg reflections over a wide range of field and temperature have been shown to be inconsistent with a single component Ginzburg-Landau theory but qualitatively agree with a two component p-wave Ginzburg-Landau theory.
|
[
{
"name": "paperpr.tex",
"string": "\\documentstyle[prl,aps]{revtex}\n%\\documentstyle[preprint,aps]{revtex}\n\n\\topmargin-2cm\n\\textheight25cm\n%\\input BoxedEPS.tex\n%\\SetOzTeXEPSFSpecial\n%\\HideDisplacementBoxes\n\n\\begin{document}\n\n\n\\preprint{Preprint}\n\\draft{}\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize \n \\csname @twocolumnfalse\\endcsname \n\n\\title{A reconstruction from small-angle neutron scattering measurements of \nthe real space magnetic field distribution in the mixed \nstate of Sr$_{2}$RuO$_{4}$.}\n\n\\author{P.~G.~Kealey$^{1,*}$,\nT.~M.~Riseman$^{1}$,\nE.~M.~Forgan$^{1}$,\nL.~M.~Galvin$^{1}$,\nA.~P.~Mackenzie$^{1}$,\nS.~L.~Lee$^{2}$, \nD.~M$^c$K.~Paul$^{3}$,\nR.~Cubitt$^{4}$,\nD.~F.~Agterberg$^{5}$,\nR.~Heeb$^{6}$,\nZ.~Q.~Mao$^{7}$,\nY.~Maeno$^{7}$.\n}\n\n\\address{$^{1}$School of Physics and Astronomy, University of \nBirmingham,\nBirmingham B15 2TT, UK. \\\\\n$^{2}$School of Physics and Astronomy, University of St.~Andrews,\nSt.~Andrews, Fife KY16 9SS, UK.\\\\ \n$^{3}$Department of Physics, University of Warwick, Coventry CV4 7AL,\nUK.\\\\\n$^{4}$Institut Laue-Langevin, 38042 Grenoble Cedex, France. \\\\\n$^{5}$National High Magnetic Field Laboratory, Florida State University, Tallahassee, FA 32306, USA.\\\\\n$^{6}$Theoretische Physik, ETH Honggerberg, CH-8093 Zurich, Switzerland. \\\\\n$^{7}$Department of Physics, Kyoto University, Kyoto 606-8052, Japan.\\\\}\n\\date{\\today}\n\\maketitle\n\\widetext\n\n\\begin{abstract}\nWe have measured the diffracted neutron scattering intensities from the square \nmagnetic flux lattice in the perovskite superconductor Sr$_2$RuO$_4$, which is thought\nto exhibit p-wave pairing with a two-component order parameter. The relative intensities\nof different flux lattice \nBragg reflections over a wide range of field and temperature have been shown to \nbe inconsistent with a single component Ginzburg-Landau theory but\nqualitatively agree with a two component p-wave Ginzburg-Landau theory.\n\n\\end{abstract}\n%\\pacs{DRAFT VERSION: NOT FOR DISTRIBUTION}\n\\pacs{PACS numbers: 61.12 Ex, 74.60 Ge, 74.70 Tx}\n]\n\\narrowtext \nThe discovery of superconductivity at temperatures near 1K in strontium \nruthenate \\cite{srofind} has excited great interest because it is a superconducting layered\nperovskite which does not contain copper. However it shows great differences \nfrom the High-T$_c$ cuprates: it is a stoichiometric {\\em undoped} \ncompound with a long \nmean free path, in which the electrons form a Fermi liquid with a \nwell-established quasi-two-dimensional Fermi surface \\cite{fermisurface}. Furthermore, it was suggested \\cite{pwave} that the \nstrongly interacting electrons pair in a triplet p-wave state (rather than the \nsinglet, mainly d-wave state which is believed to occur in hole-doped cuprates).\nClear evidence of non s-wave pairing in this compound has been provided by the \nobservation \\cite{unconv} that {\\em non}magnetic impurities strongly suppress T${_c}$, which \nextrapolates to $\\approx 1.5$\\,K in the clean limit. Strong support for\ntriplet (p-wave) \npairing is given by the results of Ishida {\\em et al.} who have measured the Knight \nshift with a field parallel to the RuO$_2$ planes \\cite{knight}; the spin susceptibility \nmeasured by the Knight shift is not suppressed below T$_c$,unlike a singlet \nsuperconductor. Also, ${\\mu}$SR measurements in the Meissner state in zero\nfield \\cite{timereverse} have revealed spontaneous \nfields, which can be generated by domain boundaries, \nsurfaces and impurities in a superconductor which breaks time-reversal symmetry \n\\cite{otherreverse}. Such states can arise most naturally with p-wave pairing, but also are \npossible with d-wave singlet pairing. \n\n\nAgterberg \\cite{agterberg1,agterberg2} argued that if the pairing was time-reversal symmetry breaking \np-wave, then in tetragonal symmetry the {\\bf d}-vector \\cite{extrayoshi} has the symmetry\n$\\hat{\\mathbf{z}}\\cdot\\exp (\\pm\\mathrm{i}\\varphi)$ (${\\varphi}$ is the azimuthal\nangle about the tetragonal c axis), and a two-component Ginzburg Landau\n(TCGL) \ntheory would be expected to describe the superconductor. In zero field, this \ngives two degenerate states which are related by time reversal; with a field \napplied in the c-direction perpendicular to the planes, one is dominant, but \nthe other is also present \\cite{heeb}. Under these conditions, a small amount of \nanisotropy in the Fermi surface would lead to a square flux lattice instead \nof a triangular one, with the orientation of the square flux line lattice (FLL) relative to the \ncrystal axes determined by the orientation of the fourfold anisotropy of the \npaired electrons. The FLL structure has been observed in \nthis material \\cite{sans} and is observed to be square over a wide range of field \nand temperature. The nearest-neighbour directions in the square FLL are at \n45${^\\circ}$ to the Ru-O-Ru directions in the crystal lattice \\cite{sanscorrection}.\n \n \nThese results are consistent with the pairing wavefunction described above. \nHowever, a square FLL is also seen in borocarbide superconductors, which \nare definitely non-p-wave \\cite{don1,don2,wilde}. Also, one can measure spontaneous fields in a \nsuperconductor by $\\mu$SR due to other causes or from other states than that \nproposed, and application of a strong field in the basal plane to observe\nthe \nKnight shift might alter the pairing state. Hence, it is important to obtain \nfurther evidence as to what kind of superconductivity occurs in strontium \nruthenate. Here we present a detailed study of the scattered neutron \nintensities from the FLL. We show that they are not consistent with a single\ncomponent Ginzburg-Landau model. Also we demonstrate how the local\n$B({\\mathbf{r}})$ may \nbe reconstructed from our data and show that the FLL structure is quite \ndifferent from the Abrikosov one.\n\t\n\nWe shall present measurements of intensities of higher-order Bragg reflections \nfrom the FLL so we consider how they are related to the FLL structure. The \nformula \\cite{christen} for the integrated intensity $I_{hk}$ of a $(h,k)$ diffracted peak of \nwavevector ${\\mathbf{q}}_{hk}$ gives: \n\n\\begin{equation} \\label{eq:intensities}\nI_{hk}\\propto \\frac{F_{hk}^2}{q_{hk}}, \n\\end{equation}\t\t\t\t\t\t\t\t\nwhere $F_{hk}$ is a spatial Fourier component of the local field $B({\\mathbf{r}})$ in the mixed \nstate:\n\n\\begin{equation} \\label{eq:fourier}\nB({\\mathbf{r}})=\\sum_{h,k}F_{hk}\\exp({\\mathrm{i}}{\\mathbf{q}}_{hk}\\cdot{\\mathbf{r}}).\n\\end{equation}\nIn the Abrikosov solution of the Ginzburg-Landau (GL) equations (as applied \nto a square lattice)~\\cite{abrikosov}, the $F_{hk}$ are given by:\n\n\\begin{equation} \\label{eq:abba}\nF_{hk}\\propto-(-1)^{(h^2+k^2+hk)}\\cdot\n\\exp\\bigg(-\\frac{\\pi}{2}(h^2+k^2)\\bigg);\n\\end{equation}\nthis rapidly falls off with $q$ (see Table \\ref{tab:tb1}).\n\nThe Abrikosov solution is only valid near $B_{c2}$. In high-$\\kappa$ superconductors, with \nthe field not close to $B_{c2}$, the London expression \\cite{london} is appropriate instead. \nThis gives $F_{hk}\\propto 1/(1+q_{hk}^2\\lambda^2)$. Note that unlike the Abrikosov solution,\nall the $F_{hk}$ are positive. Table \\ref{tab:tb1} shows that the Fourier\ncomponents fall off much less rapidly with\n$q$. However, strontium ruthenate has a value of the Ginzburg Landau parameter \n$\\kappa = \\lambda/\\xi \\sim 2.0$ for the field along the c axis, which means \nthat the London approach is not realistic except at \n{\\em very} low inductions. Therefore, to see what conventional GL theory predicts for \nthis material at lower fields, one must use the Brandt numerical solution of the GL equations \n\\cite{brandtcal}. Typical results are given in Table \\ref{tab:tb1}.\n\nNext, we consider the Agterberg TCGL solution \\cite{agterberg1,agterberg2}, which is equivalent \nto the Abrikosov one, except that there are two complex order parameters instead of \nonly one. In the mixed state with $B$ parallel to c both components are \nautomatically present because of mixed gradient terms in the free energy \nfunctional \\cite{agterberg1,agterberg2}. Typical values from this theory for $F_{hk}$,\nrelative to $F_{10}$ and the resulting SANS intensities are given in Table\n\\ref{tab:tb1}. \nIt may be seen that the two-component theory gives intensities that fall off {\\em much less rapidly}\nwith $q$ than those given by the one-component Abrikosov solution. \n\nUnder the conditions of our experiments, where the field is not close to $B_{c2}$,\nit may be argued that the Abrikosov approximation used by Agterberg is not \nappropriate. However, recently Heeb and Agterberg have solved numerically the \nGL equations at all fields for the TCGL case \n\\cite{heeb2}. We also give in Table \\ref{tab:tb1} a list of Fourier components from these \ncalculations, using values of parameters that appear to describe \nour results quite well.\n\nThe corresponding vortex structures in real space are shown in Figures\n\\ref{fig:brandt}-\\ref{fig:heeb2}. \nNote that there is a minimum field point in the two-component theory (for the\nconditions of our experiment) which lies {\\em between} the positions of the flux line \ncores, not in the centre of the square. We give results of this theory for two values of the parameter $\\nu$\n($-1< \\nu < 1$) \\cite{agterberg2} \nwhich describes the degree of fourfold anisotropy of superconducting \nelectrons ($\\nu=0$ corresponds to a cylindrical Fermi surface). We note that\nthe results do not change greatly with $\\nu$. Hence the qualitative difference \nbetween Figures \\ref{fig:brandt} and \\ref{fig:agp2} is due to the difference\nbetween TCGL and GL \ntheories rather than effects of fourfold crystal anisotropy. It may be \nthat $\\nu$ is quite small since $\\vert\\nu\\vert >0.0114$ is sufficient to\nstabilise a square FLL and align it to the crystal lattice with an orientation \ndetermined by the {\\em sign} of $\\nu$ \\cite{agterberg2}.\n\nWe now turn to measurements of the FLL structure. Single crystal Sr$_2$RuO$_4$ \nwas prepared by the floating zone technique with excess RuO$_2$ as a flux \n\\cite{maeno96}. Six plates of total mass 556 mg were cleaved from the as-grown \ncrystal and annealed for 72 hours in air at 1420${^\\circ}$C to remove defects and\nincrease T$_c$, which was 1.39K with a width (10-90\\%) of $\\approx$50 mK.\nWith the \nfield applied parallel to the c-axis at 100 mK, the value of $B_{c2}$ was \n58mT. For the small angle neutron scattering (SANS) measurements, the\nsamples\nwere mounted with conducting silver paint as an aligned mosaic with their c-axes perpendicular to a \ncopper plate, which was mounted on the mixing chamber of a dilution \nrefrigerator. This was placed between the poles of an electromagnet, \nwhich had holes parallel to the field for transmission of neutrons. The\nmagnetic field was parallel to the c-axes of the crystals within 2${^\\circ}$, and\nthe FLL was observed using long-wavelength neutrons on instrument D22 at \nthe Institut Laue Langevin. Typical wavelengths employed were 14.6\\,\\AA, \nwith a wavelength spread (FWHM) of 12\\%; the neutron beam was incident \nnearly parallel to the applied field, and the transmitted neutrons were \nregistered at a $128\\times 128$ pixel multidetector (pixel size $7.5\\times7.5$\\,mm$^2$) \nplaced 17.71\\,m beyond the sample. Typical results are shown in Fig\n\\ref{fig:fll}. In \naddition to the strongest $\\{10\\}$ reflections, the $\\{11\\}$ reflections\nare strong, and higher orders are present. The intensity of the strongest \ndiffraction spot is $<$10$^{-3}$ of the incident beam intensity, so these\nhigher \norder reflections are not due to multiple scattering. Their intensities \nare recorded in Table \\ref{tab:tb1}: it will be noted that they are much larger than \nthose given by the Abrikosov structure.\n\nTo reconstruct the $B({\\mathbf{r}})$ of the FLL corresponding to these results, we \nrequire the sign of $F_{hk}$ relative to $F_{10}$ (the FLL is centrosymmetric, \nso all the $F_{hk}$ are real). The most important component after $F_{10}$ is $F_{11}$. \nIf it has the same sign as $F_{10}$, then the $\\{11\\}$ components add in\nphase \nat the flux line cores to give a field peak that is sharper than the \nfield minimum. Measurements of the field distribution in strontium \nruthenate by $\\mu$SR \\cite{musrfind,musrgeneral} show that this is the case. \nThis sign for $F_{11}$ \nis not surprising, since all models in Table \\ref{tab:tb1} give it as positive. \nFor the small contributions of $F_{20}$ and $F_{21}$, we may assume the same signs \nas given by the Agterberg and Abrikosov solutions: taking the London \nsign makes a large difference to $B({\\mathbf{r}})$, and also can be ruled out by \n$\\mu$SR results . The reconstruction of $B({\\mathbf{r}})$ is shown in Fig. 4. Note that \nit is completely different from the Abrikosov or Brandt solutions to \nthe GL equations, and in good qualitative agreement with the TCGL \npredictions. \n\nThe results we have given so far correspond to low temperature and a \nparticular magnetic field value. In Table \\ref{tab:tb2} we present the \nvalues of the form factors $F_{10}$ and $F_{11}$ for a range of \nfields at 100mK. Also, in Fig 3 we plot versus temperature \nthe ratio of the Fourier components for the strongest two reflections \n$F_{11}/F_{10}$ at 10, 20 and 30mT. Remarkably, this ratio varies little with \nfield and temperature and does not tend to the Abrikosov value as T$\\rightarrow$T$_c$. \nNon-local effects \\cite{wilde} and deviations from GL theory in ultra pure\nsuperconductors \\cite{otherreason} should both die away at high temperatures. \nTherefore, these effects are not expected to be the cause of the flux line\nshapes we report, although they may affect the details of $B$(r) at low\ntemperatures.\n\nIn conclusion, the strength of the higher order reflections from the FLL in \nstrontium ruthenate and their temperature dependence certainly show \nthat a standard one component Ginzburg Landau model is insufficient \nto explain the observed diffraction pattern. However, our results are in good\nqualitative but not perfect agreement with a two component Ginzburg Landau\ntheory. Unconventional flux line \nshapes in this material are strong evidence for unconventional \nsuperconductivity in Sr$_2$RuO$_4$.\n\nWe thank J-L Ragazzoni of the ILL for setting up the dilution \nrefrigerator, E.H. Brandt for a copy of his code to solve the GL \nequations and G.M. Luke for communicating his results prior to \npublication. This work was supported by the U.K. E.P.S.R.C., \nCREST of Japan Science and Technology Corporation, and the neutron \nscattering was carried out at the Institut Laue-Langevin, Grenoble.\n\n\\bibliographystyle{prsty}\n\\begin{thebibliography}{99}\n\\bibitem{srofind} Y.~Maeno {\\em et al.}, Nature {\\bf 372}, 532 (1994).\n\\bibitem{fermisurface}A.~P.~Mackenzie {\\em et al.}, Phys. Rev. Lett., {\\bf 76}, 3786 \n(1996), C.Bergemann {\\em et al.}, cond-mat/9909027\n\\bibitem{pwave} T.~M.~Rice and M.~Sigrist, J.Phys: Condens. Matter {\\bf 7}, L643\n(1995).\n\\bibitem{unconv} A.~P.~Mackenzie {\\em et al.}, Phys. Rev. Lett., {\\bf 80}, 161\nand 3890 (1998).\n\\bibitem{knight} K.~Ishida {\\em et al.}, Nature {\\bf 396}, 658 (1998).\n\\bibitem{timereverse} G.~M.~Luke {\\em et al.}, Nature {\\bf 394}, 558 (1998).\n\\bibitem{otherreverse} M.~Sigrist and K.~ Ueda, Rev. Mod. Phys. {\\bf 63}, 239\n(1991).\n\\bibitem{agterberg1} D.~F.~Agterberg, Phys. Rev. Lett. {\\bf 80}, 5184 (1998).\n\\bibitem{agterberg2} D.~F.~Agterberg, Phys. Rev. B. {\\bf 58}, 14484 (1998).\n\\bibitem{extrayoshi} Y.~Maeno {\\em et al.}, Journal of Superconductivity, {\\bf 12}, 535\n(1999).\n\\bibitem{heeb} R.~Heeb and D.~F.~Agterberg, Phys. Rev. B {\\bf 59}, 7076 (1999)\n\\bibitem{sans}T.~M.~Riseman {\\em et al.}, Nature {\\bf 396}, 242-5 (1998).\n\\bibitem{sanscorrection}E.~M.~Forgan and D.~M$^c$K.~Paul, Correction to Nature,\nto be published (2000).\n\\bibitem{don1} D.~M$^c$K.~Paul {\\em et al.}. Phys. Rev. Lett. {\\bf 80} 1517 (1998).\n\\bibitem{don2} M.~R.~Eskildsen {\\em et al.}, Nature {\\bf 393}, 242 (1998) \n\\bibitem{wilde} Y.~De.~Wilde {\\em et al.}, Phys. Rev. Lett. {\\bf 78}, 4273, (1997).\n\\bibitem{christen} D.~K.~Christen {\\em et al.}, Phys. Rev. B {\\bf 15}, 4506-9\n(1977).\n\\bibitem{abrikosov} A.~A.~Abrikosov, 1957, Sov. Phys. JETP {\\bf 5} 1174 (1957),\nE.~H.~Brandt, Phys. Stat. Sol. B {\\bf 64} 257, 467 {\\bf 65}\n469 (1974).\n\\bibitem{london} M.~Tinkham, 1975, Introduction to Superconductivity, Malabar, Florida, USA:McGraw-Hill \n\\bibitem{brandtcal} E.~H.~Brandt, Phys. Rev. Lett., {\\bf 78}, 2208 (1997).\n\\bibitem{heeb2} R.~Heeb and D.~F.~Agterberg, {\\em to be published}\n\\bibitem{maeno96} Z.~Q.~Mao {\\em et al.}, to be published in Mat. Res. Bull\n(2000).\n%Y.~Maeno {\\em et al.}, J. Low temp Phys. {\\bf 105}, 1577-1588(1996).\n\\bibitem{musrfind} C.M. Aegerter {\\em et al.}, J.Phys: Cond. Mat., {\\bf 10},\n7445-51 (1998).\n\\bibitem{musrgeneral} G.~M.~Luke {\\em et al.}, {\\em to be published} (2000)\n\\bibitem{otherreason} J.~M.~Delrieu, J. Low Temp Phys., {\\bf 6}, 197-219 (1972).\n%\\bibitem{heebprivate} R.~Heeb {\\em Private communication}\n\n\\end{thebibliography}\n\n$^{*}$\\small{Email address: P.G.Kealey@bham.ac.uk}\n\n\\begin{figure}\n \\input{epsf}\n \\epsfysize 5.5cm\n \\centerline{\\epsfbox{brandt3_label.eps}}\n \\caption[~]{Contour plot of the magnetic field in a square flux line lattice\n as given by Brandt's numerical solution of the Ginzburg-Landau equations for the particular case of\n $\\kappa =2.0,B=20mT$ and $B_{c2}=58mT$. Contour lines are equally spaced.\n This result is very similar to that given by the Abrikosov solution valid near\n $H_{c2}$\n \\cite{abrikosov}.} \\label{fig:brandt}\n\\end{figure} \n\\begin{figure}\n \\input{epsf}\n \\epsfxsize 5.5cm\n \\centerline{\\epsfbox{agzero_2.eps}}\n \\caption[~]{Contour plot of the magnetic field in \na square flux line lattice as given by Agterberg's solution of his TCGL\nequations, valid near $B_{c2}$, in the case of a cylindrical Fermi\nsurface ($\\nu =0.0$).} \\label{fig:agzero}\n\\end{figure} \n\\begin{figure}\n \\input{epsf}\n \\epsfxsize 5.5cm\n \\centerline{\\epsfbox{agp2_2.eps}}\n \\caption[~]{As for Figure \\ref{fig:agzero} but with a fourfold\n distortion to the Fermi surface ($\\nu =0.2$).}\\label{fig:agp2}\n\\end{figure} \n\\begin{figure}\n \\input{epsf}\n \\epsfysize 5.5cm\n \\centerline{\\epsfbox{heebt_2b.eps}}\n \\caption[~]{Heeb and Agterberg's numerical solution to the\n TCGL equations \\cite{heeb2}, with the parameters $\\kappa =1.6, \\nu\n=0.05$\nchosen to give a good fit to our data, with an applied field of $B=20$mT\nand $B_{c2}$(100mK)=58mT.}\n\\label{fig:heeb2} \n\\end{figure}\n\\begin{figure}\n \\input{epsf}\n \\epsfysize 5.5cm\n \\centerline{\\epsfbox{prosro3_label.eps}}\n \\caption[~]{Contour plot of FLL diffraction pattern. \n A field of 20mT was applied parallel to $c$ above T$_c$; the weak\ndiffracted beams due to the flux lattice were extracted from background\nscattering by subtracting data taken above T$_c$. The axes are pixel numbers \nand the central region of the detector has been masked. The data shown is a sum\nof 5 patterns obtained by rocking the FLL up $\\pm$0.3$^\\circ$ about the [110]\naxis.} \\label{fig:fll} \n\\end{figure} \n\\begin{figure}\n \\input{epsf} \\epsfysize 6cm \n \\centerline{\\epsfbox{ratiow3_3.eps}}\n \\caption[~]{Temperature and field dependence of the ratio $F_{11}/F_{10}$.\nThe \\{1,1\\} intensity used is the direct average of all four spots, and the \\{1,0\\}\nintensity is a weighted average of the top and side spots which allows for\nthe different rocking-curve width in the vertical and\nhorizontal directions \\cite{sans} .} \\label{fig:tdep}\n\\end{figure}\n\\begin{figure}\n \\input{epsf}\n \\epsfysize 5.5cm\n \\centerline{\\epsfbox{expbr3_correct_2.eps}}\n \\caption[~]{Contour plot of the magnetic field in \nthe mixed state of SRO as reconstructed from the data represented \nin Figure~\\ref{fig:fll}, using the signs of the Fourier components as discussed \nin the text.} \\label{fig:reconstructed}\n\\end{figure}\n\\begin{table}[t]\n\\caption[~]{Calculated and experimental flux lattice Fourier components\nand intensities for $B$=20mT, $B_{c2}$=58mT .}\\label{tab:tb1}\n \\begin{center}\n \\begin{tabular}{cccc}\n $h,k$ of diffraction peak: & \\{1,1\\} & \\{2,0\\} & \\{2,1\\} \\\\\n \\hline\n $q_{hk}/q_{10}$ & 1.414 & 2.0 & 2.236\\\\\n $F_{hk}/F_{10}$ (London $\\lambda=152nm):$ & 0.53 & 0.27 & 0.22\\\\\n $F_{hk}/F_{10}$ (Abrikosov): & 0.2079 & -0.00898 & 0.00187 \\\\\n $F_{hk}/F_{10}$ (Agterberg $\\nu =0.0$): & 0.5657 & -0.1051 & 0.0353 \\\\\n $F_{hk}/F_{10}$ (Agterberg $\\nu =0.2$): & 0.7711 & -0.0667 & 0.0503 \\\\\n $F_{hk}/F_{10}$ (Heeb and Agterberg & 0.484 & -0.019 & 0.046 \\\\\n $\\nu =0.05$, $\\kappa = 1.6$): & & & \\\\\n $I_{hk}/I_{10}$ (London): & 0.199 & 0.0365 & 0.0216 \\\\\n $I_{hk}/I_{10}$ (Abrikosov): & 0.0306 & 0.00004 & 0.000002 \\\\\n $I_{hk}/I_{10}$ (GL (Brandt)): & 0.0783 & 0.00091 & 0.000298 \\\\\n $I_{hk}/I_{10}$ (Agterberg $\\nu =0.0$): & 0.2263 & 0.00552 & 0.00056 \\\\\n $I_{hk}/I_{10}$ (Heeb and Agterberg & 0.166 & 0.00095 &0.00018 \\\\\n $\\nu =0.05$, $\\kappa=1.6$): & & & \\\\ \n $I_{hk}/I_{10}$ & 0.197(2) & 0.011(3) & 0.007(1)\\\\\n (Expt. at 20 mT 100mK): & & & \\\\\n \\end{tabular}\n \\end{center}\n\\end{table}\n\\begin{table}[htbp]\n\\caption[~]{Experimental FLL Fourier components at 100mK. Errors represent \nstatistical errors; FLL disorder or static Debye Waller \nfactors may reduce $F_{hk}$ for high ${\\mathbf{q}}$.}\\label{tab:tb2}\n \\begin{center}\n \\begin{tabular}{ccc}\n Magnetic Field (mT) & $F_{10}$\\,(mT) & $F_{11}$\\,(mT) \\\\\n \\hline\n 100 & 1.14(3) & 0.66(3) \\\\\n 150 & 1.02(9) & 0.54(1) \\\\\n 200 & 0.88(1) & 0.47(1) \\\\\n 250 & 0.71(1) & 0.36(1) \\\\\n 300 & 0.58(1) & 0.29(1) \\\\\n 350 & 0.45(1) & 0.24(1) \\\\\n 400 & 0.30(2) & 0.14(1) \\\\\n \\end{tabular}\n \\end{center}\n\\end{table}\n\n\\end{document}\n\n\n\n\n\n\n"
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[
{
"name": "cond-mat0002112.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{srofind} Y.~Maeno {\\em et al.}, Nature {\\bf 372}, 532 (1994).\n\\bibitem{fermisurface}A.~P.~Mackenzie {\\em et al.}, Phys. Rev. Lett., {\\bf 76}, 3786 \n(1996), C.Bergemann {\\em et al.}, cond-mat/9909027\n\\bibitem{pwave} T.~M.~Rice and M.~Sigrist, J.Phys: Condens. Matter {\\bf 7}, L643\n(1995).\n\\bibitem{unconv} A.~P.~Mackenzie {\\em et al.}, Phys. Rev. Lett., {\\bf 80}, 161\nand 3890 (1998).\n\\bibitem{knight} K.~Ishida {\\em et al.}, Nature {\\bf 396}, 658 (1998).\n\\bibitem{timereverse} G.~M.~Luke {\\em et al.}, Nature {\\bf 394}, 558 (1998).\n\\bibitem{otherreverse} M.~Sigrist and K.~ Ueda, Rev. Mod. Phys. {\\bf 63}, 239\n(1991).\n\\bibitem{agterberg1} D.~F.~Agterberg, Phys. Rev. Lett. {\\bf 80}, 5184 (1998).\n\\bibitem{agterberg2} D.~F.~Agterberg, Phys. Rev. B. {\\bf 58}, 14484 (1998).\n\\bibitem{extrayoshi} Y.~Maeno {\\em et al.}, Journal of Superconductivity, {\\bf 12}, 535\n(1999).\n\\bibitem{heeb} R.~Heeb and D.~F.~Agterberg, Phys. Rev. B {\\bf 59}, 7076 (1999)\n\\bibitem{sans}T.~M.~Riseman {\\em et al.}, Nature {\\bf 396}, 242-5 (1998).\n\\bibitem{sanscorrection}E.~M.~Forgan and D.~M$^c$K.~Paul, Correction to Nature,\nto be published (2000).\n\\bibitem{don1} D.~M$^c$K.~Paul {\\em et al.}. Phys. Rev. Lett. {\\bf 80} 1517 (1998).\n\\bibitem{don2} M.~R.~Eskildsen {\\em et al.}, Nature {\\bf 393}, 242 (1998) \n\\bibitem{wilde} Y.~De.~Wilde {\\em et al.}, Phys. Rev. Lett. {\\bf 78}, 4273, (1997).\n\\bibitem{christen} D.~K.~Christen {\\em et al.}, Phys. Rev. B {\\bf 15}, 4506-9\n(1977).\n\\bibitem{abrikosov} A.~A.~Abrikosov, 1957, Sov. Phys. JETP {\\bf 5} 1174 (1957),\nE.~H.~Brandt, Phys. Stat. Sol. B {\\bf 64} 257, 467 {\\bf 65}\n469 (1974).\n\\bibitem{london} M.~Tinkham, 1975, Introduction to Superconductivity, Malabar, Florida, USA:McGraw-Hill \n\\bibitem{brandtcal} E.~H.~Brandt, Phys. Rev. Lett., {\\bf 78}, 2208 (1997).\n\\bibitem{heeb2} R.~Heeb and D.~F.~Agterberg, {\\em to be published}\n\\bibitem{maeno96} Z.~Q.~Mao {\\em et al.}, to be published in Mat. Res. Bull\n(2000).\n%Y.~Maeno {\\em et al.}, J. Low temp Phys. {\\bf 105}, 1577-1588(1996).\n\\bibitem{musrfind} C.M. Aegerter {\\em et al.}, J.Phys: Cond. Mat., {\\bf 10},\n7445-51 (1998).\n\\bibitem{musrgeneral} G.~M.~Luke {\\em et al.}, {\\em to be published} (2000)\n\\bibitem{otherreason} J.~M.~Delrieu, J. Low Temp Phys., {\\bf 6}, 197-219 (1972).\n%\\bibitem{heebprivate} R.~Heeb {\\em Private communication}\n\n\\end{thebibliography}"
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cond-mat0002113
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Magnetic properties of frustrated spin ladder %\footnote{Proceeding of MMM99, to appear in J. Appl. Phys.}
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"author": "T\\^oru Sakai and Nobuhisa Okazaki"
}
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The magnetic properties of the antiferromagnetic spin ladder with the next-nearest neighbor interaction, particularly under external field, are investigated by the exact diagonalization of the finite clusters and size scaling techniques. It is found that there exist two phases, the rung-dimer and rung-triplet phases, not only in the nonmagnetic ground state but also magnetized one, where the phase boundary has a small magnetization dependence. Only in the former phase, the magnetization curve is revealed to have a possible plateau at half the saturation moment, with a sufficient frustration.
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"name": "text.tex",
"string": "% ****** Start of file apssamp.tex ******\n%\n% This file is part of the APS files in the REVTeX 3.0 distribution.\n% Version 3.0 of REVTeX, November 10, 1992.\n%\n% Copyright (c) 1992 The American Physical Society.\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\documentstyle[preprint,eqsecnum,aps]{revtex}\n%\\documentstyle[preprint,aps]{revtex}\n\\documentstyle[multicol,aps,psfig]{revtex}\n%\\topmargin -8mm\n%\\oddsidemargin -8mm \\evensidemargin -7mm\n%-------------------------------------------------------------------\n\\renewcommand{\\narrowtext}{\\begin{multicols}{2}\n\\global\\columnwidth20.5pc\\noindent}\n\\renewcommand{\\widetext}{\\end{multicols}\n\\global\\columnwidth42.5pc}\n\\multicolsep = 8pt plus 4pt minus 3pt\n%-------------------------------------------------------------------\n\\begin{document}\n\\draft\n\\preprint{November 1999}\n\\title{Magnetic properties of frustrated spin ladder\n%\\footnote{Proceeding of MMM99, to appear in J. Appl. Phys.}\n}\n\\author{T\\^oru Sakai and Nobuhisa Okazaki}\n\\address\n{Faculty of Science, Himeji Institute of Technology,\n Ako, Hyogo 678-1297, Japan}\n%\\date{Received \\hspace{6cm}}\n\\date{November 1999}\n\\maketitle\n\\begin{abstract}\nThe magnetic properties of the antiferromagnetic spin ladder with \nthe next-nearest neighbor interaction, particularly under external field, \nare investigated by the exact diagonalization of the finite clusters \nand size scaling techniques. \nIt is found that there exist two phases, the rung-dimer and rung-triplet\nphases, not only in the nonmagnetic ground state but also magnetized\none, where the phase boundary has a small magnetization dependence. \nOnly in the former phase, the magnetization curve is revealed to have \na possible plateau at half the saturation moment, with a sufficient \nfrustration. \n\n\\end{abstract}\n\\pacs{PACS numbers: 75.10.Jm, 75.30.Kz, 75.50.Ee, 75.60.Ej}\n\\narrowtext\n\n\\section{Introduction}\nThe frustration of the antiferromagnetic exchange interaction \nbrings about many interesting phenomena in the quantum spin systems, \nbecause it generally enhances the quantum fluctuation. \nIt would be valuable to consider the effect of the frustration \non the spin ladder, like the materials \nSrCu$_2$O$_3$ (Ref.\\cite{azuma}), Cu$_2$(C$_2$H$_{12}$N$_2$)$_2$Cl$_4$\n(Refs.\\cite{hayward,chaboussant}) and La$_6$Ca$_8$Cu$_{24}$O$_{41}$ \n(Ref.\\cite{imai}).\nThey are strongly quantized and have the spin gap. \nWhen the next-nearest-neighbor\n(NNN) exchange interaction appears, \nthe frustration takes place in the system. \nIn the classical limit it is easily shown that the system has two \ndifferent ordered phases depending on the strength of the NNN exchange \nand the phase boundary does not change even under external magnetic \nfield. \nIn the quantum system, however, some modifications should exist \nin the ground state phase diagram, because the spin ladder has \nno long range order even at $T=0$. \nIn this paper, \nwe investigate the frustrated spin ladder by the exact diagonalization\nof the finite clusters to determine the magnetic phase diagram, \neven under external field. \nIn addition we consider the possibility of the magnetization plateau \n\\cite{oshikawa}, \nwhich is predicted by a strong coupling approach.\\cite{mila} \n\n\\section{Model and numerical method}\nThe $S=1/2$ spin ladder with NNN coupling \nis described by the Hamiltonian\n\\begin{eqnarray}\n\\label{ham}\n{\\cal H}&=&J_1\\sum_i^L({\\bf S}_{1,i} \\cdot {\\bf S}_{1,i+1}\n+{\\bf S}_{2,i} \\cdot {\\bf S}_{2,i+1})\\nonumber\\\\\n &+&J_{\\perp}\\sum_i^L({\\bf S}_{1,i} \\cdot {\\bf S}_{2,i})\\nonumber\\\\\n &+&J_2\\sum_i^L({\\bf S}_{1,i}\n\\cdot {\\bf S}_{2,i+1}+{\\bf S}_{2,i} \\cdot {\\bf S}_{1,i+1}), \n\\end{eqnarray}\nwhere $J_{1}$, $J_2$ and $J_{\\perp}$ are the coupling constants\nof the leg, NNN (diagonal) and rung exchange interactions,\nrespectively. We put $J_{\\perp}$=1 in the following.\nUsing the Lanczos algorithm we numerically solved the ground state \nof the finite clusters. \nWe also calculated the lowest energy\nof ${\\cal H}$ for $\\sum_i^L({S_{1,i}^z}+{S_{2,i}^z})=M$\n, which denotes $E(M)$. \nUsing $E(M)$, we investigate the magnetic state with \n$m\\equiv M/L$ under the external field\ndescribed by \n${\\cal H}_Z=-H\\sum_i^L({S_{1,i}^z}+{S_{2,i}^z})$.\n\n\\section{Two magnetic phases}\nConsider the nonmagnetic ground state at first. \nIn the classical limit the system has two different ordered phases \ndivided by the first-order phase boundary $J_2=J_{\\perp}/2 (=1/2)$ \nshown as a dashed line in Fig. 1. $J_1=1/2$ is also the boundary \nbecause the phase diagram should be symmetric\nunder the exchange of $J_1$ and $J_2$ (the reflection with respect to\nthe dot-dashed line in Fig. 1). \n\\begin{figure}[htb]\n\\begin{center}\n\\mbox{\\psfig{figure=fig1.eps,width=6cm,height=6cm,angle=-90}}\n\\end{center}\n\\caption{\nPhase diagram of the frustrated spin ladder with magnetization\n$m$ for $L=12$. Every phase boundary is first-order within this \nregion. The dashed line is the classical limit. \nThe dot-dashed line is the symmetric line. \n}\n\\label{fig1}\n\\end{figure}\nIn the quantum $S=1/2$ system we should distinguish the two phases \nbased on the dimer picture; the dimers along the rung \nand the diagonal, respectively. The former is realized for $J_2 \\ll\nJ_{\\perp}/2$, while the latter for $J_2 \\gg J_{\\perp}/2$. \nIn the latter phase each two spins coupled by the rung are expected to\nbehave like an effective $S=1$ (triplet) object. \nThus we call the two phases `rung-dimer' and `rung-triplet',\nrespectively. \nThe phase boundary is easily detected as a level crossing point in the\nground state even in small finite clusters. Since the boundary is almost \nindependent of $L$, we show only the result of $L=12$ as circles in Fig.\n1. \nOur study of the spin correlation function along the rung also \nsupported the above argument and suggested that the boundary is \nfirst-order. \nThe results are completely consistent with the recent analysis by the density \nmatrix renormalization group. \\cite{wang} (It also indicated the crossover \nof the phase boundary from first-order to second-order ones for \n$J_2 < 0.287 J_1$, but we don't consider such a parameter region \nin this paper.) \n\\begin{figure}[htb]\n\\begin{center}\n\\mbox{\\psfig{figure=fig2.eps,width=6cm,height=6cm,angle=-90}}\n\\end{center}\n\\caption{\nCanted N\\'eel orders of the classical system under external field $H$ \n(a) for $J_2 <1/2$ and (b) for $J_2 > 1/2$. \n}\n\\label{fig2}\n\\end{figure}\n\nEven in the magnetic state under external field the two phases \nstill can be identified by the different canted N\\'eel orders \nshown in the Figs. 2(a) and (b), respectively, in the classical system.\nThe phase boundary is the same as the nonmagnetic ground state. \nThe quantum system is gapless for $0< m < 1$ and it might be\ndifficult to distinguish the two phases by the dimer picture. \nIn this case the classical picture is useful because the gapless phase\nis characterized by the power-law decay of the dominant spin correlation\nfunction corresponding to the classical order. \nThus the quantum system should also have two phases \nlike the classical limit. \nThe same analysis as the nonmagnetic state indicated the first-order \nboundary for finite $m$. \nWe show the boundaries for $m$=1/6, 1/3, 1/2, 2/3 and 5/6 ($L$=12) \nin Fig. 1. \nThey exhibit a small $m$ dependence, although it is not so large \nthat a field-induced transition between the two phases is \nexpected to occur in any realistic situations. \nAs the magnetization increases, the boundary tends to approach \nto the classical limit for most magnetizations. \nFor $m=1/2$, however, the boundary exhibits a quite different \nbehavior and it is close to the nonmagnetic one. \nIt implies that the quantum fluctuation is enhanced by the \nfrustration particularly at $m=1/2$. \nThus we consider the possibility of another spin gap \ninduced by external field, that is observed as a plateau \nin the magnetization curve at $m=1/2$. \n\\begin{figure}[htb]\n\\begin{center}\n\\mbox{\\psfig{figure=fig3.eps,width=6cm,height=6cm,angle=-90}}\n\\end{center}\n\\caption{\n(a)Scaled plateau $L\\Delta$ for $J_1=0.4$, \n(b)central charge $c$ and critical exponent $\\eta$, \nplotted versus the NNN coupling $J_2$. \n}\n\\label{fig3}\n\\end{figure}\n\n\n\\section{Magnetization plateau}\nWe consider the magnetization plateau at $m=1/2$. \nThe plateau length \n$\\Delta \\equiv E(M+1)+E(M-1)-2E(M)$ \nis one of useful order parameters\\cite{sakai} to investigate the boundary \nbetween the gapless and plateau phases. \nSince $\\Delta$ is the low-lying energy gap, it should obey the \nrelation $\\Delta \\sim 1/L$ in the gapless phase. \nThe scaled plateau $L\\Delta$ for several $L$ is plotted versus $J_2$ \nwith fixed $J_1$ to 0.4 in Fig. 3 (a). \nIt suggested that a gapless-gapful transition occurs at $J_2 \\sim 0.2$.\nTo clarify the feature of the transition, we investigate the central charge\n$c$ of the conformal field theory (CFT)\\cite{cft} \nand the critical exponent $\\eta$. \n$\\eta$ is \ndefined by the asymptotic behavior of the spin correlation function \n$\\langle S^+_0S^-_r \\rangle \\sim (-1)^r r^{-\\eta}$.\nCFT enables us to estimate $c$ and $\\eta$ \nfrom the low-lying energy spectra of finite clusters, \nusing the forms \n$E(M)/L \\sim \\epsilon (m) -\\pi cv_s /6L^2$ and \n$\\Delta \\sim \\pi v_s \\eta / L$ $(L\\rightarrow \\infty)$, \nwhere $v_s$ is the sound velocity which is the gradient of the\ndispersion curve at the origin.\nAfter some extrapolation to the infinite length limit, \nwe show the results of $c$ and $\\eta$ for $J_1=0.4$ in Fig. 3(b). \nIt justifies that the phase boundary is of the Kosterlitz-Thouless\n(KT) transition\\cite{kt} with $c=1$ in the gapless phase and $\\eta=1$ at the \ncritical point. \nThus we determine the phase boundary as points with $\\eta=1$ \nin the $J_1$-$J_2$ plane. \nThe result of the KT line \nis shown as solid symbols in Fig. 4 together with the \nfirst-order boundary indicated as open symbols.\nFig. 4 is a complete phase diagram at $m=1/2$. \nThe plateau phase is surrounded by the KT line and first-order line. \nThe intersection of the two lines is expected to be a tri-critical\npoint. \nThe present analysis suggested that the plateau appears only in the \nrung-dimer phase. \nThe rung-triplet phase reasonably has no plateau, \nbecause it is equivalent to the uniform $S=1$ chain. \n\\begin{figure}[htb]\n\\begin{center}\n\\mbox{\\psfig{figure=fig4.eps,width=6cm,height=6cm,angle=-90}}\n\\end{center}\n\\caption{\nPhase diagram at $m=1/2$ including the plateau phase. \nThe plateau appears only in the rung-dimer phase. \n}\n\\label{fig4}\n\\end{figure}\n\nA necessary condition of the presence of the plateau in general \n1D systems was \nrigorously given\\cite{oshikawa}\n by $Q(S-m)$. \n$Q$ is the periodicity of the \nground state and $S$ is the total spin of the unit cell. \nThe present case must hold $Q=2$ in the plateau phase at $m=1/2$. \nIt suggests that the frustration \nstabilizes the structure where the singlet and triplet rung bonds \nare alternating, \nas is in the case of the zigzag ladder.\\cite{tonegawa,totsuka} \n\nFinally we present the magnetization curves for ($J_1$,$J_2$)=\n(0.5,0), (0.5,0.3) and (0.5,0.4) in Fig. 5. \nThey were obtained by the size scaling in Ref.\\cite{sakai} applied \nto the calculated energy spectra of finite systems up to $L=16$. \nThe plateau clearly appears at $m=1/2$ in the latter two cases. \n\\begin{figure}[htb]\n\\begin{center}\n\\mbox{\\psfig{figure=fig5.eps,width=6cm,height=6cm,angle=-90}}\n\\end{center}\n\\caption{\nMagnetization curves for ($J_1$,$J_2$)=(0.5,0), (0.5,0.3) and (0.5,0.4).\n}\n\\label{fig5}\n\\end{figure}\n\n\\section{Summary}\nThe antiferromagnetic spin ladder with NNN coupling is investigated \nby the exact diagonalization of finite clusters. \nIt indicated the existence of the two magnetic phases; \nthe rung-dimer and rung-triplet phases, not only for $m=0$ but \nalso in the magnetic state. \nIt is also found that the magnetization plateau possibly appears \nat $m=1/2$ only in the rung-dimer phase. \n\n%The numerical computation was done using the facility of the\n%Supercomputer Center, Institute for Solid State Physics, University of\n%Tokyo.\n\n\\begin{references}\n\\bibitem{azuma}\nM. Azuma {\\it et al.}, Phys. Rev. Lett. {\\bf 73} 3463 (1994). \n\n\\bibitem{hayward}\nC. A. Hayward, D. Poilblanc and L. P. L\\'evy,\nPhys. Rev. B {\\bf 54}, R12649 (1996).\n\n\\bibitem{chaboussant}\nG. Chaboussant {\\it et al.},\nPhys. Rev. B {\\bf 55}, 3046 (1997).\n\n\\bibitem{imai}\nT. Imai {\\it et al.}, Phys. Rev. Lett. {\\bf 81}, 220 (1998).\n\n\\bibitem{oshikawa}\nM. Oshikawa, M. Yamanaka and I. Affleck, Phys. Rev. Lett.\n{\\bf 78}, 1984 (1997).\n\n\\bibitem{mila} F.Mila, Eur.Phys.J. {\\bf B6}, 201 (1998).\n\n\\bibitem{wang}\nX. Wang, in {\\it Density-Matrix Renormalization}, edited by \nI. Peschel {\\it et al.}. (Springer, Berlin, 1999). \n\n\\bibitem{sakai}\nT. Sakai and M. Takahashi, Phys. Rev. B {\\bf 57}, R3201 (1998).\n\n\\bibitem{cft}\nJ. L. Cardy, J. Phys. {\\bf A 17}, L385 (1984);\nH. W. Bl\\\"ote, J. L. Cardy and M. P. Nightingale,\nPhys. Rev. Lett. {\\bf 56}, 742 (1986);\nI. Affleck, Phys. Rev. Lett. {\\bf 56}, 746 (1986).\n\n\\bibitem{kt}\nJ. M. Kosterlitz and D. J. Thouless, J. Phys. {\\bf C 6}, 1181 (1973).\n\n\\bibitem{tonegawa}\nT.Tonegawa, {\\it et al.}, Physica. B {\\bf\n246-247},(1998)509.\n\n\\bibitem{totsuka}\nK. Totsuka,\nPhys. Rev. B {\\bf 57}, 3454 (1998).\n\n\n\\end{references}\n\n\n\n\n\n\n\\widetext\n\\end{document}\n"
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"name": "cond-mat0002113.extracted_bib",
"string": "\\bibitem{azuma}\nM. Azuma {\\it et al.}, Phys. Rev. Lett. {\\bf 73} 3463 (1994). \n\n\n\\bibitem{hayward}\nC. A. Hayward, D. Poilblanc and L. P. L\\'evy,\nPhys. Rev. B {\\bf 54}, R12649 (1996).\n\n\n\\bibitem{chaboussant}\nG. Chaboussant {\\it et al.},\nPhys. Rev. B {\\bf 55}, 3046 (1997).\n\n\n\\bibitem{imai}\nT. Imai {\\it et al.}, Phys. Rev. Lett. {\\bf 81}, 220 (1998).\n\n\n\\bibitem{oshikawa}\nM. Oshikawa, M. Yamanaka and I. Affleck, Phys. Rev. Lett.\n{\\bf 78}, 1984 (1997).\n\n\n\\bibitem{mila} F.Mila, Eur.Phys.J. {\\bf B6}, 201 (1998).\n\n\n\\bibitem{wang}\nX. Wang, in {\\it Density-Matrix Renormalization}, edited by \nI. Peschel {\\it et al.}. (Springer, Berlin, 1999). \n\n\n\\bibitem{sakai}\nT. Sakai and M. Takahashi, Phys. Rev. B {\\bf 57}, R3201 (1998).\n\n\n\\bibitem{cft}\nJ. L. Cardy, J. Phys. {\\bf A 17}, L385 (1984);\nH. W. Bl\\\"ote, J. L. Cardy and M. P. Nightingale,\nPhys. Rev. Lett. {\\bf 56}, 742 (1986);\nI. Affleck, Phys. Rev. Lett. {\\bf 56}, 746 (1986).\n\n\n\\bibitem{kt}\nJ. M. Kosterlitz and D. J. Thouless, J. Phys. {\\bf C 6}, 1181 (1973).\n\n\n\\bibitem{tonegawa}\nT.Tonegawa, {\\it et al.}, Physica. B {\\bf\n246-247},(1998)509.\n\n\n\\bibitem{totsuka}\nK. Totsuka,\nPhys. Rev. B {\\bf 57}, 3454 (1998).\n\n\n"
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In view of the current interest in $d_{x^{2}-y^{2}}+id_{xy}$ superconductors some of their thermodynamic properties have been studied to obtain relevant information for experimental verification. The temperature dependence of the specific heat and superfluid density show marked differences in $d_{x^{2}-y^{2}}+id_{xy}$ state compared to the pure d-wave state. A second order phase transition is observed on lowering the temperature into a $d_{x^{2}-y^{2}}+id_{xy}$ state from the $d_{x^{2}-y^{2}}$ state with the opening up of a gap all over the fermi surface. The thermodynamic quantities in $d_{x^{2}-y^{2}}+id_{xy}$ state are dominated by this gap as in an s-wave superconductor as opposed to the algebraic temperature dependence in pure d-wave states coming from the low energy excitations across the node(s).
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"name": "d+id.tex",
"string": "\\documentstyle[12pt]{article}\n\\topmargin=0cm\n\\oddsidemargin=0truecm\n\\evensidemargin=0truecm\n\\textheight 8.5 in\n\\textwidth 6.5 in\n\\begin{document}\n\n\\begin{center}\n{\\Large {\\bf Thermodynamic properties of $d_{x^{2}-y^{2}}+id_{xy}$ \nSuperconductor}}\\\\ \n\\vspace{1.0cm} \n{\\bf Tulika Maitra}\\footnote{email: tulika@phy.iitkgp.ernet.in}\\\\ \n\n\\noindent Department of Physics \\& Meteorology \\\\ \nIndian Institute of Technology,\nKharagpur 721302 India \\\\ \n\n\\end{center} \n\\begin{abstract}\nIn view of the current interest in $d_{x^{2}-y^{2}}+id_{xy}$ superconductors\nsome of their thermodynamic properties have been studied to obtain relevant\ninformation for experimental verification. The temperature \ndependence of the specific heat and superfluid density show marked \ndifferences in $d_{x^{2}-y^{2}}+id_{xy}$ state compared to the pure d-wave \nstate. A second order phase transition is observed on lowering the temperature\ninto a $d_{x^{2}-y^{2}}+id_{xy}$ state from the $d_{x^{2}-y^{2}}$ state \nwith the opening up of a gap all over the fermi surface. The \nthermodynamic quantities in $d_{x^{2}-y^{2}}+id_{xy}$ state are dominated\nby this gap as in an s-wave superconductor as opposed to the algebraic\ntemperature dependence in pure d-wave states coming from the low energy \nexcitations across the node(s). \n\n\\end{abstract} \n\\noindent PACS Nos. 4.72-h, 74.20.Fg \n\\vspace{.5cm} \n\n\\noindent {\\bf Introduction} \n\\vspace{.5cm} \n\nThe series of experiments carried out over the last few years to establish\nthe nature of symmetry in high temperature superconductors with their level\nof sophistication and ingenuity have thrown up new challenges towards an\nunderstanding of the physics of these systems\\cite{ann,vanh}. New findings \nhave \ncome up with surprising regularity with the latest one being the observation\nof a plateau in the thermal conductivity by Krishana et al.\\cite{kris}\nand its subsequent interpretation in terms of the appearance of a time-reversal\nsymmetry breaking state ($d_{x^{2}-y^{2}}+id_{xy}$)\\cite{laugh}. \n\nAn understanding of the pairing mechanism that underlies the superconducting\ninstability is essential for the emergence of a microscopic \ntheory for these superconductors. Countless experiments have been performed \nand various theoretical models have been proposed to probe the symmetry of \nthe OP in these highly anisotropic \nunconventional superconductors. At present it is almost universally accepted\nthat the OP is highly anisotropic with a symmetry of the d-wave\\cite{ann}. \n\nIn the recent experiment of Krishana et. al.\\cite{kris} thermal \nconductivity \nas functions of both magnetic field and temperature has been measured on \na sample of high $T_c$ superconducting material $Bi_2Sr_2CaCu_2O_8$. \nThey observed that the thermal conductivity initially decreases with the \nincrease of magnetic field and above a particular value of the field, which\ndepends on temperature, thermal conductivity becomes independent of \nfield. These observations gave an indication that the material \nundergoes a phase transition in presence of the magnetic field. The authors \nsuggested that this magnetically induced phase might have a complex order \nparameter symmetry\nsuch as $d_{x^{2}-y^{2}}+id_{xy}$ or $d_{x^{2}-y^{2}}+is$ where the gap is \nnonzero on the entire FS. Corroboration of these results came quickly from \nother groups as well\\cite{aubi}.\n\nLaughlin\\cite{laugh} showed that in \npresence of magnetic field the new superconducting phase must have an OP \nthat violates both time reversal and parity and is of $d_{x^{2}-y^{2}}+id_{xy}$\n symmetry.\nThere were earlier predictions for such a state in the region near a grain \nboundary where the gap has sharp variation across it\\cite{sigrist} with a\nspontaneous current generated along the boundary and in a doped Mott insulator\nwith short range antiferromagnetic spin correlation\\cite{laugh2,rokh}.\n\nIt would, therefore, be interesting to study the thermodynamic behaviour of such\nsuperconductors having OP symmetry of $d_{x^2-y^2}+id_{xy}$ type to provide\nfurther experimental observations to confirm its existence.\nWe use the usual weak coupling theory to obtain the gap functions in the\nregion of parameter space where a $d+id$ state is a stable one, and calculate\nthermodynamic quantities like the specific heat and superfluid density \n(and hence the penetration depth) and contrast them with a pure d-wave state. \n\n\\vspace{.5cm} \n\\noindent {\\bf Model and Calculations} \n\\vspace{.5cm} \n\nUnlike pure d-wave OP, the $d_{x^{2}-y^{2}}+id_{xy}$ gap function has no \nnode along the FS.\nThe OP has non-zero magnitude all over but it changes sign in each quadrant \nof the Brillouin zone, while a pure s-wave OP does also have non vanishing \nmagnitude but its sign remains same throughout.\nThis non vanishing gap inhibits creation of quasiparticle excitations\nat low energies whereas in a \npure d-wave state the gapless excitations are available in large \nnumbers at low energies due to the presence of line nodes.\nTaking a tight binding model for a 2-dimensional square lattice, various \nphysical quantities have been calculated within the framework of the usual\nweak coupling theory. The effective interaction has been taken \nin the separable form\\cite{kot} and expanded in the relevant basis functions\nof the irreducible representation of $C_{4v}$. \n$$V{(\\bf k-k')}=\\sum_{i=1,2}V_{i}\\eta_{i}{(\\bf k)}\\eta_{i}{(\\bf k')}$$ \n\\noindent where \n$\\eta_{1}{(\\bf k)}=\\frac{1}{2}(cosk_{x}-cosk_{y})$ and \n$\\eta_{2}{(\\bf k)}=sink_{x}sink_{y}$ (respectively for $d_{x^{2}-y^{2}}$ and \n$d_{xy}$ symmetries). $V_{1}/8$ and\n$V_{2}/8$ are the respective coupling strengths for the near-neighbour and\nnext near-neighbour interactions.\nThe coupling strengths have been chosen in such a way as to allow both \nthe components of the OP to exist simultaneously and the superconducting\ntransition temperature of $d_{xy}$ component to be lower than that of \n$d_{x^2-y^2}$ and is in the range of the observed values.\nConsidering only the nearest neighbour hopping, the band dispersion \nis $\\epsilon_{\\bf k}=-2t(cosk_{x}+cosk_{y})$ where $t$ is the nearest \nneighbour hopping integral and expanding the OP as \n$\\Delta_{\\bf k}=\\sum_{i=1,2}\\Delta_{i}\\eta_{i}{(\\bf k)}$ for $\\eta_{i}({\\bf k})$\ndefined above (for the $d_{x^{2}-y^{2}}+id_{xy}$ symmetry), the \nstandard mean-field gap equation becomes a set of two coupled equations\n$$\\Delta_{1}=-V_{1} {\\bf \\sum_k} \\frac{\\Delta_{1}}{2E_{\\bf k}} \n\\eta_{1}^2({\\bf k})tanh\\left(\\frac{E_{\\bf k}}{2k_BT}\\right)$$\n\\noindent and $$\\Delta_{2}=-V_{2}{\\bf \\sum_k}\\frac{\\Delta_{2}}{2E_{\\bf k}}\n\\eta_{2}^2({\\bf k})tanh\\left(\\frac{E_{\\bf k}}{2k_BT}\\right).$$ \n\\noindent Here the quasiparticle spectrum in the ordered state is given by \n$E_{\\bf k}=\\sqrt{{(\\epsilon_{\\bf k}-\\mu)}^{2}+|\\Delta_{k}|^{2}}$ \nwhere $\\mu$ is the chemical potential. The coupled set of gap equations are \nsolved numerically in a selfconsistent manner \nwith the parameters $t=0.15$ eV, $V_{1}=0.445t$ eV and $V_{2}=3.202t$ eV.\nThe solutions give the expected square root temperature dependences of the \ntwo components of the order parameter $d_{x^2-y^2}$($\\Delta_{1}$) and \n$d_{xy}$($\\Delta_{2}$) and the corresponding $T_c$s ($T_{c1}$ and $T_{c2}$) \nas shown in Fig. 1. It is to be noted that the consistent solutions exist\nfor both $d_{x^2-y^2}$ and $d_{xy}$ components of the OP for a very narrow\nrange of $V_1$ and $V_2$. The $d_{xy}$ component of the OP exists\nonly when the next nearest neighbour interaction ($V_2$) is taken into account.\nIt has also been observed that the solutions have sensitive dependence on the \nvalues of the chemical potential and the next nearest neighbour \nhopping integral($t'$). To be more specific, if we change the value of chemical \npotential to $-0.25$ eV from $\\mu = 0$ with $t'=0$, the $d+id$ state ceases \nto exist, but the inclusion of the $t'$ term (with $t'=0.4t$) brings \nthe $d+id$ state back. The solutions, of course, exist only for a narrow \nrange of values of \nthe chemical potential: for instance $\\mu=-0.22$ eV to $\\mu=-0.26$ eV with\n$t'=0.4t$ has well defined solutions with $\\Delta_1(0)>\\Delta_2(0)$. \nSimilarly if we keep the value of $\\mu$ fixed at any of the above \nvalues and start changing the value of $t'$, only a very narrow range of\n$t'$ gives us a $d+id$ solution. This interplay of $t'$ and $\\mu$ is dictated\nby the location of the van Hove singularity (vHS) with respect to the \nfermi energy.\n\nThe quasiparticle spectrum along different directions in \nthe first quadrant of the Brillouin zone is shown in Fig. 2 and the finite\ngap along all $k-$points is clearly visible.\nWith the excitation spectrum thus obtained, it is straightforward to calculate\nthermodynamic quantities, namely, the specific heat and superfluid\ndensity, in the different ordered states. From the usual definition \nin terms of the \nderivative of entropy\\cite{tin}, we calculate the specific heat across the \ntransitions and show it in Fig. 3. Two sharp jumps in the specific heat curve \nare observed at the respective transition temperatures.\n\nThe superfluid density $\\rho_{s}$(T) has been calculated using the \nstandard techniques of many body theory\\cite{scal,tar}. In the presence of a\ntransverse vector potential with the chosen gauge \n$A_{y}=0$, the hopping matrix element($t_{ij}$) for the kinetic energy term in\nthe Hamiltonian$(H_{0})$ is modified by the Peierl's phase \nfactor $exp[\\frac{ie}{\\hbar c}\\int_{{\\bf r}_{j}}^{{\\bf r}_{i}}{\\bf A}.d{\\bf l}]\n$. The total current (in the linear response) $J_{x}({\\bf r_{i}})$ produced by\nthe potential consists of both the diamagnetic and paramagnetic terms and \ncan be derived by differentiating $H_{0}$ with respect to $A_{x}({\\bf r}_{i})$. \nHence\n$$j_{x}({\\bf r}_{i})=-c {\\frac{\\partial{H_{0}}}{\\partial{A_{x}}({\\bf r}_{i})}}\n={j_{x}}^{para}({\\bf r}_{i})+{j_{x}}^{dia}({\\bf r}_{i})$$ \n\\noindent where the paramagnetic current in the long \nwavelength limit in the linear response is given by\n$${\\bf j}_{x}^{para}({\\bf q})=-\\frac{i}{c}\\,\\,lim_{q\\rightarrow 0}\nlim_{\\omega\\rightarrow 0}\\int d\\tau\\theta(\\tau)e^{i\\omega\\tau} \\langle \n[j_{x}^{para}({\\bf q},\\tau),j_{x}^{para}(-{\\bf q},0)]\\rangle \n{\\bf A_{x} ({\\bf q})},$$ \n\\noindent and the diamagnetic part is given by\n$${\\bf j}_{x}^{dia}({\\bf q})= -\\frac{e^{2}}{N\\hbar^{2}c}\\sum_{{\\bf k},\\sigma}\n\\langle c^{\\dagger}_{{\\bf k},\\sigma} c_{{\\bf k},\\sigma}\\rangle{\\frac{\\partial^{2}\\epsilon_{\\bf k}}{\\partial^{2}{k_{x}}^{2}}}{\\bf A_{x}({\\bf q})}.$$\n \n\\noindent Here the averaging is done in the mean-field superconducting state.\nFig. 4 shows the variation of $\\rho_{s}$ with temperature. At low temperatures \nwhere the superconductor is in $d_{x^2-y^2}+id_{xy}$ state, the superfluid \ndensity \nexhibits an exponential decay reflecting the gapped excitations. Above the \nsecond transition temperature($T_{c2}$) \nat which the $d_{xy}$ component of the OP vanishes and the \nsuperconductor undergoes a transition to $d_{x^2-y^2}$ phase, the superfluid \ndensity curve shows a power law behaviour expected from the low energy \nquasiparticles.\n\\vspace{.5cm} \n\n\\noindent {\\bf Results and Discussion} \n\\vspace{.5cm} \n\nThe self-consistent solutions for the order parameters (Fig. 1) show that as we\ndecrease the temperature, first there is a continuous transition into a\nsuperconducting state where the OP is of $d_{x^2-y^2}$ symmetry with no\n$d_{xy}$ component. On further decreasing the temperature a second continuous\ntransition occurs and the $d_{xy}$ component appears (with a phase $\\pi/2$\nwith respect to the $d_{x^2-y^2}$ component) breaking the time reversal \nsymmetry.\nA stable $d+id$ phase does not exist unless the next nearest\nneighbour interaction is being considered. This is because the next nearest\nneighbour attraction accounts for the pairing along the (110) direction.\nThe sensitive dependence of the solutions on the chemical potential and the\nnext near neighbour hopping integral is understood by studying the nature of\nthe non-interacting density of states (DOS).\nIt has been noticed that the van Hove singularity(vHS) in the non-interacting\nDOS lies far away from the fermi level when we include the $t'$ term in the \nband keeping the chemical potential zero, but if in addition we change the\nthe chemical potential to $-0.25$ eV, the vHS moves close to \nthe fermi level. \n\nIn the $d_{x^{2}-y^{2}}+id_{xy}$ state there exists no node on the FS, a \ngap opens throughout. Hence the low energy quasiparticle \nexcitations are exponentially down in comparison to the pure d-wave state that\nhas line nodes on the FS. This is borne out from the plot of the quasiparticle \nenergy spectrum along different directions of BZ (Fig. 2).\n\nAs temperature decreases from $T_c$ corresponding to the $d_{x^2-y^2}$ state, \nthe low energy quasiparticle excitations are exponentially low in the \n$d_{x^{2}-y^{2}}+id_{xy}$ state due to the appearance of an additional OP \nof $d_{xy}$ symmetry and phased by $90$ degree with the existing \n$d_{x^{2}-y^{2}}$ OP. The thermodynamic quantities are therefore affected in \nthis new state quite severely. The temperature dependence of the specific heat \n(Fig. 3) shows the difference. The sharp jumps at transition temperatures in \nthe specific heat curve, are clear indication of second order \ntransitions\\cite{ang}. The nature of the curve has significant difference\nin the two superconducting states (pure $d_{x^2-y^2}$ and the $d+id$). \nIn the $d_{x^{2}-y^{2}}+id_{xy}$ state the specific heat\nincreases exponentially with temperature, more like the familiar s-wave \nsuperconductors whereas in the d-wave state the growth is more stiff. This in \nturn indicates that the entropy is higher in pure d-wave state than that in\n$d_{x^2-y^2}+id_{xy}$ state. So the low temperature $d+id$ phase, in a way, is \nmore ordered than the higher temperature $d_{x^{2}-y^{2}}$ phase. \n\nThe curve for the superfluid density as a function of temperature (Fig. 4)\nbehaves differently in the two superconducting phases as expected. In the \n$d_{x^{2}-y^{2}}+id_{xy}$ \nphase $\\rho_s$ falls exponentially with temperature whereas in $d_{x^{2}-y^{2}}$\nphase the descent is according to a power law. At the second transition \ntemperature($T_{c2}$), where the \ntransition occurs between the two superconducting phases, a sudden \nupturn appears in the $\\rho_{s}(T)$ curve which reflects the availability \nof quasiparticle \nexcitations due to the disappearance of the $d_{xy}$ state.\nIf we compare these results with that\nof an s-wave superconductor, we observe that the behaviour of $\\rho_{s}$ in the \n$d_{x^{2}-y^{2}}+id_{xy}$ state is qualitatively similar to that of the s-wave\nstate, with a gap all over the FS. Owing to this gap, the \nquasiparticle excitations are not easily accessible at very low temperatures\nand keeps the superfluid density almost independent of temperature at \nlow temperatures. This exponential behaviour is expected in the \nthermodynamic properties whenever there exists a gap in the excitation \nspectrum. \n \nIn conclusion, the thermodynamic properties of the $d_{x^{2}-y^{2}}+id_{xy}$\nsuperconductor are studied with a tight binding model within the \nmean-field theory. Significant differences have been observed in the nature\nof the temperature dependence of specific heat and superfluid density between\na pure d-wave state and the $d_{x^{2}-y^{2}}+id_{xy}$ state. The behaviour in\nthe latter is found to be somewhat similar to that of an s-wave superconductor.\nFurther experimental observations on the thermodynamics of this state \nwill shed light on the microscopic nature of interactions in these \nnew class of superconductors. \n\n\\vspace{.5cm} \n\\noindent {\\bf Acknowledgement} It is a pleasure to thank A. Taraphder for\nuseful discussions. \n\n\\newpage \n\\begin{thebibliography}{999}\n\\bibitem{ann} J. Annett, N. Goldenfeld and A. J. Leggett in {\\it Physical \nProperties of High Tempearture Superconductors}, vol. 5, ed. D. M. Ginsberg, \n(World Scientific) (1996). \n\n\\bibitem{vanh} Van Harlingen, Rev. Mod. Phys., {\\bf 67} 515 (1995). \n\n\\bibitem{kris} K. Krishana, et al.,Science, {\\bf 277} 83 (1997). \n\\bibitem{laugh} R. B. Laughlin, Phys. Rev. Lett., {\\bf 80} 5188 (1998).\n\\bibitem{aubi} H. Aubin, et al., Phys. Rev. Lett., {\\bf 82} 624 (1999). \n\\bibitem{sigrist} See M. Sigrist et. al., Prog. Theor. Phys., {\\bf 99} \n899 (1998) {\\it for earlier references}. \n\\bibitem{laugh2} R. B. Laughlin, Physica (Amsterdam), {\\bf 234C} 280 (1994).\n\\bibitem{rokh} D. S. Rokhsar, Phys. Rev. Lett. {\\bf 70}, 493 (1993);\n\\bibitem{kot} G. Kotliar, Phys. Rev. B {\\bf 37} 3664 (1988). \n\\bibitem{tin} M. Tinkham, {\\it Introduction to superconductivity}, (McGraw-Hill\nInc., New York, 1975). \n\\bibitem{scal} D. J. Scalapino, S. R. White and S. C. Zhang, Phys. Rev. \nLett., {\\bf 68}, 2830 (1992). \n\\bibitem{tar} B. Chattopadhya, D. M. Gaitonde and A. Taraphder, \nEurophys. Lett., {\\bf 34}, 705 (1996); Tulika Maitra\nand A. Taraphder, Physica C, {\\bf 325} 61 (1999).\n\\bibitem{ang} Angsula Ghosh and Sadhan K. Adhikari, Preprint.\n\\end{thebibliography}\n\n\\newpage\n\\center {\\Large \\bf Figure captions}\n\\vspace{0.5cm} \n\\begin{itemize} \n\n\\item[Fig. 1.] The gap parameters $\\Delta_{1}$ and $\\Delta_{2}$ (in Kelvin)\nversus temperature (in Kelvin). \n\n\\item[Fig. 2.] The quasiparticle energy spectrum ($E_{\\bf k}$) (in meV)\nalong various symmetry directions in the first quadrant of BZ (the gap \nmagnitudes have been increased ten times for visualisation). The inset shows\nhow the symmetry directions are defined in BZ.\n\n\\item[Fig. 3] The specific heat versus temperature curve in $d_{x^{2}-y^{2}}+id_{xy}$ and $d_{x^{2}-y^{2}}$ states clearly shows the difference in its\nbehaviour in these two states. The dotted line shows the normal state\nspecific heat.\n \n\\item[Fig. 4] The superfluid density is shown against temperature for \ntwo phases $d_{x^{2}-y^{2}}+id_{xy}$ and $d_{x^{2}-y^2}$.\n \n\\end{itemize} \n\n\\end{document}\n\n \n\n\n"
}
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[
{
"name": "cond-mat0002114.extracted_bib",
"string": "\\begin{thebibliography}{999}\n\\bibitem{ann} J. Annett, N. Goldenfeld and A. J. Leggett in {\\it Physical \nProperties of High Tempearture Superconductors}, vol. 5, ed. D. M. Ginsberg, \n(World Scientific) (1996). \n\n\\bibitem{vanh} Van Harlingen, Rev. Mod. Phys., {\\bf 67} 515 (1995). \n\n\\bibitem{kris} K. Krishana, et al.,Science, {\\bf 277} 83 (1997). \n\\bibitem{laugh} R. B. Laughlin, Phys. Rev. Lett., {\\bf 80} 5188 (1998).\n\\bibitem{aubi} H. Aubin, et al., Phys. Rev. Lett., {\\bf 82} 624 (1999). \n\\bibitem{sigrist} See M. Sigrist et. al., Prog. Theor. Phys., {\\bf 99} \n899 (1998) {\\it for earlier references}. \n\\bibitem{laugh2} R. B. Laughlin, Physica (Amsterdam), {\\bf 234C} 280 (1994).\n\\bibitem{rokh} D. S. Rokhsar, Phys. Rev. Lett. {\\bf 70}, 493 (1993);\n\\bibitem{kot} G. Kotliar, Phys. Rev. B {\\bf 37} 3664 (1988). \n\\bibitem{tin} M. Tinkham, {\\it Introduction to superconductivity}, (McGraw-Hill\nInc., New York, 1975). \n\\bibitem{scal} D. J. Scalapino, S. R. White and S. C. Zhang, Phys. Rev. \nLett., {\\bf 68}, 2830 (1992). \n\\bibitem{tar} B. Chattopadhya, D. M. Gaitonde and A. Taraphder, \nEurophys. Lett., {\\bf 34}, 705 (1996); Tulika Maitra\nand A. Taraphder, Physica C, {\\bf 325} 61 (1999).\n\\bibitem{ang} Angsula Ghosh and Sadhan K. Adhikari, Preprint.\n\\end{thebibliography}"
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cond-mat0002115
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Delocalization in the Anderson model due to a local measurement
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"author": "S.A. Gurvitz"
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We study a one-dimensional Anderson model in which one site interacts with a detector monitoring the occupation of that site. We demonstrate that such an interaction, no matter how weak, leads to total delocalization of the Anderson model, and we discuss the experimental consequences.
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"name": "deloc2.tex",
"string": "\\documentstyle [prl,multicol,aps,psfig]{revtex}\n\\begin{document}\n\\title{Delocalization in the Anderson model due to a local measurement}\n\\author{S.A. Gurvitz}\n\\address{Department of Particle Physics,\nWeizmann Institute of Science, Rehovot~76100, Israel}\n\\vspace{18pt}\n\\maketitle\n\\begin{abstract}\nWe study a one-dimensional Anderson model in which one site \ninteracts with a detector monitoring the occupation of that \nsite. We demonstrate that such an interaction, no matter how weak, \nleads to total delocalization of the Anderson model, and we \ndiscuss the experimental consequences.\n\\end{abstract}\n\\hspace{1.5 cm} PACS: 03.65.Bz, 73.20.Fz, 73.20.Jc \n\\vspace{18pt}\n\\begin{multicols}{1}\nConsider an electron in a one-dimensional array of $N$ coupled wells.\nThe system is described by the Anderson tunneling Hamiltonian\n\\begin{equation}\nH_A=\\sum_{j=1}^NE_jc^\\dagger_jc_j+\\sum_{j=1}^{N-1}\n(\\Omega_j c^\\dagger_{j+1}c_j+H.c.)\\ ,\n\\label{a1}\n\\end{equation}\nwhere the operator $c^\\dagger_j\\ (c_j)$ corresponds to the creation \n(annihilation) of an electron in the well $j$. \nWe assume for simplicity that each of the wells contains one \nbound state $E_j$ and is coupled only to its nearest neighbors\nwith couplings $\\Omega_j$ and $\\Omega_{j-1}$. \n(We choose $\\Omega_j$ real without loss of generality.)\n\nThe electron-wave function in this system can be written as\n$|\\Psi (t)\\rangle =\\sum_j b_j(t)c^\\dagger_j |0\\rangle$, where $b_j(t)$ \nis the probability amplitude of finding the electron in the well $j$\nat time $t$. These amplitudes are obtained from \nthe time-dependent Schr\\\"odinger equation $i\\partial_t|\\Psi (t)\\rangle \n=H_A|\\Psi (t)\\rangle$. \nIt is well known that for randomly distributed levels $E_j$ (or \nrandom couplings $\\Omega_j$) all electronic states in this structure \nare localized\\cite{ander}. Hence, if the electron initially occupies \nthe first well, $b_j(0)=\\delta_{j1}$, the probability of finding it \nin the last well, $P_N(t)=|b_N(t)|^2$ drops exponentially with $N$:\n$\\langle P_N(t\\to\\infty)\\rangle_{ensemble}\\to\\exp(-\\alpha N)$. \n\nAnderson localization is usually associated with destructive \nquantum-mechanical interference between different probability \namplitudes $b_j(t)$. This interference, however, can be affected \nby measuring the electron's position in the system \ndue to interaction of the electron with \na macroscopic detector. For instance, the continuous monitoring of one \nof the wells of a double-well system ($N=2$ in Eq.~(\\ref{a1})) \ndestroys the off-diagonal elements (coherences) \nof the electron density matrix. As a result, the latter become \nthe statistical mixture: \n$\\sigma_{jj'}(t)\\to (1/2)\\delta_{jj'}$ for $t\\to\\infty$\\cite{gur1}. \n\nIn the case of the $N$-well structure, however, the monitoring of one \nof the wells cannot determine the electron's position in the entire \nsystem. One might suppose therefore that such a local measurement cannot \ntotally destroy the interference inside the entire system and hence, \nthe electron localization. We demonstrate in this letter the contrary:\nany interaction, no matter how weak, of the electron with \na macroscopic detector placed on only {\\em one} \nwell leads to total delocalization of the electron state:\n$P_N(t\\to\\infty)= 1/N$, even when $N\\to\\infty$. \n\nAs a physical realization we consider a mesoscopic system of coupled \nquantum dots (Fig.~1), where a point contact is placed near the first dot. \nThe point contact\n\\begin{figure} \n\\psfig{figure=fig1.eps,height=5cm,width=8.5cm,angle=0}\n\\noindent\n{\\bf Fig.~1:}\nThe point-contact detector near the array of coupled dots\nwith randomly distributed energy levels, \ndescribed by the Anderson Hamiltonian (\\ref{a1}). The detector \nmonitors only the occupation of the first dot.\n\\end{figure}\n\\noindent\nis coupled to two reservoirs, emitter and collector, at different chemical \npotentials, $\\mu_L$ and \n$\\mu_R$. A current $I=eT(\\mu_L-\\mu_R)/(2\\pi)$ flows through \nthe point contact\\cite{land}, \nwhere $T$ is its transmission coefficient. If the electron occupies \nthe first dot, the transmission coefficient of the point contact \ndecreases, $T'<T$, due to the electrostatic repulsion \ngenerated by the electron. As a result, \nthe current $I'<I$ (Fig.~1a). The current \nreturns to its previous value $I$ whenever the electron occupies \nany other dot, since then it is far away from the contact (Fig.~1b). \n\nThe entire system can be described by the tunneling \nHamiltonian $H=H_A+H_{PC}+H_{int}$, where $H_A$ is given by \nEq.~(\\ref{a1}) and \n\\begin{eqnarray}\nH_{PC}&=&\\sum_lE_la_l^\\dagger a_l+\\sum_rE_ra_r^\\dagger a_r\n+\\sum_{l,r}(\\Omega_{lr}a_r^\\dagger a_l+H.c.)\n\\nonumber\\\\\nH_{int}&=&\\sum_{l,r}\\delta\\Omega_{lr}c^\\dagger_1c_1\n(a_r^\\dagger a_l+H.c.),\n\\label{a3}\n\\end{eqnarray}\nwhere $a_l^\\dagger (a_l)$ and $a_r^\\dagger (a_r)$ are the creation \n(annihilation) operators in the left and the right reservoirs, \nand $\\Omega_{lr}$ is the hopping amplitude between the \nstates $l$ and $r$ of the reservoirs. \n\nConsider an initial state where all the levels in the emitter \nand the collector are filled \nup to the Fermi energies $\\mu_L$ and $\\mu_R$, respectively, \nand the electron occupies \nthe first well. The many-body wave function describing the entire \nsystem can be written in the occupation number representation as \n\\begin{eqnarray}\n&&|\\Psi (t)\\rangle =\\sum_{j=1}^N\\left [b_j(t)c_j^\\dagger\n+\\sum_{l,r}b_{jlr}(t)c_j^\\dagger a_r^\\dagger a_l\\right.\\nonumber\\\\\n&&\\left.~~~~~~~~~~~+\\sum_{l<l',r<r'}\nb_{jll'rr'}(t)c_j^\\dagger a_r^\\dagger a_{r'}^\\dagger a_l a_{l'}+\\cdots\n\\right ]|0\\rangle\\ ,\n\\label{a4}\n\\end{eqnarray}\nwhere $b(t)$ are the probability amplitudes of finding the system \nin the states defined by the corresponding creation and \nannihilation operators. Using these amplitudes one defines \nthe reduced density matrices $\\sigma_{jj'}^{(m)} (t)$ that \ndescribe the electron and the detector, \n\\begin{eqnarray}\n&&\\sigma^{(0)}_{jj'}(t)=b_j(t)b^*_{j'}(t),~~~~\n\\sigma^{(1)}_{jj'}(t)=\\sum_{l,r}b_{jlr}(t)b^*_{j'lr}(t),~~~~\\nonumber\\\\\n&&\n\\sigma^{(2)}_{jj'}(t)=\\sum_{ll',rr'}b_{jll'rr'}(t)b^*_{j'll'rr'}(t),\\; \n~\\cdots\\\n\\label{a5}\n\\end{eqnarray}\nHere $j,j'=\\{ 1,2,\\ldots ,N\\}$ denote the occupation states of \nthe $N$-dot system. \nThe index $m$ denotes the number of electrons that have reached \nthe right-hand reservoir by time $t$. The total probability \nfor the electron to occupy the dot $j$ is\n$\\sigma_{jj}(t)=\\sum_m\\sigma^{(m)}_{jj}(t)$. The off-diagonal \ndensity-matrix element $\\sigma_{jj'}(t)=\\sum_m\\sigma^{(m)}_{jj'}(t)$\ndescribes interference between the states $E_j$ and $E_{j'}$.\n\nIn order to find the amplitudes $b(t)$, we \nsubstitute Eq.~(\\ref{a4}) into the time-dependent \nSchr\\\"odinger equation $i\\partial_t|\\Psi (t)\\rangle \n=H|\\Psi (t)\\rangle$, and use the\nLaplace transform $\\tilde b(E)=\\int_0^\\infty b(t)\\exp (iEt)dt$.\nThen we find an infinite set of algebraic equations for \nthe amplitudes $\\tilde b(E)$, given by \n\\begin{mathletters}\n\\label{a6}\n\\begin{eqnarray}\n&&(E-E_1) \\tilde{b}_{1} - \\Omega_1\\tilde b_2 \n-\\sum_{l,r} \\Omega'_{lr}\\tilde{b}_{1lr}=i\n\\label{a6a}\\\\\n&&(E-E_2) \\tilde{b}_{2} - \\Omega_1\\tilde b_1 -\\Omega_2\\tilde b_3 \n- \\sum_{l,r} \\Omega_{lr}\\tilde{b}_{2lr}=0\n\\label{a6b}\\\\\n&&(E + E_{l}-E_1 - E_r) \\tilde{b}_{1lr} - \\Omega'_{lr}\\tilde{b}_1\n-\\Omega_1\\tilde b_{2lr}\\nonumber\\\\ \n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n-\\sum_{l',r'}\\Omega'_{l'r'}\\tilde{b}_{1ll'rr'}=0\n\\label{a6c}\\\\\n&&(E + E_{l}-E_2 - E_r) \\tilde{b}_{2lr} - \\Omega_{lr}\\tilde{b}_2\n- \\Omega_1\\tilde{b}_{1lr} \\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~ \n-\\Omega_2\\tilde b_{3lr}-\\sum_{l',r'}\\Omega_{l'r'}\\tilde{b}_{2ll'rr'}=0\n\\label{a6d}\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\cdots\\, , \n\\nonumber\n\\end{eqnarray}\n\\end{mathletters}\nwhere $\\Omega'_{lr}=\\Omega_{lr}+\\delta\\Omega_{lr}$.\n \nEqs.~(\\ref{a6}) can be converted to Bloch-type equations for \nthe density matrix $\\sigma_{jj'}(t)$ without their explicit \nsolution. This technique has been derived in\n\\cite{gur1,gur2}. We explain below only the main points of \nthis procedure and the conditions for its validity.\n \nConsider, for example, Eq.~(\\ref{a6a}). \nIn order to perform the summation in the term \n$\\sum_{l,r} \\Omega'_{lr}\\tilde{b}_{1lr}$, we solve for $\\tilde b_{1lr}$\nin Eq.~(\\ref{a6c}). Then substituting the result \ninto the sum, we can rewrite Eq.~(\\ref{a6a}) as \n\\begin{eqnarray}\n&&\\left (E-E_1\n-\\int{{\\Omega'}^2_{lr}\\rho_L(E_l)\\rho_R(E_r)dE_ldE_r\n\\over E+E_l-E_1-E_r}\\right ) \n\\tilde{b}_{1}\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- \\Omega_1\\tilde b_2 \n+{\\cal F}=i\\ ,\n\\label{a7}\n\\end{eqnarray}\nwhere we have replaced the sum in Eq.~(\\ref{a6a}) by an integral \n$\\sum_{l,r}\\;\\rightarrow\\;\\int \n\\rho_{L}(E_{l})\\rho_{R}(E_{r})\\,dE_{l}dE_r\\:$,\nwith $\\rho_{L,R}$ the density of states in the emitter and collector. \nWe split this integral into its principle value and singular part. \nThe singular part yields $iD'/2$, where \n$D'=2\\pi{\\Omega'}^2\\rho_L\\rho_R (\\mu_L-\\mu_R)$, and \nthe principal part is zero, providing $\\Omega'_{lr}$ and \n$\\rho_{L,R}$ are weakly dependent on the energies $E_{l,r}$. \nNote that $(2\\pi)^2\\Omega^2\\rho_L\\rho_R=T$\\cite{bardeen},\nwhere $T$ is the tunneling transmission coefficient of the point contact. \nThus, $eD'=I'$ is the current \nflowing through the point contact\\cite{land} whenever \nthe electron occupies the first dot.\n\nThe quantity ${\\cal F}$ in Eq.~(\\ref{a7}) denotes \nthe terms in which the amplitudes $\\tilde b$ \ncannot be factored out of the integrals. \nThese terms vanish in the large-bias limit,\n$(\\mu_L-\\mu_R)\\gg\\Omega^2\\rho$. \nIndeed, all the singularities \nof the amplitude $\\tilde{b} (E,E_l,E_{l'},E_r,E_{r'})$\nin the $E_l, E_{l'}$ variables lie below the real axis. \nThis can be seen directly from Eqs.~(\\ref{a6})\nby noting that $E$ lies above the real axis in the Laplace \ntransform. Assuming that the transition amplitudes \n$\\Omega$ as well as the densities of states $\\rho_{L,R}$ are\nindependent of $E_{l,r}$, one can close the integration contour \nin the upper $E_{l,r}$-plane. Since the integrand decreases \nfaster than $1/E_{l,r}$, the resulting integrals are zero. \n\nApplying analogous considerations to the other equations of the\nsystem (\\ref{a6}) we convert Eqs.~(\\ref{a6}) directly into \nrate equations via the inverse Laplace transform. \nThe details can be found in\n\\cite{gur1,gur2,eg}. Here we present only the final equations \nfor the electron density matrix $\\sigma_{jj'}(t)$:\n\\begin{mathletters}\n\\label{a10}\n\\begin{eqnarray}\n&&\\dot\\sigma_{jj}=i\\Omega_{j-1} (\\sigma_{j,j-1}-\\sigma_{j-1,j})\n\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n+i\\Omega_j (\\sigma_{j,j+1}-\\sigma_{j+1,j}),\n\\label{a10a}\\\\\n&&\\dot\\sigma_{jj'}=i\\epsilon_{j'j}\\sigma_{jj'}+\ni\\Omega_{j'-1}\\sigma_{j,j'-1}+i\\Omega_{j'}\\sigma_{j,j'+1}\\nonumber\\\\\n&&~-i\\Omega_{j-1}\\sigma_{j-1,j'}-i\\Omega_j\\sigma_{j+1,j'}\n-\\frac{\\Gamma}{2}\\sigma_{jj'}(\\delta_{1j}+\\delta_{1j'})\\, ,\n\\label{a10b}\n\\end{eqnarray}\n\\end{mathletters}\nwhere $\\epsilon_{j'j}=E_{j'}-E_j$ and \n$\\Gamma=(\\sqrt{I/e}-\\sqrt{I'/e})^2$ is the decoherence rate,\ngenerated by interaction with the detector.\nNote that these equations have been obtained \nfrom the many-body Schr\\\"odinger equation for the entire system. \nNo stochastic assumptions have been made in their derivation.\n\nEqs.~(\\ref{a10}) are analogous to the well-known optical Bloch equations\nused to describe a multilevel \natom interacting with the quantized electromagnetic field\\cite{bloch}.\nTo our knowledge, this is their first appearance in connection with \nthe Anderson model. The equations can be rewritten in \nLindblad\\cite{lind} form as \n\\begin{equation}\n\\dot\\sigma=-i[H_A,\\sigma ]-{\\Gamma\\over 2}(Q\\sigma+\\sigma Q\n-2\\tilde Q\\sigma \\tilde Q^\\dagger)\\ ,\n\\label{lind}\n\\end{equation}\nwhere $H_A$ is given by Eq.~(\\ref{a1}) and \n$Q_{jj'}=\\tilde Q_{jj'}=\\delta_{1j}\\delta_{1j'}$.\nIf $\\Gamma=0$, Eq.~(\\ref{lind})\nis equivalent to the Schr\\\"odinger equation $i\\partial_t|\\Psi (t)\\rangle \n=H_A|\\Psi (t)\\rangle$. In this case the electron \ndensity matrix $\\sigma (t)$ displays Anderson localization, \ni.e., $\\sigma_{NN}(t\\to\\infty )\n\\sim\\exp(-\\alpha N)$. If $\\Gamma\\not =0$, however, the \nasymptotic behavior of the reduced density-matrix, \n$\\sigma_{jj'}(t\\to\\infty)$, changes dramatically: \nall eigenfrequencies (except for the zero mode) \nobtain an imaginary part due to the second \n(damping) term in Eq.~(\\ref{lind}), so that only the stationary terms \nsurvive in the limit $t\\to\\infty$. This damping is illustrated in Fig.~2 \nwhich displays the numerical\n\\begin{figure} \n\\psfig{figure=fig2.eps,height=7cm,width=8.5cm,angle=0}\n\\noindent\n{\\bf Fig.~2:}\n$P_1(t)$ and $P_4(t)$ represent the occupation of the first and \nthe last dot as a function of time. \nThe dashed lines correspond to $\\Gamma=0$ (no interaction with \nthe environment) and the solid lines correspond to $\\Gamma/\\bar\\Omega =1$.\n\\end{figure}\n\\noindent\nsolution of Eqs.~(\\ref{a10}) for $N=4$, $\\Omega_j=\\bar\\Omega$=const, and \n$E_j/\\bar\\Omega =\\{0,\\ 2,\\ 4, \\ 1\\}$. The occupation \nof the first dot, $P_1(t)=\\sigma_{11}(t)$, and the last dot, \n$P_4(t)=\\sigma_{44}(t)$, \nis shown in Fig.~2 for $\\Gamma=0$ by the dashed lines, and for \n$\\Gamma/\\bar\\Omega=1$ by the solid lines. One can clearly see \nthat all oscillations decay for $\\Gamma\\not =0$, so that \nthe density matrix reaches a stationary limit. Then we see the \nopposite of localization, as the\nprobability of finding the electron in the last dot, \n$P_4(t)$, becomes the same as the probability of finding it in \nthe first dot, $P_1(t)$. \n\nThe delocalization phenomenon, illustrated by Fig.~2, can be proven\nanalytically for any $N$. \nIndeed, let us consider Eqs.~(\\ref{a10}) in the asymptotic limit \n$t\\to\\infty$, where the electron density matrix reaches its stationary \nlimit: $\\sigma_{jj'}(t\\to\\infty ) =u_{jj'}+i\\ v_{jj'}$. \nSince for the stationary solution $\\partial_t\\sigma_{jj'}\\to 0$, \nEqs.~(\\ref{a10}) become \n\\begin{mathletters}\n\\label{a11}\n\\begin{eqnarray}\n0& = & \\epsilon_{j'j}v_{jj'}+\n\\Omega_{j'-1}v_{j,j'-1}+\\Omega_{j'}v_{j,j'+1}\n-\\Omega_{j-1}v_{j-1,j'}\\nonumber\\\\\n&&~~~~~-\\Omega_jv_{j+1,j'}\n+\\frac{\\Gamma}{2}u_{jj'}(\\delta_{1j}+\\delta_{1j'})(1-\\delta_{jj'})\\ ,\n\\label{a11a}\\\\\n0& = & \\epsilon_{j'j}u_{jj'}+\n\\Omega_{j'-1}u_{j,j'-1}+\\Omega_{j'}u_{j,j'+1}\n-\\Omega_{j-1}u_{j-1,j'}\\nonumber\\\\\n&&~~~~~-\\Omega_ju_{j+1,j'}\n-\\frac{\\Gamma}{2}v_{jj'}(\\delta_{1j}+\\delta_{1j'})(1-\\delta_{jj'})\\ .\n\\label{a11b}\n\\end{eqnarray}\n\\end{mathletters}\nEqs.~(\\ref{a11}) have the unique solution \n$v_{jj'}=0$ and $u_{jj'}=(1/N)\\delta_{jj'}$. \nThis can be obtained by solving these equations sequentially, \nstarting with $j,j'=N$, and then continuing for $j,j'=N-1,N-2,\\ldots$. \nSince $u_{jj}\\equiv\\sigma_{jj}(t\\to\\infty )$, \nwe finally obtain that \n\\begin{equation}\n\\sigma_{jj'}(t\\to\\infty)= {1\\over N}\\delta_{jj'}\\ .\n\\label{a14}\n\\end{equation} \nThis corresponds to the totally delocalized electron state. \nSince Eq.~(\\ref{a14}) represents the unique solution of\nEqs.~(\\ref{a11}), it implies that the asymptotic behavior \nof the electron density matrix is always given by Eq.~(\\ref{a14})\nfor any initial conditions. Note that this result is true only\nfor $\\Gamma\\not =0$. Otherwise the solution of Eqs.~(\\ref{a11})\nis not unique. \n\nEq.~(\\ref{a14}) tells us that an arbitrarily weak interaction with \nthe environment (detector) \nleads to delocalization in the Anderson model, \neven though this interaction affects only one of \nthe sites. In other words, Anderson \nlocalization is unstable under infinitely small decoherence. \nOne aspect of this instability is the importance of the order \nof limits $t\\to\\infty$ and $N\\to\\infty$. Taking $t\\to\\infty$ first,\nas above, gives delocalization, while taking $N\\to\\infty$ first \nwould preserve localization. In the non-interacting model, $\\Gamma =0$,\nthe order of limits is immaterial and the electron is localized. \n\nEven though a local interaction with the environment destroys the \nlocalization, the latter should affect the \ntime-dependence of the observed system. We expect \nthe delocalization time to increase exponentially with $N$ \nand to be dependent on both the decoherence rate and the localization \nlength. This matter deserves further investigation. \n\nWe would like to stress that our result is not an effect of \nfinite temperature, as is so called the hopping conductivity\\cite{mott}. \nIn the latter case, each site of the Anderson model interacts with the\nthermal bath; in our case, only one site \nis coupled to the detector (environment). If we were to let \nall the sites interact equally with the detector ($I=I'$, Fig.~1), \nwe would obtain no delocalization in our model, since $\\Gamma=0$\nin Eqs.~(\\ref{a10}), (\\ref{lind}) (see also\\cite{gur1}).\nIndeed, in this case Eq.~(\\ref{lind}) is equaivalent to the Scr\\\"odinger\nequation $i\\partial_t|\\Psi (t)\\rangle \n=H_A|\\Psi (t)\\rangle$ leading to Anderson localization. \nNote that there is no measurement when $I=I'$.\nThe origin of delocalization in our case is therefore \nthe break of coherence due to the measurement process.\n\nDelocalization of the Anderson model due to measurement \nhas been studied previously\\cite{dittr,facchi,flores}. \nYet the limit of a local and weak measurement \nhas not been achieved. In the present work we include\nthe detector in the quantum mechanical description, avoiding the use\nof the projection postulate in the course of measurement. This enables us \nto study delocalization due to local measurement and also in the limit\nof weak coupling with the measurement device.\n\nAnother experimental setup for delocalization due to a \nlocal measurement is shown schematically in Fig.~3.\nIt can be realized in atomic systems, for instance, \nin experiments with Rydberg atoms\\cite{uzi}. \nFor $N=2$ this setup is similar to a V-level system used \nfor investigation of the quantum Zeno effect\\cite{zeno}.\nThe occupation of $E_1$ is \n\\begin{figure} \n\\psfig{figure=fig3.eps,height=3cm,width=7cm,angle=0}\n\\noindent\n{\\bf Fig.~3:}\nThe 0-1 transition is driven by an intense laser field.\n$\\Omega_0$ and $\\Gamma$ represent the Rabi frequency and \nthe natural linewidth of the level $E_1$. \n\\end{figure}\n\\noindent\n monitored via spontaneous \nphoton emission, where the $0-1$ Rabi transition is generated by a laser \nfield. \n\nUsing the same derivation as \nin the previous case, Fig.~1, we obtain Bloch equations for \nthe reduced electron density matrix $\\sigma_{jj'}(t)$ where \n$j,j'={0,1,\\ldots ,N}$. The off-diagonal density-matrix elements \nare described by the same \nEq.~(\\ref{a10b}). Equation (\\ref{a10a}) for \nthe diagonal density-matrix elements, however, is modified.\nNow it reads\n\\begin{eqnarray}\n&&\\dot\\sigma_{jj}=i\\Omega_{j-1} (\\sigma_{j,j-1}-\\sigma_{j-1,j})\n\\nonumber\\\\\n%&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n&&~~~~~+i\\Omega_j (\\sigma_{j,j+1}-\\sigma_{j+1,j})\n-\\Gamma(\\delta_{j1}-\\delta_{j0})\\ \\sigma_{11}\\ .\n\\label{a15}\n\\end{eqnarray}\nThe last term in Eq.~(\\ref{a15}) describes the rates due to spontaneous \nphoton emission, Fig.~3. Here again the Bloch equations for the \nelectron density matrix can be rewritten in Lindblad form, \nEq.~(\\ref{lind}), with $Q_{jj'}=\\delta_{1j}\\delta_{1j'}$ and \n$\\tilde Q_{jj'}=\\delta_{0j}\\delta_{1j'}$.\n(For $N=2$ Eq.~(\\ref{lind}) coincides with the optical Bloch equations \nused for analysis of a V-level system\\cite{fre}). Similar to the \nprevious case, Fig.~1, Anderson localization is destroyed \nfor any value of $\\Gamma$, and the asymptotic electron distribution,\n$\\sigma_{jj}(t\\to\\infty )$, does not depend on the initial electron state. \nHere, however, the electron density matrix in the asymptotic state \nis {\\em not} a pure mixture, $\\sigma_{jj'}(t\\to\\infty )\\not =0$, \nand the probabilities $\\sigma_{jj}(t\\to\\infty )$ are not equally \ndistributed between different wells (c.f. Eq.~(\\ref{a14})).\n\nThe delocalization of the Anderson model should also \naffect its transport properties. Indeed, by connecting the first \nand the last dot in Fig.~1 to leads (reservoirs) \none can expect current to flow through the dot array \nwhenever any of the dots is monitored. Indeed, the \nstationary current through coupled dots is proportional to \nthe occupation probability of \nthe last dot, attached to the collector\\cite{gur2}. \nThe current should appear with a delay \nafter a voltage bias to the leads is switched on. \nThis time delay is precisely \nthe relaxation time needed for the electron to be\ndelocalized. \n\nAnderson localization appears not only in \nquantum mechanics, but also in \nclassical wave mechanics. Therefore the described delocalization \ndue to local interaction with an environment should have \na classical analogy. It can appear, for instance, in propagation \nof waves through coupled cavities with randomly distributed resonant\nfrequencies. A wave cannot ordinarily penetrate through such a system \ndue to the Anderson localization. Random vibration \nof one of the cavities, however, should destroy the localization, so that \nwaves begin to penetrate through the system after some time delay,\ncorresponding to the delocalization time. Such an experiment \ncan also be done using the system of transparent plates \nwith randomly varying thicknesses, described in\\cite{bk}.\n\nI am grateful to A. Buchleitner, B. Elattari, U. Smilansky\nand B. Svetitsky for very useful discussions and important \nsuggestions. \n\n\\begin{references}\n\\bibitem{ander} N.F. Mott and W.D. Twose, Adv. Phys. {\\bf 10}, 107 (1961);\nP.W. Anderson, D.J. Thouless, E. Abrahams and D.S. Fisher,\nPhys. Rev. B{\\bf 22}, 3519 (1980).\n\\bibitem{gur1} S.A. Gurvitz, Phys. Rev. B{\\bf 56}, 15215 (1997).\n\\bibitem{land} R. Landauer, IBM J. Res. Dev. {\\bf 1}, 223 (1957);\nR. Landauer, J. Phys. Condens. Matter {\\bf 1}, 8099 (1989). \n\\bibitem{gur2} S.A. Gurvitz and Ya.S. Prager, Phys. Rev. B{\\bf 53}\n(1996), 15932; S.A. Gurvitz, Phys. Rev. B{\\bf 57} (1998) 6602.\n\\bibitem{bardeen} J. Bardeen, Phys. Rev. Lett. {\\bf 6}, 57 (1961). \n\\bibitem{eg} B. Elattari and S.A. Gurvitz, Phys. Rev. Lett. {\\bf 84},\n2047 (2000).\n\\bibitem{bloch} C. Cohen-Tannoudji, J. Dupont-Roc, and\nG. Grynberg, {\\em Atom-Photon Interactions: Basic Processes\nand Applications} (Wiley, New York, 1992). \n\\bibitem{lind} G. Lindblad, Commun. Math. Phys. {\\bf 48}, 119 (1976).\n\\bibitem{mott} N.F. Mott, {\\em Metal-Insujator transitions} (Taylor \\& \nFrancis, London, 1974).\n\\bibitem{dittr} T. Dittrich and R. Graham, Europhys. Lett. 11, 589 (1990);\nPhys. Rev. A 42, 4647 (1990).\n\\bibitem{facchi} P. Facchi, S. Pascazio, and A. Scardicchio, Phys. Rev. Lett.\n83, 61 (1999). \n\\bibitem{flores} J.C Flores, Phys. Rev. B{\\bf60}, 30 (1999).\n\\bibitem{uzi} R. Bl\\\"umel, A. Buchleitner, R. Graham, L. Sirko, \nU. Smilansky, and H. Walther, Phys. Rev. A{\\bf 44}, 4521 (1991). \n\\bibitem{zeno} W.M. Itano, D.J. Heinzen, J.J. Bollinger, and \nD.j. Vineland, Phys. Rev. A {\\bf 41}, 2295 (1990).\n\\bibitem{fre} V. Frerichs and A. Schenzle, Phys. Rev. A{\\bf 44}, 1962 (1991).\n\\bibitem{bk} M.~V. Berry and S.~Klein, Eur. J. Phys. {\\bf 18}, 222 (1997).\n\\end{references}\n\\end{multicols}\n\\end{document} \n\n\n\n"
}
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[
{
"name": "cond-mat0002115.extracted_bib",
"string": "\\bibitem{ander} N.F. Mott and W.D. Twose, Adv. Phys. {\\bf 10}, 107 (1961);\nP.W. Anderson, D.J. Thouless, E. Abrahams and D.S. Fisher,\nPhys. Rev. B{\\bf 22}, 3519 (1980).\n\n\\bibitem{gur1} S.A. Gurvitz, Phys. Rev. B{\\bf 56}, 15215 (1997).\n\n\\bibitem{land} R. Landauer, IBM J. Res. Dev. {\\bf 1}, 223 (1957);\nR. Landauer, J. Phys. Condens. Matter {\\bf 1}, 8099 (1989). \n\n\\bibitem{gur2} S.A. Gurvitz and Ya.S. Prager, Phys. Rev. B{\\bf 53}\n(1996), 15932; S.A. Gurvitz, Phys. Rev. B{\\bf 57} (1998) 6602.\n\n\\bibitem{bardeen} J. Bardeen, Phys. Rev. Lett. {\\bf 6}, 57 (1961). \n\n\\bibitem{eg} B. Elattari and S.A. Gurvitz, Phys. Rev. Lett. {\\bf 84},\n2047 (2000).\n\n\\bibitem{bloch} C. Cohen-Tannoudji, J. Dupont-Roc, and\nG. Grynberg, {\\em Atom-Photon Interactions: Basic Processes\nand Applications} (Wiley, New York, 1992). \n\n\\bibitem{lind} G. Lindblad, Commun. Math. Phys. {\\bf 48}, 119 (1976).\n\n\\bibitem{mott} N.F. Mott, {\\em Metal-Insujator transitions} (Taylor \\& \nFrancis, London, 1974).\n\n\\bibitem{dittr} T. Dittrich and R. Graham, Europhys. Lett. 11, 589 (1990);\nPhys. Rev. A 42, 4647 (1990).\n\n\\bibitem{facchi} P. Facchi, S. Pascazio, and A. Scardicchio, Phys. Rev. Lett.\n83, 61 (1999). \n\n\\bibitem{flores} J.C Flores, Phys. Rev. B{\\bf60}, 30 (1999).\n\n\\bibitem{uzi} R. Bl\\\"umel, A. Buchleitner, R. Graham, L. Sirko, \nU. Smilansky, and H. Walther, Phys. Rev. A{\\bf 44}, 4521 (1991). \n\n\\bibitem{zeno} W.M. Itano, D.J. Heinzen, J.J. Bollinger, and \nD.j. Vineland, Phys. Rev. A {\\bf 41}, 2295 (1990).\n\n\\bibitem{fre} V. Frerichs and A. Schenzle, Phys. Rev. A{\\bf 44}, 1962 (1991).\n\n\\bibitem{bk} M.~V. Berry and S.~Klein, Eur. J. Phys. {\\bf 18}, 222 (1997).\n"
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cond-mat0002116
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Incorporation of Density Matrix Wavefunctions in Monte Carlo Simulations: Application to the Frustrated Heisenberg Model
|
[
{
"author": "M. S. L. du Croo de Jongh"
},
{
"author": "J. M. J. van Leeuwen and W. van Saarloos"
}
] |
We combine the Density Matrix Technique (DMRG) with Green Function Monte Carlo (GFMC) simulations. Both methods aim to determine the groundstate of a quantum system but have different limitations. The DMRG is most successful in 1-dimensional systems and can only be extended to 2-dimensional systems for strips of limited width. GFMC is not restricted to low dimensions but is limited by the efficiency of the sampling. This limitation is crucial when the system exhibits a so--called sign problem, which on the other hand is not a particular obstacle for the DMRG. We show how to combine the virtues of both methods by using a DMRG wavefunction as guiding wave function for the GFMC. This requires a special representation of the DMRG wavefunction to make the simulations possible within reasonable computational time. As a test case we apply the method to the 2--dimensional frustrated Heisenberg antiferromagnet. By supplementing the branching in GFMC with Stochastic Reconfiguration (SR) we get a stable simulation with a small variance also in the region where the fluctuations due to minus sign problem are maximal. The sensitivity of the results to the choice of the guiding wavefunction is extensively investigated. We analyse the model as a function of the ratio of the next--nearest to nearest neighbor coupling strength which is a measure for the frustration. In agreement with earlier calculations it is found from the DMRG wavefunction that for small ratios the system orders as a N\'eel type antiferromagnet and for large ratios as a columnar antiferromagnet. The spin stiffness suggests an intermediate regime without magnetic long range order. The energy curve indicates that the columnar phase is separated from the intermediate phase by a first order transition. The combination of DMRG and GFMC allows to substantiate this picture by calculating also the spin correlations in the system. We observe a pattern of the spin correlations in the intermediate regime which is in--between dimerlike and plaquette type ordering, states that have recently been suggested. It is a state with strong dimerization in one direction and weaker dimerization in the perpendicular direction and thus it lacks the the square symmetry of the plaquette state.
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[
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"name": "paper.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% \n% Information available from: jmjvanl@lorentz.leidenuniv.nl\n% (telephone number (71) 5275500 Fax: (71) 5275511 (The Netherlands))\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Paper intended for Physical Review B (2000)\n% Submitted 2/2/00 \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%\\documentstyle[aps,preprint,epsfig]{revtex}\n%\\documentstyle[aps,prl]{revtex}\n\\documentstyle[prb,aps,multicol,epsfig,array]{revtex}\n%\\setlength{\\topmargin}{-2.5cm}\n%\\setlength{\\textheight}{25cm}\n%\\setlength{\\textwidth}{15.5cm}\n%\\setlength{\\oddsidemargin}{1mm}\n%\\usepackage{epsfig}\n%\\usepackage{times}\n%\\usepackage{mathtime} \n\\begin{document}\n\\title{Incorporation of Density Matrix Wavefunctions in Monte Carlo \nSimulations: Application to the Frustrated Heisenberg Model}\n\\author{M. S. L. du Croo de Jongh, J. M. J. van Leeuwen and W. van Saarloos}\n\\address{Instituut--Lorentz, Leiden University, P. O. Box 9506, \n2300 RA Leiden, The Netherlands} \n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nWe combine the Density Matrix Technique (DMRG) with Green \nFunction Monte Carlo (GFMC) simulations.\nBoth methods aim to determine the groundstate of a quantum system but have\ndifferent limitations. The DMRG is most successful in 1-dimensional\nsystems and can only be extended to 2-dimensional systems \nfor strips of limited width. GFMC is not restricted to low dimensions\nbut is limited by the efficiency of the sampling. This limitation is crucial\nwhen the system exhibits a so--called sign problem, which on the other\nhand is not a particular obstacle for the DMRG.\nWe show how to combine the virtues of both methods by using a DMRG\nwavefunction as guiding wave function for the GFMC. This requires a\nspecial representation of the DMRG wavefunction to make the simulations\npossible within reasonable computational time. As a test case we apply\nthe method to the 2--dimensional frustrated Heisenberg antiferromagnet.\nBy supplementing the branching in GFMC with Stochastic Reconfiguration (SR) \nwe get a stable simulation with a small variance also in the region\nwhere the fluctuations due to minus sign problem are maximal. The \nsensitivity of the results to the choice of the guiding wavefunction is\nextensively investigated.\n \nWe analyse the model as a function of the ratio of the next--nearest \nto nearest neighbor coupling strength which is a measure for the \nfrustration. In agreement with earlier calculations it is found from \nthe DMRG wavefunction that for small ratios\nthe system orders as a N\\'eel type antiferromagnet and for large ratios\nas a columnar antiferromagnet. The spin stiffness suggests an intermediate\nregime without magnetic long range order. The energy curve indicates that \nthe columnar phase is separated from the intermediate phase by a first \norder transition. The combination of DMRG and GFMC allows to substantiate\nthis picture by calculating also the spin correlations in the system.\nWe observe a pattern of the spin correlations in the intermediate\nregime which is in--between dimerlike and plaquette type ordering,\nstates that have recently been suggested.\nIt is a state with strong dimerization in one direction and weaker \ndimerization in the perpendicular direction and thus it lacks the\nthe square symmetry of the plaquette state.\n\n\\end{abstract}\n\\vspace*{4mm}\n\nPACS numbers: 75.40.Mg, 75.10.Jm, 02.70.Lq\n\n\\begin{multicols}{2}\n\\section{Introduction}\n\nThe Density Matrix Technique (DMRG) has proven to be a \nvery efficient method to determine\nthe groundstate properties of low dimensional systems \\cite{Wh1}. \nFor a quantum chain it produces extremely accurate values for the \nenergy and the correlation functions. In two\ndimensional systems the calculational effort increases rapidly with the\nsize of the system. The most favorable geometry is that of a long small\nstrip. In practice the width of the strip is limited to around 8 to 10 lattice\nsites. Greens Function Monte Carlo (GFMC)\nis not directly limited by the size of the system but by\nthe efficiency of the importance sampling. When the system has a\nminus sign problem the statistics is ruined in the long run and accurate\nestimates are impossible. Many proposals \\cite{sign} have been made to\nalleviate or avoid the minus sign problem with varying success, but\nall of them introduce uncontrollable errors in the sampling. In the DMRG\ncalculation of the wavefunction the minus sign problem is not manifestly\npresent. In all proposed cures of the minus sign problem the errors decrease\nwhen the guiding wavefunction approaches the groundstate.\n\nThe idea of this paper is that DMRG wavefunctions are much better, also \nfor larger systems, than the educated guesses which usually feature as\nguiding wave functions. Moreover DMRG is a general\ntechnique to construct a wavefunction without knowing too much about the\nnature of the groundstate, with the possibility to systematically increase\nthe accuracy. Thus DMRG wavefunctions would do very well when they could be\nused as guiding functions in the importance sampling of the GFMC.\nThere is a complicating factor which prevents a straightforward\nimplementation of this idea due to the fact that interesting systems\nare so large that it is impossible to use a wavefunction via a look--up table.\nThe value of the wavefunction in a configuration has to be calculated by an\nin--line algorithm. This has limited the guiding wavefunctions to simple \nexpressions which are fast to evaluate. Consequently such guiding \nwavefunctions are not an accurate representation of the true \ngroundstate wavefunction, in particular if the physics of the groundstate\nis not well understood. In this paper we describe a method to read out\nthe DMRG wavefunction in an efficient way by using a \nspecial representation of the DMRG wavefunction. \n\nA second problem is \nthat a good guiding wavefunction alleviates the minus sign problem, \nbut cannot remove it as long as it is not exact. We resolve this dilemma\nby applying the method of Stochastic Reconfiguration which has recently \nbeen proposed by Sorella \\cite{Sor}. The viability of our method is \ntested for the frustrated Heisenberg model.\n\nThe behavior of the Heisenberg antiferromagnet has been intruiging for a \nlong time and still is in the center of research.\nThe groundstate of the antiferromagnetic 1-dimensional \nchain with nearest neighbor coupling is exactly known. In higher dimensions \nonly approximate theories or simulation results are available. The source of \nthe complexity of the groundstate are the large quantum fluctuations\nwhich counteract the tendency of classical ordering. The unfrustrated\n2--dimensional Heisenberg antiferromagnet orders in a N\\'eel state and\nby numerical methods the properties of this state can be analyzed accurately\n\\cite{San}. The situation is worse when the interactions are \ncompeting as in a 2-dimensional square lattice with antiferromagnetic \nnearest neighbor $J_1$ and next nearest neighbor $J_2$ coupling. \nThis spin system with\ncontinuous symmetry can order in 2 dimensions at zero temperature, but \nit is clear that the magnetic order is frustrated by the opposing \ntendencies of the two types of interaction. The ratio $J_2/J_1$ is a \nconvenient parameter for the frustration. For small values the system \norders antiferromagnetically in a N\\'eel type arrangement, which \naccomodates the nearest neighbor interaction. For large ratios a magnetic \norder in alternating columns of aligned spins (columnar phase) will prevail; \nin this regime the roles of the two couplings are reversed: the nearest\nneighbor interaction frustrates the order imposed by the next nearest neighbor\ninteraction. In between, for ratios of the order of 0.5, the frustration is\nmaximal and it is not clear which sort of groundstate results. This problem\nhas been attacked by various methods but not yet by DMRG and\nonly very recently by GFMC \\cite{Sor2}.\nThis paper addresses the issue by studying the spin correlations.\n\nA simple road to the answer is not possible since the behavior of \nthe system with frustration presents some fundamental problems. \nThe most severe obstacle is that frustration implies a sign problem which\nprevents the straightforward use of the GFMC simulation technique. \nMoreover the frustration substantially complicates the structure of the \ngroundstate wavefunction. Generally frustration encourages the formation \nof local structures such as dimers and plaquettes which are at odds, but not\nincompatible, with long range magnetic order. These correlation patterns\nare the most interesting part of the intermediate phase and the main\ngoal of this investigation. \n\nMany attempts have been made to clarify the situation. Often \nsimple approximations such as mean--field or spin--wave theory\ngive useful information about the qualitative behavior of the phase diagram.\nA fairly sophistocated mean--field theory using the Schwinger boson \nrepresentation does not give an intermediate phase \\cite{Duc}.\nGiven the complexity of the phase diagram and the subtlety of the effects\nit is not clear whether such approximate methods can give in this case \na reliable clue to the qualitative behavior of the system. \n\nExact calculations have been performed on small systems up to \nsize $6 \\times 6$ by Schulz et al. \\cite{Sch}.\nAlthough this information is very accurate and unbiased to possible phases,\nthe extrapolation to larger systems is a long way, the more so \nin view of indications\nthat the anticipated finite size behavior only applies for larger systems. \nAnother drawback of these small systems is that the groundstate is assumed\nto have the full symmetry of the lattice. Therefore the symmetry\nbreaking, associated with the formation of dimers, ladders or plaquettes,\nwhich is typical for the intermediate state, can not be observed directly.\n\nMore convincing are the systematic series expansion as reported \nrecently by Kotov et al. \\cite{Kot},\\cite{Kot2} and by Singh et al. \n\\cite{Sin}, which bear on an infinite system. They start with an independent \ndimers (plaquettes) and study the series expansion in the coupling \nbetween the dimers (plaquettes). By the choice of the state, around which\nthe perturbation expansion is made, the type of spatial symmetry breaking\nis fixed. These studies favor in the intermediate\nregime the dimer state over the plaquette state. Their dimer state has \ndimers organized in ladders in which the\nchains and the rungs have nearly equal strength. So the system breaks the \ntranslational invariance only in one direction. The energy differences\nare however small and the series is finite, so further investigation is\nuseful. Our simulations yield correlations in good agreement with theirs,\nbut do not confirm the picture of translational invariant ladders. Instead\nwe find an additional weaker symmetry breaking {\\it along} the ladders, such\nthat we come closer to the plaquette picture.\n\nVery recently Capriotti and Sorella \\cite{Sor2} have carried out a GFMC \nsimulation for $J_2 = 0.5J_1$ and have studied the susceptibilities for\nthe orientational and translational symmetry breaking. They conclude\nthat the groundstate is a plaquette state with full\nsymmetry between the horizontal and vertical direction.\n\nFrom the purely theoretical side the problem has been discussed by Sachdev and\nRead \\cite{Sac} on the basis of a large spin expansion. From their analysis a\nscenario emerges in which the N\\'eel phase disappears upon increasing\nfrustration in a continuous way. Then a gapped spatial--inhomogeneous\nphase with dimerlike correlations appears. For even higher \nfrustration ratios a first order transition takes place to the columnar phase.\nAlthough this scenario is qualitative, without precise location of the\nphase transition points, it definitively excludes dimer formation\nin the magnetically ordered N\\'eel and columnar phase. It is remarkable\nthat two quite different order parameters (the magnetic order and the\ndimer order) disappear simultaneously and continuously on opposite sides \nof the phase transition. In this scenario, this is taken as an indication of\nsome kind of duality of the two phases. \n \nGiven all these predictions it is of utmost interest to further study the \nnature of the intermediate state. Due to the smallness of the differences in \nenergy between the various possibilities, the energy will not be the ideal \ntest for the phase diagram. Therefore we have decided to focus directly \non the spin correlations as a function of the ratio $J_2/J_1$.\nIn this paper we first investigate the 2--dimensional frustrated Heisenberg \nmodel by constructing the DMRG wave function of the groundstate for long \nstrips up to a width of 8 sites.\nThe groundstate energy and the spin stiffness which are calculated,\nconfirm the overal picture described above, but the results are not\naccurate enough to allow for a conclusive extrapolation to larger systems.\nThen we study an open 10x10 lattice by means of the GFMC technique using\nDMRG wavefunctions as guiding wavefunction for the importance sampling.\nThe GFMC are supplemented with Stochastic Reconfiguration as proposed\nby Sorella \\cite{Sor} as an extension of the Fixed Node technique \\cite{Cep}.\nThis method avoids\nthe minus sign problem by replacing the walkers regularly by a new set\nof positive sign with the same statistical properties. The first observation\nis that GFMC improves the energy of the DMRG in a substantial and systematic\nway as can be tested in the unfrustrated model where sufficient information\nis available from different sources. Secondly the spin correlations \nbecome more accurate and less dependent on the technique used for constructing\nthe DMRG wavefunction. The DMRG technique is focussed on the energy of \nthe system and less on the correlations. The GFMC probes mostly the local \ncorrelations of the system as all the moves are small and correspond to local\nchanges of the configurations. With these spin correlations we investigate\nthe phase diagram for various values of the frustration ratio $J_2/J_1$.\n\nThe paper begins with the definition of the model to avoid ambiguities. Then\na short description of our implementation of\nthe DMRG method is given. We go into more detail\nabout the way how the constructed wavefunctions can be used as guiding\nwavefunctions in the GFMC simulation. This is a delicate problem since the full\nconstruction of a DMRG wavefunction takes several hours on a workstation.\nTherefore we separate off the construction of the wavefunction and cast it in\na form where the configurations can be obtained from each other by matrix\noperations on a vector. So the length of the computation of the wavefunction\nin a configuration scales with the square of the number of states \nincluded in the DMRG wavefunction. But even then the actual \nconstruction of the value of the wavefunction in a given configuration \nis so time consuming that utmost effiency must be reached in obtaining\nthe wavefunction for successive configurations. The remaining\nsections are used to outline the GFMC and the Stochastic Reconfiguration\nand to discuss the results. We concentrate on the correlation\nfunctions since we see them as most significant for the structure of the\nphases. We give first a global evaluation of the correlation function patterns \nfor a wide set of frustration ratios and then focus on a number of points \nto see the dependence on the guiding wavefunction and to deduce the trends.\nThe paper closes with a discussion and a comparison with other results \nin the literature.\n\n\\section{The Hamiltonian}\n\nThe hamiltonian of the system refers to spins on a square lattice.\n\\begin{equation} \\label{a1}\n{\\cal H} = J_1 \\sum_{(i,j)} {\\bf S}_i \\cdot {\\bf S}_j + J_2\n\\sum_{[i,j]} {\\bf S}_i \\cdot {\\bf S}_j. \n\\end{equation} \nThe ${\\bf S}_i$ are spin $\\frac{1}{2}$ operators and \nthe sum is over pairs of nearest neigbors $(i,j)$ and over pairs of\nnext nearest neighbors $[i,j]$ on a quadratic lattice. Both coupling \nconstants $J_1$ and $J_2$ are supposed to be positive.\n$J_1$ tries to align the nearest neigbor spin in an \nantiferromagnetic way and $J_2$ tries to do the same with the next nearest\nneighbors. So the spin system is frustrated, implying an intrinsic minus \nsign in the simulations that cannot be gauged away by a rotation of the\nspin operators.\n\\begin{figure}\n \\centering \\epsfxsize=6cm\n \\epsffile{heisenbergj1j2.eps}\n \\caption{The interaction constants $J_1$ and $J_2$}\n \\label{fig0}\n\\end{figure}\n\nIn order to prepare for the representation of the hamiltonian we express\nthe spin components in spin raising and lowering operators\n\\begin{equation} \\label{a2}\n{\\bf S}_i \\cdot {\\bf S}_j = \\frac{1}{2} (S^+_i S^-_j + S^-_i S^+_j)\n + S^z_i S^z_j.\n\\end{equation}\nWe will use the $z$ component representation of the spins and a complete\nstate of the spins will be represented as\n\\begin{equation} \\label{a3}\n| R \\rangle = |s_1, s_2, \\cdots , s_N \\rangle,\n\\end{equation}\nwhere the $s_j$ are eigenvalues of the $S^z_j$ operator. \nThe diagonal matrix elements of the hamiltonian \nare in the representation (\\ref{a3}) given by\n\\begin{equation} \\label{a4}\n\\langle R | {\\cal H} | R \\rangle =J_1 \\sum_{(i,j)} s_i s_j + \nJ_2 \\sum_{[i,j]} s_i s_j. \n\\end{equation}\nThe off-diagonal elements are between two nearby configurations \n$R'$ and $R$. $R'$ is the same as $R$ except at a pair of nearest \nneighbors sites $(i,j)$ or next nearest neighbor sites $[i,j]$, for\nwhich the spins $s_i$ and $s_j$ are opposite. In $ R'$ the pair is turned\nover by the hamiltonian. Then\n\\begin{equation} \\label{a5}\n\\langle R' | {\\cal H} | R \\rangle = \\frac{1}{2} J_1 \\hspace*{1cm}\n{\\rm or} \\hspace*{1cm} \\langle R' | {\\cal H} | R \\rangle = \\frac{1}{2} J_2,\n\\end{equation}\ndepending on whether a nearest or a next nearest pair is flipped.\n\n\\section{The DMRG Procedure}\n\nThe DMRG procedure approximates the groundstate wavefunction by searching\nthrough various representations in bases of a given dimension $m$ \\cite{Wh1}.\nHere we take the standard method (with two connecting sites) for granted\nand make the preparations for the extraction of the wavefunction.\n\\begin{figure}[h]\n \\centering \\epsfxsize=\\linewidth\n \\epsffile{procedure_finite.eps}\n \\caption{The DMRG procedure with one connecting site}\n \\label{fig1}\n\\end{figure}\n\nThe system is mapped onto a 1-dimensional chain (see Fig. \\ref{fig1}) and\nseparated into two parts: a {\\it left} and {\\it right} hand part.\nThey are connected by one site. Each part is represented in\na basis of at most $m$ states. With a representation of all the operators\nin the hamiltonian in these bases one can find the groundstate of the\nsystem. We thus have several representations of the groundstate\ndepending on the way in which the system is divided up into subsystems. \nThe point is to see how these representations are connected and\nhow they possibly can be improved. We take a representation for the\nright hand parts and improve those on the left.\nSo we assume that for a given division we have the groundstate of the\nwhole system and we want to\nenlarge the left hand side at the expense of the right hand side. The\nfirst step is to include the connecting site in the left hand part. This\nenlarges the basis for the left hand side from $m$ to $2m$ and a selection\nhas to be made of $m$ basis states. This goes with the help of the density\nmatrix for the left hand side as induced by the wavefunction for the whole\nsystem. For later use we write out the basic equations for the density\nmatrix in the configuration representation. Let, at a certain stage in the\ncomputation, $| \\Phi \\rangle$ be the approximation to the groundstate.\nThe configurations of the right hand part and the left hand part are \ndenoted by $R_r$ and $R_l$.\nThen the density matrix for the left hand part reads\n\\begin{equation} \\label{b1}\n\\langle R_l |\\rho | R'_l \\rangle = \\sum_{R_r} \\langle R_l, R_r | \\Phi \\rangle\n \\langle \\Phi | R'_l, R_r \\rangle.\n\\end{equation}\nIn practice we do not solve the eigenvalues of the density matrix in\nthe configuration representation, but in a projection on a smaller basis. \nWhite \\cite{Wh1} has shown that the best way to represent the state \n$| \\Phi \\rangle$ is to select the $m$ eigenstates $| \\alpha \\rangle$ \nwith the largest eigenvalue\n\\begin{equation} \\label{b2}\n\\sum_{R'_l} \\langle R_l |\\rho | R'_l \\rangle \\langle R'_l | \\alpha \\rangle =\n\\lambda_\\alpha \\langle R_l | \\alpha \\rangle.\n\\end{equation}\n\nThe next step is to break up the right hand part into a connecting\nsite and a remainder. With the basis for this remainder and the newly\nacquired basis for the left hand part we can again compute the groundstate \nof the whole system as indicated in the lower part of the figure. Now we are \nin the same position as we started, with the difference that the connecting\nsite has moved one position to the right. Thus we may repeat the cycle\ntill the right hand part is so small that it can exactly be represented\nby $m$ states or less. Then we have constructed for the left hand part\na new set of bases, all containing $m$ states, for system parts of variable \nlength. Next we reverse the roles of {\\it left} and {\\it right} \nand move back in order to improve the bases for the right hand parts \nwith the just constructed bases for the left hand part.\n\nThe process may be iterated till it converges towards a steady state. The \ngreat virtue of the method is that it is variational. In each step the\nenergy will lower till it saturates. In 1-dimensional system the method\nhas proven to be very accurate \\cite{Wh1}. So one wonders what the \nmain trouble is in higher dimensions. \n\\begin{figure}[h]\n \\centering \\epsfxsize=\\linewidth\n \\epsffile{straight_meander2.eps}\n \\caption{Two 1--dimensional paths through the system: ``straight'' (a)\n and ``meandering'' (b).}\n \\label{fig2}\n\\end{figure}\n\nIn Fig. \\ref{fig2} we have drawn 2 possible ways to map the\nsystem on a 1-dimensional chain. One sees that if we divide again \nthe chain into a left hand part and a right hand part and a connecting\nsite, quite a few sites of the left hand part\nare nearest or next nearest neighbors of sites of the right hand part. So \nthe coupling between the two parts of the chain is not only through the\nconnecting site but also through sites which are relatively far away from\neach other in the 1-dimensional path. The operators for the spins on these\nsites are not as well represented as those of the connecting site, which \nis fully represented by the two possible spin states. Yet the correlations\nbetween the interacting sites count as much for the energy of the system\nas those interacting with the connecting site. One may say that the further\naway two interacting sites are in the 1-dimensional chain the poorer their\ninfluence is accounted for. This consideration explains in part why open\nsystems can be calculated more accurately than closed systems, even in \n1-dimensional systems.\n\nIt is an open question which map of the 2-dimensional onto a 1-dimensional\nchain gives the best representation of the groundstate of the system. \nAlso other divisions of the system\nthan those suggested by a map on a 1--dimensional chain are possible and \nwe have been experimenting with arrangements which reflect better the\n2--dimensional character of the lattice\\cite{Luc}. They are\npromising but the software for these is not as sophisticated as the \none developed by White \\cite{Wh2} for the 1-dimensional chain. \nWe therefore have restricted our calculations to the two paths shown here. \nThe second choice. the ``meandering'' path, was motivated\nby the fact that it has the strongest correlated sites most nearby in the\nchain and this choice was indeed justified by a lower energy for a\ngiven dimension $m$ of the representation than for the ``straight'' path.\n\nThe DMRG calculations as well as the corresponding GFMC simulations \nare carried out for both paths. The meandering path has to be\npreferred over the straight path as the DMRG wavefunctions \ngenerally give a better energy value and the simulations suffer less from \nfluctuations. Nevertheless we have also investigated the straight path,\nsince the path chosen leaves its imprints on the resulting\ncorrelation pattern and the paths break the symmetries in different ways.\nBoth paths have an orientational preference. In open systems the \ntranslational symmetry is broken anyway, but the meandering path\nhas in addition a staggering in the horizontal direction. This together\nwith the horizontal nearest neighbor sites appearing in the meandering \npath gives a preference for horizontal dimerlike correlations in this path.\nOn the other hand the straight path prefers the dimers in the vertical\ndirection. Comparing the results of the two choices, allows us to draw further \nconclusions on the nature of the intermediate state.\n\n\\section{Extracting configurations from the DMRG wavefunction}\n\nIt is clear that the wavefunction which results from a DMRG-procedure is\nquite involved and it is not simple to extract its value for a given\nconfiguration. We assume now that the DMRG wavefunction\nhas been obtained by some procedure and we will give below an \nalgorithm to obtain efficiently the value for an arbitrary configuration\n(see also \\cite{Luc} for an alternative description).\n\nThe first step is the construction of a set of representations for the\nwavefunction in terms of two parts (without a connecting site in between). \nLet the left hand part contain $l$ sites and the other part $N-l$ sites.\nWe denote the $m$ basis states of the left hand part by the index $\\alpha$\nand those of the right hand part by $\\bar{\\alpha}$.\nThe eigenstates of the two parts are closely linked and related as follows\n\\begin{equation} \\label{c1}\n\\left\\{ \\begin{array}{rcl}\n\\langle R_l | \\alpha \\rangle & = & \\displaystyle\n\\frac{1}{\\sqrt{\\lambda_\\alpha}} \\sum_{R_r} \\langle \\Phi | R_l, R_r \\rangle \n\\langle R_r | \\bar{\\alpha} \\rangle, \\\\*[3mm]\n\\langle R_r | \\bar{\\alpha} \\rangle & = & \\displaystyle \n\\frac{1}{\\sqrt{\\lambda_\\alpha}} \\sum_{R_l} \n\\langle \\alpha | R_l \\rangle \\langle R_l, R_r | \\Phi \\rangle. \n\\end{array} \\right.\n\\end{equation}\nIt means that for every eigenvalue $\\lambda_\\alpha$ there is and eigenstate\n$\\alpha$ for the left hand part and an $\\bar{\\alpha}$ for the right hand\ndensity matrix. The proof of (\\ref{c1}) follows from insertion in the\ndensity matrix eigenvalue equation (\\ref{b2}).\n\nThe second step is a relation for the groundstate wavefunction in terms of\nthese eigenfunctions. Generally we have\n\\begin{equation} \\label{c2}\n\\langle R_l, R_r | \\Phi \\rangle = \\sum_{\\alpha, \\bar{\\beta}}\n\\langle R_l | \\alpha \\rangle \\langle R_r | \\bar{\\beta} \\rangle \n\\langle \\alpha \\bar{\\beta} | \\Phi \\rangle, \n\\end{equation}\nwhile due to (\\ref{c1}) we find\n\\begin{equation} \\label{c3}\n\\begin{array}{rcl}\n\\langle \\alpha \\bar{\\beta} | \\Phi \\rangle & = & \\displaystyle \n\\sum_{R_l, R_r} \\langle \\alpha |\nR_l \\rangle \\langle \\bar{\\beta} | R_r \\rangle \\langle R_l, R_r | \\Phi \\rangle\n\\\\*[2mm] \n & = & \\displaystyle\\sqrt{\\lambda_\\alpha} \\sum_{R_r} \\langle \\bar{\\beta} \n| R_r \\rangle \\langle R_r | \\bar{\\alpha} \\rangle \\ = \\delta_{\\alpha, \\beta} \n\\sqrt{\\lambda_\\alpha}.\n\\end{array}\n\\end{equation}\nThus we can represent the groundstate as\n\\begin{equation} \\label{c4}\n\\langle R_l, R_r | \\Phi \\rangle = \\sum_\\alpha \\sqrt{\\lambda^l_\\alpha} \n\\langle R_l |\\alpha \\rangle_l \\langle R_r | \\bar{\\alpha} \\rangle_{N-l}.\n\\end{equation}\nFor this part of the problem we have to compute and store the set \nof $m$ eigenvalues $\\lambda^l_\\alpha$ for each division $l$.\nWe point out again that we have on the left hand side the wavefunction\nand on the right hand side representations for given division $l$,\nwhich all lead to the same wavefunction.\nThe last step is to see the connection between these representations.\n\nAs intermediary we consider a representation of the wavefunction with\none site $s_l$ separating the spins $s_1 \\cdots s_{l-1}$ on the left hand side\nfrom $s_{l+1} \\cdots s_N$ on the right hand side. Using the same basis as\nin (\\ref{c4}) we have\n\\begin{equation} \\label{c5}\n\\begin{array}{l}\n\\langle s_1 \\cdots s_{l-1}, s_l, s_{l+1} \\cdots s_N | \\Phi \\rangle = \\\\*[2mm]\n\\sum_{\\alpha, \\alpha'} \\langle s_1 \\cdots s_{l-1} | \\alpha \\rangle\n\\phi^l_{\\alpha,\\alpha'} (s_l) \\langle s_{l+1} \\cdots s_N |\\bar{\\alpha'}\\rangle.\n\\end{array}\n\\end{equation}\nWe compare this representation in two ways with (\\ref{c4}). First we\ncombine the middle site with the left hand part. This leads to $m$ states\nwhich can be expressed as linear combinations of the states of the \nenlarged segment\n\\begin{equation} \\label{c6}\n\\sum_\\alpha \\langle s_1 \\cdots s_{l-1} | \\alpha \\rangle \n\\phi^l_{\\alpha,\\alpha'} (s_l) = \\sum_{\\alpha''} \n\\langle s_1 \\cdots s_l | \\alpha'' \\rangle T^l_{\\alpha'', \\alpha'}.\n\\end{equation}\nIn fact this relation is the very essence of the DMRG procedure. The \nwave function in the larger space is projected on the eigenstates of the\nthe density matrix of that space. Since the process of zipping back \nforth has converged there is indeed a fixed\nrelation (\\ref{c6}). However when we insert (\\ref{c6}) into (\\ref{c5}) and\ncompare it with (\\ref{c4}) we conclude that the matrix $T$ must be diagonal\n\\begin{equation} \\label{c7}\n T^l_{\\alpha'', \\alpha'} = \\delta_{\\alpha'',\\alpha'} \n\\sqrt{\\lambda^l_{\\alpha'}}.\n\\end{equation}\nThis leads to the recursion relation\n\\begin{equation} \\label{c8}\n\\langle s_1 \\cdots s_l | \\alpha' \\rangle = \\sum_{\\alpha} \n \\langle s_1 \\cdots s_{l-1} | \\alpha \\rangle A^l_{\\alpha,\\alpha'}(s_l)\n\\end{equation}\nwith \n\\begin{equation} \\label{c9}\n A^l_{\\alpha,\\alpha'}(s_l) = \\phi^l_{\\alpha,\\alpha'} (s_l)/\n\\sqrt{\\lambda^l_{\\alpha'}}.\n\\end{equation}\nThe second combination concerns the contraction of the middle site with the\nright hand part. This leads to the recursion relation\n\\begin{equation} \\label{c10}\n\\langle s_l \\cdots s_N | \\bar{\\alpha} \\rangle = \\sum_{\\alpha'} \n B^{l-1}_{\\alpha, \\alpha'}(s_l) \\langle s_{l+1} \\cdots s_N | \\bar{\\alpha}' \n\\rangle\n\\end{equation}\nwith\n\\begin{equation} \\label{c11}\nB^{l-1}_{\\alpha, \\alpha'}(s) = \\phi^l_{\\alpha,\\alpha'} (s)/\n\\sqrt{\\lambda^{l-1}_\\alpha} .\n\\end{equation}\nThe $A$ and $B$ matrices are the essential ingredients of the\ncalculation of the wavefunction. From (\\ref{c11}) and (\\ref{c9}) follows that\nthey are related as\n\\begin{equation} \\label{c12}\nB^{l-1}_{\\alpha, \\alpha'} (s) =\\sqrt{\\lambda^l_{\\alpha'}\\, /\\,\n\\lambda^{l-1}_\\alpha} \\,A^l_{\\alpha,\\alpha'} (s).\n\\end{equation}\n\nBy the recursion relations the basis states are expressed as products of\n$m \\times m$ matrices. The determination of the DMRG wavefunction and\nthe matrices $A$ (or $B$) is part of the determination of the DMRG\nwavefunction which is indeed lengthy but fortunately no part of the \nsimulation. The matrices can be stored and contain the information\nto calculate the wavefunction for any configuration. The value of the \nwavefunction is now obtained as the product of matrices acting on a\nvector. Thus the calculational effort scales with $m^2$. Using \nrelation (\\ref{c12}) one reconfirms by direct calculation that the \nwavefunction is indeed independent of the division $l$.\n\nWhen the simulation is in the \nconfiguration $R$, all the $\\langle R_l | \\alpha \\rangle_l$ and the \n$\\langle R_r | \\bar{\\alpha} \\rangle_{N-l}$ are calculated and stored, with\nthe purpose to calculate the wavefunctions more efficiently for the\nconfigurations $R'$ which are connected to $R$ by the hamiltonian and which\nare the candidates for a move. The structure of of these nearby states\nis $R' = s_1 \\cdots s_{l_2} \\cdots s_{l_1} \\cdots s_N \\quad (l_2 > l_1)$. \nSo we have that for $R'$ the representation\n\\begin{equation} \\label{c13}\n\\langle R' | \\Phi \\rangle = \\sum_\\alpha \\sqrt{\\lambda^{l_2}_\\alpha}\n\\langle s_1 \\cdots s_{l_2} \\cdots s_{l_1} | \\alpha \\rangle\n\\langle s_{l_2 + 1} \\cdots s_N | \\bar{\\alpha} \\rangle\n\\end{equation}\nholds. Now we see the advantage of having the wavefunction stored for all the\ndivisions. The second factor in (\\ref{c13}) is already tabulated; \nthe first factor involves a number\nof matrix multiplications equal to the distance in the chain of the two\nspins $l_1$ and $l_2$ till one reaches a tabulated function. One can use\nthe tables for a certain number of moves but after a while it starts to pay\noff to make a fresh list. \n\n\\section{Green Function Monte Carlo simulations}\n\nThe GFMC technique employs the operator\n\\begin{equation} \\label{d1}\n{\\cal G} = 1 - \\epsilon {\\cal H}\n\\end{equation}\nand uses the fact that the groundstate $| \\Psi_0 \\rangle$ results from \n\\begin{equation} \\label{d2} \n| \\Psi_0 \\rangle \\sim {\\cal G}^n | \\Phi \\rangle, \\quad \\quad \\quad\n \\epsilon \\ll 1, \\quad \\quad \\quad n \\epsilon \\gg 1\n\\end{equation}\nwhere in principle $| \\Phi \\rangle $ may be any function which is \nnon-orthogonal to the groundstate. In view of the possible symmetry breaking,\nthe overlap is a point of serious concern on which we come back in the \ndiscussion. In practice we will use the best\n$| \\Phi \\rangle $ that we can construct conveniently by the DMRG--procedure\ndescribed above. The closer $| \\Phi \\rangle $ is to the groundstate the \nsmaller the number of factors $n$ in the product needs to be in order \nto find the groundstate. \nEvaluating (\\ref{d2}) in the spin representation gives for the projection on \nthe trial wavefunction the following long product\n\\begin{equation} \\label{d3}\n\\langle \\Phi|\\Psi_0 \\rangle \\simeq \\sum_{\\bf R} \\langle \\Phi |R_M \\rangle \n\\left[ \\prod^M_{i=1}\n\\langle R_i | {\\cal G} |R_{i-1} \\rangle \\right] \\langle R_0 | \\Phi \\rangle.\n\\end{equation}\nHere the sum is over paths ${\\bf R} = (R_M, \\cdots R_1,R_0)$ which will be\ngenerated by a Markov process.\nThe Markov process involves a transition probability \n$T(R_i \\leftarrow R_{i-1})$ and the averaging process uses a \nweight $m(R)$. Its is natural to connect the\ntransition probabilities to the matrix elements of the Greens Function \n${\\cal G}$. But here comes the sign problem into the game: \nthe transition probabilities have to be positive (and normalized). \nSo we put the transition rate proportional\nto the absolute value of the matrix element of the Greens Function\n\\begin{equation} \\label{d4}\nT(R \\leftarrow R') = \\frac{|\\langle R |{\\cal G} | R'\\rangle|}\n{\\sum_{R''} | \\langle R'' | {\\cal G} | R' \\rangle |}.\n\\end{equation}\nThis implies that we have to use a sign function $s(R,R')$\n\\begin{equation} \\label{d5}\ns(R,R') = \\frac{\\langle R |{\\cal G} | R'\\rangle}\n {|\\langle R |{\\cal G} | R'\\rangle|}\n\\end{equation}\nand a weight factor\n\\begin{equation} \\label{d6}\nm(R) = \\sum_{R'} | \\langle R' | {\\cal G} | R \\rangle |.\n\\end{equation}\nAll these factors together form the matrixelement of the Green Function\n\\begin{equation} \\label{d7}\n\\langle R |{\\cal G} | R'\\rangle = T(R \\leftarrow R') s(R,R') m(R').\n\\end{equation}\nIf the matrix elements of the Greens Function were all positive, or could\nbe made positive by a suitable transformation,\nwe would not have to introduce the sign function. We leave\nits consequences to the next section.\nBy the representation (\\ref{d7}) we can write the contribution of the path\nas a product of transition probabilities, signs and local weights. The \ntransition probabilities control the growth of the\nMarkov chain. The signs and weights constitute the weight of a path\n\\begin{equation} \\label{d8}\nM({\\bf R}) = m_f (R_M) \\left[ \\prod^M_{i=1} s(R_i, R_{i-1}) m(R_{i-1})\\right]\nm_i (R_0).\n\\end{equation}\nThe initial and final weight have to be chosen such that the weight of the\npaths corresponds to the expansion (\\ref{d3}). For the innerproduct\n$\\langle \\Phi | \\Psi_0 \\rangle$ we get\n\\begin{equation} \\label{d9}\nm_i (R) = \\langle R | \\Phi \\rangle, \\quad \\quad \\quad\nm_f (R) = \\langle \\Phi | R \\rangle.\n\\end{equation}\nWith this final weight we have projected the groundstate on the trial\nwave. This allows us to calculate the so--called mixed averages.\nFor that purpose we define the local estimator ${\\cal O}$\n\\begin{equation} \\label{d10}\nO(R) = \\frac{\\langle \\Phi | {\\cal O} | R \\rangle}\n{\\langle \\Phi | R \\rangle}\\, ,\n\\end{equation}\nwhich yields the mixed average\n\\begin{equation} \\label{d11}\n\\langle {\\cal O} \\rangle_m \\equiv \\frac{\\langle \\Phi |{\\cal O} | \n\\Psi_0 \\rangle}{\\langle \\Phi | \\Psi_0 \\rangle} =\n\\frac{\\sum_{{\\bf R}} O(R_M) M({\\bf R})}{\\sum_{{\\bf R}} M({\\bf R})}.\n\\end{equation}\nFor operators not commuting with the hamiltonian the mixed average is\nan approximation to the groundstate average. Later on we will improve on it.\n\nIn this raw form the GFMC would hardly work because all paths are generated\nwith equal weight. One can do better by importance sampling in which one\ntransforms the problem to a Greens Function with matrix elements\n\\begin{equation} \\label{d12}\n\\langle R | \\tilde{{\\cal G}} | R' \\rangle = \\frac{\\langle \\Phi | R \\rangle\n\\langle R | {\\cal G} | R' \\rangle}{ \\langle \\Phi | R' \\rangle}.\n\\end{equation}\nGenerally this can be seen as a similarity transformation on the \noperators and from now on\neverywhere operators with a tilde are related to their counterpart without a \ntilde as in (\\ref{d12}). It gives only a minor change in the formulation.\nThe transition rates are based on the matrix elements of $\\tilde{{\\cal G}}$\nand so are the signs and weights. Thus we have a set of definitions like\n(\\ref{d4})--(\\ref{d6}) with everywhere a tilde on top. It leads also to \na change of the initial and final weight\n\\begin{equation} \\label{d13}\n\\tilde{m}_i (R) = |\\langle \\Phi | R \\rangle|^2, \\quad \\quad \\quad \n\\tilde{m}_f (R) = 1.\n\\end{equation}\nBy chosing these weights the formula (\\ref{d10}) for the average still\napplies with a weight $\\tilde{M} ({\\bf R})$ made up as in (\\ref{d8}) with\nthe weights and signs with a tilde.\nUsing the tilde operators the local estimator (\\ref{d10}) reads\n\\begin{equation} \\label{d14}\nO(R) = \\sum_{R'} \\langle R' |\\tilde{{\\cal O}} | R \\rangle.\n\\end{equation}\n\nWe will speak about the various paths in terms of independent walkers\nthat sample these paths.\nAs some walkers become more important than others in the process, it is\nwise to improve the variance by branching, which we will discuss later with\nthe sign problem. Before we embark on the discussion of the\nsign problem we want to summarize a number of aspects of the GFMC simulation\nrelevant to our work.\n\\begin{itemize}\n\\item The steps in the Markov process are small, only the ones induced\nby one term of the hamiltonian feature in a transition to a new state. This \nmakes the subsequent states quite correlated. So many steps have to be\nperformed before a statistical independent configuration is reached; on the\naverage a number of the order of the number of sites.\n\\item In every configuration the wave function for a number of neighboring \nstates (the ones which are reachable by the Greens Function), has to be\nevaluated. This is a time consuming operation and it makes the simulation\nquite slow, because out of the possibilities (of order $N$) only one\nis chosen and all the information gathered on the others is virtually useless.\n\\item The necessity to choose a small $\\epsilon$ in the Greens Function seems\na further slow down of the method, but it can be avoided by the technique\nof continuous time steps developed by Ceperley and Trivedi \\cite{Tri}.\nIn this method the\npossibility of staying in the same configuration (the diagonal element of the\nGreens Function) is eliminated and replaced by a waiting time before a move\nto another state is made. (For further details in relation to the present \npaper we refer to \\cite{Luc})\n\\item The average (\\ref{d10}) can be improved by replacing it by\n\\begin{equation} \\label{d15}\n\\langle {\\cal O} \\rangle_{im} = 2 \\frac{\\langle \\Phi | {\\cal O} | \\Psi_0\n\\rangle}{\\langle \\Phi | \\Psi_0 \\rangle} - \\frac{\\langle \\Phi | {\\cal O}\n| \\Phi \\rangle}{\\langle \\Phi | \\Psi_0 \\rangle}\n\\end{equation}\nof which the error with respect to the true average is of second order in\nthe deviation of $|\\Phi \\rangle$ from $|\\Psi_0 \\rangle$. For conserved\noperators, such as the energy, this correction is not needed since the\nthe mixed average gives already the correct value.\n\\end{itemize}\n\n\\section{The sign problem and its remedies}\n\nIn the hamiltonian (\\ref{a1}) the $z$ component of the spin operator keeps\nthe spin configuration invariant, whereas the $x$ and $y$ components change\nthe configuration. The typical change is that a pair of nearest or next\nnearest neighbors is spin reversed. Inspecting the Greens Function it means\nthat all changes to another configuration involve a minus sign! Thus the\nGreens Function is as far as possible from the ideal of positive matrix \nelements. The diagonal terms are positive, but they always are positive for \nsufficiently small $\\epsilon$. Importance sampling can remove minus signs\nin the transition rates, when the ratio of the guiding wavefunction involves\nalso a minus sign. For $J_2 \\neq 0$ no guiding wavefunction can remove the\nminus sign problem completely. In Fig. 3 we show a loop of two nearest\nneighbor spin flips followed by a flip in a next nearest neighbor pair, \nsuch that the starting configuration is restored. The product of\nthe ratios of the guiding wavefunction drops out in this loop, but the\nproduct of the three matrix elements has a minus sign. So at least \none of the transitions must involve a minus sign.\n\\begin{figure}[h]\n \\centering \\epsfxsize=\\linewidth\n \\epsffile{sign-problem.eps}\n \\caption{Illustration of the sign problem in the frustrated Heisenberg \nmodel. The shown sequence of spin flips always involves a sign that \ncan not be gauged away by a different choice of guiding wavefunctions}\n \\label{fig3}\n\\end{figure}\n\nFor unfrustrated systems these loops do not exist and one can remove\nthe minus sign by a transformation of the spin operators\n\\begin{equation} \\label{a6}\nS^x_i \\rightarrow -S^x_i, \\quad \\quad \\quad\nS^y_i \\rightarrow -S^y_i, \\quad \\quad \\quad\nS^z_i \\rightarrow S^z_i\n\\end{equation}\nwhich leave the commutation operators invariant. Applying this transformation\non every other spin (the white fields of a \ncheckerboard) all flips involving a pair of nearest neighbors then give a\npositive matrix element for the Greens Function. So when $J_2=0$ the\nappearant sign problem is transformed away. \nFor sufficiently small $J_2$, Marshall \\cite{Mar} has shown\nthat the wave function of the system has only positive components \n(after the ``Marshall'' sign flip (\\ref{a6})).\nSo the minus sign problem is not due to the wave function but to the \nfrustration. (For the Hubbard model it is the guiding wave function which\nmust have minus signs due to the Pauli principle, while the bare transition\nprobabilities can be taken positive).\n\nDue to the minus sign the weight of a long path picks up a arbitrary sign.\nGenerally the weights are also growing along a path. Thus if various paths\nare traced out by a number of independent walkers, the average over the\npaths or the walkers becomes a sum over large terms of both\nsigns, or differently phrased: the average becomes small with respect to the\nvariance; the signal gets lost in the noise.\n\nCeperley and Alder \\cite{Cep} constructed a method, Fixed Node Monte\nCarlo (FNMC), which avoids the minus sign problem\nat the expense of introducing an approximation. Their method is designed\nfor continuum systems and handling fermion wavefunctions. They argued that\nthe configuration space in which the wavefunction has a given sign, say\npositive, is sufficient for exploring the properties of the groundstate,\nsince the other half of the configuration space contains identical information.\nThus they designed a method in which the walkers remain in one domain of\na given sign, essentially by forbidding to cross the nodes of the wavefunction.\nThe approximation is that one has to take the nodal structure of the guiding\nwavefunction for granted and one cannot improve on that, at least not without\nsophistocation (nodal release). The method is variational in the sense that\nerrors in the nodal structure always raise the groundstate energy.\n\nIt seems trivial to take over this idea to the lattice but it is not. The\nreason is that in continuum systems one can make smaller steps when a walker\napproaches a node without introducing errors. In a lattice system the\nconfiguration space is discrete; so the location of the node is not\nstrictly defined. The important part is that, loosely speaking the nodes\nare between configurations and one cannot make smaller moves than displacing\na particle over a lattice distance or flip a pair of spins. Van Bemmel et al.\n\\cite{Bem} adapted the FNMC concept to lattice systems preserving \nits variational character. This extension to the lattice suffers from the\nsame shortcoming as the method of Ceperley and Alder: the ``nodal''\nstructure of the guiding wavefunction is given and cannot be improved by the\nMonte Carlo process. Recently Sorella \\cite{Sor} proposed a modification which\novercomes this drawback. It is based on two ingredients.\n\nSorella noticed that the following effective hamiltonian yields also an upper\nbound to the energy:\n\\begin{equation} \\label{f1}\n\\begin{array}{rcl}\n\\langle R | \\tilde{{\\cal H}}_{\\rm eff} | R' \\rangle & = & \n\\left\\{ \\begin{array}{rcl}\n\\langle R | \\tilde{{\\cal H}} | R' \\rangle & {\\rm if} & \n\\langle R | \\tilde{{\\cal H}} | R' \\rangle < 0 \\\\*[2mm]\n-\\gamma \\langle R | \\tilde{{\\cal H}} | R' \\rangle & {\\rm if} & \n\\langle R | \\tilde{{\\cal H}} | R' \\rangle >0 \\quad (\\gamma \\geq 0)\n\\end{array} \\right. \\\\*[6mm]\n\\langle R | \\tilde{{\\cal H}}_{\\rm eff} | R \\rangle & = & \n\\langle R | \\tilde{{\\cal H}} | R \\rangle + (1 + \\gamma) V_{\\rm sf} (R)\n\\end{array}\n\\end{equation}\nHere the ``sign flip'' potential is the same as that of ten Haaf et al.\n\\cite{Haa} and given by\n\\begin{equation} \\label{e2}\nV_{\\rm sf} (R) = \\sum_{R'_{\\rm na}} \\langle R'_{\\rm na} | \n\\tilde{{\\cal H}} | R \\rangle\n\\end{equation}\nwhere the subscript ``na'' (not--allowed) on $R'$ restricts the summation \nto the moves for which the\nmatrix element of the hamiltonian is positive (\\ref{f1}). \n\nIf the guiding\nwavefunction were to coincide with the true wavefunction, the simulation\nof the effective hamiltonian, which is sign free by construction, yields\nexact averages. So one may expect that good guiding wavefunctions lead to\ngood upperbounds for the energy. This upperbound increases with $\\gamma$,\nindicating that $\\gamma = 0$ seems the best choice, which is the effective \nhamiltonian of ten Haaf et al.\\cite{Haa}. That hamiltonian however is a \ntruncated version of the true hamiltonian in which all the dangerous moves \nare eliminated. The sign flip potential must correct this truncation by\nsuppressing the probability that the walker \nwill stay in a configuration with a large potential. \n\nThe second ingredient uses the fact that\nthe hamiltonian (\\ref{f1}) explores a larger phase space and therefore\ncontains more information than the truncated one. Parallel to the\nsimulation of the effective hamiltonian one can calculate the weights for\nthe true hamiltonian. As we saw in the summary\nof the GFMC method, forcefully made positive transition rates still\ncontain the correct weights when supplemented with sign functions.\nFor the true weights of the transition probabilities\nas given in (\\ref{f1}), the ``sign function'' must be chosen as\n\\begin{equation} \\label{f2}\ns(R,R') = \\left\\{ \\begin{array}{lcl}\n1 & {\\rm if} & \\langle R | \\tilde{{\\cal H}} | R' \\rangle >0 \\\\*[4mm]\n-1/\\gamma & {\\rm if} & \\langle R | \\tilde{{\\cal H}} | R' \\rangle < 0 \\\\*[4mm]\n\\displaystyle \\frac{1 - \\epsilon \\langle R |\\tilde{{\\cal H}} |R \\rangle}\n{1 - \\epsilon \\langle R |\\tilde{{\\cal H}}_{\\rm eff} |R \\rangle} & \n{\\rm if} & R = R'\n\\end{array} \\right.\n\\end{equation}\nWith these ``signs'' in the weights a proper average can be calculated, but\nthese averages suffer from the sign problem, the more so the smaller $\\gamma$\nis as one sees from (\\ref{f2}). So some intermediate value of $\\gamma$\nhas to be chosen. Fortunately \nthe results are not too sensitively dependent on $\\gamma$; the value\n$\\gamma=0.5$ is a good compromise and has been taken in our simulations. \n\nIn any simulation some walkers obtain a large weight and others a small one.\nTo lower the variance branching is regularly applied, which means a \nmultiplication of the heavily weighted walkers in favor of the removal of\nthose with small weight. It is not difficult to do this in an unbiased way.\nSorella \\cite{Sor} proposed to use the branching much more effectively in \nconjunction with the signs defined in (\\ref{f2}). The average sign is an \nindicator of the usefulness of the set of walkers. Start with a set of \nwalkers with positive sign. When the average sign becomes\nunacceptably low, the process is stopped and a reconfiguration takes place.\nThe walkers are replaced by another set with positive weights only, such that\na number of measurable quantities gives the same average. The more observables\nare included the more faithful is the replacement. The construction of the\nequivalent set requires the solution of a set of linear equations. \nWith the new set of walkers one continues the simulation on\nthe basis of the effective hamiltonian and one keeps track of the true\nweights with signs. The reconfiguration on the basis of some observables\ngives at the same time a measure for these obervables. Thus measurement\nand reconfiguration go together. As the number\nof observables that can be included is limited some biases are necessarily\nintroduced. Sorella showed that the error in the\nenergy of the guiding wave function is easily reduced by a factor of 10,\nwhereas reduction by the FNMC of ten Haaf et al. \\cite{Haa} rather gives\nonly a factor 2. \n\\begin{figure}[h] \n \\centering \\epsfxsize=\\linewidth\n \\epsffile{newener.eps}\n \\caption{The energy as function of the frustration ratio}\n \\label{fig4}\n\\end{figure}\n\n\\section{Results for the DMRG}\n\nIn this section we give a brief summary of the results of a pure \nDMRG--calculation. Extensive details can be found in \\cite{Luc}.\nThe systems are strips of widths up to $W=8$ and of various lengths $L$.\nThey are periodic in the small direction and open in the long direction.\nThe periodicity enables us to study the spin stiffness. We have chosen \nopen boundaries in the long direction to avoid the errors in the DMRG\nwavefunction due to periodic boundaries. Since we have good control \nof the scaling behavior in $L$ we extrapolate to \n$L \\rightarrow \\infty$ \\cite{Luc}. In the small direction\nwe are restricted to $W=2,4,6$ and 8 as odd values are not compatible\nwith the antiferromagnetic character of the system. For wider system sizes\nthe number of states which has to be taken into account exceeds the \npossiblities of the present workstations. Our criterion is that the value\nof the energy does not drift anymore appreciably upon the inclusion of more\nstates. This does not mean that the wavefunction is virtually exact, \nsince the energy is a rather insensitive probe for the wavefunction.\nFor instance correlation functions still improve from the\ninclusion of more states.\nIn Fig. \\ref{fig4} we present the energy as function of the ratio $J_2/J_1$,\nfor strip widths 4,6 and 8 together with the best extrapolation to\ninfinite width systems. \nThe figure strongly suggests that the infinite system undergoes a\nfirst order phase transition around a value 0.6.\nThis can be attributed to the transition to a columnar order (lines of \nopposite magnetisation). It is impossible to deduce more information \nfrom such an energy curve as other phase transitions are likely to be \ncontinuous with small differences in energy between the phases.\n\nThe spin stiffness can be calculated with the DMRG--wavefunction for\nsystems which are periodic in at least one direction \\cite{Luc}.\n\\begin{figure}[h] \n \\centering \\epsfxsize=\\linewidth\n \\epsffile{rhosh.eps}\n \\caption{The stiffness ${\\bf \\rho}_s$ as function of the frustration ratio.\nFinite size extrapolations put the region where $\\rho_s$ vanishes between\n0.38 and 0.62 \\cite{Sch}}\n \\label{Fig. 4 }\n\\end{figure}\nThe result of the computation is plotted in Fig. 5.\nOne observes a substantial decrease of $\\rho_s$ in the frustrated\nregion indicating the appearance of a magnetically disordered phase. \nIn contrast to the energy the data \ndo not allow a meaningful extrapolation to large \nwidths. The lack of clear finite size scaling behavior in the regime of \nsmall values of $W$ prevents to draw firm conclusions on the disappearence \nof the stiffness in the middle regime.\n\nFor the correlation functions following from the DMRG wavefunction we refer\nto \\cite{Luc}.\n\n\\section{Results for GFMC with SR}\n\nWe now come to the crux of this study: the simulations of the system with\nGFMC, using the DMRG wavefunctions to guide the importance sampling.\nAll the simulations have been carried out for $10 \\times 10$ lattice \nwith open boundaries. Standardly we have 6000 walkers and we run the \nsimulations for about $10^4$ measurements. These measuring points are\nnot fully independent and the variance is determined by chopping up the\nsimulations into 50-100 groups, often carried out in parallel on different\ncomputers. We first give an overall assessment of the correlation \nfunction pattern and then analyze some values of the ratio $J_2/J_1$. \n\nIn the first series\nwe have used the guiding wavefunction on the basis of the meandering\npath Fig. \\ref{fig2}(b), because it gives a better energy than the \nstraight option (a) . The number of basis states \nis $m=75$, which is small enough to carry out the\nsimulations with reasonable speed and large enough that trends begin\nto manifest themselves. Measurements of a number of correlation functions are \nmade in conjunction with Stochastic Reconfiguration as described in section 7.\nThe details of these calculations are given in Table \\ref{tab0}.\nNote that the DMRG guiding wavefunction gives a better energy for\nthe meandering path than for the straight path for values of\n$J_2/J_1$ up to 0.6. From 0.7 on this difference is virtually absent.\nThis undoubtly has to do with the change to the columnar state which\ncan equally well be realized by both paths. The value of $\\epsilon$ has\nbeen chosen as a compromise: independent measurements require a large\n$\\epsilon$ but the minus sign problem requires to apply often Stochastic \nReconfiguration i.e. a small $\\epsilon$. One sees that in the heavily \nfrustrated region the $\\epsilon$ must be taken small. In fact the more\ndetailed calculations for $J_2 = 0.3J_1$ and $J_2 = 0.5J_1$ were carried \nout with $\\epsilon = 0.01$.\n\nIn Fig. 6 and 7 we have plotted a sequence of visualizations of the \ncorrelations. From top to bottom (zig--zag)\nthey give the correlations for the values of $J_2/J_1$. In order\nto highlight the differences a distinction is made between correlations\nwhich are above average (solid lines) and below average (broken lines).\nAll nearest neighbor spin correlations shown are negative. \nIn all the pictures one sees the influence of the boundaries on the spin\ncorrelations. Only 1/4 of the lattice has been pictured, the other\nsegments follow by symmetry. The upper right corner, which\ncorresponds to the center of the lattice, is the most significant for the\nbehavior of the bulk. The overall trend is that spatial variations\nin the correlation functions occur in growing size with $J_2/J_1$.\nOn the side of low $J_1/J_2$ (N\\'eel phase)\none sees dimer patterns in the horizontal direction, they\nturn over to vertical dimers (around $J_2 = 0.7 J_1$) and rapidly\ndisappear in the columnar phase.\nThis is again support for the fact that the columnar phase is separated\nfrom the intermediate state by a first order phase transition.\n\nOpen boundary conditions have the disadvantage of boundary effects, which\nmake it more difficult to distinguish between spontaneous and induced \nbreaking of the translational symmetry. On the other hand for open \nboundaries, dimers, plaquettes or any other interruption of the \ntranslational symmetry have a natural reference frame. \nThe correlations are not only influenced by the boundaries of the system,\nalso the guiding DMRG--wavefunction leaves its imprint on the results.\nThis is mainly due to the fact that we have only mixed estimators for the\ncorrelation functions, which show a mix of the guiding wavefunction and the\ntrue wavefunction. The improved estimator, used in these pictures, corrects\nfor this effect to linear order in the deviation.\\footnote{Forward walking\nallows to make a pure estimate of the correlations, but requires much\nmore calculations. \\cite{Sor2}} The ladder like structure\nin the DMRG path is reflected in a ladder like pattern in the correlations\nas an inspection of the correlations in the DMRG wavefunctions (not shown\nhere) reveals. But ladders are clearly also present in the GFMC results \nshown in the pictures. \n\nIn order to eliminate the influence of the guiding \nwavefunction we scrutinize some of values of $J_2/J_1$ in more detail,\nby inspecting how the results depend on the size of the basis in the\nDMRG wavefunction and on the choice of the DMRG path. Since we are\nmostly interested in the behavior of the infinite lattice, we discuss mainly\nthe behavior of the correlations in and around the central plaquette. \nSo we study a sequence of DMRG wavefunctions for $m=32$, 75, 100, 128 \nand 150(200) and carry out for each of them extensive GFMC simulations.\nFirst we look to the case $J_2=0$, which is easy because we know \nthat it must be N\\'eel ordered and therefore it serves as a \ncheck on the calculations. Then we take $J_2=0.3 J_1$ which is the\nmost difficult case since it is likely to be close to a phase transition.\nFinally we inspect $J_2 = 0.5 J_1$ where \nwe are fairly sure that some dimerlike phase is realized. \n\n\\subsection{$J_2=0$}\n\nFor the unfrustrated Heisenberg model we have several checkpoints for\nour calculations. We can find to a high degree of accuracy the groundstate\nenergy and we are sure that the N\\'eel phase is homogeneous, i.e. that \nthe correlations show no spatial variation other than that of the \nantiferromagnet.\nWe have two ways of estimating the energy of a $10 \\times 10$ lattice.\nThe first method is based on finite size interpolation. From \nDMRG calculations \\cite{Luc} we have an exact value for a\n$4 \\times 4$ lattice, an accurate value for the $6 \\times 6$ lattice and\na good value for the $8 \\times 8$ lattice. There is also the very \naccurate calculation of Sandvik \\cite{San} for an infinitely large lattice,\nyielding the value of $e_0=-0.669437(5)$. \nThe leading finite size correction goes as $1/L$. \nIncluding also a $1/L^2$ term we have esimated the value for a $10 \\times 10$\nlattice as 0.629(1) and incorporated this value in Table \\ref{tab1}(a). \nWe stress that this is an {\\it interpolation} for which the value of Sandvik \nis the most important input. \n\nThe second method is less well founded and uses the experience that \nDMRG energy estimates can be improved considerably by {\\it extrapolating} \nto zero truncation error. When plotted as function of this\ntruncation error the energy is often remarkably linear. In Table \n\\ref{tab1}(b) we give for a series of bases $m=32, 75, 100, 128$\nand 150, the values of the truncation error and the corresponding DMRG energy\nper site together with the extrapolation on the basis of linear behavior.\\footnote{The value for $m=150$ is not in line with the others. This can be\nexplained by the fact that the construction of this DMRG wavefunction was\nslightly different from the others in which the basis was built up gradually.}\nNote that the two estimates are compatible. In Table \\ref{tab1}(b) we have \nalso listed the values of the GFMC simulations for the corresponding values \nof $m$. They do agree quite well with these estimates in particular with the \none based on finite size scaling. We point out that one \nwould have to go very far in the number of states in the DMRG calculation to\nobtain an accuracy that is easily obtained with GFMC. Thus the combination\nof GFMC and DMRG does really better than the individual components.\nOne might wonder why there is still a drift to lower energy values in\nthe GFMC simulations (which is also present in the tables to come).\nThe reason is that the DMRG wavefunction is strictly zero outside a \ncertain domain of configurations, because the truncation of the basis\ninvolves also the elimination of certain combinations of conserved \nquantities of the constituing parts. The domain of the wavefunction grows\nwith the size of the basis.\n\nTurning now to the correlations it seems that they are homogeneous in \nthe center of the lattice for $J_2=0$. However a closer inspection\nreveals small differences. In Table \\ref{tab2} we list the asymmetries in\nthe horizontal and vertical directions of the spin correlations in and around\nthe central plaquette as function of the number of states.\nIf we number the spins on the lattice as \n${\\bf S}_{n,m}$ with $1 \\leq n,m \\leq 10$, the central plaquette \nhas the coordinates (5,5), (5,6), (6,5) and (6,6). We then define the \nasymmetry parameters $\\Delta_x$ and $\\Delta_y$ as\n\\end{multicols}\n\\begin{equation} \\label{g1}\n\\left\\{ \\begin{array}{l}\n\\Delta_x = %\\displaystyle\n\\frac{1}{4} \\langle {\\bf S}_{4,5} \\cdot {\\bf S}_{5,5} +\n{\\bf S}_{4,6} \\cdot {\\bf S}_{5,6} + {\\bf S}_{6,5} \\cdot {\\bf S}_{7,5} +\n{\\bf S}_{6,6} \\cdot {\\bf S}_{7,6} \\rangle -\\frac{1}{2} \\langle\n{\\bf S}_{5,5} \\cdot {\\bf S}_{6,5} + {\\bf S}_{5,6} \\cdot {\\bf S}_{6,6}\n\\rangle \\\\*[2mm]\n\\Delta_y = %\\displaystyle\n\\frac{1}{4} \\langle {\\bf S}_{5,4} \\cdot {\\bf S}_{5,5} +\n{\\bf S}_{6,4} \\cdot {\\bf S}_{6,5} + {\\bf S}_{5,6} \\cdot {\\bf S}_{5,7} +\n{\\bf S}_{6,6} \\cdot {\\bf S}_{6,7} \\rangle -\\frac{1}{2} \\langle\n{\\bf S}_{5,5} \\cdot {\\bf S}_{5,6} + {\\bf S}_{6,5} \\cdot {\\bf S}_{6,6} \\rangle\n\\end{array} \\right.\n\\end{equation}\n\\begin{multicols}{2}\nSo $\\Delta_x$ is the average value of the \ncorrelations on the 4 horizontal bonds which are connected to the central\nplaquette minus the average of the values on the 2 horizontal bonds in\nthe plaquette. Similarly $\\Delta_y$ corresponds to the vertical direction.\nThe values for the asymmetry in Table \\ref{tab2} in the vertical direction \nare so small that they have no significance. Note that the\nanticipated decrease in $\\Delta_x$ is slow in DMRG and therefore also\nslow in the mixed estimator of the GFMC. The improved estimator (\\ref{d15})\nhowever is truely an improvement! So one sees that all the observed small\ndeviations from the homogeneous state will disappear with the increase of \nthe number of states in the basis of the DMRG wavefunction. (In general the\naccuracy of the correlations is determined by that of the GFMC simulations.\nWe get as variance a number of the order 0.01, implying twice that value\nfor the improved estimator) $\\;$ The vanishing of $\\Delta_x$ and $\\Delta_y$\nalso prove that finite size effects are small in the center of the \n$10 \\times 10$ lattice. \nFrom these data we may conclude that the GFMC can make up for\nthe errors in the DMRG wavefunction for a relative low number of basis states.\nWe have not carried out a similar series for the straight path since\nthis will certainly show no dimers as will become clear from the following\ncases.\n\n\\subsection{$J_2 = 0.3 J_1$}\n\nThis case is the most difficult to analyze since it is expected to be close\nto a continuous phase transition from the N\\'eel state to a \ndimerlike state. As is known \\cite{Ost} the DMRG structure of the wavefunction\nis not very adequate to cope with the long--range correlation in the\nspins typical for a critical point. In Table \\ref{tab3} we have presented the\nsame data as in Table \\ref{tab2} but now for $J_2=0.3$.\nThere is no pattern in the energy as function of the truncation error\n$\\delta$. The decrease of the energy as function of the size of the \nbasis $m$ is in the DMRG wavefunctions is not saturated.\nThe GFMC simulations lead to a notably lower energy and they\ndo hardly show a leveling off as function of the basis of the \nguiding wavefunction. All these points\nare indicators that the DMRG wavefunction is rather far from convergence\nand that more accurate data would require a much larger basis. As far as the \nstaggering in the correlations is concerned the values for $\\Delta_x$ are\nsignificant, also because the simulation results generally increase the \nvalues. Those for $\\Delta_y$ are not small enough to be considered as noise.\nGiven the fact that most authors locate the\nphase transition at higher values $J_2 \\simeq 0.4 J_1$ we would expect\nboth $\\Delta$'s to vanish. So either the dimerlike state is realized for\nvalues as low as $J_2 = 0.3 J_1$ or dimer formation already starts \nin the N\\'eel state. \n\nTo get more insight in the nature of the groundstate we have also carried\nout the same set of simulations on the straight path (a) in Fig. \\ref{fig2}. \nThis guiding wavefunction shows virtually no formation of dimers in \nany direction as can be observed from Table \\ref{tab4}.\nIn spite of the fact that the trends indicated in the table have not come to\nconvergence one may draw a few conclusions from the comparison of the\ntwo sets of simulations. The overal impression is that the meandering\nguiding wavefunction represents a groundstate of a different symmetry\nas compared to the straight path guiding wavefunction.\nThe meandering wavefunction prefers dimers in the horizontal direction\nand the straight wavefunction leads to some dimerization in the vertical\ndirection. The difference also shows up in the energy, it is\nnot only large on the DMRG level but it also persists at the GFMC level. \nWe see similar trends in the next case.\n\n\\subsection{$J_2 = 0.5 J_1$}\n\nBy any estimate this value of the next nearest neighbor coupling leads to\na dimerlike state if it exists at all. No accurate data are available \non the energy of the $10 \\times 10$ system to compare to our results.\nIn Table \\ref{tab5} we list the data for a set of DMRG wavefunctions \nwith bases $m=32$, 75, 100, 128, 150 and 200.\nThe DMRG values of the energy (with exception of the value for $m=32$) can\nbe extrapolated to zero truncation error with the limiting value\n$E_0 = -48.4(1)$, which corresponds very well with the level in the\nGFMC values for larger sizes of the basis.\nThis indicates again that the GFMC simulations can\nmake up for the shortcoming of the DMRG wavefunction. One would indeed have\nto enlarge the basis to $m$ of the order of 1000 in order to achieve the \nvalue of the energy of the simulations which use DMRG guiding \nwavefunctions with a basis of the order of 100. \n\nThe staggering in the correlations expressed by the quantities $\\Delta_x$ \nfor the horizontal direction and $\\Delta_y $ for the vertical direction,\nhas values that are significant. If one looks to the contributions \nof the DMRG wavefunction and the GFMC simulation separately, one observes\nthat the overall values do agree quite well, with the tendency that the\nGFMC simulations lowers the staggerring in the horizontal direction and\nslightly increases it in the vertical direction.\nSo we may conclude that indeed in the groundstate of the $J_2=0.5 J_1$ system, \nthe correlations of the spins are not translation invariant but show \na staggering. However these results neither confirm the picture that\nthe dimerstate is the lowest (as suggested by Kotov et al. \\cite{Kot})\nnor that the plaquettestate is the groundstate (as concluded by Capriotti \nand Sorella \\cite{Sor2}). We comment on these discrepancies further in the\ndiscussion.\n\nAgain it is worthwhile to compare these results with a simulation on the\nbasis of the straight path (a) in Fig. \\ref{fig2}. \nHere it is manifest that the straight path prefers to have the\ndimers in the vertical direction. Again the impression is that the\nstraight path leads to a different symmetry as compared to the meandering\npath. It is not only the different preference in the main direction of \nthe dimers, also the secondary dimerization in the perpendicular direction,\nnotably in the meandering case, is not present in the straight case.\nThe fairly large difference in energy on the DMRG level becomes quite small\non the GFMC level.\n \n\\section{Discussion}\n\nWe have presented a method to employ the DMRG wavefunctions as guiding\nwavefunctions for a GFMC simulation of the groundstate. Generally \nthe combination is much better than the two individual methods. The\nGFMC simulations considerably improve the DMRG wavefunction. In the\nintermediate regime the properties of the GFMC simulations depend \non the guiding wavefunction as the results for two different DMRG guiding\nwavefunctions show. \n\nThe method has been used to observe spin correlations in the frustrated\nHeisenberg model on a square lattice. In this discussion we focus \non the intermediate region where the model is most frustrated\nand which is the ``piece de resistance'' of the present research.\nWe see patterns of strongly correlated nearest neighbor spins, to be\ncalled dimers. To indicate what me mean by strong and weak we give the\nvalues in and around the central square of the $10 \\times10$ lattice, \nfor the case $J_2 = 0.5 J_1$. In Fig. \\ref{comp}(c) we have given the\nvalues of the central square extrapolated to an infinite lattice.\n\\begin{figure}[h]\n \\centering \\epsfig{file=kotov-sorella-thispaper.eps,width=\\linewidth}\n \\caption[]{The correlation pattern for the nearest spins for $J_2 = 0.5J_1$;\n(a) according to Kotov et al. \\cite{Kot2}: a dimer pattern in which the \nstrength of the correlation is indicated; (b) according to Capriotti and \nSorella \\cite{Sor2}: a plaquette state and (c) according to this paper: an\nintermediate pattern in which the translational invariance is broken in\nboth directions but with unequal strength. The values indicated are those\nbased on the meandering path and the improved estimator.}\n\\label{comp}\n\\end{figure}\nThe values are based on the improved estimator and it is interesting to\nsee the trends. The horizontal strong correlation of -0.42 is the result\nof the DMRG value -0.44 and the GFMC value -0.43, while the weak bond\n-0.15 is the result of the DMRG value -0.09 and the GFMC value -0.12. \nThus the GFMC weakens the order parameter $D_x$ associated with the \nstaggering. For the vertical direction\nthere is hardly a change from DMRG to GFMC. One has to go to the next \ndecimal to see the difference. The strong bond equals -0.368 and is \ncoming from the DMRG value -0.375 and the GFMC\nvalue -0.371, while the improved weak bond of -0.271 is the resulting value\nof -0.275 for DMRG and -0.273 for GFMC. \n\nBefore we comment on this result we discuss the influence of the choice\nof the guiding wave function. We note that for both points $J_2 = 0.3 J_1 $\nand $J_2 = 0.5 J_1 $ the two choices for the DMRG wavefunction give\ndifferent results. First of all the main staggering is for the meandering\npath (b) of Fig. \\ref{fig2} in the horizontal direction, while the \nstraight path (a) of Fig. \\ref{fig2} prefers the dimers in the vertical \ndirection. There is\nnot much difference in the values of the strong and weak correlations.\nSecondly the straight path shows no appreciable staggering in the other\ndirection, so one may wonder whether the observed effect for the meandering\npath is real. In our opinion this difference has to do with the effect that \nthe DMRG wavefunction ``locks in'' on a certain symmetry. The straight path\nyields a groundstate which is truely dimerlike in the sense that it is\ntranslational invariant in the direction perpendicular to the dimers.\nThe meandering path locks in on a different groundstate which holds the\nmiddle between a dimerlike and a plaquettelike state. The GFMC simulations\ncannot overcome this difference in symmetry, likely because the two\nlowest states with different symmetry are virtually orthogonal. On the\nDMRG level there is a large difference in energy between the two states,\nfavoring the meandering path strongly, on the GFMC level this difference\nhas become very small. With this observation in mind we compare\nour result with other findings.\n\nThe results of the series expansions \\cite{Kot}, \\cite{Kot2} and \n\\cite{Sin} are shown in Fig. \\ref{comp}(a).\nTheir correlations organize themselves in spinladders.\nThe correlations on the rungs of the ladder are $-0.45 \\pm 0.5$ which compares\nwell with our strongest horizontal correlation and this holds also for\nthe weak horizontal correlation (--0.12 vs --0.15). The most noticeble\ndifference is the value of our weak correlation in the vertical direction\n(--0.27 vs --0.36) while the strong correlation (--0.37 vs --0.36) agrees.\nThere is no real conflict between our result and theirs since \nthe symmetry they find is fixed by the state around which the series\nexpansion is made. So our claim is only that our state with different \nsymmetry is the lower one. In fact in the paper of Singh et al. \\cite{Sin},\nit is noted that the susceptibility to a staggering operator in the \nperpendicular direction (our $\\Delta_y$) becomes very large in the dimer \nstate for $J_2 = 0.5 J_1$ which we take as an indication of the nearby \nlower state. The analytical calculations in \\cite{Kot} and \\cite{Kot2}\nhowever do not support the existence of the state we find.\n\nNeither do we find support for the plaquette state found in \\cite{Sor2},\nwhich we have sketched in Fig. \\ref{comp}(b).\nThe evidence of this investigation is based on the boundedness of the\nsusceptibility for the operator which breaks the orientational symmetry and\nthe divergence of the susceptibility for the order parameter breaking\ntranslational invariance (corresponding to $\\Delta_x$). They have not\nseparately investigated the values of $\\Delta_x$ and $\\Delta_y$ since\ntheir groundstate has the symmetry of the lattice and one would \nfind automatically the same answer.\nThey conclude that in absence of orientational order parameter and with the\npresence of the translational order parameter the state must be plaquettelike.\nWe believe that their result is influenced by the guiding wavefunction\nfor which the one-step Lanczos approximation is taken. This wavefunction\ncertainly has the symmetry of the square and again GFMC cannot find\na groundstate with a different symmetry. \n\nFinally we comment on the fact that we find the dimerization already \nfor values as low as $J_2 = 0.3 J_1$ at least for the meandering path.\nAs we have mentioned earlier the results as function of the number of\nstates have not sufficiently converged to make a firm conclusion, the\nmore so since there is a large difference between DMRG and GFMC. Still\nit could be an indication that the phase transition from the N\\'eel\nstate to the dimer state takes place for lower values than the\nestimated $J_2 = 0.38 J_1$ \\cite{Sch}.\n \nThus many questions are left over, amongst others how the order parameters\nbehave as function of the frustation ratio in the intermediate region. We\nfeel that the combination of DMRG and GFMC is a good tool to investigate\nthese issues since they demonstrate {\\it ad oculos} the correlations in the\nintermediate state.\n\n{\\bf Acknowledgement} \nThe authors are indebted to Steve White for making his software available.\nOne of us (M. S. L. duC. de J.) gratefully acknowledges the hospitality\nof Steve for a stay at Irvine of 3 months, where the basis \nof this work was laid. The authors have also benefitted from \nilluminating discussions with Subir Sachdev and Jan Zaanen. The authors\nwant to acknowledge the efficient help of Michael Patra with the \nsimulations on the cluster of PC's of the Instituut-Lorentz.\n\n\\begin{thebibliography}{99}\n\\bibitem[1]{Wh1} S. R. White, Phys. Rev. Lett. {\\bf 69} (1992) 2863;\\\\ \nS. R. White, Phys. Rev. B {\\bf 48}(1993) 10345. \n\\bibitem[2]{sign} For a recent discussion of the sign problem in various\ncases see P. Henelius and A. W. Sandvik cond--mat/0001351 and references there\nin.\n\\bibitem[3]{Sor} S. Sorella, Phys. Rev. Lett. {\\bf 80} (1998) 4558.\n\\bibitem[4]{San} A. W. Sandvik, Phys. Rev. {\\bf B56} (1998) 11678.\n\\bibitem[5]{Sor2} L. Capriotti and S. Sorella cond--mat/9911161.\n\\bibitem[6]{Duc} M. S. L. du Croo de Jongh and P. H. J. Denteneer,\nPhys. Rev.B {\\bf 55} (1997) 2713.\n\\bibitem[7]{Sch} H. J. Schulz, T. A. L. Ziman and D. Poilblanc \nJ. Phys. I {\\bf 6} (1996) 675.\n\\bibitem[8]{Kot} V. Kotov, J. Oitmaa, O. P Sushkov and Z. Weihong, Phys. Rev.\nB {\\bf 60} (1999) 14613. \n\\bibitem[9]{Kot2} V. Kotov, J. Oitmaa, O. P Sushkov and Z. Weihong, \ncond--mat/9912228.\n\\bibitem[10]{Sin} R. R. P. Singh, Z. Weihong, C. J. Hamer and J. Oitmaa,\nPhys. Rev. B {\\bf 60} (1999) 7278.\n\\bibitem[11]{Sac} \nN. Read and S. Sachdev, Phys. Rev. Lett. {\\bf 62}, 1694 (1989).\\\\\nN. Read and S. Sachdev, Phys. Rev. B {\\bf 42}, 4568 (1990).\\\\\nN. Read and S. Sachdev, Phys. Rev. Lett. {\\bf 66}, 1773 (1991).\\\\\nS. Sachdev and N. Read, Int. J. Mod. Phys. B {\\bf 5}, 219 (1991).\\\\\nS. Sachdev, Quantum Phase Transitions, Cambridge University Press,\nCambridge (1999).\n\\bibitem[12]{Cep} D. M. Ceperley and B. J. Alder Phys. Rev. Lett. \n{\\bf 45}(1980) 566; \\\\D. M. Ceperley, Rev. Mod. Phys. {\\bf 67} (1995) 279.\n\\bibitem[13]{Luc} M. S. L. du Croo de Jongh, Thesis Leiden University 1999, cond-mat/9908200.\n\\bibitem[14]{Wh2} We are grateful to Steve White for making his software\navailable to us.\n\\bibitem[15]{Tri} N. Trivedi and D. M. Ceperley, Phys. Rev. {\\bf B 50} (1990) \n4552.\n\\bibitem[16]{Mar} W. Marshall, Proc. Royal Soc. Londen Ser. \n{\\bf A 232} (1955) 48.\n\\bibitem[17]{Bem} H. J. M. van Bemmel, D. F. B. ten Haaf, W. van Saarloos,\nJ. M. J. van Leeuwen and G. An, Phys. Rev. Lett. {\\bf 72} (1994) 2442. \n\\bibitem[18]{Haa} D. F. B. ten Haaf, H. J. M. van Bemmel, J. M. J. van Leeuwen,\nW. van Saarloos and D. M. Ceperley, Phys. Rev. {\\bf B51} (1995) 13039.\n\\bibitem[19]{Ost} S \\\"Ostlund and S. Rommer,\n Phys. Rev. Lett. {\\bf 75}, 3537 (1995);\\\\\nS.Rommer and S. \\\"Ostlund, Phys. Rev. B {\\bf 55} (1997) 2164.\n\n\\end{thebibliography}\n\\end{multicols}\n\\vspace*{1cm}\n\n\\begin{table}[h]\n \\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n \\hline\n & & \\multicolumn{2}{|c|}{} & \\multicolumn{2}{|c|}{} \\\\*[-3mm]\n & & \\multicolumn{2}{|c|}{Straight} & \\multicolumn{2}{|c|}{Meander} \\\\*[1mm]\n\\cline{2-6}\n$J_2$ & $\\epsilon$& $E_{\\rm DMRG}$ & $E_{\\rm GFMC}$ & $E_{\\rm DMRG}$ & $E_{\\rm GFMC}$ \\\\*[1mm]\n\\hline\n0.0 & 0.3 & -61.30 & -62.33(8) & -61.84 & -62.54(4) \\\\*[1mm]\n0.1 & 0.06 & -57.96 & & -58.53 & -59.25(2) \\\\*[1mm]\n0.2 & 0.04 & -54.75 & -56.08(11) & -55.48 & -56.22(4) \\\\*[1mm]\n0.3 & 0.02 & -51.75 & -53.17(4) & -52.50 & -53.38(3) \\\\*[1mm]\n0.4 & 0.02 & -49.00 & -50.51(8) & -49.92 & -50.60(5) \\\\*[1mm]\n0.5 & 0.014 & -46.68 & -47.76(6) & -47.78 & -48.34(4) \\\\*[1mm]\n0.6 & 0.015 & -45.41 & & -46.03 & -46.40(3) \\\\*[1mm]\n0.7 & 0.015 & -45.67 & & -45.60 & -46.00(2) \\\\*[1mm]\n0.8 & 0.02 & -49.16 & & -49.13 & -49.60(9) \\\\*[1mm]\n0.9 & 0.02 & -53.61 & & -53.70 & -54.52(2) \\\\*[1mm]\n1.0 & 0.02 & -58.46 & -59.71(9) & -58.64 & -59.80(8)\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab0} For each degree of frustration the imaginary \ntime interval $\\epsilon$, the energy of the guiding state \n$E_{\\rm DMRG}$ and that of the GFMC state $E_{\\rm GFMC}$ are listed.}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|l|}\n\\hline\n & \\\\*[-3mm]\n$L$ & $e_0(L \\times L)$ \\\\*[1mm]\n\\hline\n & \\\\*[-3mm]\n4 & -0.5740 \\\\*[1mm]\n6 & -0.6031 \\\\*[1mm]\n8 & -0.6188 \\\\*[1mm]\n10 & -0.629(1) \\\\*[1mm]\n$\\infty$ & -0.669437(5) \\\\*[1mm]\n\\hline\n\\end{tabular} \\quad \\quad \\quad\n\\begin{tabular}{|l|l|l|l|}\n\\hline\n & & & \\\\*[-3mm]\n \\# states & trunc. error & $ e_0$ (DMRG) & $ e_0$ (GFMC)\\\\*[1mm]\n\\hline\n & & & \\\\*[-3mm]\n32 & 21.2 $\\times 10^{-5}$ & -0.6084 & -0.6192(1) \\\\*[1mm]\n75 & 12.0 $\\times 10^{-5}$ & -0.6184 & -0.6254(5) \\\\*[1mm]\n100 & 10.5 $\\times 10^{-5}$ & -0.6201 & -0.625(2) \\\\*[1mm]\n128 & 8.7 $\\times 10^{-5}$ & -0.6214 & -0.6269(6) \\\\*[1mm]\n150 & 9.6 $\\times 10^{-5}$ & -0.6231 & -0.6277(5) \\\\*[1mm]\n$2^N$ & 0 & -0.631(3) & \\\\*[1mm]\n\\hline\n\\end{tabular}\\\\*[4mm]\n\\hspace*{-2cm} (a) \\hspace*{7cm} (b)\n\\caption{\\label{tab1} Interpolation (a) and extrapolation (b) \nestimates of the energy per site of a $10 \\times 10$ lattice}\n\\end{center}\n\\end{table}\n\\vspace*{1cm}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|r|r|r|}\n\\hline \n & \\multicolumn{3}{|c|}{} & \\multicolumn{3}{|c|}{} \\\\*[-3mm]\n\\# states & \\multicolumn{3}{|c|}{$\\Delta_x$} & \n\\multicolumn{3}{|c|}{$\\Delta_y$} \\\\*[1mm]\n\\cline{2-7} \n & & & & & & \\\\*[-3mm]\nm & DMRG & GFMC & Improved & DMRG & GFMC & Improved \\\\*[1mm]\n32 & 0.14373 & 0.09981 & 0.05589 & -0.00060 & 0.00078 & 0.00216 \\\\*[1mm]\n75 & 0.07291 & 0.05668 & 0.04045 & 0.00081 & 0.00601 & 0.01121 \\\\*[1mm]\n100 & 0.06432 & 0.04255 & 0.03088 & 0.00030 & 0.00173 & 0.00316 \\\\*[1mm]\n128 & 0.05619 & 0.03734 & 0.01849 & 0.00091 & -0.00040 & -0.00173 \\\\*[1mm]\n150 & 0.05044 & 0.03612 & 0.02221 & 0.00079 & 0.00261 & 0.00442 \\\\\n\\hline \n\\end{tabular}\\\\*[4mm]\n\\caption{\\label{tab2} Values for the asymmetry in the center for $J_2 = 0$.\nAs discussed in the text the error in the improved estimator values is of\nthe order 0.02, which means that for $m=128$ and higher the values are\nstatistically indistinguishable from zero.}\n\\end{center}\n\\end{table}\n\\vspace*{1cm}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n & \\multicolumn{4}{|c|}{} & \\multicolumn{3}{|c|}{} \\\\*[-3mm]\n\\# states & \\multicolumn{4}{|c|}{{\\rm DMRG}} & \n\\multicolumn{3}{|c|}{{\\rm GFMC}} \\\\*[2mm]\n\\cline{2-8} \n & & & & & & & \\\\*[-3mm]\n$m$ & $\\delta * 10^5$ & $E_{{\\rm DMRG}} $ & $\\Delta_x $ & \n$ \\Delta_y $ & $E_{{\\rm GFMC}} $ & $\\Delta_x $ & $ \\Delta_y $ \\\\*[2mm]\n\\hline\n & & & & & & &\\\\*[-3mm]\n32 & 19.0 & -51.609 & 0.27784 & 0.00295 & -52.81(43) & 0.363 & -0.009 \\\\*[2mm]\n75 & 10.6 & -52.581 & 0.15462 & 0.00616 & -53.29(05) & 0.207 & 0.011 \\\\*[2mm]\n100 & 9.4 & -52.707 & 0.14709 & 0.00943 & -53.32(33) & 0.145 & 0.009 \\\\*[2mm]\n128 & 10.6 &-52.821 & 0.13042 & 0.00577 & -54.01(04) & 0.254 & 0.063 \\\\*[2mm]\n150 & 10.4 &-52.888 & 0.12564 & 0.00737 & -54.10(12) & 0.236 & 0.103 \\\\*[1mm] \n\\hline\n\\end{tabular} \\\\*[4mm]\n\\caption{\\label{tab3} Energies and asymmetries for the case \n$J_2 = 0.3 J_1$ as function of the number of basis states $m$. $\\delta$ is the\ntruncation error. The asymmetries $\\Delta_x $ and $\\Delta_y$ for the\nGFMC simulations are \ncalculated with the improved estimator. The guiding wavefunction is obtained\nfrom the meandering path (b) in Fig. \\ref{fig2}. The statistical error in\n$\\Delta_x$ and $\\Delta_y$ is of the order 0.02}\n\\end{center}\n\\end{table} \n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n & \\multicolumn{4}{|c|}{} & \\multicolumn{3}{|c|}{} \\\\*[-3mm]\n\\# states & \\multicolumn{4}{|c|}{{\\rm DMRG}} & \\multicolumn{3}{|c|}{{\\rm GFMC}} \\\\*[2mm]\n\\cline{2-8} \n & & & & & & & \\\\*[-3mm]\n$m$ & $\\delta * 10^5$ & $E_{{\\rm DMRG}} $ & $\\Delta_x $ & \n$ \\Delta_y $ & $E_{{\\rm GFMC}} $ & $\\Delta_x $ & $ \\Delta_y $ \\\\*[2mm]\n\\hline\n & & & & & & &\\\\*[-3mm]\n32 & 30.0 & -50.672 & 0.00032 & 0.01657 & -52.15(11) & 0.061 & 0.047 \\\\*[2mm]\n75 & 18.9 & -51.733 & -0.00295 & 0.00426 & -53.21(10) & -0.030 & 0.036 \\\\*[2mm]\n100 & 19.9 & -52.066 & 0.00349 & 0.00492 & -53.84(72) & 0.061 & 0.079 \\\\*[2mm]\n128 & 24.6 & -52.302 & 0.00139 & 0.00791 & -53.50(19)& 0.079 & 0.027 \\\\*[2mm]\n150 & 25.7 & -52.455 & 0.00222 & 0.00780 & -53.52(10) & 0.022 & 0.065 \\\\\n\\hline \n\\end{tabular} \\\\*[4mm]\n\\caption{\\label{tab4} Comparison of the energies and the \nvalues for the asymmetry in the center for the DMRG wavefunction based\non the first (straight) path (a) in Fig. \\ref{fig2} and the associated \nGFMC simulation; $J_2 = 0.3 J_1$}\n\\end{center}\n\\end{table}\n\\vspace*{1cm}\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n & \\multicolumn{4}{|c|}{} & \\multicolumn{3}{|c|}{} \\\\*[-3mm]\n\\# states & \\multicolumn{4}{|c|}{{\\rm DMRG}} & \\multicolumn{3}{|c|}{{\\rm GFMC}} \\\\*[2mm]\n\\cline{2-8} \n & & & & & & & \\\\*[-3mm]\n$m$ & $\\delta * 10^5$ & $E_{{\\rm DMRG}} $ & $\\Delta_x $ & \n$ \\Delta_y $ & $E_{{\\rm GFMC}} $ & $\\Delta_x $ & $ \\Delta_y $ \\\\*[2mm]\n\\hline\n & & & & & & &\\\\*[-3mm]\n32 & 11.8 & -47.116 & 0.43245 & 0.14667 & -47.55(29) & 0.295 & 0.065 \\\\*[2mm]\n75 & 17.4 & -47.771 & 0.38954 & 0.13059 & -48.22(04) & 0.339 & 0.070 \\\\*[2mm]\n100 &12.4 & -47.924 & 0.39364 & 0.07877 & -48.37(22) & 0.310 & 0.110 \\\\*[2mm]\n128 & 8.4 & -48.014 & 0.37317 & 0.08246 & -48.32(05) & 0.336 & 0.139 \\\\*[2mm]\n150 & 8.3 & -48.088 & 0.35819 & 0.07983 & -48.33(12) & 0.324 & 0.112 \\\\*[2mm]\n200 & 7.6 & -48.153 & 0.34590 & 0.09973 & -48.43(05) & 0.272 & 0.094 \\\\*[1mm]\n\\hline\n\\end{tabular} \\\\*[4mm]\n\\caption{\\label{tab5} Energies and asymmetries for $J_2=0.5 J_1$ with \nguiding wavefunction based on the meandering path (b) in Fig. \\ref{fig2}}\n\\end{center}\n\\end{table}\n\\vspace*{1cm}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n & \\multicolumn{4}{|c|}{} & \\multicolumn{3}{|c|}{} \\\\*[-3mm]\n\\# states & \\multicolumn{4}{|c|}{{\\rm DMRG}} & \\multicolumn{3}{|c|}{{\\rm GFMC}} \\\\*[2mm]\n\\cline{2-8} \n & & & & & & & \\\\*[-3mm]\n$m$ & $\\delta * 10^5$ & $E_{{\\rm DMRG}} $ & $\\Delta_x $ & \n$ \\Delta_y $ & $E_{{\\rm GFMC}} $ & $\\Delta_x $ & $ \\Delta_y $ \\\\*[2mm]\n\\hline\n & & & & & & &\\\\*[-3mm]\n32 & 69.4 & -45.756 & 0.00172 & 0.24701 & -47.45(08) & 0.074 & 0.185 \\\\*[2mm]\n75 & 26.2 & -46.718 & 0.00171 & 0.34950 & -47.81(25) & -0.025 & 0.302 \\\\*[2mm]\n100 & 21.2 & -46.993 & 0.00063 & 0.33131 & -48.16(06) & -0.003 & 0.350 \\\\*[2mm]\n128 & 24.6 & -47.231 & -0.00029 & 0.32994 & -48.31(08)& 0.013 & 0.291 \\\\*[2mm]\n150 & 25.7 & -47.379 & 0.00215 & 0.32458 & -48.33(06) & -0.026 & 0.257 \\\\\n\\hline \n\\end{tabular} \\\\*[4mm]\n\\caption{\\label{tab6} Same as Table \\ref{tab5} but\nnow for the ``straight'' path Fig (a) \\ref{fig2}}\n\\end{center}\n\\end{table}\n\n\\begin{figure} \n \\epsfig{file=result.10x10.m75.e0.25.J20.meander2.ss.eps,width=6cm}\n \\hfill\n \\epsfig{file=result.10x10.m75.e0.06.J20.1.meander2.ss.eps,width=6cm} \\vspace{0.5cm}\\\\\n \\epsfig{file=result.10x10.m75.e0.04.J20.2.meander2.ss.eps,width=6cm}\n \\hfill\n \\epsfig{file=result.10x10.m75.e0.02.J20.3.meander2.ss.eps,width=6cm}\\vspace{0.5cm}\\\\\n \\epsfig{file=result.10x10.m75.e0.02.J20.4.meander2.ss.eps,width=6cm}\n \\hfill\n \\epsfig{file=result.10x10.m75.e0.014.J20.5.meander2.ss.eps,width=6cm}\n \\hfill \\centering \\\\*[4mm]\n\\caption[]{The relative correlation strengths on $10 \\times 10$ lattice.\n All other nearest neighbour correlations can be obtained by\n reflection these picture in the two dashed lines. The DMRG guiding\n state follows the meandering sequence of Fig. \\ref{fig2}(b). \n More explanation is given in the\n text. Reading zig zag from top left to bottom right, the values for $J_2$\n are $J_2=0,\\dots,0.5$ in steps of $0.1$. }\n\\label{fig:variousmeander2}\n\\end{figure}\n\n\\begin{figure}\n \\epsfig{file=result.10x10.m75.e0.014.J20.5.meander2.ss.eps,width=6cm}\n \\hfill\n \\epsfig{file=result.10x10.m75.e0.015.J20.6.meander2.ss.eps,width=6cm} \\vspace{0.5cm}\\\\\n \\epsfig{file=result.10x10.m75.e0.015.J20.7.meander2.ss.eps,width=6cm}\n \\hfill\n \\epsfig{file=result.10x10.m75.e0.02.J20.8.meander2.ss.eps,width=6cm}\\vspace{0.5cm}\\\\\n \\epsfig{file=result.10x10.m75.e0.02.J20.9.meander2.ss.eps,width=6cm}\n \\hfill\n \\epsfig{file=result.10x10.m75.e0.02.J21.0.meander2.ss.eps,width=6cm}\n \\hfill \\centering \\\\*[4mm]\n\\caption[]{The continuation of figure \\ref{fig:variousmeander2}; the\n relative correlation strengths on $10 \\times 10$ lattice.\n $J_2=0.5,\\dots,1.0$ in steps of $0.1$. }\n\\label{fig:variousmeander2b}\n\\end{figure}\n\n\\end{document}\n"
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{
"name": "cond-mat0002116.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem[1]{Wh1} S. R. White, Phys. Rev. Lett. {\\bf 69} (1992) 2863;\\\\ \nS. R. White, Phys. Rev. B {\\bf 48}(1993) 10345. \n\\bibitem[2]{sign} For a recent discussion of the sign problem in various\ncases see P. Henelius and A. W. Sandvik cond--mat/0001351 and references there\nin.\n\\bibitem[3]{Sor} S. Sorella, Phys. Rev. Lett. {\\bf 80} (1998) 4558.\n\\bibitem[4]{San} A. W. Sandvik, Phys. Rev. {\\bf B56} (1998) 11678.\n\\bibitem[5]{Sor2} L. Capriotti and S. Sorella cond--mat/9911161.\n\\bibitem[6]{Duc} M. S. L. du Croo de Jongh and P. H. J. Denteneer,\nPhys. Rev.B {\\bf 55} (1997) 2713.\n\\bibitem[7]{Sch} H. J. Schulz, T. A. L. Ziman and D. Poilblanc \nJ. Phys. I {\\bf 6} (1996) 675.\n\\bibitem[8]{Kot} V. Kotov, J. Oitmaa, O. P Sushkov and Z. Weihong, Phys. Rev.\nB {\\bf 60} (1999) 14613. \n\\bibitem[9]{Kot2} V. Kotov, J. Oitmaa, O. P Sushkov and Z. Weihong, \ncond--mat/9912228.\n\\bibitem[10]{Sin} R. R. P. Singh, Z. Weihong, C. J. Hamer and J. Oitmaa,\nPhys. Rev. B {\\bf 60} (1999) 7278.\n\\bibitem[11]{Sac} \nN. Read and S. Sachdev, Phys. Rev. Lett. {\\bf 62}, 1694 (1989).\\\\\nN. Read and S. Sachdev, Phys. Rev. B {\\bf 42}, 4568 (1990).\\\\\nN. Read and S. Sachdev, Phys. Rev. Lett. {\\bf 66}, 1773 (1991).\\\\\nS. Sachdev and N. Read, Int. J. Mod. Phys. B {\\bf 5}, 219 (1991).\\\\\nS. Sachdev, Quantum Phase Transitions, Cambridge University Press,\nCambridge (1999).\n\\bibitem[12]{Cep} D. M. Ceperley and B. J. Alder Phys. Rev. Lett. \n{\\bf 45}(1980) 566; \\\\D. M. Ceperley, Rev. Mod. Phys. {\\bf 67} (1995) 279.\n\\bibitem[13]{Luc} M. S. L. du Croo de Jongh, Thesis Leiden University 1999, cond-mat/9908200.\n\\bibitem[14]{Wh2} We are grateful to Steve White for making his software\navailable to us.\n\\bibitem[15]{Tri} N. Trivedi and D. M. Ceperley, Phys. Rev. {\\bf B 50} (1990) \n4552.\n\\bibitem[16]{Mar} W. Marshall, Proc. Royal Soc. Londen Ser. \n{\\bf A 232} (1955) 48.\n\\bibitem[17]{Bem} H. J. M. van Bemmel, D. F. B. ten Haaf, W. van Saarloos,\nJ. M. J. van Leeuwen and G. An, Phys. Rev. Lett. {\\bf 72} (1994) 2442. \n\\bibitem[18]{Haa} D. F. B. ten Haaf, H. J. M. van Bemmel, J. M. J. van Leeuwen,\nW. van Saarloos and D. M. Ceperley, Phys. Rev. {\\bf B51} (1995) 13039.\n\\bibitem[19]{Ost} S \\\"Ostlund and S. Rommer,\n Phys. Rev. Lett. {\\bf 75}, 3537 (1995);\\\\\nS.Rommer and S. \\\"Ostlund, Phys. Rev. B {\\bf 55} (1997) 2164.\n\n\\end{thebibliography}"
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cond-mat0002117
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Role of Secondary Motifs in Fast Folding Polymers: A Dynamical Variational Principle
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[
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"author": "Amos Maritan$^1$"
},
{
"author": "Cristian Micheletti$^1$ and Jayanth R. Banavar$^2$"
}
] |
A fascinating and open question challenging biochemistry, physics and even geometry is the presence of highly regular motifs such as $\alpha$-helices in the folded state of biopolymers and proteins. Stimulating explanations ranging from chemical propensity to simple geometrical reasoning have been invoked to rationalize the existence of such secondary structures. We formulate a dynamical variational principle for selection in conformation space based on the requirement that the backbone of the native state of biologically viable polymers be rapidly accessible from the denatured state. The variational principle is shown to result in the emergence of helical order in compact structures.
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"name": "paper.tex",
"string": "\\documentstyle[prb,aps]{revtex}\n\\begin{document}\n\\input{psfig}\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname\n\n\\title{Role of Secondary Motifs in Fast Folding Polymers: A Dynamical\nVariational Principle}\n\n\\author{Amos Maritan$^1$, Cristian Micheletti$^1$ and Jayanth R. Banavar$^2$}\n\\vskip 0.3cm\n\\address{(1) International School for Advanced Studies (S.I.S.S.A.) - INFM,\nVia Beirut 2-4, 34014 Trieste, Italy and \\\\\nthe Abdus Salam International Centre for Theoretical Physics.}\n\\address{(2) Department of Physics and Center for Materials Physics,\n104 Davey Laboratory, The Pennsylvania State University, University\nPark, Pennsylvania 16802}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nA fascinating and open question challenging biochemistry, physics and\neven geometry is the presence of highly regular motifs such as\n$\\alpha$-helices in the folded state of biopolymers and proteins.\nStimulating explanations ranging from chemical propensity to simple\ngeometrical reasoning have been invoked to rationalize the existence\nof such secondary structures. We formulate a dynamical variational\nprinciple for selection in conformation space based on the requirement\nthat the backbone of the native state of biologically viable polymers\nbe rapidly accessible from the denatured state. The variational\nprinciple is shown to result in the emergence of helical order in\ncompact structures.\n\\end{abstract}\n]\n\n\nA fundamental problem in every day life is that of packing\nwith examples ranging from fruits in a grocery,\nclothes and personal belongings in a suitcase,\natoms and colloidal particles in crystals and glasses, and\namino acids in the folded state of proteins. The\nsimplest problem in packing consists of determining the spatial\narrangement that accomodates the highest packing density of its\nconstituent entities with the result being a crystalline structure.\nBesides packing considerations, dynamical effects\nplay a significant role when rapid packing/unpacking is entailed, as\nin the formation of amorphous glasses where crystallization is\ndynamically thwarted or in the more familiar suitcase problem.\n\nFast packing has been recognized as a central issue for biopolymers,\nsuch as proteins, since the early work of Levinthal \\cite{Levin}.\nFurther, the native conformations display extremely regular motifs,\nsuch as $\\alpha$-helices or $\\beta$-sheets. In this Letter we\npostulate a direct connection between the dynamics of rapid folding\nand the emergence of secondary motifs in the native state\nconformations. In fact, an intuitive approach to rapid and\nreproducible folding might be to create neat patterns of lower\ndimensional manifolds than the physical space and bend and curl them\ninto the final folded state. For proteins, secondary structures such\nas $\\alpha$-helices and $\\beta$-sheets are indeed patterns in low\ndimensions.\n\nThere are two key aspects distinguishing a protein from a generic\nheteropolymer: the specially selected sequence of amino acids and the\nthree-dimensional structure that it folds reversibly into. For a given\ntarget native structure, the selection mechanism in sequence space is\nthe principle of minimal frustration \\cite{7}. The chosen sequences\nare such that their target native states are reached through a\nfunnel-like landscape \\cite{9} which facilitates the harmonious\nfitting together of pieces to form the whole.\n\nThe three-dimensional structure impacts on the functionality of the\nprotein and a fascinating issue is the elucidation of the selection\nmechanism in conformation space that picks out certain viable\nstructures from the innumerable ones with a given compactness. Earlier\nstudies have shown that there is a direct link between viable native\nconformations and high designability \\cite{LG}. Recently \\cite{19}, it\nwas observed that the natural folds of proteins have a much larger\ndensity of nearby structures than generic (artificial) conformations\nof the same character and that the exceedingly large geometrical\naccessibility of natural proteins may be related to the presence of\nsecondary motifs.\n\nThe realization that proteins have secondary structures arose with\nearly crystallographic studies and the brilliant deduction of Pauling\net al. \\cite{2} of the ability of an $\\alpha$-helix of the correct\npitch to accomodate hydrogen bonds, thus promoting its stability.\nInspired by the findings of Pauling, helix-coil transition models have\nbeen used to study the thermodynamics of helix formation\n\\cite{hc}. The models encompass features that ensure the helical\nnature of the low-energy states by assuming first that that monomers\ncan be in a helical state and by then introducing co-operative\ninteractions that favor helical regions. It is interesting to note,\nhowever, that the number of hydrogen bonds is nearly the same when a\nsequence is in an unfolded structure in the presence of a polar\nsolvent or in its native state rich in secondary structure content\n\\cite{4}. It has also been suggested that the $\\alpha$-helix is an\nenergetically favorable conformation for main-chain atoms but the\nside-chain suffers from a loss of entropy \\cite{4,18}. Nelson {\\em et\nal.} \\cite{21} have shown both numerically and experimentally that\nnon-biological oligomers fold reversibly like proteins into a specific\nthree-dimensional structure with high helical content driven only by\nsolvophobic interactions. Recent studies have attempted to explain\nthe emergence of secondary structure from geometrical principles\nrather than invoking detailed chemistry. Despite the concerted efforts\nof several groups, a simple general explanation remains elusive. In\nparticular, the work of Yee {\\em et al.} \\cite{3}, Hunt {\\em et al.}\n\\cite{4}, and Socci {\\em et al.} \\cite{5} have shown that compactness\nalone can only account for a small secondary structure content. These\nfacts are also corroborated by the recent study of the kinetics of\nhomopolymer collapse, where no evidence was found for the formation of\nlocal regular structures \\cite{halper}.\n\nWe propose a selection mechanism in structure space in the form of a\nvariational principle postulating that, {\\em among all possible native\nconformations, a protein backbone will attain only those which are\noptimal under the action of evolutionary pressure favouring rapid\nfolding}. Our goal is to elucidate the role played by the bare native\nbackbone independent of the selection in sequence space and hence of\nthe (imperfectly-known) inter-amino-acid potentials. We therefore\nchoose to employ a Go-like model \\cite{12} with no other interaction\nthat promotes or disfavours secondary structures. The model is a\nsequence-independent limiting case of minimal frustration \\cite{7}\nwhich, for a given target native state conformation, favours the\nformation of native contacts -- the energy of a sequence in a\nconformation is simply obtained as the negative of the number of\ncontacts in common with the target conformation. We will consider two\nnon-consecutive amino acids to be in contact if their separation is\nbelow a cutoff $r_0 = 6.5$ \\AA (the results are qualitatively similar\nwhen slightly different values of $r_0$ in the range $6-8$ \\AA are\nchosen).\n\nThe energy of structure $\\Gamma$ in the Go model is given by\n\\begin{equation}\nH(\\Gamma) = - {1 \\over 2} \\sum_{i, j} \\Delta_{i,j}(\\Gamma)\n\\Delta_{i,j}(\\Gamma_0)\n\\label{eqn:ham}\n\\end{equation}\n\n\\noindent where the sum is taken over all pairs of amino acids,\n$\\Gamma_0$ is the target structure, $\\Delta_{i,j}(\\Gamma)$ is the\ncontact map of structure $\\Gamma$:\n\\begin{equation}\n\\Delta_{ij}(\\Gamma) = \\{\n\\begin{array}{l l}\n1 &\\ \\ \\ { R_{ij}<r_0 \\mbox{ and } |i-j| >2;}\\\\\n0 & \\ \\ \\ \\mbox{ otherwise, }\n\\end{array}\n\\end{equation}\n\n\\noindent where $R_{ij}$ is the distance of amino acids $i$ and $j$.\n\n The polypeptide chain is modelled as a chain of beads subject to\nsteric constraints \\cite{17,19}. We adopted a discrete\nrepresentation similar to the one of Covell and Jernigan \\cite{17},\nin which each bead occupies a site of an FCC lattice with lattice\nspacing equal to 3.8 \\AA. Such a representation is able to describe\nthe backbone of natural proteins to better that 1 \\AA \\ rmsd per\nresidue (equal to the best experimental resolution) and preserves typical\ntorsional angles. All discretized structures were subject to a\nsuitable constraint: any two non-consecutive residues cannot be closer\nthan $4.65$ \\AA\\ due to excluded volume effects and the distance\nbetween consecutive residues can fluctuate between 2.6 \\AA $< d <$ 4.7\n\\AA. Such constraints were determined by an analysis of the\ncoarse-grainings of several proteins of intermediate length ($\\approx\n100$ residues). In order to enforce a realistic global compactness\nfor a backbone of length $L$, the number of contacts in all the target\nstructures considered was chosen \\cite{cont} to be around $N=1.9L$\nwhile, locally, no residue was allowed to make contact with four or\nmore consecutive residues.\n\n\nIn order to assess the validity of the variational principle, it is\nnecessary to evaluate the typical time, $t(\\Gamma_0)$, taken to fold\ninto a given target structure, $\\Gamma_0$, followed by a selection of\nthe structures $\\Gamma_0$, that have the smallest folding times. To\ndo this, an initial set of ten conformations was generated by\ncollapsing a loose chain starting from random initial conditions. In\neach case, we modified the random initial conformation by using Monte\nCarlo dynamics: we move up to 3 consecutive beads to unoccupied\ndiscrete positions that do not violate any of the physical constraints\nand accept the moves according to the standard Metropolis rule. The\nenergy is given by eq. (\\ref{eqn:ham}), while the temperature for the\nMC dynamics was set to 0.35. This value was chosen in preliminary runs\nso that it was higher than the temperature \\cite{7} below which the\nsequence is trapped in metastable states but comparable to the folding\ntransition temperature so that conformations with significant overlap\nwith the native state are sampled in thermal equilibrium.\n\nFor each structure, as a measure of the folding time we took the\nmedian over various attempts (typically 41) of the total number of\nMonte Carlo moves necessary to form a pre-assigned fraction of native\ncontacts, typically 66\\%, starting from a random conformation. Our\nresults were unaltered on increasing this fraction to 75\\%; indeed,\nthis fraction could be progressively increased towards 100\\% with\nsuccessive generations without increase in the computational cost\nsince better and better folders are obtained.\n\n\nA new generation of ten structures is created by ``hybridizing'' pairs\nof structures of the previous generation ensuring that structures with\nsmall folding times are hybridized more and more frequently as the\nnumber of generations, $g$, increases \\cite{14}. To do this, each of\nthe two distinct parent structures to be hybridized, $\\Gamma_1$ and\n$\\Gamma_2$ are chosen with probability proportional to\n$\\exp[-(g-1)*ft)/1000]$, where $g$ is the index of the current\ngeneration (initially equal to 1), $ft$ is the median folding\ntime. Then, a hybrid map is created by taking the union of the two\nparent maps:\n\\begin{equation}\n\\Delta_{ij}^{Union} = \\max(\\Delta_{ij}(\\Gamma_1),\\Delta_{ij}(\\Gamma_2))\\ .\n\\end{equation}\n\n\\noindent Because it is not guaranteed that $\\Delta^{Union}$\ncorresponds to a three-dimensional structure obeying the same physical\nconstraints as $\\Gamma_1$ and $\\Gamma_2$, the corresponding hybrid\n$\\Gamma$ is constructed by taking one of the two parent structures (or\nalternatively a random one) as the starting conformation and carrying\nout MC dynamics favouring the formation of each of the contacts in the\nunion map (i.e. using eq. (1) with $\\Delta_{ij}(\\Gamma_0)$ substituted\nby $\\Delta_{ij}^{Union}$). The dynamics is carried out starting from a\ntemperature of $0.7$ and then decreasing it gradually over a\nsufficiently long time (typically thousands of MC steps) to achieve\nthe maximum possible overlap with the union map, while simultaneously\nmaintaining the realistic compactness. The resulting hybrid structure\nis typically midway between the two parent structures, in that it\ninherits native contacts from both of them. We adopted the following\ndefinition in order to obtain an objective and unbiased way to\nquantitatively estimate the presence of secondary content: a given\nresidue, $i$ was defined to belong to a secondary motif if, for some\n$j$, one of these conditions held:\n\n\\begin{eqnarray}\na)\\ \\Delta_{i-1,j-1}&=&\\Delta_{i,j}=\\Delta_{i+1,j+1}=\\Delta_{i,j+1}\\nonumber \\\\\n &=& \\Delta_{i+1,j+2}=\\Delta_{i-1,j}=1;\\nonumber \\\\\nb)\\ \\Delta_{i+1,j-1}&=&\\Delta_{i,j}=\\Delta_{i-1,j+1}=\\Delta_{i,j+1}\\nonumber \\\\\n &=& \\Delta_{i+1,j}=\\Delta_{i-1,j+2}=1.\\nonumber\n\\end{eqnarray}\n\n\\noindent The former [latter] identifies the presence of helices and\nparallel [anti-parallel] $\\beta$ sheets in natural proteins, which can\nbe identified by the visual inspection of contact matrices and appears\nas thick bands parallel or orthogonal to the diagonal.\n\n\n\n\\noindent The upper plot of Fig. 1 shows the decrease of the typical\nfolding time over the generations for chains of length 25, while the\nmiddle panel shows the accompanying increase in the number of residues\nin secondary motifs (secondary content). The bottom panel shows a\nmilder decrease of the contact order (i.e. a larger number of\nshort-range contacts) as the generations evolved, in agreement with\nthe experimental findings of Plaxco {\\em et al.}\\cite{15}\n\n\n\n\n\nOne of the optimal structures of length 25 is shown in Figure\n\\ref{fig:fig2}a. Due to the absence of any chirality bias in our\nstructure space exploration, the helix does not have a constant\nhandedness. The signature of the secondary motifs in the optimal\nstructures is clearly visible in the contact maps of Figure\n\\ref{fig:fig3}, which are not sensitive to structure chirality.\nStrikingly, the variational principle selects conformations with\nsignificant secondary content as those facilitating the fastest\nfolding. The correlation of the emergence of secondary structures with\ndecrease of folding times is shown in the plot of\nFig. \\ref{fig:trend}. We verified that the hybridization procedure is\nnot biased towards low contact order by iterating it for various\ngenerations and hybridizing the structures at random. Even after\ndozens of generations, the generated structures had secondary contents\nof about 1/3-1/4 of the true extremal structures.\n\n\nThe very high secondary content in optimal conformations was found to\nbe robust against changes in chain length or compactness of the target\nstructure. On requiring that the structure be more compact, bundles of\nhelices emerge [see Fig. \\ref{fig:fig2}b] along with an increase in\ncontact order, signalling the presence of some longer range contacts,\nwhich are necessitated in order to accomodate the shorter radius of\ngyration. It is noteworthy that our calculations lead predominantly\nto $\\alpha$-helices and not $\\beta$ sheets, a fact accounted for by\nthe demonstration that steric overlaps and the associated loss of entropy\nlead to the destabilization of helices in favor of sheets \\cite{18},\nthe appearance of such sheets only in sufficiently long proteins\n\\cite{22} and the much slower folding rate of $\\beta$-sheets\ncompared to $\\alpha$-helices \\cite{slow}. It is remarkable that the same\nrequirement of rapid folding is sufficient to lead to a selection in\nboth sequence and structure space underscoring the harmony in the\nevolutionary design of proteins. The results and strategies presented\nhere ought to be applicable in protein-engineering contexts, for\nexample by ensuring optimal dynamical accessibility of the backbone of\nproteins. A systematic collection of the rapidly-accessible\nstructures of various length should also lead to the creation of\nunbiased libraries of protein folds.\n\n\n\n\n{\\bf Acknowledgements} This work was supported by INFM, INFN sez. di\nTrieste, NASA, NATO and The Donors of the Petroleum Research Fund\nadministered by the American Chemical Society. We thank F. Seno and A.\nTrovato for useful discussions.\\\\\n\n\n\\begin{references}\n\n\\bibitem{Levin} C. Levinthal, J. Chim. Phys. {\\bf 65}, 44 (1968).\n\n\\bibitem{7} J. D. Bryngelson and P. G. Wolynes, {\\em Proc. Natl.\nAcad. Sci. USA} {\\bf 84}, 7524-7528 (1987); J. D. Bryngelson,\nJ. N. Onuchic, J. N. Socci and P. G. Wolynes, {\\em Proteins:\nStruc. Funct. Genet.} {\\bf 21}, 167-195 (1995).\n\n\\bibitem{9} P. E. Leopold, M. Montal and J. N. Onuchic, {\\em\nProc. Natl. Acad. Sci. USA} {\\bf 89}, 8721-8725 (1992); P. G. Wolynes\nJ. N. Onuchic and D. Thirumalai, {\\em Science} {\\bf 267}, 1619-1620\n(1995); J. N. Onuchic, Z. Luthey Schulten and P. G. Wolynes, {\\em\nAnn. Rev. Phys. Chem.} {\\bf 48}, 545-600 (1997); K. A. Dill and\nH. S. Chan, {\\em Nature Structural Biology} {\\bf 4}, 10-19 (1997).\n\n\\bibitem{LG} H. Li, R. Helling, C. Tang and N. Wingreen, {\\em Science}\n{\\bf 273}, 666-669 (1996); N. E. G. Buchler and R. A. Goldstein, {\\em\nProteins: Struc. Funct. Genet.} {\\bf 34}, 113-124 (1999);\nC. Micheletti, A. Maritan, J. R. Banavar and F. Seno, {\\em\nPhys. Rev. Lett.} {\\bf 80}, 5683 (1998); C. Micheletti, A. Maritan and\nJ. R. Banavar, {\\em J. Chem. Phys.} {\\bf 110}, 9730 (1999).\n\n\n\n\\bibitem{19} C. Micheletti, J. R. Banavar, A. Maritan and F. Seno,{\\em\nPhys. Rev. Lett.} {\\bf 82}, 3372-3375 (1999).\n\n\\bibitem{2} L. Pauling, R. B. Corey and H. R. Branson,\n{\\it Proc. Nat. Acad. Sci.} {\\bf 37}, 205-208 (1951).\n\n\\bibitem{hc} B. H. Zimm and J. Bragg, {\\em J. Chem. Phys.}, {\\bf 31},\n526 (1959); O. B. Ptitsyn and A. M. Skvortsov, {\\em Biophys.} {\\bf\n10}, 1007 (1965); I. M. Lifshitz, A. Y. Grosberg and A. R. Khokhlov, {\\em\nRev. Mod. Phys.}, {\\bf 50}, 683 (1978).\n\n\n\\bibitem{4} N. G. Hunt, L. M. Gregoret and F. E. Cohen,\n{\\it J. Mol. Biol.} {\\bf 241}, 214-225 (1994).\n\n\\bibitem{18} R. Aurora, T. P. Creamer, R. Srinivasan and G. D. Rose,\n{\\em J. Mol. Biol.} {\\bf 272}, 1413-1416 (1997).\n\n\\bibitem{21} J. C. Nelson, J. G. Saven, J. S. Moore, and P. G. Wolynes,\n{\\em Science} {\\bf 277}, 1793-1796 (1997).\n\n\\bibitem{3} D. P. Yee, H. S. Chan, T. F. Havel and K. A. Dill,\n{\\it J. Mol. Biol.} {\\bf 241}, 557-573 (1994).\n\n\\bibitem{5} N. D. Socci, W. S. Bialek, and J. N. Onuchic,\n{\\it Phys. Rev. E} {\\bf 49}, 3440-3443 (1994).\n\n\\bibitem{halper} A. Halperin and P. M. Goldbart, Phys. Rev. E, in\npress (cond-mat/9905306).\n\n\\bibitem{12} N. Go, {\\em Macromolecules} {\\bf 9}, 535-541 (1976).\n\n\n\\bibitem{17} D. G. Covell and R. Jernigan {\\it Biochem.} {\\bf 29},\n3287 (1990).\n\n\\bibitem{cont} The native state structures of monomeric\nproteins of length between 50 and 200 show an excellent correlation of\nthis form, when two non-consecutive amino acids along the sequence are\ndefined to be in contact when they are within 6.5 \\AA\\ of each\nother.\n\n\\bibitem{14} J. H. Holland, {\\em Adaptation in natural and artificial\nsystems}, MIT press ed. (1992).\n\n\\bibitem{15} K. M. Plaxco, K. T. Simons and D. Baker, {\\it\nJ. Mol. Biol.} {\\bf 277}, 985-994 (1998).\n\n\\bibitem{22} A. P. Capaldi and S. E. Radford, S. E., {\\em\nCurr. Op. Str. Biol.} {\\bf 8}, 86-92 (1998).\n\n\\bibitem{slow} V. Mu\\~noz, E. R. Henry, J. Hofrichter and W. A. Eaton,\n{\\em Proc. Natl. Acad. Sci. USA}, {\\bf 95}, 5872 (1998).\n\\end{references}\n\n\\begin{figure}\n\\label{fig:fig1}\n\\centerline{\\psfig{figure=fig1.eps,width=3.5in}}\n\\caption{Evolution of the median folding time (measured in Monte Carlo\nsteps), secondary structure content and contact order as a function of\nthe number of generations in the optimization algorithm for compact\nstructures of length $L=25$. The dashed curve denotes an average over\nall ten structures in a given generation, whereas the solid curve\nshows the behaviour of the structure at each generation with the\nfastest median folding time. Analogous results are obtained for other\nruns and for other values of $L$. The dramatic decrease of folding\ntime is accompanied by an equally significant increase in the\nsecondary content.}\n\\end{figure}\n\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig2.eps,width=3.5in}}\n\\caption{ a) RASMOL plot of a structure with very low median folding\ntime and $L=25$. b) Structure with very low median folding time,\n$L=25$ and higher compactness (all target conformations were\nconstrained to have a radius of gyration smaller than 6.5\n\\AA). Optimal compact structures correspond to helices packed\ntogether, as observed in naturally occurring proteins.}\n\\label{fig:fig2}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig3.eps,width=3.5in}}\n\\caption{The panel on the left [right] shows the contact map of a\nstructure with a very low [average] median folding time. The signature\nof helices in map (a) is shown by the thick bands parallel to the\ndiagonal, while no such patterns are observed in the matrix (b).}\n\\label{fig:fig3}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig4.eps,width=3.5in,height=3.0in}}\n\\caption{Scatter plot of folding time versus secondary content for\nstructures of length 25 collected over several generation of the\noptimization algorithm.}\n\\label{fig:trend}\n\\end{figure}\n\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n"
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"string": "% Psfig/TeX Release 1.8\n% dvips version\n%\n% All psfig/tex software, documentation, and related files\n% in this distribution of psfig/tex are \n% Copyright 1987, 1988, 1991 Trevor J. Darrell\n%\n% Permission is granted for use and non-profit distribution of psfig/tex \n% providing that this notice is clearly maintained. The right to\n% distribute any portion of psfig/tex for profit or as part of any commercial\n% product is specifically reserved for the author(s) of that portion.\n%\n% *** Feel free to make local modifications of psfig as you wish,\n% *** but DO NOT post any changed or modified versions of ``psfig''\n% *** directly to the net. Send them to me and I'll try to incorporate\n% *** them into future versions. If you want to take the psfig code \n% *** and make a new program (subject to the copyright above), distribute it, \n% *** (and maintain it) that's fine, just don't call it psfig.\n%\n% Bugs and improvements to trevor@media.mit.edu.\n%\n% Thanks to Greg Hager (GDH) and Ned Batchelder for their contributions\n% to the original version of this project.\n%\n% Modified by J. Daniel Smith on 9 October 1990 to accept the\n% %%BoundingBox: comment with or without a space after the colon. Stole\n% file reading code from Tom Rokicki's EPSF.TEX file (see below).\n%\n% More modifications by J. Daniel Smith on 29 March 1991 to allow the\n% the included PostScript figure to be rotated. The amount of\n% rotation is specified by the \"angle=\" parameter of the \\psfig command.\n%\n% Modified by Robert Russell on June 25, 1991 to allow users to specify\n% .ps filenames which don't yet exist, provided they explicitly provide\n% boundingbox information via the \\psfig command. Note: This will only work\n% if the \"file=\" parameter follows all four \"bb???=\" parameters in the\n% command. This is due to the order in which psfig interprets these params.\n%\n% 3 Jul 1991\tJDS\tcheck if file already read in once\n% 4 Sep 1991\tJDS\tfixed incorrect computation of rotated\n%\t\t\tbounding box\n% 25 Sep 1991\tGVR\texpanded synopsis of \\psfig\n% 14 Oct 1991\tJDS\t\\fbox code from LaTeX so \\psdraft works with TeX\n%\t\t\tchanged \\typeout to \\ps@typeout\n% 17 Oct 1991\tJDS\tadded \\psscalefirst and \\psrotatefirst\n%\n\n% From: gvr@cs.brown.edu (George V. Reilly)\n%\n% \\psdraft\tdraws an outline box, but doesn't include the figure\n%\t\tin the DVI file. Useful for previewing.\n%\n% \\psfull\tincludes the figure in the DVI file (default).\n%\n% \\psscalefirst width= or height= specifies the size of the figure\n% \t\tbefore rotation.\n% \\psrotatefirst (default) width= or height= specifies the size of the\n% \t\t figure after rotation. Asymetric figures will\n% \t\t appear to shrink.\n%\n% \\psfigurepath#1\tsets the path to search for the figure\n%\n% \\psfig\n% usage: \\psfig{file=, figure=, height=, width=,\n%\t\t\tbbllx=, bblly=, bburx=, bbury=,\n%\t\t\trheight=, rwidth=, clip=, angle=, silent=}\n%\n%\t\"file\" is the filename. If no path name is specified and the\n%\t\tfile is not found in the current directory,\n%\t\tit will be looked for in directory \\psfigurepath.\n%\t\"figure\" is a synonym for \"file\".\n%\tBy default, the width and height of the figure are taken from\n%\t\tthe BoundingBox of the figure.\n%\tIf \"width\" is specified, the figure is scaled so that it has\n%\t\tthe specified width. Its height changes proportionately.\n%\tIf \"height\" is specified, the figure is scaled so that it has\n%\t\tthe specified height. Its width changes proportionately.\n%\tIf both \"width\" and \"height\" are specified, the figure is scaled\n%\t\tanamorphically.\n%\t\"bbllx\", \"bblly\", \"bburx\", and \"bbury\" control the PostScript\n%\t\tBoundingBox. If these four values are specified\n% *before* the \"file\" option, the PSFIG will not try to\n% open the PostScript file.\n%\t\"rheight\" and \"rwidth\" are the reserved height and width\n%\t\tof the figure, i.e., how big TeX actually thinks\n%\t\tthe figure is. They default to \"width\" and \"height\".\n%\tThe \"clip\" option ensures that no portion of the figure will\n%\t\tappear outside its BoundingBox. \"clip=\" is a switch and\n%\t\ttakes no value, but the `=' must be present.\n%\tThe \"angle\" option specifies the angle of rotation (degrees, ccw).\n%\tThe \"silent\" option makes \\psfig work silently.\n%\n\n% check to see if macros already loaded in (maybe some other file says\n% \"\\input psfig\") ...\n\\ifx\\undefined\\psfig\\else\\endinput\\fi\n\n%\n% from a suggestion by eijkhout@csrd.uiuc.edu to allow\n% loading as a style file:\n\\edef\\psfigRestoreAt{\\catcode`@=\\number\\catcode`@\\relax}\n\\catcode`\\@=11\\relax\n\\newwrite\\@unused\n\\def\\ps@typeout#1{{\\let\\protect\\string\\immediate\\write\\@unused{#1}}}\n\\ps@typeout{psfig/tex 1.8}\n\n%% Here's how you define your figure path. Should be set up with null\n%% default and a user useable definition.\n\n\\def\\figurepath{./}\n\\def\\psfigurepath#1{\\edef\\figurepath{#1}}\n\n%\n% @psdo control structure -- similar to Latex @for.\n% I redefined these with different names so that psfig can\n% be used with TeX as well as LaTeX, and so that it will not \n% be vunerable to future changes in LaTeX's internal\n% control structure,\n%\n\\def\\@nnil{\\@nil}\n\\def\\@empty{}\n\\def\\@psdonoop#1\\@@#2#3{}\n\\def\\@psdo#1:=#2\\do#3{\\edef\\@psdotmp{#2}\\ifx\\@psdotmp\\@empty \\else\n \\expandafter\\@psdoloop#2,\\@nil,\\@nil\\@@#1{#3}\\fi}\n\\def\\@psdoloop#1,#2,#3\\@@#4#5{\\def#4{#1}\\ifx #4\\@nnil \\else\n #5\\def#4{#2}\\ifx #4\\@nnil \\else#5\\@ipsdoloop #3\\@@#4{#5}\\fi\\fi}\n\\def\\@ipsdoloop#1,#2\\@@#3#4{\\def#3{#1}\\ifx #3\\@nnil \n \\let\\@nextwhile=\\@psdonoop \\else\n #4\\relax\\let\\@nextwhile=\\@ipsdoloop\\fi\\@nextwhile#2\\@@#3{#4}}\n\\def\\@tpsdo#1:=#2\\do#3{\\xdef\\@psdotmp{#2}\\ifx\\@psdotmp\\@empty \\else\n \\@tpsdoloop#2\\@nil\\@nil\\@@#1{#3}\\fi}\n\\def\\@tpsdoloop#1#2\\@@#3#4{\\def#3{#1}\\ifx #3\\@nnil \n \\let\\@nextwhile=\\@psdonoop \\else\n #4\\relax\\let\\@nextwhile=\\@tpsdoloop\\fi\\@nextwhile#2\\@@#3{#4}}\n% \n% \\fbox is defined in latex.tex; so if \\fbox is undefined, assume that\n% we are not in LaTeX.\n% Perhaps this could be done better???\n\\ifx\\undefined\\fbox\n% \\fbox code from modified slightly from LaTeX\n\\newdimen\\fboxrule\n\\newdimen\\fboxsep\n\\newdimen\\ps@tempdima\n\\newbox\\ps@tempboxa\n\\fboxsep = 3pt\n\\fboxrule = .4pt\n\\long\\def\\fbox#1{\\leavevmode\\setbox\\ps@tempboxa\\hbox{#1}\\ps@tempdima\\fboxrule\n \\advance\\ps@tempdima \\fboxsep \\advance\\ps@tempdima \\dp\\ps@tempboxa\n \\hbox{\\lower \\ps@tempdima\\hbox\n {\\vbox{\\hrule height \\fboxrule\n \\hbox{\\vrule width \\fboxrule \\hskip\\fboxsep\n \\vbox{\\vskip\\fboxsep \\box\\ps@tempboxa\\vskip\\fboxsep}\\hskip \n \\fboxsep\\vrule width \\fboxrule}\n \\hrule height \\fboxrule}}}}\n\\fi\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% file reading stuff from epsf.tex\n% EPSF.TEX macro file:\n% Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.\n% Revised by Don Knuth, 3 Jan 1990.\n% Revised by Tomas Rokicki to accept bounding boxes with no\n% space after the colon, 18 Jul 1990.\n% Portions modified/removed for use in PSFIG package by\n% J. Daniel Smith, 9 October 1990.\n%\n\\newread\\ps@stream\n\\newif\\ifnot@eof % continue looking for the bounding box?\n\\newif\\if@noisy % report what you're making?\n\\newif\\if@atend % %%BoundingBox: has (at end) specification\n\\newif\\if@psfile % does this look like a PostScript file?\n%\n% PostScript files should start with `%!'\n%\n{\\catcode`\\%=12\\global\\gdef\\epsf@start{%!}}\n\\def\\epsf@PS{PS}\n%\n\\def\\epsf@getbb#1{%\n%\n% The first thing we need to do is to open the\n% PostScript file, if possible.\n%\n\\openin\\ps@stream=#1\n\\ifeof\\ps@stream\\ps@typeout{Error, File #1 not found}\\else\n%\n% Okay, we got it. Now we'll scan lines until we find one that doesn't\n% start with %. We're looking for the bounding box comment.\n%\n {\\not@eoftrue \\chardef\\other=12\n \\def\\do##1{\\catcode`##1=\\other}\\dospecials \\catcode`\\ =10\n \\loop\n \\if@psfile\n\t \\read\\ps@stream to \\epsf@fileline\n \\else{\n\t \\obeyspaces\n \\read\\ps@stream to \\epsf@tmp\\global\\let\\epsf@fileline\\epsf@tmp}\n \\fi\n \\ifeof\\ps@stream\\not@eoffalse\\else\n%\n% Check the first line for `%!'. Issue a warning message if its not\n% there, since the file might not be a PostScript file.\n%\n \\if@psfile\\else\n \\expandafter\\epsf@test\\epsf@fileline:. \\\\%\n \\fi\n%\n% We check to see if the first character is a % sign;\n% if so, we look further and stop only if the line begins with\n% `%%BoundingBox:' and the `(atend)' specification was not found.\n% That is, the only way to stop is when the end of file is reached,\n% or a `%%BoundingBox: llx lly urx ury' line is found.\n%\n \\expandafter\\epsf@aux\\epsf@fileline:. \\\\%\n \\fi\n \\ifnot@eof\\repeat\n }\\closein\\ps@stream\\fi}%\n%\n% This tests if the file we are reading looks like a PostScript file.\n%\n\\long\\def\\epsf@test#1#2#3:#4\\\\{\\def\\epsf@testit{#1#2}\n\t\t\t\\ifx\\epsf@testit\\epsf@start\\else\n\\ps@typeout{Warning! File does not start with `\\epsf@start'. It may not be a PostScript file.}\n\t\t\t\\fi\n\t\t\t\\@psfiletrue} % don't test after 1st line\n%\n% We still need to define the tricky \\epsf@aux macro. This requires\n% a couple of magic constants for comparison purposes.\n%\n{\\catcode`\\%=12\\global\\let\\epsf@percent=%\\global\\def\\epsf@bblit{%BoundingBox}}\n%\n%\n% So we're ready to check for `%BoundingBox:' and to grab the\n% values if they are found. We continue searching if `(at end)'\n% was found after the `%BoundingBox:'.\n%\n\\long\\def\\epsf@aux#1#2:#3\\\\{\\ifx#1\\epsf@percent\n \\def\\epsf@testit{#2}\\ifx\\epsf@testit\\epsf@bblit\n\t\\@atendfalse\n \\epsf@atend #3 . \\\\%\n\t\\if@atend\t\n\t \\if@verbose{\n\t\t\\ps@typeout{psfig: found `(atend)'; continuing search}\n\t }\\fi\n \\else\n \\epsf@grab #3 . . . \\\\%\n \\not@eoffalse\n \\global\\no@bbfalse\n \\fi\n \\fi\\fi}%\n%\n% Here we grab the values and stuff them in the appropriate definitions.\n%\n\\def\\epsf@grab #1 #2 #3 #4 #5\\\\{%\n \\global\\def\\epsf@llx{#1}\\ifx\\epsf@llx\\empty\n \\epsf@grab #2 #3 #4 #5 .\\\\\\else\n \\global\\def\\epsf@lly{#2}%\n \\global\\def\\epsf@urx{#3}\\global\\def\\epsf@ury{#4}\\fi}%\n%\n% Determine if the stuff following the %%BoundingBox is `(atend)'\n% J. Daniel Smith. Copied from \\epsf@grab above.\n%\n\\def\\epsf@atendlit{(atend)} \n\\def\\epsf@atend #1 #2 #3\\\\{%\n \\def\\epsf@tmp{#1}\\ifx\\epsf@tmp\\empty\n \\epsf@atend #2 #3 .\\\\\\else\n \\ifx\\epsf@tmp\\epsf@atendlit\\@atendtrue\\fi\\fi}\n\n\n% End of file reading stuff from epsf.tex\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% trigonometry stuff from \"trig.tex\"\n\\chardef\\letter = 11\n\\chardef\\other = 12\n\n\\newif \\ifdebug %%% turn me on to see TeX hard at work ...\n\\newif\\ifc@mpute %%% don't need to compute some values\n\\c@mputetrue % but assume that we do\n\n\\let\\then = \\relax\n\\def\\r@dian{pt }\n\\let\\r@dians = \\r@dian\n\\let\\dimensionless@nit = \\r@dian\n\\let\\dimensionless@nits = \\dimensionless@nit\n\\def\\internal@nit{sp }\n\\let\\internal@nits = \\internal@nit\n\\newif\\ifstillc@nverging\n\\def \\Mess@ge #1{\\ifdebug \\then \\message {#1} \\fi}\n\n{ %%% Things that need abnormal catcodes %%%\n\t\\catcode `\\@ = \\letter\n\t\\gdef \\nodimen {\\expandafter \\n@dimen \\the \\dimen}\n\t\\gdef \\term #1 #2 #3%\n\t {\\edef \\t@ {\\the #1}%%% freeze parameter 1 (count, by value)\n\t\t\\edef \\t@@ {\\expandafter \\n@dimen \\the #2\\r@dian}%\n\t\t\t\t %%% freeze parameter 2 (dimen, by value)\n\t\t\\t@rm {\\t@} {\\t@@} {#3}%\n\t }\n\t\\gdef \\t@rm #1 #2 #3%\n\t {{%\n\t\t\\count 0 = 0\n\t\t\\dimen 0 = 1 \\dimensionless@nit\n\t\t\\dimen 2 = #2\\relax\n\t\t\\Mess@ge {Calculating term #1 of \\nodimen 2}%\n\t\t\\loop\n\t\t\\ifnum\t\\count 0 < #1\n\t\t\\then\t\\advance \\count 0 by 1\n\t\t\t\\Mess@ge {Iteration \\the \\count 0 \\space}%\n\t\t\t\\Multiply \\dimen 0 by {\\dimen 2}%\n\t\t\t\\Mess@ge {After multiplication, term = \\nodimen 0}%\n\t\t\t\\Divide \\dimen 0 by {\\count 0}%\n\t\t\t\\Mess@ge {After division, term = \\nodimen 0}%\n\t\t\\repeat\n\t\t\\Mess@ge {Final value for term #1 of \n\t\t\t\t\\nodimen 2 \\space is \\nodimen 0}%\n\t\t\\xdef \\Term {#3 = \\nodimen 0 \\r@dians}%\n\t\t\\aftergroup \\Term\n\t }}\n\t\\catcode `\\p = \\other\n\t\\catcode `\\t = \\other\n\t\\gdef \\n@dimen #1pt{#1} %%% throw away the ``pt''\n}\n\n\\def \\Divide #1by #2{\\divide #1 by #2} %%% just a synonym\n\n\\def \\Multiply #1by #2%%% allows division of a dimen by a dimen\n {{%%% should really freeze parameter 2 (dimen, passed by value)\n\t\\count 0 = #1\\relax\n\t\\count 2 = #2\\relax\n\t\\count 4 = 65536\n\t\\Mess@ge {Before scaling, count 0 = \\the \\count 0 \\space and\n\t\t\tcount 2 = \\the \\count 2}%\n\t\\ifnum\t\\count 0 > 32767 %%% do our best to avoid overflow\n\t\\then\t\\divide \\count 0 by 4\n\t\t\\divide \\count 4 by 4\n\t\\else\t\\ifnum\t\\count 0 < -32767\n\t\t\\then\t\\divide \\count 0 by 4\n\t\t\t\\divide \\count 4 by 4\n\t\t\\else\n\t\t\\fi\n\t\\fi\n\t\\ifnum\t\\count 2 > 32767 %%% while retaining reasonable accuracy\n\t\\then\t\\divide \\count 2 by 4\n\t\t\\divide \\count 4 by 4\n\t\\else\t\\ifnum\t\\count 2 < -32767\n\t\t\\then\t\\divide \\count 2 by 4\n\t\t\t\\divide \\count 4 by 4\n\t\t\\else\n\t\t\\fi\n\t\\fi\n\t\\multiply \\count 0 by \\count 2\n\t\\divide \\count 0 by \\count 4\n\t\\xdef \\product {#1 = \\the \\count 0 \\internal@nits}%\n\t\\aftergroup \\product\n }}\n\n\\def\\r@duce{\\ifdim\\dimen0 > 90\\r@dian \\then % sin(x+90) = sin(180-x)\n\t\t\\multiply\\dimen0 by -1\n\t\t\\advance\\dimen0 by 180\\r@dian\n\t\t\\r@duce\n\t \\else \\ifdim\\dimen0 < -90\\r@dian \\then % sin(-x) = sin(360+x)\n\t\t\\advance\\dimen0 by 360\\r@dian\n\t\t\\r@duce\n\t\t\\fi\n\t \\fi}\n\n\\def\\Sine#1%\n {{%\n\t\\dimen 0 = #1 \\r@dian\n\t\\r@duce\n\t\\ifdim\\dimen0 = -90\\r@dian \\then\n\t \\dimen4 = -1\\r@dian\n\t \\c@mputefalse\n\t\\fi\n\t\\ifdim\\dimen0 = 90\\r@dian \\then\n\t \\dimen4 = 1\\r@dian\n\t \\c@mputefalse\n\t\\fi\n\t\\ifdim\\dimen0 = 0\\r@dian \\then\n\t \\dimen4 = 0\\r@dian\n\t \\c@mputefalse\n\t\\fi\n%\n\t\\ifc@mpute \\then\n \t% convert degrees to radians\n\t\t\\divide\\dimen0 by 180\n\t\t\\dimen0=3.141592654\\dimen0\n%\n\t\t\\dimen 2 = 3.1415926535897963\\r@dian %%% a well-known constant\n\t\t\\divide\\dimen 2 by 2 %%% we only deal with -pi/2 : pi/2\n\t\t\\Mess@ge {Sin: calculating Sin of \\nodimen 0}%\n\t\t\\count 0 = 1 %%% see power-series expansion for sine\n\t\t\\dimen 2 = 1 \\r@dian %%% ditto\n\t\t\\dimen 4 = 0 \\r@dian %%% ditto\n\t\t\\loop\n\t\t\t\\ifnum\t\\dimen 2 = 0 %%% then we've done\n\t\t\t\\then\t\\stillc@nvergingfalse \n\t\t\t\\else\t\\stillc@nvergingtrue\n\t\t\t\\fi\n\t\t\t\\ifstillc@nverging %%% then calculate next term\n\t\t\t\\then\t\\term {\\count 0} {\\dimen 0} {\\dimen 2}%\n\t\t\t\t\\advance \\count 0 by 2\n\t\t\t\t\\count 2 = \\count 0\n\t\t\t\t\\divide \\count 2 by 2\n\t\t\t\t\\ifodd\t\\count 2 %%% signs alternate\n\t\t\t\t\\then\t\\advance \\dimen 4 by \\dimen 2\n\t\t\t\t\\else\t\\advance \\dimen 4 by -\\dimen 2\n\t\t\t\t\\fi\n\t\t\\repeat\n\t\\fi\t\t\n\t\t\t\\xdef \\sine {\\nodimen 4}%\n }}\n\n% Now the Cosine can be calculated easily by calling \\Sine\n\\def\\Cosine#1{\\ifx\\sine\\UnDefined\\edef\\Savesine{\\relax}\\else\n\t\t \\edef\\Savesine{\\sine}\\fi\n\t{\\dimen0=#1\\r@dian\\advance\\dimen0 by 90\\r@dian\n\t \\Sine{\\nodimen 0}\n\t \\xdef\\cosine{\\sine}\n\t \\xdef\\sine{\\Savesine}}}\t \n% end of trig stuff\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\def\\psdraft{\n\t\\def\\@psdraft{0}\n\t%\\ps@typeout{draft level now is \\@psdraft \\space . }\n}\n\\def\\psfull{\n\t\\def\\@psdraft{100}\n\t%\\ps@typeout{draft level now is \\@psdraft \\space . }\n}\n\n\\psfull\n\n\\newif\\if@scalefirst\n\\def\\psscalefirst{\\@scalefirsttrue}\n\\def\\psrotatefirst{\\@scalefirstfalse}\n\\psrotatefirst\n\n\\newif\\if@draftbox\n\\def\\psnodraftbox{\n\t\\@draftboxfalse\n}\n\\def\\psdraftbox{\n\t\\@draftboxtrue\n}\n\\@draftboxtrue\n\n\\newif\\if@prologfile\n\\newif\\if@postlogfile\n\\def\\pssilent{\n\t\\@noisyfalse\n}\n\\def\\psnoisy{\n\t\\@noisytrue\n}\n\\psnoisy\n%%% These are for the option list.\n%%% A specification of the form a = b maps to calling \\@p@@sa{b}\n\\newif\\if@bbllx\n\\newif\\if@bblly\n\\newif\\if@bburx\n\\newif\\if@bbury\n\\newif\\if@height\n\\newif\\if@width\n\\newif\\if@rheight\n\\newif\\if@rwidth\n\\newif\\if@angle\n\\newif\\if@clip\n\\newif\\if@verbose\n\\def\\@p@@sclip#1{\\@cliptrue}\n\n\n\\newif\\if@decmpr\n\n%%% GDH 7/26/87 -- changed so that it first looks in the local directory,\n%%% then in a specified global directory for the ps file.\n%%% RPR 6/25/91 -- changed so that it defaults to user-supplied name if\n%%% boundingbox info is specified, assuming graphic will be created by\n%%% print time.\n%%% TJD 10/19/91 -- added bbfile vs. file distinction, and @decmpr flag\n\n\\def\\@p@@sfigure#1{\\def\\@p@sfile{null}\\def\\@p@sbbfile{null}\n\t \\openin1=#1.bb\n\t\t\\ifeof1\\closein1\n\t \t\\openin1=\\figurepath#1.bb\n\t\t\t\\ifeof1\\closein1\n\t\t\t \\openin1=#1\n\t\t\t\t\\ifeof1\\closein1%\n\t\t\t\t \\openin1=\\figurepath#1\n\t\t\t\t\t\\ifeof1\n\t\t\t\t\t \\ps@typeout{Error, File #1 not found}\n\t\t\t\t\t\t\\if@bbllx\\if@bblly\n\t\t\t\t \t\t\\if@bburx\\if@bbury\n\t\t\t \t\t\t\t\\def\\@p@sfile{#1}%\n\t\t\t \t\t\t\t\\def\\@p@sbbfile{#1}%\n\t\t\t\t\t\t\t\\@decmprfalse\n\t\t\t\t \t \t\\fi\\fi\\fi\\fi\n\t\t\t\t\t\\else\\closein1\n\t\t\t\t \t\t\\def\\@p@sfile{\\figurepath#1}%\n\t\t\t\t \t\t\\def\\@p@sbbfile{\\figurepath#1}%\n\t\t\t\t\t\t\\@decmprfalse\n\t \t\t\\fi%\n\t\t\t \t\\else\\closein1%\n\t\t\t\t\t\\def\\@p@sfile{#1}\n\t\t\t\t\t\\def\\@p@sbbfile{#1}\n\t\t\t\t\t\\@decmprfalse\n\t\t\t \t\\fi\n\t\t\t\\else\n\t\t\t\t\\def\\@p@sfile{\\figurepath#1}\n\t\t\t\t\\def\\@p@sbbfile{\\figurepath#1.bb}\n\t\t\t\t\\@decmprtrue\n\t\t\t\\fi\n\t\t\\else\n\t\t\t\\def\\@p@sfile{#1}\n\t\t\t\\def\\@p@sbbfile{#1.bb}\n\t\t\t\\@decmprtrue\n\t\t\\fi}\n\n\\def\\@p@@sfile#1{\\@p@@sfigure{#1}}\n\n\\def\\@p@@sbbllx#1{\n\t\t%\\ps@typeout{bbllx is #1}\n\t\t\\@bbllxtrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@sbbllx{\\number\\dimen100}\n}\n\\def\\@p@@sbblly#1{\n\t\t%\\ps@typeout{bblly is #1}\n\t\t\\@bbllytrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@sbblly{\\number\\dimen100}\n}\n\\def\\@p@@sbburx#1{\n\t\t%\\ps@typeout{bburx is #1}\n\t\t\\@bburxtrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@sbburx{\\number\\dimen100}\n}\n\\def\\@p@@sbbury#1{\n\t\t%\\ps@typeout{bbury is #1}\n\t\t\\@bburytrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@sbbury{\\number\\dimen100}\n}\n\\def\\@p@@sheight#1{\n\t\t\\@heighttrue\n\t\t\\dimen100=#1\n \t\t\\edef\\@p@sheight{\\number\\dimen100}\n\t\t%\\ps@typeout{Height is \\@p@sheight}\n}\n\\def\\@p@@swidth#1{\n\t\t%\\ps@typeout{Width is #1}\n\t\t\\@widthtrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@swidth{\\number\\dimen100}\n}\n\\def\\@p@@srheight#1{\n\t\t%\\ps@typeout{Reserved height is #1}\n\t\t\\@rheighttrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@srheight{\\number\\dimen100}\n}\n\\def\\@p@@srwidth#1{\n\t\t%\\ps@typeout{Reserved width is #1}\n\t\t\\@rwidthtrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@srwidth{\\number\\dimen100}\n}\n\\def\\@p@@sangle#1{\n\t\t%\\ps@typeout{Rotation is #1}\n\t\t\\@angletrue\n%\t\t\\dimen100=#1\n\t\t\\edef\\@p@sangle{#1} %\\number\\dimen100}\n}\n\\def\\@p@@ssilent#1{ \n\t\t\\@verbosefalse\n}\n\\def\\@p@@sprolog#1{\\@prologfiletrue\\def\\@prologfileval{#1}}\n\\def\\@p@@spostlog#1{\\@postlogfiletrue\\def\\@postlogfileval{#1}}\n\\def\\@cs@name#1{\\csname #1\\endcsname}\n\\def\\@setparms#1=#2,{\\@cs@name{@p@@s#1}{#2}}\n%\n% initialize the defaults (size the size of the figure)\n%\n\\def\\ps@init@parms{\n\t\t\\@bbllxfalse \\@bbllyfalse\n\t\t\\@bburxfalse \\@bburyfalse\n\t\t\\@heightfalse \\@widthfalse\n\t\t\\@rheightfalse \\@rwidthfalse\n\t\t\\def\\@p@sbbllx{}\\def\\@p@sbblly{}\n\t\t\\def\\@p@sbburx{}\\def\\@p@sbbury{}\n\t\t\\def\\@p@sheight{}\\def\\@p@swidth{}\n\t\t\\def\\@p@srheight{}\\def\\@p@srwidth{}\n\t\t\\def\\@p@sangle{0}\n\t\t\\def\\@p@sfile{} \\def\\@p@sbbfile{}\n\t\t\\def\\@p@scost{10}\n\t\t\\def\\@sc{}\n\t\t\\@prologfilefalse\n\t\t\\@postlogfilefalse\n\t\t\\@clipfalse\n\t\t\\if@noisy\n\t\t\t\\@verbosetrue\n\t\t\\else\n\t\t\t\\@verbosefalse\n\t\t\\fi\n}\n%\n% Go through the options setting things up.\n%\n\\def\\parse@ps@parms#1{\n\t \t\\@psdo\\@psfiga:=#1\\do\n\t\t {\\expandafter\\@setparms\\@psfiga,}}\n%\n% Compute bb height and width\n%\n\\newif\\ifno@bb\n\\def\\bb@missing{\n\t\\if@verbose{\n\t\t\\ps@typeout{psfig: searching \\@p@sbbfile \\space for bounding box}\n\t}\\fi\n\t\\no@bbtrue\n\t\\epsf@getbb{\\@p@sbbfile}\n \\ifno@bb \\else \\bb@cull\\epsf@llx\\epsf@lly\\epsf@urx\\epsf@ury\\fi\n}\t\n\\def\\bb@cull#1#2#3#4{\n\t\\dimen100=#1 bp\\edef\\@p@sbbllx{\\number\\dimen100}\n\t\\dimen100=#2 bp\\edef\\@p@sbblly{\\number\\dimen100}\n\t\\dimen100=#3 bp\\edef\\@p@sbburx{\\number\\dimen100}\n\t\\dimen100=#4 bp\\edef\\@p@sbbury{\\number\\dimen100}\n\t\\no@bbfalse\n}\n% rotate point (#1,#2) about (0,0).\n% The sine and cosine of the angle are already stored in \\sine and\n% \\cosine. The result is placed in (\\p@intvaluex, \\p@intvaluey).\n\\newdimen\\p@intvaluex\n\\newdimen\\p@intvaluey\n\\def\\rotate@#1#2{{\\dimen0=#1 sp\\dimen1=#2 sp\n% \tcalculate x' = x \\cos\\theta - y \\sin\\theta\n\t\t \\global\\p@intvaluex=\\cosine\\dimen0\n\t\t \\dimen3=\\sine\\dimen1\n\t\t \\global\\advance\\p@intvaluex by -\\dimen3\n% \t\tcalculate y' = x \\sin\\theta + y \\cos\\theta\n\t\t \\global\\p@intvaluey=\\sine\\dimen0\n\t\t \\dimen3=\\cosine\\dimen1\n\t\t \\global\\advance\\p@intvaluey by \\dimen3\n\t\t }}\n\\def\\compute@bb{\n\t\t\\no@bbfalse\n\t\t\\if@bbllx \\else \\no@bbtrue \\fi\n\t\t\\if@bblly \\else \\no@bbtrue \\fi\n\t\t\\if@bburx \\else \\no@bbtrue \\fi\n\t\t\\if@bbury \\else \\no@bbtrue \\fi\n\t\t\\ifno@bb \\bb@missing \\fi\n\t\t\\ifno@bb \\ps@typeout{FATAL ERROR: no bb supplied or found}\n\t\t\t\\no-bb-error\n\t\t\\fi\n\t\t%\n%\\ps@typeout{BB: \\@p@sbbllx, \\@p@sbblly, \\@p@sbburx, \\@p@sbbury} \n%\n% store height/width of original (unrotated) bounding box\n\t\t\\count203=\\@p@sbburx\n\t\t\\count204=\\@p@sbbury\n\t\t\\advance\\count203 by -\\@p@sbbllx\n\t\t\\advance\\count204 by -\\@p@sbblly\n\t\t\\edef\\ps@bbw{\\number\\count203}\n\t\t\\edef\\ps@bbh{\\number\\count204}\n\t\t%\\ps@typeout{ psbbh = \\ps@bbh, psbbw = \\ps@bbw }\n\t\t\\if@angle \n\t\t\t\\Sine{\\@p@sangle}\\Cosine{\\@p@sangle}\n\t \t{\\dimen100=\\maxdimen\\xdef\\r@p@sbbllx{\\number\\dimen100}\n\t\t\t\t\t \\xdef\\r@p@sbblly{\\number\\dimen100}\n\t\t\t \\xdef\\r@p@sbburx{-\\number\\dimen100}\n\t\t\t\t\t \\xdef\\r@p@sbbury{-\\number\\dimen100}}\n%\n% Need to rotate all four points and take the X-Y extremes of the new\n% points as the new bounding box.\n \\def\\minmaxtest{\n\t\t\t \\ifnum\\number\\p@intvaluex<\\r@p@sbbllx\n\t\t\t \\xdef\\r@p@sbbllx{\\number\\p@intvaluex}\\fi\n\t\t\t \\ifnum\\number\\p@intvaluex>\\r@p@sbburx\n\t\t\t \\xdef\\r@p@sbburx{\\number\\p@intvaluex}\\fi\n\t\t\t \\ifnum\\number\\p@intvaluey<\\r@p@sbblly\n\t\t\t \\xdef\\r@p@sbblly{\\number\\p@intvaluey}\\fi\n\t\t\t \\ifnum\\number\\p@intvaluey>\\r@p@sbbury\n\t\t\t \\xdef\\r@p@sbbury{\\number\\p@intvaluey}\\fi\n\t\t\t }\n%\t\t\tlower left\n\t\t\t\\rotate@{\\@p@sbbllx}{\\@p@sbblly}\n\t\t\t\\minmaxtest\n%\t\t\tupper left\n\t\t\t\\rotate@{\\@p@sbbllx}{\\@p@sbbury}\n\t\t\t\\minmaxtest\n%\t\t\tlower right\n\t\t\t\\rotate@{\\@p@sbburx}{\\@p@sbblly}\n\t\t\t\\minmaxtest\n%\t\t\tupper right\n\t\t\t\\rotate@{\\@p@sbburx}{\\@p@sbbury}\n\t\t\t\\minmaxtest\n\t\t\t\\edef\\@p@sbbllx{\\r@p@sbbllx}\\edef\\@p@sbblly{\\r@p@sbblly}\n\t\t\t\\edef\\@p@sbburx{\\r@p@sbburx}\\edef\\@p@sbbury{\\r@p@sbbury}\n%\\ps@typeout{rotated BB: \\r@p@sbbllx, \\r@p@sbblly, \\r@p@sbburx, \\r@p@sbbury}\n\t\t\\fi\n\t\t\\count203=\\@p@sbburx\n\t\t\\count204=\\@p@sbbury\n\t\t\\advance\\count203 by -\\@p@sbbllx\n\t\t\\advance\\count204 by -\\@p@sbblly\n\t\t\\edef\\@bbw{\\number\\count203}\n\t\t\\edef\\@bbh{\\number\\count204}\n\t\t%\\ps@typeout{ bbh = \\@bbh, bbw = \\@bbw }\n}\n%\n% \\in@hundreds performs #1 * (#2 / #3) correct to the hundreds,\n%\tthen leaves the result in @result\n%\n\\def\\in@hundreds#1#2#3{\\count240=#2 \\count241=#3\n\t\t \\count100=\\count240\t% 100 is first digit #2/#3\n\t\t \\divide\\count100 by \\count241\n\t\t \\count101=\\count100\n\t\t \\multiply\\count101 by \\count241\n\t\t \\advance\\count240 by -\\count101\n\t\t \\multiply\\count240 by 10\n\t\t \\count101=\\count240\t%101 is second digit of #2/#3\n\t\t \\divide\\count101 by \\count241\n\t\t \\count102=\\count101\n\t\t \\multiply\\count102 by \\count241\n\t\t \\advance\\count240 by -\\count102\n\t\t \\multiply\\count240 by 10\n\t\t \\count102=\\count240\t% 102 is the third digit\n\t\t \\divide\\count102 by \\count241\n\t\t \\count200=#1\\count205=0\n\t\t \\count201=\\count200\n\t\t\t\\multiply\\count201 by \\count100\n\t\t \t\\advance\\count205 by \\count201\n\t\t \\count201=\\count200\n\t\t\t\\divide\\count201 by 10\n\t\t\t\\multiply\\count201 by \\count101\n\t\t\t\\advance\\count205 by \\count201\n\t\t\t%\n\t\t \\count201=\\count200\n\t\t\t\\divide\\count201 by 100\n\t\t\t\\multiply\\count201 by \\count102\n\t\t\t\\advance\\count205 by \\count201\n\t\t\t%\n\t\t \\edef\\@result{\\number\\count205}\n}\n\\def\\compute@wfromh{\n\t\t% computing : width = height * (bbw / bbh)\n\t\t\\in@hundreds{\\@p@sheight}{\\@bbw}{\\@bbh}\n\t\t%\\ps@typeout{ \\@p@sheight * \\@bbw / \\@bbh, = \\@result }\n\t\t\\edef\\@p@swidth{\\@result}\n\t\t%\\ps@typeout{w from h: width is \\@p@swidth}\n}\n\\def\\compute@hfromw{\n\t\t% computing : height = width * (bbh / bbw)\n\t \\in@hundreds{\\@p@swidth}{\\@bbh}{\\@bbw}\n\t\t%\\ps@typeout{ \\@p@swidth * \\@bbh / \\@bbw = \\@result }\n\t\t\\edef\\@p@sheight{\\@result}\n\t\t%\\ps@typeout{h from w : height is \\@p@sheight}\n}\n\\def\\compute@handw{\n\t\t\\if@height \n\t\t\t\\if@width\n\t\t\t\\else\n\t\t\t\t\\compute@wfromh\n\t\t\t\\fi\n\t\t\\else \n\t\t\t\\if@width\n\t\t\t\t\\compute@hfromw\n\t\t\t\\else\n\t\t\t\t\\edef\\@p@sheight{\\@bbh}\n\t\t\t\t\\edef\\@p@swidth{\\@bbw}\n\t\t\t\\fi\n\t\t\\fi\n}\n\\def\\compute@resv{\n\t\t\\if@rheight \\else \\edef\\@p@srheight{\\@p@sheight} \\fi\n\t\t\\if@rwidth \\else \\edef\\@p@srwidth{\\@p@swidth} \\fi\n\t\t%\\ps@typeout{rheight = \\@p@srheight, rwidth = \\@p@srwidth}\n}\n%\t\t\n% Compute any missing values\n\\def\\compute@sizes{\n\t\\compute@bb\n\t\\if@scalefirst\\if@angle\n% at this point the bounding box has been adjsuted correctly for\n% rotation. PSFIG does all of its scaling using \\@bbh and \\@bbw. If\n% a width= or height= was specified along with \\psscalefirst, then the\n% width=/height= value needs to be adjusted to match the new (rotated)\n% bounding box size (specifed in \\@bbw and \\@bbh).\n% \\ps@bbw width=\n% ------- = ---------- \n% \\@bbw new width=\n% so `new width=' = (width= * \\@bbw) / \\ps@bbw; where \\ps@bbw is the\n% width of the original (unrotated) bounding box.\n\t\\if@width\n\t \\in@hundreds{\\@p@swidth}{\\@bbw}{\\ps@bbw}\n\t \\edef\\@p@swidth{\\@result}\n\t\\fi\n\t\\if@height\n\t \\in@hundreds{\\@p@sheight}{\\@bbh}{\\ps@bbh}\n\t \\edef\\@p@sheight{\\@result}\n\t\\fi\n\t\\fi\\fi\n\t\\compute@handw\n\t\\compute@resv}\n\n%\n% \\psfig\n% usage : \\psfig{file=, height=, width=, bbllx=, bblly=, bburx=, bbury=,\n%\t\t\trheight=, rwidth=, clip=}\n%\n% \"clip=\" is a switch and takes no value, but the `=' must be present.\n\\def\\psfig#1{\\vbox {\n\t% do a zero width hard space so that a single\n\t% \\psfig in a centering enviornment will behave nicely\n\t%{\\setbox0=\\hbox{\\ }\\ \\hskip-\\wd0}\n\t%\n\t\\ps@init@parms\n\t\\parse@ps@parms{#1}\n\t\\compute@sizes\n\t%\n\t\\ifnum\\@p@scost<\\@psdraft{\n\t\t%\n\t\t\\special{ps::[begin] \t\\@p@swidth \\space \\@p@sheight \\space\n\t\t\t\t\\@p@sbbllx \\space \\@p@sbblly \\space\n\t\t\t\t\\@p@sbburx \\space \\@p@sbbury \\space\n\t\t\t\tstartTexFig \\space }\n\t\t\\if@angle\n\t\t\t\\special {ps:: \\@p@sangle \\space rotate \\space} \n\t\t\\fi\n\t\t\\if@clip{\n\t\t\t\\if@verbose{\n\t\t\t\t\\ps@typeout{(clip)}\n\t\t\t}\\fi\n\t\t\t\\special{ps:: doclip \\space }\n\t\t}\\fi\n\t\t\\if@prologfile\n\t\t \\special{ps: plotfile \\@prologfileval \\space } \\fi\n\t\t\\if@decmpr{\n\t\t\t\\if@verbose{\n\t\t\t\t\\ps@typeout{psfig: including \\@p@sfile.Z \\space }\n\t\t\t}\\fi\n\t\t\t\\special{ps: plotfile \"`zcat \\@p@sfile.Z\" \\space }\n\t\t}\\else{\n\t\t\t\\if@verbose{\n\t\t\t\t\\ps@typeout{psfig: including \\@p@sfile \\space }\n\t\t\t}\\fi\n\t\t\t\\special{ps: plotfile \\@p@sfile \\space }\n\t\t}\\fi\n\t\t\\if@postlogfile\n\t\t \\special{ps: plotfile \\@postlogfileval \\space } \\fi\n\t\t\\special{ps::[end] endTexFig \\space }\n\t\t% Create the vbox to reserve the space for the figure\n\t\t\\vbox to \\@p@srheight true sp{\n\t\t\t\\hbox to \\@p@srwidth true sp{\n\t\t\t\t\\hss\n\t\t\t}\n\t\t\\vss\n\t\t}\n\t}\\else{\n\t\t% draft figure, just reserve the space and print the\n\t\t% path name.\n\t\t\\if@draftbox{\t\t\n\t\t\t% Verbose draft: print file name in box\n\t\t\t\\hbox{\\frame{\\vbox to \\@p@srheight true sp{\n\t\t\t\\vss\n\t\t\t\\hbox to \\@p@srwidth true sp{ \\hss \\@p@sfile \\hss }\n\t\t\t\\vss\n\t\t\t}}}\n\t\t}\\else{\n\t\t\t% Non-verbose draft\n\t\t\t\\vbox to \\@p@srheight true sp{\n\t\t\t\\vss\n\t\t\t\\hbox to \\@p@srwidth true sp{\\hss}\n\t\t\t\\vss\n\t\t\t}\n\t\t}\\fi\t\n\n\n\n\t}\\fi\n}}\n\\psfigRestoreAt\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002117.extracted_bib",
"string": "\\bibitem{Levin} C. Levinthal, J. Chim. Phys. {\\bf 65}, 44 (1968).\n\n\n\\bibitem{7} J. D. Bryngelson and P. G. Wolynes, {\\em Proc. Natl.\nAcad. Sci. USA} {\\bf 84}, 7524-7528 (1987); J. D. Bryngelson,\nJ. N. Onuchic, J. N. Socci and P. G. Wolynes, {\\em Proteins:\nStruc. Funct. Genet.} {\\bf 21}, 167-195 (1995).\n\n\n\\bibitem{9} P. E. Leopold, M. Montal and J. N. Onuchic, {\\em\nProc. Natl. Acad. Sci. USA} {\\bf 89}, 8721-8725 (1992); P. G. Wolynes\nJ. N. Onuchic and D. Thirumalai, {\\em Science} {\\bf 267}, 1619-1620\n(1995); J. N. Onuchic, Z. Luthey Schulten and P. G. Wolynes, {\\em\nAnn. Rev. Phys. Chem.} {\\bf 48}, 545-600 (1997); K. A. Dill and\nH. S. Chan, {\\em Nature Structural Biology} {\\bf 4}, 10-19 (1997).\n\n\n\\bibitem{LG} H. Li, R. Helling, C. Tang and N. Wingreen, {\\em Science}\n{\\bf 273}, 666-669 (1996); N. E. G. Buchler and R. A. Goldstein, {\\em\nProteins: Struc. Funct. Genet.} {\\bf 34}, 113-124 (1999);\nC. Micheletti, A. Maritan, J. R. Banavar and F. Seno, {\\em\nPhys. Rev. Lett.} {\\bf 80}, 5683 (1998); C. Micheletti, A. Maritan and\nJ. R. Banavar, {\\em J. Chem. Phys.} {\\bf 110}, 9730 (1999).\n\n\n\n\n\\bibitem{19} C. Micheletti, J. R. Banavar, A. Maritan and F. Seno,{\\em\nPhys. Rev. Lett.} {\\bf 82}, 3372-3375 (1999).\n\n\n\\bibitem{2} L. Pauling, R. B. Corey and H. R. Branson,\n{\\it Proc. Nat. Acad. Sci.} {\\bf 37}, 205-208 (1951).\n\n\n\\bibitem{hc} B. H. Zimm and J. Bragg, {\\em J. Chem. Phys.}, {\\bf 31},\n526 (1959); O. B. Ptitsyn and A. M. Skvortsov, {\\em Biophys.} {\\bf\n10}, 1007 (1965); I. M. Lifshitz, A. Y. Grosberg and A. R. Khokhlov, {\\em\nRev. Mod. Phys.}, {\\bf 50}, 683 (1978).\n\n\n\n\\bibitem{4} N. G. Hunt, L. M. Gregoret and F. E. Cohen,\n{\\it J. Mol. Biol.} {\\bf 241}, 214-225 (1994).\n\n\n\\bibitem{18} R. Aurora, T. P. Creamer, R. Srinivasan and G. D. Rose,\n{\\em J. Mol. Biol.} {\\bf 272}, 1413-1416 (1997).\n\n\n\\bibitem{21} J. C. Nelson, J. G. Saven, J. S. Moore, and P. G. Wolynes,\n{\\em Science} {\\bf 277}, 1793-1796 (1997).\n\n\n\\bibitem{3} D. P. Yee, H. S. Chan, T. F. Havel and K. A. Dill,\n{\\it J. Mol. Biol.} {\\bf 241}, 557-573 (1994).\n\n\n\\bibitem{5} N. D. Socci, W. S. Bialek, and J. N. Onuchic,\n{\\it Phys. Rev. E} {\\bf 49}, 3440-3443 (1994).\n\n\n\\bibitem{halper} A. Halperin and P. M. Goldbart, Phys. Rev. E, in\npress (cond-mat/9905306).\n\n\n\\bibitem{12} N. Go, {\\em Macromolecules} {\\bf 9}, 535-541 (1976).\n\n\n\n\\bibitem{17} D. G. Covell and R. Jernigan {\\it Biochem.} {\\bf 29},\n3287 (1990).\n\n\n\\bibitem{cont} The native state structures of monomeric\nproteins of length between 50 and 200 show an excellent correlation of\nthis form, when two non-consecutive amino acids along the sequence are\ndefined to be in contact when they are within 6.5 \\AA\\ of each\nother.\n\n\n\\bibitem{14} J. H. Holland, {\\em Adaptation in natural and artificial\nsystems}, MIT press ed. (1992).\n\n\n\\bibitem{15} K. M. Plaxco, K. T. Simons and D. Baker, {\\it\nJ. Mol. Biol.} {\\bf 277}, 985-994 (1998).\n\n\n\\bibitem{22} A. P. Capaldi and S. E. Radford, S. E., {\\em\nCurr. Op. Str. Biol.} {\\bf 8}, 86-92 (1998).\n\n\n\\bibitem{slow} V. Mu\\~noz, E. R. Henry, J. Hofrichter and W. A. Eaton,\n{\\em Proc. Natl. Acad. Sci. USA}, {\\bf 95}, 5872 (1998).\n"
}
] |
cond-mat0002118
|
Extensive Chaos in the Nikolaevskii Model
|
[
{
"author": "Hao-wen Xi$^1$"
},
{
"author": "Ra\\'ul Toral$^2$"
},
{
"author": "J. D. Gunton$^3$ and Michael I. Tribelsky$^4$"
}
] |
We carry out a systematic study of a novel type of chaos at onset (``soft-mode turbulence") based on numerical integration of the simplest one dimensional model. The chaos is characterized by a smooth interplay of different spatial scales, with defect generation being unimportant. The Lyapunov exponents are calculated for several system sizes for fixed values of the control parameter $\epsilon$. The Lyapunov dimension and the Kolmogorov-Sinai entropy are calculated and both shown to exhibit extensive and microextensive scaling. The distribution functional is shown to satisfy Gaussian statistics at small wavenumbers and small frequency.\\
|
[
{
"name": "prl.tex",
"string": "\\documentstyle[aps,prl,epsfig,amssymb,balanced,times]{revtex}\n\\begin{document}\n\\title{Extensive Chaos in the Nikolaevskii Model}\n\\author{Hao-wen Xi$^1$, Ra\\'ul Toral$^2$, J. D. Gunton$^3$ and\nMichael I. Tribelsky$^4$}\n\\address{$^1$Department of Physics and Astronomy,\nBowling Green State University, Bowling Green OH 43403\\\\\n$^2$Instituto Mediterr\\'aneo de Estudios Avanzados (IMEDEA),\nCSIC-UIB, E-07071 Palma de Mallorca, Spain\\\\ $^3$Department of\nPhysics, Lehigh University, Bethlehem, PA 18015\\\\ $^4$Department of\nApplied Physics, Faculty of Engineering, Fukui University, Bunkyo\n3-9-1, Fukui 910-8507, Japan\\\\}\n\\maketitle\n\\pacs{PACS numbers: 05.45.+b, 47.20.Ky, 47.27.Eq, 47.52.+j}\n\\begin{abstract}\nWe carry out a systematic study of a novel type of chaos at onset\n(``soft-mode turbulence\") based on numerical integration of the\nsimplest one dimensional model. The chaos is characterized by a\nsmooth interplay of different spatial scales, with defect\ngeneration being unimportant. The Lyapunov exponents are calculated\nfor several system sizes for fixed values of the control parameter\n$\\epsilon$. The Lyapunov dimension and the Kolmogorov-Sinai entropy\nare calculated and both shown to exhibit extensive and\nmicroextensive scaling. The distribution functional is shown to\nsatisfy Gaussian statistics at small wavenumbers and small\nfrequency.\\\\\n\\end{abstract}\n\n\\begin{twocolumns}\nSpatiotemporal chaos (STC) is a subject of considerable\nexperimental and theoretical importance and occurs in a wide\nvariety of driven, dissipative systems\\cite{CH93,PM90,HG98}. Such\nchaotic behavior in spatially extended systems is extremely\ndifficult to characterize quantitatively, as the dynamics involves\na large number of degrees of freedom. The most common and useful\ntool for the characterization of chaos is given by the Lyapunov\nexponents $\\{\\lambda_i\\}$. Knowledge of this Lyapunov spectrum\npermits one to estimate the number of effective degrees of freedom\nof the system (i.e., the dimension of the attractor), using for\nexample the Kaplan-Yorke\\cite{KY79} formula for the Lyapunov\ndimension $D(L)$, where $L$ is the linear system size. It also\npermits one to test the important concept of extensivity of chaos,\ndefined as the case in which $\\lim_{L\\rightarrow\\infty} D(L)\\sim\nL^{d}$, where $d$ is the spatial dimension of the\nsystem\\cite{CH93,HS89}. An interpretation of extensive chaos is\nthat the whole system can then be thought of in some sense as the\nunion of almost independent subsystems. This was originally\nproposed by Ruelle\\cite{DR82}, who argued that widely separated\nsubsystems of a turbulent system should be weakly correlated, so\nthat the spectrum of Lyapunov exponents would be the union of\nexponents associated with each of the subsystems. The question is\nclosely related to the fundamental problem of ergodicity of\nnonequilibrium systems. If the chaos is extensive and each\nsubsystem evolves in time practically independently of the others,\nthen in a steady (non-transient) chaotic state the time average is\nequivalent to the ensemble average and the system should be\nergodic. Much work has focused on attempting to characterize\nspatiotemporal dynamics in these terms (see, e.g.,\n\\cite{VY89,LLPP93,CH95,WH99,COB99}). However, in spite of the\nfundamental importance of the question practically all the results\nare related just to a few discrete coupled map\nlattices\\cite{LPR92,HC93,PP98} and two continuous systems, namely\nthe complex Ginzburg-Landau (CGL) and Kuramoto-Sivashinsky (KS)\nequations (see, e.g., refs.\\cite{HC93,PM85,EG95}). This is partly\ndue to the computational complexity of the problem but primarily to\nthe lack of simple models exhibiting the requisite chaotic\nbehavior. It would therefore be of considerable interest to\ncharacterize quantitatively other types of STC.\n\nRecently attention was drawn to the existence of a new wide class\nof systems displaying such a behavior\\cite{RBR95,KHH96,TT96,MIT97}.\nTheir properties are qualitatively different from those of the CGL\nand KS models. In contrast to both these models the chaos is\nassociated with smooth, random long-wavelength modulations of a\nshort-wavelength pattern, with defect generation being unimportant.\nThe short-wavelength pattern arises due to a single supercritical\nbifurcation of the Turing type such as occurs in\nRayleigh-B\\'{e}nard convection. The long-wavelength modes belong to\na Goldstone branch of the spectrum originated in a broken\ncontinuous symmetry. The symmetry makes the system degenerate to\nthe extent that instead of a single, unique spatially uniform\nstate, it has a {\\it continuous family} of equivalent spatially\nuniform states, which may be obtained from each other by the\nsymmetry transformation. This symmetry, which is additional to the\ntrivial groups of translations and rotations, can be one of many\ndifferent types. For this reason the STC in question is quite a\ncommon phenomenon and occurs, for example, in electroconvection in\nliquid crystals\\cite{RBR95,KHH96}, in convection in a fluid with\nstress-free boundary conditions\\cite{SZ82,BB84,XLG97,N}, etc., see\nref.\\cite{MIT97} for further discussion. The chaos observed in such\ncases may be interpreted as a macroscopic dynamical analog of\nsecond order phase transitions, where the order parameter is\nrelated to the amplitudes of turbulent modes. Due to this analogy\nit has been called {\\it soft-mode turbulence} (SMT)\\cite{KHH96}.\nThe simplest model exhibiting SMT was introduced by\nNikolaevskii\\cite{VN89,BN93} to describe longitudinal seismic waves\nin viscoelastic media. In what follows we exploit the simplicity of\nthis model to shed light on general features of this new type of\nSTC.\n\nWe present a detailed systematic study of the Nikolaevskii model,\nincluding calculation of the Lyapunov exponents, fractal dimension,\netc. We show that the Lyapunov dimension and Kolmogorov-Sinai\nentropy are extensive quantities, which supports the validity of\nthe Ruelle's ideas for SMT. We also show that the power spectrum\ncan be described by Gaussian statistics for small wavenumbers and\nfrequencies, in agreement with a general argument of Hohenberg and\nShraiman\\cite{HS89}.\n\nThe model is defined by the following partial differential equation\nfor the real scalar field $v(x,t)$ (longitudinal mode of the\ndisplacement velocity in the original formulation)\\cite{VN89,BN93}:\n\\begin{equation}\n\\frac{\\partial v}{\\partial t} + \\frac{\\partial^{2}}{\\partial x^{2}}\n[\\epsilon - (1 + \\frac{\\partial^{2}}{\\partial x^{2}})^{2}]v +\nv\\frac{\\partial v}{\\partial x}= 0\n\\end{equation}\nwith $0\\leq x \\leq L$ and periodic boundary conditions. This model\nhas two control parameters, $\\epsilon$ (the distance from onset)\nand $L$, in contrast to, e.g., the KS model, where the only\nnon-trivial control parameter is the system size $L$. An essential\nfeature of the model is that even at small $\\epsilon$ it cannot be\nreduced to any $\\epsilon$-free form\\cite{MIT97,BM92}, which makes\nthe hierarchy of characteristic scales much more complex than those\nin the KS and CGL models\\cite{TV96,MT97}. Eq. (1) may be regarded\nas a generalized Burgers equation and shares with it the same group\nof symmetry, namely trivial symmetries under shifts of the\nspatiotemporal coordinate system, and nontrivial invariance with\nrespect to the Galilean transformation $v(x,t)\\rightarrow\nv(x-v_{o}t,t)+v_{o}$, where $v_{o}$ is an arbitrary constant. The\nGalilean invariance plays the role of the above specified\nadditional symmetry, generating for Eq. (1) the continuous family\nof solutions $v=v_{o}$. The Nikolaevskii equation admits a\ncontinuous set of spatially periodic, stationary solutions, whose\ninstability has been proved analytically\\cite{TV96}. Computer\nsimulations\\cite{TT96} of an equivalent version of this model for\nan order parameter $u(x,t)\\equiv\\int v(x,t)dx$ showed that even at\nextremely small $\\epsilon=10^{-4}$, the system exhibits STC.\nHowever, this simulation did not provide any quantitative results\nabout the STC. The only result of this kind is in ref.\\cite{KM97},\nin which just a single quantitative characteristic, namely the\ndependence of the mean amplitude of chaotic patterns on the control\nparameter, was studied. In this Letter we span the gap in our\nknowledge of this new type of STC, providing a detailed\nquantitative description of its most important properties based on\nnumerical integration of Eq. (1). The simulations were carried out\nusing the pseudo-spectral method combined with a fourth-order\npredictor-corrector integrator, for several different values of $L$\nand two values of $\\epsilon$ (0.2 and 0.5, respectively)\\cite{NID}.\nThe Lyapunov exponents were calculated by linearizing the equation\nalong the trajectory, performing a re-orthonormalization after a\nfew integration steps to prevent the largest Lyapunov exponent from\nswamping all the others\\cite{PC89}. A typical pattern $v(x,t)$ as a\nfunction of space and time in the steady chaotic regime is shown in\nFig.~1. The time averaged power spectrum $\\langle |\\hat\nv(k)|^{2}\\rangle $ obtained over a time period of $T=10^{4}$ for\nsystem size $L=78$ with $\\epsilon=0.2$ is shown in Fig.~2. As can\nbe seen, the dominant modes occur in the vicinity of $k=\\pm 1$.\nNote, however, the smaller peak near zero wavenumber, which arises\nfrom coupling between unstable short-wavelength modes centered\nabout $k=\\pm 1$ and the slowly decaying modes from the Goldstone\nbranch of the spectrum centered at $k = 0$ (using terminology based\nupon the linear stability analysis of the spatially uniform\nsolution). The power spectrum for $\\hat u(k)=\\hat v(k)/ik$ (shown\nin the insert in Fig.~2) has the dominant peak near $k=0$, as was\noriginally demonstrated in\\cite{TT96}. One would also expect on\ngeneral grounds\\cite{HS89} each Fourier transform variable $\\hat\nv(k,\\omega)$ to be governed by a Gaussian probability distribution\nfunctional, $\\exp(-D(|\\hat {v}(k,\\omega)|^{2})$, for small $k$ and\n$\\omega$. We have verified this for several values of $k$ and\n$\\omega$. In particular, for $\\omega=0.6$ the Gaussian distribution\nholds for $0 < k < 2$ (see Fig. 3).\n\n\\begin{figure}\n\\epsfxsize = 0.45\\textwidth\n\\epsfysize = 0.4\\textwidth\n\\makebox{\\epsfbox{figure1.eps}}\n\\caption{Typical space-time configuration for $v(x,t)$\nin the steady chaotic regime is shown here. The configuration shown has\nevolved from random initial condition within a system size $L=78$\nand $\\epsilon=0.5$.\\label{fig1}}\n\\end{figure}\n\n\\begin{figure}\n\\epsfxsize = 0.45\\textwidth\n\\epsfysize = 0.35\\textwidth\n\\makebox{\\epsfbox{figure2.eps}}\n\\caption{Time average power spectra $\\langle|\\hat {v}(k)|^{2}\\rangle$\nand $\\langle|\\hat {u}(k)|^{2}\\rangle$ in $k$-space are plotted. The\nsystem size is $L=78$ and control parameter $\\epsilon=0.2$. Note\nthat the power spectrum has symmetry with respect to $ k\\rightarrow\n-k$.\\label{fig2}}\n\\end{figure}\n\nAs has already been emphasized, the key question for any system\nexhibiting STC is whether it can be represented as a union of\nweakly correlated subsystems. If this is the case, then the\nspectrum of Lyapunov exponents for the entire system should be {\\it\nintensive} in the sense that $\\lambda_{i}$ is a function only of\nthe intensive index $i/V$, i.e., $\\lambda_{i}=f(i/V)$, where V\nstands for the volume of the system\\cite{HG98}. The question is not\ntrivial for the type of STC considered here, because of the\nimportance of the long-wavelength modes and the divergence of the\ntwo point correlation length $\\xi_{2}$\\cite{MT99} as $\\epsilon\n\\rightarrow 0$\\cite{MIT97}. To answer this question, a detailed\nstudy of the Lyapunov spectrum for the Nikolaevskii model was\nconducted. The results are shown in Fig. 4, where the number of\nLyapunov exponents greater than a particular value $\\lambda_{i}$,\nscaled by the system size $L$, is plotted versus $\\lambda_{i}$ for\n$\\epsilon=0.5$. A similar curve is found at $\\epsilon=0.2$, but the\nmaximum positive eigenvalue is now smaller (one expects this\neigenvalue to vanish as $\\epsilon \\rightarrow 0$.) The intensive\nnature of the Lyapunov density is evident. In this case, one also\nexpects the fractal dimension $D(L)$ of an attractor to be\nextensive for large enough $L$ (extensive chaos), as was first\nshown by Manneville\\cite{PM85} for chaotic solutions of the\nKuramoto-Sivashinsky equation. We have checked this for the\nNikolaevskii model using the Kaplan-Yorke formula\\cite{KY79} for\nthe Lyapunov dimension,\n\\begin{equation}\nD(L) = K + \\sum_{i=1}^{K} \\lambda_{i}/|\\lambda_{k+1}|\n\\end{equation}\nwhere the integer $K$ is the largest integer such that the sum of\nthe first $K$ Lyapunov exponents is nonnegative. We have also\ncalculated the Kolmogorov-Sinai entropy $H(L)$ from the definition\n\\begin{equation}\nH(L) = \\sum_{i=1}^{i_{+}} \\lambda_{i}\n\\end{equation}\nwhere the sum is over the positive Lyapunov exponents. The\nKolmogorov-Sinai entropy\\cite{PC89} is a measure of the mean rate\nof information production in a system, or the mean rate of growth\nof uncertainty in a system subjected to small perturbations. We\nfind that for large enough $L$ both $D(L)$ and $H(L)$ are\nextensive. Our results for the Lyapunov dimension $D(L)$ are shown\nin Figure 5 for $\\epsilon=0.5$. The same behavior is found at\n$\\epsilon=0.2$, but with a different slope (naturally the slope of\nthe straight line D(L) at $\\epsilon=0.2$ is smaller than that at\n$\\epsilon=0.5$). In addition, we find that the upper index $i_{+}$\nin (3), which corresponds to the smallest positive Lyapunov\nexponent, is also proportional to $L$.\n\n\\begin{figure}\n\\epsfxsize = 0.45\\textwidth\n\\epsfysize = 0.35\\textwidth\n\\makebox{\\epsfbox{figure3.eps}}\n\\caption{Probability density distribution of\npower spectrum $\\hat {v}(k)$ for $k=1.5$ with system size $L=78$\nand $\\epsilon=0.5$. Notice that the validity of the Gaussian\ndistribution for $\\hat v(k)$ (see the main text) implies an\nexponential distribution $f(z)=D{\\rm e}^{-D z}$ for the variable\n$z\\equiv |\\hat {v}(k)|^{2}$. In this case we find the value\n$D\\approx 47$.\\label{fig3}}\n\\end{figure}\n\nThe important characteristic of STC is the dimension correlation\nlength $\\xi_{\\delta}$\\cite{CH93,HG98}. This length is defined as\n$\\xi_{\\delta} \\equiv \\delta^{-1/d}$, where $\\delta \\equiv\nlim_{L\\rightarrow \\infty} D(L)/L^{d}$. It can be thought of as the\n``radius\" of a volume that contains one degree of freedom, or, as\nthe linear size of the subsystem described above. The value of this\ndimension correlation length for Eq.~(1) is $\\xi_{\\delta}= 3.0$ for\n$\\epsilon=0.5$ and $\\xi_{\\delta}= 3.3$ for $\\epsilon=0.2$. In\ncontrast to $\\xi_{\\delta}$ the above-mentioned two point\ncorrelation length $\\xi_2$ governs the spatial decay of the\ncorrelation function. In general these two lengths are\ndifferent\\cite{HG98}. We found that $\\xi_{2} \\cong 4.9$ for the\nNikolaevskii model at $\\epsilon = 0.5$ and $\\xi_{2}\n\\cong 5.6$ at $\\epsilon=0.2$.\n\nFinally, Tajima and Greenside\\cite{TG99} have recently found for\nthe one dimensional Kuramoto-Sivashinsky model that $D(L)$ is also\n\"microextensive.\" Namely, they found that if one increases L by a\nsmall amount $\\delta L$, with $\\delta L\\ll \\xi_{\\delta}$ ($\\delta\nL=0.8$ in our simulation), one finds that $D(L)$ satisfies the same\nlinear relationship as that characterizing extensive chaos. We have\nexamined this for two different domains of $L$ for our model and\nfound that microextensivity holds for both $D(L)$ and $H(L)$. Our\nresults for microextensivity for $D(L)$ are shown in the insert in\nFig. 5. \n\n\\begin{figure}\n\\epsfxsize = 0.45\\textwidth\n\\epsfysize = 0.35\\textwidth\n\\makebox{\\epsfbox{figure4.eps}}\n\\caption{Here $N_{>i}$ is the number of Lyapunov exponents greater than\na particular value $\\lambda_{i}$. We plot $N_{>i}/L$ (scaled by the\nsystem size $L$) vs $\\lambda_{i}$ in the case $\\epsilon=0.5$.\\label{fig4}}\n\\end{figure}\n\nIn conclusion, our detailed study of this new type of STC based on\nthe numerical integration of the Nikolaevskii model shows that for\nsufficiently large system size the chaos is both extensive and\nmicroextensive. We also found that the system satisfies Gaussian\nstatistics at sufficiently small wavenumbers and frequencies. We\nbelieve these results are quite general and reflect intrinsic\nfeatures of this type of STC, rather than specific peculiarities of\nthe model.\n\nThere are several interesting questions to investigate in the limit\n$\\epsilon \\rightarrow 0$, including the dependence of quantities\nsuch as the correlation lengths and Lyapunov exponents on\n$\\epsilon$ as well as the possible scaling of the power spectrum.\nThis study is in progress and will be reported elsewhere.\n\n\\begin{figure}\n\\epsfxsize = 0.45\\textwidth\n\\epsfysize = 0.35\\textwidth\n\\makebox{\\epsfbox{figure5.eps}}\n\\caption{Lyapunov dimension $D(L)$ vs system size $L$ for $\\epsilon=0.5$.\n$\\delta L=15.5$ for extensive, and $\\delta L=0.8$ for\nmicroextensive.\\label{fig5}}\n\\end{figure}\n\nWe would like to thank Henry Greenside for a helpful discussion.\nThis work was supported by a grant from NATO CRG.CRG972822, by NSF\nGrant DMR9810409, projects PB94-1167 and PB97-0141-C02-01 (Spain)\nand the Grant-in-Aid for Scientific Research (No. 11837006) from\nthe Ministry of Education, Science, Sports and Culture (Japan). We\nalso wish to acknowledge an allocation of time on the Pittsburgh\nSupercomputer Center, where some of this work was carried out.\n\n\\begin{references}\n\\bibitem{CH93} M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. {\\bf 65}\n851 (1993).\n\n\\bibitem{PM90} {\\it Dissipative Structures and Weak Turbulence}, P. Manneville,\nAcademic Press (1990).\n\n\\bibitem{HG98} H. Greenside, \"Spatiotemporal Chaos in Large Systems:\nThe Scaling of Complexity with Size\", in Proceedings of the CRM\nWorkshop, {\\it Semi-Analytic Methods for the Navier Stokes\nEquations}, ed. K. Coughlin (1998), and references therein.\n\n\\bibitem{KY79} J. L. Kaplan, and J. A. Yorke, in {\\it Lecture Notes in Mathematics},\n228, Springer-Verlag (1979).\n\n\\bibitem{HS89} P. C. Hohenberg, and B. I. Shraiman, Physica D {\\bf 37}, 109\n(1989).\n\n\\bibitem{DR82} D. Ruelle, Commun. Math. Phys. {\\bf 87}, 287\n(1982).\n\n\\bibitem{VY89} V. Yakhot, Phys. Rev. A {\\bf 24}, 642 (1989).\n\n\\bibitem{LLPP93} V. S. 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Buka, Europhys. Lett. {\\bf 30}, 37 (1995).\n\n\\bibitem{KHH96} S. Kai, K. Hayashi, and Y. Hidaka, J. Phys.\nChem. {\\bf 100}, 19007 (1996).\n\n\\bibitem{TT96} M. I. Tribelsky, and K. Tsuboi, Phys. Rev.\nLett. {\\bf 76}, 1631 (1996).\n\n\\bibitem{MIT97} M. I. Tribelskii, Usp. Fiz. Nauk. {\\bf 167},\n167 (1997) [Phys. Usp. {\\bf 40}, 159 (1997)].\n\n\\bibitem{SZ82} E.D. Siggia, and A. Zippelius, Phys. Rev. Lett. {\\bf 47}, 835 (1981).\n\n\\bibitem{BB84} F. H. Busse and E.W. Bolton, J. Fluid. Mech. {\\bf 146}, 115\n(1984).\n\n\\bibitem{XLG97} H. W. Xi, X. J. Li and J. D. Gunton, Phys.\nRev. Lett. {\\bf 78}, 1046 (1997).\n\n\\bibitem{N} It is relevant to mention that the\nconsideration of convection with the stress-free boundary\nconditions\\cite{SZ82,BB84} was the very first analysis of pattern\nformation in the degenerate systems. However, at that time the\ndramatic destabilization of steady patterns obtained in this case\nwas regarded as a peculiarity of the particular problem,\nrather than a generic attribute of the degeneracy.\n\n\\bibitem{VN89} V.N. Nikolaevskii, in {\\it Recent Advances in\nEngineering Science}, ed. S. L. Koh and C. G. Speciale, Lecture\nNotes in Engineering, No. 39, Springer-Verlag (1989), p. 210.\n\n\\bibitem{BN93}I. A. Beresnev and V. N. Nikolaevskii, Physica D {\\bf 66},\n1 (1993).\n\n\\bibitem{BM92} B. A. Malomed, Phys. Rev. A {\\bf 45}, 1009 (1992).\n\n\\bibitem{TV96} M. I. Tribelsky and M. G. Velarde, Phys. Rev. E {\\bf\n54}, 4973 (1996).\n\n\\bibitem{MT97} M. I. Tribelsky, Inter. Jour. Bifur. Chaos\n{\\bf 7}, 997 (1997).\n\n\\bibitem{KM97} I. L. Kliakhandler and B. A. Malomed, Phys. Lett.\nA {\\bf 231}, 191 (1997).\n\n\\bibitem{NID} Although the original numerical study\\cite{TT96}\nwas carried out in terms of the order parameter $u(x,t)$, we have\nfound that Eq.~(1) is more convenient for numerical study as it\nexhibits greater numerical stability. The reason is that the\nconservation law $d\\bar v/dt=0$ holding for this equation (where\nthe bar denotes a spatial average) helps to stabilize the code. In\nour simulations the value $\\bar v=0$ was imposed by the initial,\nrandom, conditions. For the space discretization we have found that\n$\\Delta x=0.31$ provides a good enough resolution.\n\n\\bibitem{PC89}T. S. Parker and L. O. Chua, {\\it Practical Numerical\nAlgorithms for Chaotic Systems} (Springer-Verlag, 1989).\n\n\\bibitem{MT99}At small $\\epsilon$ in this model the two point\ncorrelation function ($\\langle v(x,t)v(x+x',t)\\rangle$, where the\n$\\langle\\ldots\\rangle$ denotes the time average) is generally\noscillatory with a spatial period of the oscillations close to\n$2\\pi$ and a slow decay of its amplitude over a scale $\\xi_{2}$.\nThus, the model has two correlation length scales, with $\\xi_2$\nbeing the larger of the two.\n\n\\bibitem{TG99} S. Tajima and H. S. Greenside, \"Microextensive\nChaos of a Spatially Extended System\", March Meeting of APS,\nAtlanta, Georgia (1999).\n\\end{references}\n\n\\end{twocolumns}\n\n\\end{document}\n"
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{
"name": "cond-mat0002118.extracted_bib",
"string": "\\bibitem{CH93} M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. {\\bf 65}\n851 (1993).\n\n\n\\bibitem{PM90} {\\it Dissipative Structures and Weak Turbulence}, P. Manneville,\nAcademic Press (1990).\n\n\n\\bibitem{HG98} H. Greenside, \"Spatiotemporal Chaos in Large Systems:\nThe Scaling of Complexity with Size\", in Proceedings of the CRM\nWorkshop, {\\it Semi-Analytic Methods for the Navier Stokes\nEquations}, ed. K. Coughlin (1998), and references therein.\n\n\n\\bibitem{KY79} J. L. Kaplan, and J. A. Yorke, in {\\it Lecture Notes in Mathematics},\n228, Springer-Verlag (1979).\n\n\n\\bibitem{HS89} P. C. Hohenberg, and B. I. Shraiman, Physica D {\\bf 37}, 109\n(1989).\n\n\n\\bibitem{DR82} D. Ruelle, Commun. Math. Phys. {\\bf 87}, 287\n(1982).\n\n\n\\bibitem{VY89} V. Yakhot, Phys. Rev. A {\\bf 24}, 642 (1989).\n\n\n\\bibitem{LLPP93} V. S. L'vov, V. V. Lebedev, M. Paton, and I. Procaccia,\nNonlinearity {\\bf 6}, 25 (1993).\n\n\n\\bibitem{CH95} C. C. Chow, and T. Hwa, Physica D {\\bf 84}, 494 (1995).\n\n\n\\bibitem{WH99} R. W. Wittenberg, and P. Holmes, Chaos {\\bf 9}, 452 (1999).\n\n\n\\bibitem{COB99} R. Carretero-Gonzalez, S. Orstavik, J. Huke, D. S.\nBroomhead, and J. Stark, Chaos {\\bf 9}, 466 (1999).\n\n\n\\bibitem{LPR92} R. Livi, A. Politi, and S. Ruffo, J. Phys. A {\\bf 25}, 4813 (1992).\n\n\n\\bibitem{HC93} H. Chate, Europhys. Lett. {\\bf 21}, 419 (1993).\n\n\n\\bibitem{PP98}A. Pikovsky, and A. Politi, Nonlinearity {\\bf 11}, 1049 (1998).\n\n\n\\bibitem{PM85} P. Manneville, \"Liapounov exponents for the\nKuramoto-Sivashinsky Model\", in {\\it Macroscopic Modeling of\nTurbulent Flows}, ed. O. Pironneau, Lecture Notes in Physics, No.\n230 Springer-Verlag (1985), p. 319.\n\n\n\\bibitem{EG95} D. A. Egolf, and H. S. Greenside, Phys. Rev. Lett.\n{\\bf 74}, 1751 (1995).\n\n\n\\bibitem{RBR95} H. Richter, N. Kloepper, A. Hertrich,\nand A. Buka, Europhys. Lett. {\\bf 30}, 37 (1995).\n\n\n\\bibitem{KHH96} S. Kai, K. Hayashi, and Y. Hidaka, J. Phys.\nChem. {\\bf 100}, 19007 (1996).\n\n\n\\bibitem{TT96} M. I. Tribelsky, and K. Tsuboi, Phys. Rev.\nLett. {\\bf 76}, 1631 (1996).\n\n\n\\bibitem{MIT97} M. I. Tribelskii, Usp. Fiz. Nauk. {\\bf 167},\n167 (1997) [Phys. Usp. {\\bf 40}, 159 (1997)].\n\n\n\\bibitem{SZ82} E.D. Siggia, and A. Zippelius, Phys. Rev. Lett. {\\bf 47}, 835 (1981).\n\n\n\\bibitem{BB84} F. H. Busse and E.W. Bolton, J. Fluid. Mech. {\\bf 146}, 115\n(1984).\n\n\n\\bibitem{XLG97} H. W. Xi, X. J. Li and J. D. Gunton, Phys.\nRev. Lett. {\\bf 78}, 1046 (1997).\n\n\n\\bibitem{N} It is relevant to mention that the\nconsideration of convection with the stress-free boundary\nconditions\\cite{SZ82,BB84} was the very first analysis of pattern\nformation in the degenerate systems. However, at that time the\ndramatic destabilization of steady patterns obtained in this case\nwas regarded as a peculiarity of the particular problem,\nrather than a generic attribute of the degeneracy.\n\n\n\\bibitem{VN89} V.N. Nikolaevskii, in {\\it Recent Advances in\nEngineering Science}, ed. S. L. Koh and C. G. Speciale, Lecture\nNotes in Engineering, No. 39, Springer-Verlag (1989), p. 210.\n\n\n\\bibitem{BN93}I. A. Beresnev and V. N. Nikolaevskii, Physica D {\\bf 66},\n1 (1993).\n\n\n\\bibitem{BM92} B. A. Malomed, Phys. Rev. A {\\bf 45}, 1009 (1992).\n\n\n\\bibitem{TV96} M. I. Tribelsky and M. G. Velarde, Phys. Rev. E {\\bf\n54}, 4973 (1996).\n\n\n\\bibitem{MT97} M. I. Tribelsky, Inter. Jour. Bifur. Chaos\n{\\bf 7}, 997 (1997).\n\n\n\\bibitem{KM97} I. L. Kliakhandler and B. A. Malomed, Phys. Lett.\nA {\\bf 231}, 191 (1997).\n\n\n\\bibitem{NID} Although the original numerical study\\cite{TT96}\nwas carried out in terms of the order parameter $u(x,t)$, we have\nfound that Eq.~(1) is more convenient for numerical study as it\nexhibits greater numerical stability. The reason is that the\nconservation law $d\\bar v/dt=0$ holding for this equation (where\nthe bar denotes a spatial average) helps to stabilize the code. In\nour simulations the value $\\bar v=0$ was imposed by the initial,\nrandom, conditions. For the space discretization we have found that\n$\\Delta x=0.31$ provides a good enough resolution.\n\n\n\\bibitem{PC89}T. S. Parker and L. O. Chua, {\\it Practical Numerical\nAlgorithms for Chaotic Systems} (Springer-Verlag, 1989).\n\n\n\\bibitem{MT99}At small $\\epsilon$ in this model the two point\ncorrelation function ($\\langle v(x,t)v(x+x',t)\\rangle$, where the\n$\\langle\\ldots\\rangle$ denotes the time average) is generally\noscillatory with a spatial period of the oscillations close to\n$2\\pi$ and a slow decay of its amplitude over a scale $\\xi_{2}$.\nThus, the model has two correlation length scales, with $\\xi_2$\nbeing the larger of the two.\n\n\n\\bibitem{TG99} S. Tajima and H. S. Greenside, \"Microextensive\nChaos of a Spatially Extended System\", March Meeting of APS,\nAtlanta, Georgia (1999).\n"
}
] |
cond-mat0002119
|
Theory of Four Wave Mixing of Matter Waves from a Bose-Einstein Condensate
|
[
{
"author": "Marek Trippenbach$^{\\,1}$"
},
{
"author": "Y.\\ B.\\ Band$^{\\,1}$ and P.\\ S.\\ Julienne$^{\\,2}$"
}
] |
A recent experiment [Deng et al., Nature 398, 218(1999)] demonstrated four-wave mixing of matter wavepackets created from a Bose-Einstein condensate. The experiment utilized light pulses to create two high-momentum wavepackets via Bragg diffraction from a stationary Bose-Einstein condensate. The high-momentum components and the initial low momentum condensate interact to form a new momentum component due to the nonlinear self-interaction of the bosonic atoms. We develop a three-dimensional quantum mechanical description, based on the slowly-varying-envelope approximation, for four-wave mixing in Bose-Einstein condensates using the time-dependent Gross-Pitaevskii equation. We apply this description to describe the experimental observations and to make predictions. We examine the role of phase-modulation, momentum and energy conservation (i.e., phase-matching), and particle number conservation in four-wave mixing of matter waves, and develop simple models for understanding our numerical results.
|
[
{
"name": "fwm_final.tex",
"string": "\\documentstyle[pra,aps,preprint]{revtex}\n%\\documentstyle[pra,aps,twocolumn]{revtex}\n\\include{epsf}\n\\begin{document}\n\\tightenlines\n\\title{Theory of Four Wave Mixing of Matter Waves from a Bose-Einstein\nCondensate}\n\n\\author{Marek Trippenbach$^{\\,1}$, Y.\\ B.\\ Band$^{\\,1}$ and P.\\ S.\\\nJulienne$^{\\,2}$}\n\n\\address{${}^{1}$ Departments of Chemistry and Physics, \\\\ Ben-Gurion\nUniversity of the Negev, Beer-Sheva, Israel 84105 \\\\ ${}^{2}$ Atomic\nPhysics Division, A267 Physics\\\\ National Institute of Standards and\nTechnology, Gaithersburg, MD 20899}\n\n\\maketitle\n\n\\begin{abstract} A recent experiment [Deng et al., Nature 398, 218(1999)]\ndemonstrated four-wave mixing of matter wavepackets created from\na Bose-Einstein condensate. The experiment utilized light pulses to\ncreate two high-momentum wavepackets via Bragg diffraction from a\nstationary Bose-Einstein condensate. The high-momentum components\nand the initial low momentum condensate interact to form a new\nmomentum component due to the nonlinear self-interaction of the\nbosonic atoms. We develop a three-dimensional quantum mechanical\ndescription, based on the slowly-varying-envelope approximation, for\nfour-wave mixing in Bose-Einstein condensates using the time-dependent\nGross-Pitaevskii equation. We apply this description to describe the\nexperimental observations and to make predictions. We examine the role of\nphase-modulation, momentum and energy conservation (i.e., phase-matching),\nand particle number conservation in four-wave mixing of matter waves,\nand develop simple models for understanding our numerical results.\n\\end{abstract}\n\n\\pacs{PACS Numbers: 3.75.Fi, 67.90.+Z, 71.35.Lk}\n\n\\section{Introduction}\n\n \nNonlinear optics has been made possible by the nonlinear nature of the\ninteraction between light and matter and by the development of intense\nlight sources that can probe the nonlinear regime of this interaction. \nNonlinear optical processes include three- and four-wave mixing (4WM)\nprocesses (e.g., second harmonic generation and third harmonic\ngeneration). In 4WM three waves (or light pulses) mix to produce a\nfourth. In this paper we detail our studies of 4WM of coherent matter\nwaves. Trippenbach {\\it et al.}~\\cite{Tripp98} proposed a 4WM\nexperiment using three colliding Bose-Einstein condensate (BEC)\nwavepackets with different momenta. Deng {\\it et al.}~\\cite{Deng}\nsuccessfully demonstrated 4WM in an experiment with three BEC\nwavepackets, which interact in a nonlinear manner to make a fourth BEC\nwavepacket. Here we greatly elaborate on and further develop the\ntheory and describe numerical simulations of the 4WM output that agree\nwell with the experimental measurements of ~\\cite{Deng}.\n\nThe experimental study of nonlinear atom optics is made possible by\nthe advent of Bose-Einstein condensation of dilute atomic\ngases~\\cite{atom_BECs,reviews} and the atom ``laser''~\\cite{Mewesoc},\na source of coherent matter-waves analogous to the output of optical\nlasers. A set of optical light pulses incident on a parent condensate\nwith momentum ${\\bf P}_1 = {\\bf 0}$ can, by Bragg\nscattering~\\cite{Kozuma99}, create two new daughter BEC wavepackets\nwith momenta ${\\bf P}_{2}$ and ${\\bf P}_{3}$. Four-wave mixing in a\nsingle spin-component condensate occurs as a result of the nonlinear\nself-interaction term in the Hamiltonian for a BEC when three such BEC\nwavepackets with momenta ${\\bf P}_1$, ${\\bf P}_2$, and ${\\bf P}_3$\ncollide and interact. The nonlinear self-interaction can generate a\nnew BEC wavepacket with a new momentum ${\\bf P}_4 = {\\bf P}_1 - {\\bf\nP}_2 + {\\bf P}_3$.\n\nThe possibility of nonlinear effects in atom optics has been long\nrecognized~\\cite{Lenz}. Goldstein {\\it et al.}~\\cite{Goldstein95}\nproposed that phase conjugation of matter waves should be possible in\nanalogy to this phenomenon in nonlinear optics, including the case of\nmultiple spin-component condensates~\\cite{Goldstein99}. They\nconsidered the case where a ``probe'' BEC wavepacket interacts with\ntwo counter-propagating ``pump'' wavepackets to generate a fourth that\nis phase conjugate to the probe, where the probe is weak and causes\nnegligible depletion of the pump. Law {\\it et al.}~\\cite{Law98} also\nsuggested analogies between interactions in multiple spin-component\ncondensates and four-wave mixing. Goldstein and\nMeystre~\\cite{Goldstein99b} develop a theory of 4WM in multicomponent\nBECs based on an algebraic angular momentum approach to obtain the\nmodes of the coupled operator equations. Our treatment for a single\nspin-component condensate is based on the time-dependent\nGross-Pitaevskii equation (GPE), which has proved to be highly\nsuccessful in describing the properties of a variety of actual BEC\nexperiments~\\cite{reviews}. Thus, our treatment is for a zero\ntemperature condensate. It also can describe 4WM with or without the\npresence of a trapping potential.\n\nThe nature of 4WM in BEC collisions of matter waves is unlike 4WM for\noptical wavepacket collisions in dispersive media\n~\\cite{Hellwarth,Maker,Yariv}. The nonlinearity in the case of\nBEC is introduced by collisions rather than by interaction with an\nexternal medium, and the momentum and energy constraints imposed are\ndifferent in the two cases. The kinetic energy of massive particle\nwaves is quadratic in the wavevector of the particles and given by\n$(\\hbar{\\bf k})^{2}/2m$, whereas the energy of a photon is linear in\nthe vacuum wavevector of the photon, ${\\bf k}$, and is given by\n$\\hbar c|{\\bf k}|$. Moreover, the momentum of massive particle waves\nis linear in the wavevector of the particles and given by $\\hbar{\\bf\nk}$, whereas for light in a dispersive medium, it is proportional to\nthe product of the frequency of the light, $\\omega = c|{\\bf k}|$ and\nthe refractive index, $n(\\omega)$, where the refractive index depends\nupon frequency (and the propagation direction in non-isotropic media). \nHence, conservation of energy does not in general guarantee\nconservation of momentum in optical 4WM. Clearly, complications\ninvolving the properties of an additional medium does not arise in the\nBEC case. In any case, the creation of new BEC wavepackets in 4WM is\nlimited to cases when momentum, energy and particle number\nconservation are simultaneously satisfied.\n\nIn this paper we develop a general three-dimensional (3D) description\nof four-wave mixing in single-spin-component Bose-Einstein condensates\nusing a mean-field approach similar to the time-dependent GPE, also\nknown as the nonlinear Schr\\\"{o}dinger equation \\cite{reviews}. We\nintroduce the slowly-varying-envelope approximation (SVEA), a very\npowerful tool that not only gives insight into the nature of 4WM but\nalso gives a set of four coupled equations for the four interacting\nBEC waves that are more computationally tractable for numerical\nsimulations of the time-dependent dynamics. Section \\ref{theory}\nexplains the experimental situation we have in mind and develops the\nbasic theoretical methods. Section \\ref{NS} describes the results of\nour numerical calculations and compares these to the NIST experiment\n\\cite{Deng}. Finally, in Sec.~\\ref{conclusions} we present a summary\nand conclusion.\n\n\\section{Theory of Matter-Wave Four-Wave Mixing}\\label{theory}\n\nIn this section we describe the theoretical tools used in our study of\n4WM of matter waves. Section \\ref{SecBragg} reviews how high momentum\ncomponents of a BEC can be formed using optical Bragg pulses to\nprepare the initial configuration for the ``half collision'' event. \nSection \\ref{SecScales} specifies the parameters that describe the\nstrength of the various physical effects that play a role in 4WM:\ndiffraction, potential energy, nonlinear self-energy, and\ncollisions between the different momentum wavepackets. This Section\nalso describes how to transform between 1D, 2D and 3D calculations\ninvolving the GPE. This is important because, without the\nslowly-varying-envelope approximation (SVEA) that we introduce below,\nfull 3D calculations are too computationally expensive to carry out\nfor the actual experimental conditions. Hence, the SVEA must be\nexplicitly checked in 2D against the full GP solution. Section\n\\ref{SecSVEA} describes the details of the SVEA approximation for 4WM.\nThen Section \\ref{simple} introduces a simple estimate for the 4WM\noutput. Finally, Section \\ref{el_scat} shows how the effect of\nelastic scattering between atoms in different momentum wavepackets can\nbe accounted for. This process causes loss of atoms from the\nwavepackets and lowers the 4WM output.\n\nLet us consider three BEC wavepackets moving with central momenta\n${\\bf P}_1$, ${\\bf P}_2$, and ${\\bf P}_3$. Such moving wavepackets\ncan be created, for example, by optically-induced Bragg diffraction of\na condensate \\cite{Kozuma99}. If these three wavepackets overlap\nspatially, the self-energy of the atoms can produce matter-wave\n4WM, just as the third-order Kerr type nonlinearity can produce\noptical 4WM in nonlinear media. One can imagine a number of\nscenarios in which 4WM can occur in matter-wave interactions. One\ncan consider a ``whole collision'' in which three initially\nseparated BEC wavepackets collide together at the same time, or a\n``half collision'' in which the wavepackets are initially formed in\nthe same condensate at (nearly) the same time. Although we considered\nthe ``whole collision'' case in Ref.~\\cite{Tripp98}, the ``half\ncollision'' case is easier to realize experimentally \\cite{Deng} using\nthe above-mentioned Bragg diffraction technique~\\cite{Kozuma99}. In\nwhat follows, we consider only this configuration, in which the three\nwavepackets initially overlap because they have been created as copies\nof the initial condensate. These wavepackets have different\nnon-vanishing central momenta and therefore they fly apart from one\nanother after they have been created.\n\nFig.~\\ref{f1}a shows the basic configuration in momentum space of the\nwavepackets which we consider here. Two daughter condensate\nwavepackets with momenta ${\\bf P}_2$ and ${\\bf P}_3$ are created from\na parent condensate with mean momentum ${\\bf P}_1=0$. Fig.~\\ref{f2}a\nshows these three momenta in the lab frame in which the experiment is\ncarried out at two different times: during the early stage of the\n``half collision'' when they still overlap spatially, and at a later\ntime when they have spatially separated into four distinct\nwavepackets. We let ${\\bf P}_3$ lie along the $x$-axis of the\ncoordinate system, and ${\\bf P}_2$ make some angle $\\theta$ with\nrespect to the $x$-axis. Nonlinear 4WM creates a fourth wavepacket\nwith momentum ${\\bf P}_4={\\bf P}_1 - {\\bf P}_2 + {\\bf P}_3$. We\ndemonstrate below in Sec.~\\ref{SecSVEA} that four-wave mixing of\nmatter waves is only possible if there exists a coordinate frame in\nwhich the mixing is degenerate, that is, all four ${\\bf P}'_i$ values\nin this frame have the same magnitude. Fig.~\\ref{f2}b shows the\ndegenerate frame corresponding to a moving frame with velocity ${\\bf\nV}_{deg} = ({\\bf P}_1 + {\\bf P}_3)/(2m)$, where $m$ is the atomic\nmass. The total momentum is zero in the degenerate frame, and the\nwavepackets move in oppositely moving pairs. The angle\n$\\theta^{\\prime}$ between the vectors ${\\bf P}_2^{\\prime}$ and ${\\bf\nP}_3^{\\prime}$ is arbitrary. In the laboratory frame, the angle\n$\\theta$ is given by $\\theta = \\theta^{\\prime}/2$, and the length of\nthe vector ${\\bf P}_2$ is given by $|{\\bf P}_2| = |{\\bf\nP}_3|\\cos(\\theta)$. Fig.~\\ref{f1}b shows a set of different possible\nvalues of ${\\bf P}_2$.\n\n\\subsection{Bragg Pulse Creation of High Momentum Components}\n\\label{SecBragg}\n\nWe assume that the condensate has only a single spin-component, and\nthat its dynamics can be described by the GPE, which is known to \nprovide an excellent account of condensate properties \\cite{reviews}:\n\\begin{equation}\n i\\hbar \\frac{\\partial\\Psi({\\bf r},t)}{\\partial t} = (T_{{\\bf r}}+\n V({\\bf r},t) + NU_0|\\Psi|^2) \\Psi({\\bf r},t), \\label{GP}\n\\end{equation}\nwhere $T_{{\\bf r}} = \\frac{-\\hbar^2}{2m} \\nabla_{\\bf r}^2$ is the\nkinetic energy operator, $V({\\bf r},t)$ is the external potential\nimposed on the atoms, $NU_{0} = N\\frac{4\\pi a_{0}\\hbar^{2}}{m}$ is the\natom-atom interaction strength that is proportional to the $s$-wave\nscattering length $a_{0}$ (assumed to be positive), $m$ is the atomic\nmass, and $N$ is the total number of atoms. The numerical methods for\nsolving the GPE are described below in Sec.~\\ref{NS}.\n\nFirst, we use the GPE to obtain the ground state condensate in the\ntrapping potential at time $t=0$, $\\Psi({\\bf r},t=0)$. This\ncondensate wavefunction is centered around ${\\bf r}={\\bf 0}$, and\nnormalized to unity. We assume, as is the case in the NIST\nexperiments~\\cite{Deng}, that the trapping potential $V({\\bf r},t)$ is\nturned off at $t=0$ and that the condensate is allowed to evolve under\nthe influence of only the mean-field interaction until time $t_1$. \nThis includes the special case $t_1=0$. We could equally well treat\nthe case of leaving the trap on, and we would obtain similar results. \nEq.~(\\ref{GP}) determines the evolved condensate wavefunction,\n$\\Psi({\\bf r},t_1)$. After this period of free evolution, the Bragg\npulses are applied to create the wavepackets with momenta ${\\bf\nP}_{1}$, ${\\bf P}_{2}$ and ${\\bf P}_{3}$. The momentum differences\n$|{\\bf P}_{i} - {\\bf P}_{j}|$ are much larger than the momentum spread\nof the initial parent BEC wavepacket. The experimental time scale\n$\\delta t$ for creating these wavepackets is short ($\\approx$ 70\n$\\mu$s) compared to the time scale on which the wavepackets evolve.\nThe state at $t_2=t_1+\\delta t$ provides the initial condition for\nsubsequent evolution of these three wavepackets as they undergo\nnonlinear evolution.\n\nThe initial state at $t_2$ immediately after the Bragg pulse sequences\ncan be approximated in a number of ways. In principle one could set\nup a set of coupled GPEs for the ground and excited atomic state\ncomponents and explicitly include the effect of coupling the light\nfield to the excited electronic state. A simpler approach would be to\ncarry out an adiabatic elimination of the excited state and develop an\neffective light-shift potential in which the ground state atoms move. \nIf such approaches are carried out in this case, they show that the\nlight acts as a ``sudden'' perturbation such that each of the\nwavepackets with central momenta ${\\bf P}_{1}$, ${\\bf P}_{2}$ and\n${\\bf P}_{3}$ is to a very good approximation simply a ``copy'' of the\nparent condensate at $t=t_1$ \\cite{Marya}. Thus, the initial\ncondition immediately after the application of the Bragg pulses can be\napproximated as being comprised of three BEC wavepackets,\n\\begin{equation} \n \\Psi({\\bf r},t_2) = \\Psi({\\bf r},t_1) \\sum_{i=1}^{3} f_{i}^{1/2}\n \\exp(i{\\bf P}_{i}\\cdot{\\bf r}/\\hbar), \\label{in_con}\n\\end{equation} \nwhere $f_i=N_i/N$ is the fraction of atoms in wavepacket $i$, and\n$\\sum_{i=1}^{3} f_{i} = 1$ so the norm of $\\Psi$ remains unity.\n\nAfter the formation of the wavepackets with momenta ${\\bf P}_{1}$,\n${\\bf P}_{2}$ and ${\\bf P}_{3}$, the initial wavefunction in\nEq.~(\\ref{in_con}) evolves, and the wavepackets with the different\nmomenta separate. During this separation,the nonlinear term in the\nGPE generates a wavepacket with central momentum ${\\bf P}_{4} = {\\bf\nP}_{1} - {\\bf P}_{2} + {\\bf P}_{3}$, as long as the constraints\ndiscussed in relation to Figs.~\\ref{f1} and \\ref{f2} are\nsatisfied. Energy and momentum are conserved during the wavepacket\nevolution. This can be readily checked by verifying that $dE(t)/dt =\n0$ and $d{\\bf P}(t)/dt = 0$, where\n\\begin{equation} \n E(t) = \\langle \\Psi(t) |(T_{{\\bf r}}+ \\frac{1}{2} U_0|\\Psi|^2)\n |\\Psi(t) \\rangle, \\label{energy} \n\\end{equation} \nis the energy per particle and\n\\begin{equation} \n {\\bf P}(t) = -i\\hbar \\langle \\Psi(t)|\\mbox{\\boldmath $\\nabla$}|\\Psi(t) \n \\rangle \\ ,\n\\label{momentum}\n\\end{equation} \nis the momentum per particle. We have verified numerically that \nenergy and momentum are indeed conserved in our calculations described \nin Section \\ref{NS}.\n\n\\subsection{Characteristic Time Scales, and Dimensionless \nParameters}\\label{SecScales}\n\nIn this subsection we discuss characteristic time scales that can be\nused to estimate the importance of the various effects occurring\nduring the dynamics for a particular set of experimental parameters. \nIt is convenient to use the Thomas--Fermi (TF) \napproximation~\\cite{reviews} to give\nquantitative estimates of the size of the condensate and the time\nscales characterizing the dynamics. In the TF approximation, one\nneglects the kinetic energy operator in the time-independent nonlinear\nSchr\\\"{o}dinger equation,\n\\begin{equation} \n\\mu \\Psi = (T_{{\\bf r}}+ V({\\bf r},t) + NU_0|\\Psi|^2) \\Psi ,\n\\label{TIGP}\n\\end{equation}\nwhere $\\mu$ is the chemical potential, to obtain the following\nanalytical expression for the wavefunction: $|\\Psi({\\bf r})|^2 =\n\\frac{\\mu-V({\\bf r})}{NU_0}$ for ${\\bf r}$ such that $V({\\bf r}) \\le\n\\mu$ and $\\Psi({\\bf r})=0$ otherwise. The TF approximation is valid\nfor sufficiently large numbers of atoms $N$. It is convenient to define\nthe geometric average of the oscillator frequencies for an asymmetric\nharmonic potential as $\\bar{\\omega} =\n(\\omega_x\\omega_y\\omega_z)^{1/3}$. The size of the condensate is then\ngiven by the TF radius $r_{TF} = \\sqrt{2\\mu/(m\\bar{\\omega})}$, where\nthe TF approximation to the chemical potential $\\mu$ is determined by\nthe normalization of the wavefunction to unity and is given by $\\mu =\n\\frac{1}{2} \\left( \\frac{15 U_0 N}{4\\pi}\\right)^{2/5}\n(m\\bar{\\omega}^2)^{3/5}$. Hence, the TF radius $r_{TF}$ scales with\n$N$ as $N^{1/5}$. The size of the TF wavepacket in the $i=$ $x$, $y$, \nand $z$ directions is $r_{TF}(i)= (\\bar{\\omega}/\\omega_i)r_{TF}$.\n\nIn order to estimate the importance of the various terms in the GPE,\nwe set $V=0$ for free wavepacket evolution and rewrite Eq.~(\\ref{GP})\nin terms of characteristic time scales $t_{DF}$ for diffraction, and\n$t_{NL}$ for the nonlinear interaction, in the following manner\n\\cite{Tripp98,Trip1,Trip2}:\n\\begin{equation}\n \\frac{\\partial\\Psi}{\\partial t} = i\\left[ \\frac{r_{TF}^2}{t_{DF}} \\,\n (\\frac{\\partial^2 }{\\partial x^2} + \\frac{\\partial^2\n }{\\partial y^2} + \\frac{\\partial^2 }{\\partial z^2}) - \n \\frac{1}{t_{NL}} \\,\\frac{|\\Psi|^2}{|\\Psi_{m}|^{2}} \\right] \\Psi.\n\\label{GP_reduced}\n\\end{equation}\nThe diffraction time and the nonlinear interaction time are given by\n$t_{DF} = 2m r_{TF}^2/\\hbar$, $t_{NL}=(NU_0 |\\Psi_{m}|^2\n/\\hbar)^{-1}$, respectively. Here $|\\Psi_{m}|^2$ is the maximum value\nof $|\\Psi({\\bf r})|^2$, i.e., $|\\Psi_{m}|^2 = |\\Psi({\\bf 0})|^2$;\nhence in the TF approximation, $t_{NL}^{-1}=\\mu/\\hbar$. The smaller\nthe characteristic time, the larger is the corresponding term in\nthe GPE. We also define the collision duration time $t_{col} =\n(2r_{TF})/v$, where ${\\bf v}=({\\bf P}_{3}-{\\bf P}_{1})/m$ is the\ninitial relative velocity of wavepackets 1 and 3. Thus, $t_{col}$ is\nthe time it takes the wavepackets 1 and 3 to move so that they just\ntouch at their TF radii, and therefore no longer overlap. The ratio\n$t_{col}/t_{NL}$ gives an indication of the strength of the\nnonlinearity during the collision. The larger the ratio of\n$t_{col}/t_{NL}$, the stronger the effects of the nonlinearity during\nthe overlap of the wavepackets. These characteristic times stand in\nthe ratios $t_{DF}:t_{col}:t_{NL} = 1 : \\frac{\\lambda}{2 \\pi r_{TF}} :\n\\frac{r_{TF}}{6 a_{0} N}$, where $\\lambda$ is the De Broglie\nwavelength associated with the wavepacket velocity $v$. Experimental\ncondensates with $t_{col}/t_{NL} \\gg 1$ can be readily achieved. \nThus, the nonlinear term will have time to act while the BEC\nwavepackets remain physically overlapped during a collision. Another\nrelevant time scale in the dynamics is the characteristic condensate\nexpansion time, $t_{exp} = \\bar{\\omega}^{-1}$. In the typical\nexperiments modeled below, $t_{DF} \\gg t_{exp} > t_{col} > t_{NL}$.\n\nIn addition to time scales, there are several natural length scales\nthat are important: the size $r_{TF}$ of the condensate, the scale\n$(\\Delta k)^{-1}$ of phase variation across the parent condensate as\nit expands and develops a momentum spread $\\hbar \\Delta k$ due to the\nmean field potential, and the scale $(k')^{-1}$ of phase variation due to\nthe fast imparted momentum $P'=\\hbar k'$, where $P'$ is the common\nmagnitude of the momentum for the packets in the degenerate frame\n(Fig.~\\ref{f1}). These stand in the relation $(k')^{-1} \\ll\n(\\Delta k)^{-1} \\ll r_{TF}$. The grid spacings in numerical\ncalculations are determined by the necessity to resolve the\nwavefunction on its fastest scale of variation. Thus, using the form\nof Eq.~(\\ref{in_con}) for $\\Psi$ requires a grid smaller than\n$(k')^{-1}$. This requirement limits practical calculations to 2\ndimensions (2D). We will introduce an approximation in the next\nsection that allows three dimensional (3D) calculations by eliminating\nthe rapidly varying phase factors from the equations to be solved.\n\nWe find it convenient to use reduced dimensionless variables to\ncalculate the dynamics. The most commonly used set of reduced\ndimensionless variables in BEC problems involves using ``trap units''\n\\cite{reviews}. Here however, except for determining the initial\nconditions at $t=0$, the trap potential is turned off, and trap units\nare not particularly relevant. Since we do both 2D and 3D\ncalculations, some care is needed in developing a set of units. The\nprimary requirement to simulate 3D experiments with a 2D model is that\nthe relations between the characteristic timescales, $t_{DF}$,\n$t_{col}$ and $t_{NL}$, are as determined by experiment. We have done\nthis by scaling the solution of the $d$--dimensional time-dependent\nGPE by a $d$--dimensional volume so that the coefficient of the\nnonlinear term depends only on the dimension and the chemical\npotential $\\mu$. By scaling the condensate wavefunction as $\\Psi =\n\\bar{\\Psi}/\\sqrt{r_{TF}^{d}}$, the $d$--dimensional time-dependent GPE\nfor a harmonic potential with frequencies $\\omega_j$, $j=1 \\dots d$\ncan be written as\n\\begin{equation}\n i\\hbar\\frac{\\partial\\bar{\\Psi}\\left({\\bf r}\\right)}{\\partial t} =\n -\\frac{\\hbar^{2}}{2m}\\sum_{j=1}^{d}\n \\frac{\\partial^{2}\\bar{\\Psi}}{\\partial x_{j}^{2}} +\n \\left(\\sum_{j=1}^{d}\\frac{1}{2}m\\omega_{j}^{2}x_{j}^{2}\\right)\n \\bar{\\Psi}\\left({\\bf r}\\right) \n + \\left(\\frac{\\pi^{d/2}}{\\Gamma(2+\\frac{d}{2})}\\right)\\mu_{{\\rm TF}}\n \\left|\\bar{\\Psi}\\left({\\bf r}\\right)\\right|^{2}\n \\bar{\\Psi}\\left({\\bf r}\\right) \\ .\n \\label{GPd}\n\\end{equation}\nHere $\\bar{\\Psi}$ is dimensionless for any $d$, and the known\n$\\mu_{TF}$ for the 3D problem can be transferred to an equivalent\ntime-dependent GPE for a 2D calculation. Furthermore, if we\ndefine the reduced unit of length, $x_R$, to be $x_R = r_{TF}$, define\nthe unit of time, $t_R$, such that $ t_R = m x_R^2/(2\\hbar)$, and use\nthe normalization condition: $\\int |\\bar{\\Psi}|^2 \\,\nd^d {\\bf r}/x_R^d = 1$, we preserve the ratios between the most\nimportant time scales of the problem. The nonlinear time scale,\n$t_{NL}$ depends only on $\\mu_{TF}$ and is independent of dimension. The\nspecific relations between the 3D nonlinear coupling parameter\n$U_0^{3D}$ multiplying $\\left|\\bar{\\Psi}\\left({\\bf r}\\right)\\right|^{2}\n\\bar{\\Psi}\\left({\\bf r}\\right)$ in Eq.~(\\ref{GPd}) and $U_0^{1D}$ and\n$U_0^{2D}$, the respective self-energy parameters in 1D and 2D are:\n$U_0^{1D} = \\frac{5}{2\\pi}U_0^{3D}$, and $U_0^{2D} =\n\\frac{15}{16}U_0^{3D}$. These values for $U_0^{d}$ insure that the\nchemical potential $\\mu_{TF}$ (and all the time scales) are the same as in\n3D.\n\n\\subsection{Slowly Varying Envelope Approximation}\\label{SecSVEA}\n\nLet us consider the case when the total wavefunction consists of four\nwavepackets moving with different central momenta ${\\bf P}_i= \\hbar\n{\\bf k}_i, i=1,\\ldots,4$. We write the wavefunction as\n\\begin{equation}\n\\Psi({\\bf r},t) = \\sum_{i=1}^{4} \\Phi_i({\\bf r},t)\\,\\exp{[i({\\bf\nk}_i{\\bf r} - \\omega_i t)]}\\,,\n\\label{SVEApsi}\n\\end{equation}\nin order to separate out explicitly the fast oscillating phase factors\nrepresenting central momentum $\\hbar {\\bf k}_i$ and kinetic energy\n$E_i = \\hbar\\omega_i = \\hbar^2 k_i^2/2m$. The slowly varying\nenvelopes $\\Phi_i({\\bf r},t)$ vary in time and space on much longer\nscales than the phases. The number of atoms in each wavepacket is\n$N_i= N \\int_V|\\Phi_i({\\bf r},t)|^2 d^3{\\bf r}$, and\n$\\sum_{i=1}^{4}N_{i}=N$ is a constant. Although the slowly varying\nenvelope $\\Phi_4({\\bf r},t=0)$ is unpopulated initially, it evolves\nand becomes populated as a result of the 4WM process. If we\nsubstitute the expanded form of the wavefunction in\nEq.~(\\ref{SVEApsi}) into the GPE, collect terms multiplying the same\nphase factors, multiply by the complex conjugate of the appropriate\nphase factors, and neglect all terms that are not phase matched (phase\nmatched terms have stationary phases, do not oscillate, and satisfy\nEqs.~(\\ref{mom_cons}-\\ref{energy_cons}) below), we obtain a set of\ncoupled equations for the slowly varying envelopes $\\Phi_i({\\bf\nr},t)$:\n\\begin{eqnarray} \n\\left( \\frac{\\partial}{\\partial t} + (\\hbar{\\bf k}_i/m) \\cdot\n\\mbox{\\boldmath $\\nabla$} +\n\\frac{i}{\\hbar}(-\\frac{\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) )\n\\right) \\Phi_i({\\bf r},t) &=& -\\frac{i}{\\hbar} N U_0 \n\\sum_{i^*jj^*} \\delta ({\\bf k}_i+{\\bf k}_{i^*}-{\\bf k}_j -{\\bf\nk}_{j^*}) \\times \\nonumber \\\\\n&& \\delta(\\omega_i + \\omega_{i^*} - \\omega_j - \\omega_{j^*})\n \\times \\nonumber \\\\\n&& \\Phi_{j^*}({\\bf r},t) \\Phi_{i^*}^{*}({\\bf r},t) \\Phi_{j}({\\bf r},t) \\ ,\n\\label{SVEA}\n\\end{eqnarray}\nwhere the delta-functions represent Kronecker delta-functions that are\nunity when the argument vanishes. Mixing between different momentum\ncomponents can result from the nonvanishing nonlinear terms in\nEq.~(\\ref{SVEA}), which satisfy the phase matching constraints\nrequired by momentum and energy conservation:\n\\begin{eqnarray}\n{\\bf k}_i + {\\bf k}_{i^*} - {\\bf k}_j - {\\bf k}_{j^*} &=&0,\n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\label{mom_cons} \\\\\nk^2_i + k^2_{i^*} - k^2_j - k^2_{j^*} &=& 0 \\ .\n\\label{energy_cons}\n\\end{eqnarray}\nEach of the indices $i,i^*,j,j^*$ may take any value between 1 and 4. \nEqs.~(\\ref{mom_cons}) and (\\ref{energy_cons}) are automatically\nsatisfied in two cases: (a) $i=i^*=j=j^*$ (all indices are equal), or\n(b) $j=i \\ne j^*=i^*$ (two pairs of equal indices). The corresponding\nterms describe what is called in nonlinear optics cross and self\nmodulation terms respectively. The cross and self phase modulation\nterms do not involve particle exchange between different momentum\ncomponents. In the absence of the trapping potential they modify both\namplitude and phase of the wavepacket through the mean field\ninteraction. Particle exchange between different momentum wavepackets\noccurs only when all four indices in Eq.~(\\ref{SVEA}) are different,\nand conservation of momentum and energy of the atoms participating in\nthe exchange process occurs. A set of {\\it coupled} equations\ninvolving wave mixing between the various momentum components is\ntherefore obtained.\n\nThe momentum conservation of Eq.~(\\ref{mom_cons}) implies ${\\bf k}_i +\n{\\bf k}_{i^*} = {\\bf k}_j + {\\bf k}_{j^*} = $ {\\boldmath $\\kappa$}. \nIt is always possible to construct a special reference frame, which we\ncall the {\\em degenerate frame}, where {\\boldmath $\\kappa$}$=0$. \nConsequently, in this frame ${\\bf k}_i =- {\\bf k}_{i^*}$ and ${\\bf\nk}_j =- {\\bf k}_{j^*}$. In addition energy conservation in\nEq.~(\\ref{energy_cons}) imposes the condition $|{\\bf k}_j| = |{\\bf\nk}_i|$ in the degenerate frame. In this frame all four momenta are\nequal in magnitude and can be divided into two pairs of opposite\nvectors. This explains the use of the conjugated pairs of symbols\n$(i,i^*)$ and $(j,j^*)$ in our notation. The total number of\nparticles, in all wavepackets, is a conserved quantity. The\ngeometrical configuration of the wavepacket momenta in the degenerate\nframe are illustrated in Fig.~\\ref{f2}b. In the figure we see two\npairs of conjugate wavepackets (1,3) and (2,4). All four momenta are\nequal in magnitude and momenta ${\\bf P}_1^{\\prime}$ and ${\\bf\nP}_3^{\\prime}$ are opposite as are the momenta ${\\bf P}_2^{\\prime}$\nand ${\\bf P}_4^{\\prime}$. The angle $\\theta$ depicted in the figure\nis completely arbitrary. However, $\\theta \\approx 0$ is not allowed,\nsince the wavepackets would no longer be distinguishable. \nFig.~\\ref{f1}b shows a range of possible ${\\bf P}_2$ values for\nwavepackets in the lab frame that satisfy the phase-matching\nconditions in Eqs.~(\\ref{mom_cons}) and (\\ref{energy_cons}). These \nconditions only allow $|{\\bf P}_2|=|{\\bf P}_3| \\cos{(\\theta)}$.\n\n4WM can be viewed as a process in which one particle is annihilated in\neach wavepacket belonging to an initially populated pair of\nwavepackets and simultaneously one particle is created in each of two\nwavepackets of another pair, one of which is initially populated and\nthe other (wavepacket 4) is initially unpopulated. Hence, using\nFig.~\\ref{f2}b in the moving degenerate frame, 4WM removes one atom\nfrom each of the ``pump'' wavepackets 1 and 3, and places one atom in\nthe ``probe'' wavepackets 2 and one atom in the 4WM output wavepacket\n4. This picture is a consequence of the nature of the nonlinear terms\nin the four SVEA equations. It is this bosonic stimulation of\nscattering that mimics the stimulated emission of photons from an\noptical nonlinear medium.\n\nThe full SVEA equations for 4WM are explicitly given by:\n\\begin{eqnarray}\n&& \\left( \\frac{\\partial}{\\partial t} \n+ (\\hbar {\\bf k_1}/m) \\cdot\\mbox{\\boldmath $\\nabla$} \n+ \\frac{i}{\\hbar}(\\frac{-\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) )\n\\right) \\Phi_1({\\bf r},t) = \\nonumber \\\\\n&& -\\frac{i}{\\hbar} N U_0 \n(|\\Phi_1|^2 +2|\\Phi_2|^2 + 2|\\Phi_{3}|^2 + 2|\\Phi_{4}|^2) \\Phi_1 \n- \\frac{i}{\\hbar} N U_0\\Phi_4 \\Phi_2 \\Phi_3^* \\ ,\n\\label{SVEA1}\n\\end{eqnarray}\n\\begin{eqnarray}\n&& \\left( \\frac{\\partial}{\\partial t} \n+ (\\hbar {\\bf k_2}/m) \\cdot \\mbox{\\boldmath $\\nabla$} \n+ \\frac{i}{\\hbar}(\\frac{-\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) ) \\right)\n\\Phi_2({\\bf r},t) = \\nonumber \\\\\n&& -\\frac{i}{\\hbar} N U_0 \n(|\\Phi_2|^2 +2|\\Phi_1|^2 + 2|\\Phi_{3}|^2 + 2|\\Phi_{4}|^2) \\Phi_2 \n- \\frac{i}{\\hbar} N U_0\\Phi_4^* \\Phi_1 \\Phi_3 \\ ,\n\\label{SVEA2}\n\\end{eqnarray}\n\\begin{eqnarray}\n&& \\left( \\frac{\\partial}{\\partial t} \n+ (\\hbar {\\bf k_3}/m) \\cdot \\mbox{\\boldmath $\\nabla$} \n+ \\frac{i}{\\hbar}(\\frac{-\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) ) \\right)\n\\Phi_{3}({\\bf r},t) = \\nonumber \\\\\n&& -\\frac{i}{\\hbar} N U_0 \n(|\\Phi_3|^2 +2|\\Phi_1|^2 + 2|\\Phi_{2}|^2 + 2|\\Phi_{4}|^2) \\Phi_3 \n- \\frac{i}{\\hbar} N U_0\\Phi_4 \\Phi_1^* \\Phi_2 \\ ,\n\\label{SVEA3}\n\\end{eqnarray}\n\\begin{eqnarray}\n&& \\left( \\frac{\\partial}{\\partial t} \n+ (\\hbar {\\bf k_4}/m) \\cdot \\mbox{\\boldmath $\\nabla$} \n+ \\frac{i}{\\hbar}(\\frac{-\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) ) \\right)\n\\Phi_{4}({\\bf r},t) = \\nonumber \\\\\n&& -\\frac{i}{\\hbar} N U_0 \n(|\\Phi_4|^2 +2|\\Phi_1|^2 + 2|\\Phi_{2}|^2 + 2|\\Phi_{3}|^2) \\Phi_4 \n- \\frac{i}{\\hbar} N U_0\\Phi_1 \\Phi_2^* \\Phi_3 \\ .\n\\label{SVEA4}\n\\end{eqnarray}\nThe left hand side of these equations describes the motion of the\nwavepackets due to their kinetic and potential energies. The right\nhand side describes the effect of the phase matched nonlinear\ninteraction terms. The last term on the right hand side of each of\nthe SVEA equations is a source term which either creates or destroys\natoms in the wavepacket being propagated. The other terms on the\nright hand side of the equations account for the self- and cross-phase\nmodulation. These phase modulation terms provide an effective\npotential for each wavepacket that accelerates the atoms in it and\nmodifies its internal momentum distribution.\n\nBefore we propagate the SVEA equations, the initial wavefunction of\nthe parent condensate is determined using the time-dependent GPE.\nFirst, the propagation is in imaginary time to obtain the initial\neigenstate in the presence of the magnetic potential. Then, after\nturning off the magnetic potential, the free evolution in the absence\nof a trapping potential is calculated to provide the initial condition\nin Eq.~(\\ref{in_con}). This free evolution causes a spatially varying\nphase to develop across the condensate as it expands in the absence of\nthe trapping potential. Given the initial condition, the SVEA\nequations can be used to propagate the envelope function of each\nwavepacket, using the same numerical method used to propagate the\nordinary time-dependent GPE.\n\n\n\\subsection{Simple Approximations and Scaling with $N$} \\label{simple}\n\nAn estimate of the number of atoms that will be transferred to the 4WM\nwavepacket can be developed as follows. To get the small signal\ngrowth at early times, multiply both sides of the dynamical equation \nfor the rate of change of $\\Phi_{4}$ , where for simplicity we keep \nonly the 4WM term on the right hand side of the equation,\n\\begin{equation}\n \\frac{\\partial\\Phi_4}{\\partial t} = -\\frac{i}{\\hbar} N U_0 \\Phi_1\n \\Phi_2^* \\Phi_3 \\ , \\label{approx1}\n\\end{equation}\nby a small time increment $\\delta t$ to get the growth $\\delta\n\\Phi_{4}$ in $\\Phi_{4}$ during $\\delta t$:\n\\begin{equation}\n \\delta \\Phi_{4} \\approx -i (f_1f_2f_3)^{1/2} \\frac{NU_0}{\\hbar}\n |\\Psi|^{2} \\Psi \\delta t \\approx -i (f_1f_2f_3)^{1/2} \\frac{\\delta\n t}{t_{NL}} \\Psi \\,.\n\\end{equation}\nHere $f_i=N_i/N$ is the initial fraction of atoms in wavepacket $i$, \nand we assume that $\\Phi_i=f_i^{1/2}\\Psi$ at early times, because the\nthree wavepackets initially satisfy this relation. Since most of the\ngrowth takes place in the center of the packets where $\\Psi$ is the\nlargest, the factor $ N U_0|\\Psi|^2/\\hbar$ is approximated by\n$1/t_{NL} = N U_0|\\Psi({\\bf 0})|^2/\\hbar$. Upon squaring this\nequation, and integrating over all space, the total growth in the 4WM\noutput $\\delta f_4$ is\n\\begin{equation}\n \\delta f_4 = \\frac{\\delta N_4}{N} \\approx f_1f_2f_3 \\left\n (\\frac{\\delta t}{t_{NL}} \\right )^2 \\,. \\label{EarlyTime}\n\\end{equation}\nThus, the 4WM signal should grow quadratically at early times.\nIf we take $\\delta t$ to be the total interaction time $t_{col}$ \ndefined in Section \\ref{SecScales}, then an estimate of the total \n4wm output fraction is\n\\begin{equation}\n f_4 = \\frac{N_4(t_{col})}{N} \\approx f_1f_2f_3 \\left\n (\\frac{t_{col}}{t_{NL}} \\right )^2 \\,.\n\\end{equation} \nThis should be an upper bound on the 4WM output, since the mutual\ninteraction of the packets due to the self- and cross-phase modulation\nterms (the self- and cross-interaction energy terms), and their\nseparation from one another when $t \\approx t_{col}$, will lower the\noutput. Using the TF approximation, $1/t_{NL}=\\mu/\\hbar \\sim N^{2/5}$\nand $t_{col}=2r_{TF}/v \\sim N^{1/5}$. Thus, the output fraction\n$\\frac{N_4}{N} \\sim (N^{1/5}N^{2/5})^2$ scales as $N^{6/5}$. This\nscaling, which was discussed in reference \\cite{Deng}, will be checked\nin our numerical calculations below.\n\n\n\\subsection{Elastic scattering loss}\\label{el_scat}\n\nAtoms from two {\\it different} momentum wavepackets can undergo\n$s$-wave elastic scattering that removes the atoms from the packets\nand scatters them into $4\\pi$ steradians \\cite{BTBJ}. This becomes\nimportant when the mean-free-path $\\ell_{mfp}$ becomes comparable to\nor smaller than the condensate size, $r_{TF}$. The mean-free-path is\n$\\ell_{mfp} = (\\sigma {\\bar n})^{-1}$, where $\\sigma = 8\\pi a_0^2$ is\nthe elastic scattering cross section and ${\\bar n}$ is the mean\ndensity. Profuse elastic scattering of this type has been recently\nobserved \\cite{SK-K}. This mechanism can also affect the 4WM process\nsince loss of atoms from the moving packets reduce the nonlinear\nsource terms in the SVEA equations. Although the cloud of elastically\nscattered atoms can not be simply described by the mean-field picture,\nthe loss of atoms from the wavepackets due to this elastic scattering\nmechanism can be described in terms of the SVEA. This is because each\nmomentum component is treated separately, and the loss terms due to\nelastic scattering can be added to the SVEA equations.\n\nThe elastic scattering loss is incorporated by adding loss terms to\nthe right hand side of the envelope equations in the form of\nimaginary potentials that are proportional to the density of the\n``other'' momentum component involved in the elastic scattering. The\nfull SVEA equations for 4WM, including the effects of elastic\nscattering loss \\cite{BTBJ}, are given by:\n\\begin{eqnarray}\n&& \\left( \\frac{\\partial}{\\partial t} \n+ (\\hbar{\\bf k_1}/m) \\cdot \\mbox{\\boldmath $\\nabla$}\n+ \\frac{i}{\\hbar}(\\frac{-\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) ) \\right)\n\\Phi_1({\\bf r},t) = \\nonumber \\\\\n&& -\\frac{i}{\\hbar} N U_0 \n(|\\Phi_1|^2 +2|\\Phi_2|^2 + 2|\\Phi_{3}|^2 + 2|\\Phi_{4}|^2) \\Phi_1 \n- \\frac{i}{\\hbar} N U_0\\Phi_4 \\Phi_2 \\Phi_3^* \\nonumber \\\\\n&&- \\frac{(\\hbar|{\\bf k}_1-{\\bf k}_2|/m) \\sigma N}{2} |\\Phi_2|^2 \\Phi_1 \n- \\frac{(\\hbar|{\\bf k}_1-{\\bf k}_3|/m) \\sigma N}{2} |\\Phi_3|^2 \\Phi_1 \n- \\frac{(\\hbar|{\\bf k}_1-{\\bf k}_4|/m) \\sigma N}{2} |\\Phi_4|^2 \\Phi_1 \\ ,\n\\label{SVEA1el}\n\\end{eqnarray}\n\\begin{eqnarray}\n&& \\left( \\frac{\\partial}{\\partial t} \n+ (\\hbar{\\bf k_2}/m) \\cdot \\mbox{\\boldmath $\\nabla$} \n+ \\frac{i}{\\hbar}(\\frac{-\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) ) \\right)\n\\Phi_2({\\bf r},t) = \\nonumber \\\\\n&& -\\frac{i}{\\hbar} N U_0 \n(|\\Phi_2|^2 +2|\\Phi_1|^2 + 2|\\Phi_{3}|^2 + 2|\\Phi_{4}|^2) \\Phi_2 \n- \\frac{i}{\\hbar} N U_0\\Phi_4^* \\Phi_1 \\Phi_3 \\nonumber \\\\\n&&- \\frac{(\\hbar|{\\bf k}_2-{\\bf k}_2|/m) \\sigma N}{2} |\\Phi_1|^2 \\Phi_2 \n- \\frac{(\\hbar|{\\bf k}_2-{\\bf k}_3|/m) \\sigma N}{2} |\\Phi_3|^2 \\Phi_2 \n- \\frac{(\\hbar|{\\bf k}_2-{\\bf k}_4|/m) \\sigma N}{2} |\\Phi_4|^2 \\Phi_2 \\ ,\n\\label{SVEA2el}\n\\end{eqnarray}\n\\begin{eqnarray}\n&& \\left( \\frac{\\partial}{\\partial t} \n+ (\\hbar{\\bf k_3}/m) \\cdot \\mbox{\\boldmath $\\nabla$} \n+ \\frac{i}{\\hbar}(\\frac{-\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) ) \\right)\n\\Phi_{3}({\\bf r},t) = \\nonumber \\\\\n&& -\\frac{i}{\\hbar} N U_0 \n(|\\Phi_3|^2 +2|\\Phi_1|^2 + 2|\\Phi_{2}|^2 + 2|\\Phi_{4}|^2) \\Phi_3 \n- \\frac{i}{\\hbar} N U_0\\Phi_4 \\Phi_1^* \\Phi_2 \\nonumber \\\\\n&&- \\frac{(\\hbar|{\\bf k}_3-{\\bf k}_1|/m) \\sigma N}{2} |\\Phi_1|^2 \\Phi_3 \n- \\frac{(\\hbar|{\\bf k}_3-{\\bf k}_2|/m) \\sigma N}{2} |\\Phi_2|^2 \\Phi_3 \n- \\frac{(\\hbar|{\\bf k}_3-{\\bf k}_4|/m) \\sigma N}{2} |\\Phi_4|^2 \\Phi_3 \\ ,\n\\label{SVEA3el}\n\\end{eqnarray}\n\\begin{eqnarray}\n&& \\left( \\frac{\\partial}{\\partial t} \n+ (\\hbar{\\bf k_4}/m) \\cdot \\mbox{\\boldmath $\\nabla$} \n+ \\frac{i}{\\hbar}(\\frac{-\\hbar^{2}}{2m}\\nabla^{2} + V({\\bf r},t) ) \\right)\n\\Phi_{4}({\\bf r},t) = \\nonumber \\\\\n&& -\\frac{i}{\\hbar} N U_0 \n(|\\Phi_4|^2 +2|\\Phi_1|^2 + 2|\\Phi_{2}|^2 + 2|\\Phi_{3}|^2) \\Phi_4 \n- \\frac{i}{\\hbar} N U_0\\Phi_1 \\Phi_2^* \\Phi_3 \\nonumber \\\\\n&&- \\frac{(\\hbar|{\\bf k}_4-{\\bf k}_1|/m) \\sigma N}{2} |\\Phi_1|^2 \\Phi_4 \n- \\frac{(\\hbar|{\\bf k}_4-{\\bf k}_2|/m) \\sigma N}{2} |\\Phi_2|^2 \\Phi_4 \n- \\frac{(\\hbar|{\\bf k}_4-{\\bf k}_3|/m) \\sigma N}{2} |\\Phi_3|^2 \\Phi_4 \\ .\n\\label{SVEA4el}\n\\end{eqnarray}\nThere are three elastic scattering loss terms for each SVE momentum\ncomponent $\\Phi_i$ arising from the interaction of each momentum\ncomponent with the other three momentum components. The factor of\n$\\frac{1}{2}$ in the loss terms is due to the fact that these are\nequations for the amplitudes, not the densities.\n\nThe density dependence of the elastic scattering loss terms is\nidentical to that of the mean-field interaction terms since both terms\nare due to elastic scattering. It is of interest to compare the\nstrength (size of the coefficient) of the loss term due to elastic\nscattering with the nonlinear term in the GPE. The nonlinear term has\na coefficient $U_0 / \\hbar = 4\\pi \\hbar a_0/m$, whereas the loss term\nfor interaction of packets $i$ and $j$ has a coefficient $\\frac{1}{2}\nv \\sigma = 4\\pi \\hbar |{\\bf k}_i-{\\bf k}_j| a_0^2/m$, where $v$ is the\nrelative velocity. The ratio ${\\cal R}= (\\frac{1}{2} v \\sigma)/(U_0 /\n\\hbar)$ of loss to mean-field terms for packets 1 and 3 in\nFig.~\\ref{f1} is\n\\begin{equation}\n {\\cal R} = 2|{\\bf k}_1| a_0 \\ . \n \\label{ratio}\n\\end{equation} \nThis ratio is about $0.06$ for the NIST 4WM experiment \\cite{Deng}.\n\n\\section{Numerical Simulations} \\label{NS}\n\n\\subsection{Experimental Configuration}\n\nIn the NIST experiment \\cite{Deng}, the initial sodium $F, M_{F} = 1,\n-1$ condensate is comprised of magnetically confined atoms in a TOP\n(time-orbiting-potential) trap without a discernible non-condensed\nfraction. The trap is adiabatically expanded to reduce the trap\nfrequencies in the $x$, $y$ and $z$ directions to 84, 59 and 42 Hz (the\nfrequency ratios are $\\omega_x:\\omega_y:\\omega_z = 1:1/\\sqrt{2}:1/2$). \nAfter adiabatic expansion, the trap is switched off by removing the\nconfining magnetic fields. The condensate freely expands during a\ndelay time $t_1=600$ $\\mu$s, after which a sequence of two Bragg\npulses of $589$ nm wavelength creates the two moving wavepackets 2 and\n3. Each 30 $\\mu$s Bragg pulse is composed of two linearly polarized\nlaser beams detuned from the $3S_{1/2}, F = 1, M_{F} = -1\n\\,\\rightarrow \\, 3P_{3/2}, F=2, M_{F} = 2$ transition by about\n$\\Delta/2\\pi = -2$ GHz to suppress spontaneous emission and scattering\nof the optical waves by the atoms. The frequency difference between\nthe two laser beams of a single Bragg pulse is chosen to fulfill a\nfirst-order Bragg diffraction condition that changes the momentum\nstate of the atoms without changing their internal state. The first\nBragg pulse is composed of two mutually perpendicular laser beams of\nfrequencies $\\nu_{\\alpha}$ and $\\nu_{\\beta}= \\nu_{\\alpha}- 50$ kHz,\nand wavevectors and ${\\bf k}_{\\alpha} = k \\hat{{\\bf x}}$ and\n$k_{\\beta} = k \\hat{{\\bf y}}$. This pulse sequence causes a fraction\n$f_2$ of the BEC atoms to acquire momentum ${\\bf P}_{2} = \\hbar({\\bf\nk}_{\\alpha}-{\\bf k}_{\\beta}) = \\hbar k (\\hat{{\\bf x}} + \\hat{{\\bf\ny}})$. A second set of Bragg pulses is applied 20 ms after the end of\nthe first Bragg pulse sequence. This pulse is composed of two\ncounter-propagating laser beams with frequencies $\\nu_{\\alpha}$ and\n$\\nu_{\\beta}= \\nu_{\\alpha}- 100$ kHz, and wavevectors and ${\\bf\nk}_{\\alpha} = k \\hat{{\\bf x}}$ and ${\\bf k}_{\\beta} = -k \\hat{{\\bf\nx}}$. This pulse sequence causes a fraction $f_3$ of the BEC atoms to\nacquire momentum ${\\bf P}_{3} = \\hbar({\\bf k}_{\\alpha}-{\\bf\nk}_{\\beta}) = 2\\hbar k \\hat{{\\bf x}}$. Thus, there are three initial\ncondensate wavepackets with momenta ${\\bf P}_{1} = {\\bf 0}$, ${\\bf\nP}_{2}$ and ${\\bf P}_{3}$ as shown in Fig.~\\ref{f1}. The respective\nwavepacket populations, $f_1=1-f_2-f_3$, $f_2$, and $f_3$, have a\ntypical ratio $f_1:f_2:f_3 = 7:3:7$.\n\nThe number of atoms could be varied between around $3\\times10^5$ and\n$3\\times 10^6$. As a typical example, we take $N=1.5\\times10^6$ atoms\nin the trap. Taking $a_0=2.8$ nm \\cite{Tiesinga96}, the nonlinear\ntime is $t_{NL} = 96.2$ $\\mu$s. The Thomas Fermi radius is $r_{TF} =\n20.3$ $\\mu$m. Since the separation velocity defined in Section\n\\ref{SecScales} is $v=0.0691$ m/s for light of wavelength $589$ nm,\nthe physical separation time $t_{col} =\\frac{2 r_{TF}}{v} = 687$\n$\\mu$s in the NIST experiment, and indeed is longer than the nonlinear\ntime. The characteristic condensate expansion time, $t_{exp} =\n\\bar{\\omega}^{-1} = 1.89$ ms for a trap with\n$\\bar{\\omega}=2\\pi\\frac{84}{\\sqrt{2}}$ s$^{-1}$. The characteristic\ndiffraction time $t_{DF} = 2m r_{TF}^2/\\hbar = 300$ ms provides by far\nthe longest time scale in the dynamics. Thus, there is negligible\ndiffraction on the time scale of the experiment.\n\n\\subsection{Simulations of the NIST Experiments}\\label{results}\n\n\nOur solution to the time-dependent GPE uses a standard split-operator\nfast Fourier transform method to propagate an initial state forward in\ntime\\cite{SplitFFT}. The initial state $\\Psi({\\bf r},t=0)$ of the\ncondensate in the trap is found by iteratively propagating in\nimaginary time. Fig.~\\ref{f3} shows examples of a 3D\nparent condensate wavefunction $\\Psi(x,y,z,t)$ for two different\ntimes. The $t=0$ solution shows the wavefunction in the harmonic\ntrap, and the $t = t_1=600$ $\\mu$s solution shows the wavefunction\nafter 600 $\\mu$s of free evolution without a trap potential. Although\nthe $t=0$ wavefunction in Fig.~\\ref{f3}a has a constant phase (taken\nto be 0), it is apparent from Fig.~\\ref{f3}b that the evolution leads\nto the development of phase modulation across the condensate, i.~e.,\nthe wavefunction develops a spatially dependent phase, and therefore\nan imaginary part of the wavefunction. This is due to the evolution\nof the condensate under the influence of the mean field term, $N U_0\n|\\Psi({\\bf r},t)|^2$, when the trapping potential is no longer\npresent. An analytic form for the spatially dependent phase which\nevolves can be obtained in the Castin-Dum model \\cite{CD}. As we show\nbelow, this phase modulation is important for 4WM. There is very little\nphysical expansion of the condensate after 600 $\\mu$s, since the\ncondensate densities $|\\Psi({\\bf r},t)|^2$ are nearly the same for the\nwavefunctions in Figs.~\\ref{f3}a and \\ref{f3}b. However,\nFig.~\\ref{f4} shows that the acceleration due to the mean field is\nalready quite evident in the momentum distribution at $t=600$ $\\mu$s,\nwhich is much broader than that at $t=0$. The two peaks near $k=\\pm 5\nr_{TF}^{-1}$ in the $t=t_1=600$ $\\mu$s distribution indicate the\nformation of accelerated condensate particles which will lead to\ncondensate expansion at later times.\n\nOur treatment for applying Bragg pulses uses the model given by\nEq.~(\\ref{in_con}). This approximation neglects detailed dynamics\nduring the application of the Bragg pulses. Each initial wavepacket\n$i$ at time $t_2$ after the Bragg pulses is a copy of the parent\ncondensate wavefunction at $t=t_1$ with population fraction\n$f_i=N_i/N$. Unless stated otherwise, we will always use the ratio\n$f_1:f_2:f_3 = 7:3:7$ of population fractions as typical of the NIST\nexperiment \\cite{Deng}. We let the three BEC wavepackets evolve for\n$t > t_2 \\approx t_1$ using three different versions of the\ntime-dependent GPE. Two of them are 2D versions, and one is the\n3D-SVEA version. The 2D-full version uses the GPE, Eq.~(\\ref{GP}), to\nevolve the initial state $\\Psi$ in Eq.~(\\ref{in_con}). The 2D-SVEA\nversion uses the SVEA form in Eqs.~(\\ref{SVEA1})-(\\ref{SVEA4}) for the\nevolution. A typical 2D calculation used a grid of discrete $x,y$\npoints within a box $5r_{TF}$ wide in the $x$ and $y$ directions\ncentered on $x=y=0$. In order to resolve the rapid phase variations\ndue to the $e^{i({\\bf k} \\cdot {\\bf r})}$ factor, the 2D-full\ncalculation required an $x,y$ grid of up to $4096\\times 4096$ points. \nOn the other hand, the 2D-SVEA only requires a $128\\times 128$ $x,y$\ngrid to achieve comparable accuracy. The 3D-SVEA calculations added a\n$4r_{TF}$ wide box in the $z$ direction, and an $x,y,z$ grid of\n$128\\times 128\\times 64$ was sufficient.\n\nFig.~\\ref{f5} compares the 4WM output fraction $f_4(t)\\equiv N_4(t)/N$\nfor the three different types of calculation for the case of\n$N=1.5\\times 10^6$ atoms. The 2D-full and 2D-SVEA calculations give\nthe same results within numerical accuracy and can not be\ndistinguished on the graph. We take this to be a strong justification\nof the SVEA, and a strong indication that it will be equally\ntrustworthy in the 3D calculations. In both 2D and 3D cases, the\noutput grows quadratically at early time, as predicted by\nEq.~(\\ref{EarlyTime}). The arrows indicate the characteristic\nnonlinear time $t_{NL}$ and the collision time $t_{col}$. In\naddition, the figure shows $t_{col}(x)=t_{col}/\\sqrt{2}$. The latter\nis the time it takes wavepackets 1 and 2 to move so that they just\ntouch at their Thomas-Fermi radii in the $x$ direction. At that time\nwavepackets 1 and 2 no longer have significant overlap with each\nother, although they still have some overlap with wavepacket 3. As\nthe wavepackets begin to move apart, the output saturates near $t-t_2\n\\approx t_{col}(x)/2$ and approaches its final value when $t - t_{2}\n\\approx t_{col}$. There is a significant difference between the\n3D-SVEA and 2D-SVEA output fraction. The 4WM output is lower for the\n3D case. This is because the nonlinear 4WM process depends on the\nspatial overlap of the moving wavepackets. The packets are not as\nwell-overlapped geometrically in 3D as in the 2D model. Henceforth,\nall our calculations are 3D-SVEA ones, unless stated otherwise.\n\nFig.~\\ref{f6} shows a sequence of contour images of the time evolution\nof the wavepackets from the time the trap is turned off at $t=0$ to\nthe time of separation of the four wavepackets. The contours show the\n$z$-integrated column density, $\\sum_{i=1}^4 \\int \\Phi_i(x,y,z,t)|^2\ndz$, from the 3D-SVEA calculation. (The constructive and destructive\ninterference fringes in the wavepacket overlap region due to the\n$e^{i{\\bf k} \\cdot {\\bf r}}$ phase factors is not shown since it would\nrequire very high resolution to represent it with sufficient\naccuracy). Panel (a) shows the eigenstate density in the harmonic\ntrap. Panel (b) shows the wavepacket at $t = t_2$ just after the\nBragg pulses have fired. Since there is negligible expansion in the\ndensity profile during the initial 600 $\\mu$s of free evolution, the\nwavepacket is very similar to that in panel (a). However, we learned\nfrom Fig.~\\ref{f3} that a phase modulation has developed across the\nwavepacket. This does not show up in the density profile. Panel (c)\nfor $t-t_2 =190$ $\\mu$s indicates some initial motion by the moving\nwavepackets. In panel (d) the spread of the three wavepackets due to\ntheir different momenta is evident, and in panel (e) the separation of\nthe 4WM wavepacket is clearly apparent. Panel (e) shows the four\nwavepackets after almost complete separation at $t-t_2 =760$ $\\mu$s,\nwhich is larger than $t_{col} = 687 \\mu$s.\n\nFig.~\\ref{f7} compares the output fraction $N_4(t)/N$ versus time for\nthree different initial total atom numbers, $N=0.2\\times 10^6$,\n$1.5\\times 10^{6}$ and $5.0\\times 10^{6}$, and $t_1 = 600$ $\\mu$s. \nAgain, at early times the quadratic dependence of the fraction as a\nfunction of time is clearly evident. After a quadratic rise at early\ntime, the output saturates and even undergoes oscillations before\nfinally settling down to a final value when $t > t_{col}$. The\noscillations of $N_4(t)/N$ in time develop and become more pronounced\nas the initial number of atoms increases. These are due to\nback-transfer from the $i=$ 2 and 4 packets to the $i=$ 1 and 3\npackets due to the mutual coupling between the packets. A closer\nexamination of the detailed time evolution shows that the transfer\noccurs on the trailing edge of the wavepackets where they are still\nsubstantially overlapped. When $N$ is large enough, the wavepackets\nexperience significant distortion in shape by the time they separate. \nThe output fraction $N_4(t)/N$ clearly increases with $N$.\n\nFig.~\\ref{f8} shows the output fraction $N_4(t)/N$ versus time for\n$1.5\\times 10^{6}$ atoms for four different values of the free\nevolution time $t_1 = 0$ $\\mu$s, 600 $\\mu$s, 1200 $\\mu$s, and 1800\n$\\mu$s. The self-phase modulation resulting from the nonlinear\nself-energy interaction reduces the 4WM output as $t_1$ increases. \nThis is analogous to the destruction of third harmonic generation due\nto self- and cross-phase modulation in nonlinear optics \\cite{Band90},\nand occurs because the phase modulation destroys the phase matching\nthat is necessary for 4WM to develop. For $t > t_{col}$,\nthe number of atoms in the different wavepackets no longer change,\nsince the wavepackets are well separated (exchange of the number of\nbosonic atoms between wavepackets can no longer occur when the terms\nin the dynamical equations responsible for 4WM vanish). From these\ncalculations it seems clear that 4WM should be much stronger if the\ntrap is left on instead of being turned off. These calculations\nindicate that the 4WM output of the NIST experiment \\cite{Deng} might\nbe as much as a factor of two higher if there had not been 600 $\\mu$s\nof free evolution before the Bragg pulses were applied.\n\nWe expect the 4WM output will be larger if the wavepackets stay\ntogether for a longer interaction time $t_{col}$. The interaction\ntime can be changed by changing the velocity of the wavepackets. \nFig.~\\ref{f9} plots $N_{4}(t)/N$ versus time for $1.5\\times 10^6$\natoms for the original case shown in Figs.~\\ref{f7} and \\ref{f8} and\nfor two new cases where the interaction times are changed by factors\nof 0.7 and 2. This is achieved in the code by scaling the momentum\nwavevectors by factors of $1/0.7$ and $1/2$ respectively. Our\ncalculations show that the 4WM output is reduced by a factor of 0.6 in\nthe first case and increased by a factor of 2 in the second. In\nprinciple, velocities of the wavepackets can be controlled by changing\nthe frequencies and angle of the two Bragg pulses that create an\noutcoupled wavepacket \\cite{Kozuma99}. Thus, some degree of control\nover the 4WM output should be possible by varying the interaction\ntime.\n\nFig.~\\ref{f10} shows $f_3(t)$ and $f_4(t)$ for the case of a weak\n$i=2$ ``probe'' with initial population fraction $0.001$ incident on\ntwo strong $i=$ 1 and 3 ``pump'' wavepackets with population fractions\n0.4995. This is analogous to the phase conjugation process envisioned\nin reference \\cite{Goldstein95}. Here bosonic stimulation, which\nremoves 2 atoms from the ``pump'' packets 1 and 3 and puts them in\npackets 2 and 4, results in a strong amplification of packet 2, which\ngrows in atom number 8-fold as the 4WM signal grows.\n\nFig.~\\ref{f11} shows 4WM output fraction $N_4/N$ after the\nhalf-collision is over ($t>t_{col}$) as a function of $N$, plotted in\na log--log plot. The figure shows the results for both the 2D-SVEA\nand 3D-SVEA calculations. The dashed lines show the 4WM output for\nsmall $N$ scales well with $N^{6/5}$, as estimated from the simple\nmodel in Section \\ref{simple}. The scaling with $N^{6/5}$ for small\n$N$ is clearly evident in both 2D and 3D results. The latter is\nuniformly lower than the former, due to the smaller overlap of the\nwavepackets in 3D because of geometrical reasons, but saturates a\nlittle more slowly with increasing $N$ than the former. At the higher\n$N$ values typical of Na condensates, this scaling from the simple\nmodel seriously overestimates the output, which begins to saturate\nwith increasing $N$.\n\nFig.~\\ref{f12} shows three curves giving the fraction of atoms in the\n4WM output wavepacket as a function of the initial total number of\natoms $N$ as calculated by (1) 2D-SVEA and (2) 3D-SVEA simulations\nwithout including elastic scattering loss, and as calculated by (3) a\n3D-SVEA simulation including elastic scattering loss. In one set of\ncalculations we used a ratio of atoms in the three initial wavepackets\nof $N_1:N_2:N_3 = 7:3:7$. These calculations produce the three smooth\ncurves in Figure \\ref{f12}. In another set of calculations, we used\nthe measured final fractions from the NIST experiment \\cite{Deng} to\ndetermine the initial ratios $N_1:N_2:N_3$, rather than taking the\nnominal values $7:3:7$. The open circles in Figure \\ref{f12}, which\nno longer fall on a smooth line, show the 3D-SVEA without elastic\nscattering for these cases with experimental scatter in initial\nconditions. The relatively small deviation of the points from the\nsolid curve for the 3D-SVEA without elastic scattering show that the\ncalculations with the $7:3:7$ ratio is useful for generating a smooth\ncurve to compare to experimental data.\n\nThe effect of including loss from the BEC wavepackets due to elastic\nscattering collisions was modeled using\nEqs.~(\\ref{SVEA1el})-(\\ref{SVEA4el}). The 4WM output reduction in\nFigure \\ref{f12} due to elastic scattering ranges from 6 per cent to\n16 per cent in going from $10^{5}$ to $10^{6}$ atoms, and becomes more\npronounced for large values of $N$, with the loss due to elastic\nscattering reaching 36 per cent for $5\\times 10^6$ atoms. Elastic\nscattering of atoms from the different momentum wavepackets removes\natoms from the four BEC wavepackets, and it thereby also lowers the\nnonlinear coupling term that gives rise to the 4WM. Although the\nmean-free-path for elastic collisions is on the order of 10 times\n$r_{TF}$ for $1.5\\times 10^6$ atoms, there are a sufficient number of\ncollisions to make a noticable reduction in the nonlinear output.\n\nFinally, Fig.~\\ref{f13} compares our 3D-SVEA calculation, with\ncorrections due to elastic scattering, to the observed output 4WM\nfraction in the NIST experiment \\cite{Deng}. The overall agreement is\ngood, given the approximations in the model and the scatter in the\nexperimental data. The calculated curve tends to be slightly larger\nthan the mean of the measured points, and in particular, does not seem\nto saturate as fast at large $N$ as the experimental data. Since\nsystematic error bars were not given for the data, it is difficult to\nknow whether this slight disagreement is significant. There are\nclearly approximations in the theory, such as using the GPE method or\nignoring the dynamics during the application of the Bragg pulses. \nThere also are effects in the experiment that might have a bearing on\nthe comparison. For example, Fig.~2b of reference \\cite{Deng}\nreported a best case of 10.6 per cent 4WM output for $N=1.7\\times\n10^6$ atoms, although a lower figure near 6 per cent reported in\nFig.~3 of reference \\cite{Deng} was more typical. The 10.6 per cent\noutput would disagree with our calculations on the high side. This\nindicates that there is sufficient uncertainty in the quantitative\naspects of the experiment to warrant a more systematic experimental\nexploration of the 4WM signal. Other possible sources of differences\nbetween theory and experiment include micromotion of the initial BEC\nin the time-orbiting-trap, laser misalignment, and a small finite\ntemperature component of the BEC.\n\n\\section{Summary and Conclusions and Outlook} \\label{conclusions}\n\nWe have developed a full description of four-wave mixing (4WM) using a\nmean-field treatment of Bose-Einstein condensates. The\nslowly-varying-envelope approximation is a powerful tool that reduces\nthe numerical grid requirements for calculating the time-dependent\ndynamics of fast-moving wavepackets with velocities greater than a\nphoton recoil velocity. We find that elastic scattering loss between\natoms in the fast wavepackets removes enough atoms from the\nwavepackets to affect the 4WM output. The quantum mechanical 3D\ncalculations presented here show good agreement with experiment.\n\nIn spite of the strong analogy between atom and optical 4WM, there are\nfundamental differences. In optical 4WM, the energy-momentum\ndispersion relation is different than in the massive boson case. \nBecause we neither create nor destroy atoms, the only 4WM processes\nallowed for matter waves are particle number conserving. This is not\nthe case for optical 4WM where, for example, in frequency tripling\nthree photons are annihilated and one is created. Particle, energy\nand momentum conservation limit all matter 4WM processes to\nconfigurations that can be viewed as degenerate 4WM in an appropriate\nmoving frame.\n\nWe have considered 4WM using condensates of the same internal states. \nThe internal states of the atoms can be changed by using Raman\ntransitions. Thus, one can envision scattering atoms in one internal\nstate from the matter-wave grating formed by atoms in a different\ninternal hyperfine state. It is also possible to study the details of\n4WM between mixed atomic species. We are in the process of carrying\nout such calculations. Quantum correlations created by the nonlinear\nprocess could lead to the study of non-classical matter-wave fields,\nanalogous to squeezed and other non-classical states of light. It is\nof interest to investigate such cases. By varying the magnetic field\nto allow a Feshbach resonance to change the $U_0$ coupling parameter,\n4WM can be modified dynamically during the dynamics that occur as the\nwavepacket fly apart, thus increasing or decreasing 4WM output. Such\nstudies are also feasible.\n\nIt is possible to modify the mean-field description of 4WM, and more\ngenerally, Bragg scattering of BECs, by generalizing the GP equation\nto allow incorporation of momentum dependence of the nonlinear\nparameters, thereby putting the treatment of elastic and inelastic\nscattering on a firm footing. This will be presented elsewhere\n\\cite{BTiesBJ}.\n\n\\bigskip\n\n\\begin{acknowledgments} This work was supported in part by grants from\nthe US-Israel Binational Science Foundation, the James Franck Binational\nGerman-Israel Program in Laser-Matter Interaction (YBB) and the U.S.\nOffice of Naval Research (PSJ). We are grateful to Eduard Merzlyakov for\nassisting with the 3D computations carried out on the Israel Supercomputer\nCenter Cray computer. We thank Ed Hagley, Lu Deng, William D. Phillips,\nMarya Doery and Keith Burnett for stimulating discussions on the subject.\n\\end{acknowledgments}\n\n\n\n\\begin{references}\n\n\\bibitem{Tripp98} M.\\ Trippenbach, Y.\\ B.\\ Band, and P.\\ S.\\ Julienne,\nOptics Express {\\bf 3}, 530 (1998).\n\n\\bibitem{Deng} L.\\ Deng, E.\\ W.\\ Hagley, J.\\ Wen, M. Trippenbach, Y.\\\nB.\\ Band, P.\\ S.\\ Julienne, J.E. Simsarian, K. Helmerson, S.L. Rolston,\nand W.D. Phillips, Nature (London) {\\bf 398}, 218 (1999).\n\n\\bibitem{atom_BECs}\nM.H. 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Rev.\nLett. {\\bf 78}, 3276-3279 (1997).\n\n\\bibitem{Wallis} H. Wallis, A. Rohrl, M. Naraschewski and A. Schenzle,\nPhys. Rev. {\\bf A55}, 2109 (1997).\n\n\\bibitem{Trip1} M. Trippenbach and Y.\\ B.\\ Band, Phys. Rev. {\\bf A56},\n4242 (1997)\n\n\\bibitem{Trip2} M. Trippenbach and Y.\\ B.\\ Band, Phys. Rev. {\\bf A57},\n4791 (1998).\n\n\\bibitem{BTBJ}\nY.\\ B.\\ Band, M.\\ Trippenbach, J.\\ P.\\ Burke, and P.\\ S.\\ Julienne, \n``Elastic scattering loss of atoms from colliding Bose-Einstein condensate \nwavepackets\", Phys. Rev. Lett. (submitted).\n\n\\bibitem{SK-K}\nA.P. Chikkatur, A. Goerlitz, D.M. Stamper-Kurn, S. Gupta, S. Inouye,\nD.E. Pritchard, and W. Ketterle, to be published.\n\n\\bibitem{Tiesinga96}\nE. Tiesinga, C. J. Williams, P. S. Julienne, K. M Jones, P. D. Lett,\nand W. D. Phillips, J. Res. Natl. Inst. Stand. Technol. {\\bf 101},\n505 (1996).\n\n\\bibitem{SplitFFT} \nJ. A. Fleck, J. R. Morris and M. D. Feit, Appl. Opt. \n{\\bf 10}, 129 (1976); M. D. Feit and J. A. Fleck, Appl. Opt.{\\bf 17}, \n3390 (1978); Appl. Opt. {\\bf 18}, 2843 (1979).\n\n\\bibitem{CD} Y. Castin and R. Dum,Phys. Rev. Lett {\\bf 77}, 5315 (1996).\n\n\\bibitem{Band90}\nY. B. Band, Phys. Rev. {\\bf A42}, 5530 (1990).\n\n\\bibitem{BTiesBJ}\nY.\\ B.\\ Band, E.\\ Tiesinga, J.\\ P.\\ Burke, and P.\\ S.\\ Julienne, \nunpublished (2000).\n\n\\end{references}\n\n\n\n\\begin{figure} \n\\centerline{\\epsfxsize=4.25in\\epsfbox{f1fwm.eps}}\n\\caption {Momentum space view of the wavepackets participating in the\nfour-wave mixing process. (a) Conservation of momentum in the\nlaboratory frame. (b) A set of possible wavepackets in the laboratory\nframe with momenta that satisfy the phase-matching conditions in\nSection \\ref{SecSVEA}, namely, $|{\\bf P}_2|=|{\\bf P}_3| \\cos{\\theta}$.}\n\\label{f1}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f2fwm.eps}}\n\\caption {(a) Lab frame view of the four-wave mixing process, showing\nthe four wavepackets at early time while they are still interacting\nand at late time after they have separated. (b) Degenerate frame view\nof the same cases as in (a).}\n\\label{f2}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f3fwm.eps}}\n\\caption {(a) Cuts along the $x$, $y$ and $z$ axes of the parent\ncondensate wavefunction $\\Psi(x,y,z,t=0)$ for $N=1.5\\times 10^6$ atoms\nin a trap with harmonic frequencies of 84 Hz, 59.4 Hz, and 42 Hz in\nthe respective $x$, $y$, and $z$ directions. The arrows show the TF\nradii $r_{TF}(i)$ in the $i=x,y,z$ directions. The curves labeled\n``$x$'', ``$y$'', and ``$z$'' respectively represent\n$\\mathrm{Re}[\\Psi(x,0,0,0)]$, $\\mathrm{Re}[\\Psi(0,y,0,0)]$, and\n$\\mathrm{Re}[\\Psi(0,0,z,0)]$; $\\mathrm{Im}[\\Psi(x,y,z,0)]$ is\nidentically zero for each case. (b) Cuts along the $x$ axis of\n$\\mathrm{Re}[\\Psi(x,0,0,t = t_1)]$ and $\\mathrm{Im}[\\Psi(x,0,0,t =\nt_1)]$ for $t_1 = 600\\ \\mu$s.}\n\\label{f3}\n\\end{figure}\n\n\\newpage\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f4fwm.eps}}\n\\caption {Cut in the $k_x$ direction $(k_y=k_z=0)$ of the squared\nmomentum distribution $|\\Psi({\\bf k},t)|^2$ for the wavefunctions in\nFig.~\\ref{f3} for $t_1=0$ and $t_1=600$ $\\mu$s. }\n\\label{f4}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f5fwm.eps}}\n\\caption {Comparison of $N_{4}(t)/N$ versus $t-t_2$ for 2D and 3D\ncalculations for $1.5\\times 10^6$ atoms. The trap is the same as in\nFig.~\\ref{f3}. The Bragg pulses are applied 600 $\\mu$s after the\ntrapping potential is turned off and are over at time $t_2$.}\n\\label{f5}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f6fwm.eps}}\n\\caption {Contour plots of integrated column density from the 3D-SVEA\ncalculations vs $x$ and $y$ for $N=1.5\\times 10^{6}$ and the same trap\nas for Fig.~\\ref{f3}. Panels (a) through (f) show the time\ndevelopment of the wavepackets from the from the time the trap is\nturned off until the wavepackets physically separate.}\n\\label{f6}\n\\end{figure}\n\n\\newpage\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f7fwm.eps}}\n\\caption {Comparison of $N_{4}(t)/N$ versus $t-t_2$ for $0.2\\times\n10^6$, $1.5\\times 10^6$ and $5.0\\times 10^6$ atoms. The trap is the\nsame as in Fig.~\\ref{f3}. The Bragg pulses are applied 600 $\\mu$s\nafter the trapping potential is turned off.}\n\\label{f7}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f8fwm.eps}}\n\\caption {Comparison of $N_{4}(t)/N$ versus $t-t_2$ for $1.5\\times\n10^6$ atoms. The different curves show cases where the Bragg pulses\nare applied at $t_1 = 0$, $600$, $1200$ and $1800$ $\\mu$s after the\ntrapping potential is turned off ($t_2 \\approx t_1$). The trap is the\nsame as in Fig.~\\ref{f3}.}\n\\label{f8}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f9fwm.eps}}\n\\caption {Comparison of $N_{4}(t)/N$ versus $t-t_2$ for $1.5\\times\n10^6$ atoms. The trap is the same as in Fig.~\\ref{f3}. The Bragg\npulses are applied 600 $\\mu$s after the trapping potential is turned\noff. The three different curves are for the cases where the\nseparation times are scaled by factors of 0.7, 1, and 2 by scaling the\nseparation velocities by $1/0.7$, 1, and $1/2$.}\n\\label{f9}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f10fwm.eps}}\n\\caption {Growth of $N_{4}(t)/N$ and $N_{4}(t)/N$ versus $t-t_2$ for\nthe case where a weak probe wavepacket 2 with initial population\nfraction 0.001 encounters strong ``pump'' wavepackets with initial\nfractions 0.4995. The trap is the same as in Fig.~\\ref{f3}. The\nBragg pulses are applied 600 $\\mu$s after the trapping potential is\nturned off.}\n\\label{f10}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f11fwm.eps}}\n\\caption {$N_{4}/N$ dependence dependence on the total number of\natoms, $N$, calculated in 2D and 3D. The dashed lines show the\n$N^{6/5}$ dependence predicted by the simple theory in subsection\n\\ref{simple}. The trap is the same as in Fig.~\\ref{f3}. The Bragg\npulses are applied 600 $\\mu$s after the trapping potential is turned\noff.}\n\\label{f11}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f12fwm.eps}}\n\\caption {Fraction of atoms in the 4WM output wavepacket, $N_{4}/N$,\nversus the total number of initial atoms, $N$, calculated in 2D, 3D\nand 3D with inclusion of elastic scattering loss as discussed in\nSec.~{\\protect \\ref{el_scat}}. The open circles represent\ncalculations using experimental data {\\protect \\cite{Deng}} to\ndetermine the ratios $N_1:N_2:N_3$ rather than taking the nominal\nvalues $N_1:N_2:N_3 = 7:3:7$. The trap is the same as in\nFig.~\\ref{f3}. The Bragg pulses are applied 600 $\\mu$s after the\ntrapping potential is turned off.}\n\\label{f12}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=4.25in\\epsfbox{f13fwm.eps}}\n\\caption {Fraction of atoms in the 4WM output wavepacket, $N_{4}/N$,\nversus the total number of initial atoms, $N$, calculated in 3D\nwithout and with inclusion of elastic scattering loss as discussed in\nSec.~{\\protect \\ref{el_scat}}. The dots are experimental data\n{\\protect \\cite{Deng}}. The trap is the same as in Fig.~\\ref{f3}. \nThe Bragg pulses are applied 600 $\\mu$s after the trapping potential\nis turned off.}\n\\label{f13}\n\\end{figure}\n\n\n\\end{document}\n\n"
}
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[
{
"name": "cond-mat0002119.extracted_bib",
"string": "\\bibitem{Tripp98} M.\\ Trippenbach, Y.\\ B.\\ Band, and P.\\ S.\\ Julienne,\nOptics Express {\\bf 3}, 530 (1998).\n\n\n\\bibitem{Deng} L.\\ Deng, E.\\ W.\\ Hagley, J.\\ Wen, M. Trippenbach, Y.\\\nB.\\ Band, P.\\ S.\\ Julienne, J.E. Simsarian, K. Helmerson, S.L. Rolston,\nand W.D. Phillips, Nature (London) {\\bf 398}, 218 (1999).\n\n\n\\bibitem{atom_BECs}\nM.H. Anderson {\\it et al.}, Science {\\bf 269}, 198 (1995); \nK. B. Davis {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 3969 (1995); \nC.C. Bradley, {\\it et al.}, Phys. Rev. Lett. {\\bf 78}, 985 (1997); \nsee also C.C. Bradley {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 1687 \n(1995).\n\n\n\\bibitem{reviews} See reviews of BEC by F. Dalfovo, S. Giorgini, L. P.\nPitaevskii and S. Stringari, Rev. of Mod. Phys. {\\bf 71}, 463\n(1999) and A. S. Parkins and D. F. Walls, Phys. Reports {\\bf 303}, 1\n(1998).\n\n\n\\bibitem{Mewesoc} M.\\ O.\\ Mewes, et al., Phys. Rev. Lett. {\\bf 78},\n582 (1997); B. P. Anderson and M A Kasevich, Science {\\bf 282}, 1686\n(1998); E Hagley {\\it et al.}, Science {\\bf 283}, 1706 (1999); I.\nBloch, T. W. H\\\"{a}nsch and T. Esslinger, Phys. Rev. Lett. {\\bf\n82}, 3008 (1999).\n\n\n\\bibitem{Kozuma99} M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R.\nLutwak, K. Helmerson, S. L. Rolston, and W. D. Phillips, Phys. Rev. \nLett. {\\bf 82}, 871 (1999).\n\n\n\\bibitem{Lenz} G. Lenz, P. Meystre and E.W. Wright, Phys. Rev. Lett. 71,\n3271 (1993).\n\n\n\\bibitem{Goldstein95} E. Goldstein, K. Pl\\\"{a}ttner, and P. Meystre,\nQuantum Semiclass. Opt. {\\bf 7}, 743 (1995); E. Goldstein, K.\nPl\\\"{a}ttner, and P. Meystre, J. Res. Nat. Inst. Stand. Technol. \n{\\bf 101}, 583 (1996)\n\n\n\\bibitem{Goldstein99} E. Goldstein and P. Meystre, Phys. Rev. A\n{\\bf 59}, 1509 (1999).\n\n\n\\bibitem{Law98} C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. \nLett. {\\bf 81}, 5257 (1998).\n\n\n\\bibitem{Goldstein99b} E. Goldstein and P. Meystre, Phys. Rev. A\n{\\bf 59}, 3896 (1999).\n\n\n\\bibitem{Hellwarth} R. W. Hellwarth, Prog. Quant. Electr. {\\bf 5}, 1\n(1977).\n\n\n\\bibitem{Maker} P. D. Maker and R. W. Terhune, Phys. Rev. {\\bf A137},\n801 (1965).\n\n\n\\bibitem{Yariv} A. Yariv and D. M. Pepper, Opt. Lett. {\\bf 1}, 16 (1977).\n\n\n\\bibitem{Marya} M. Doery, private communication (1999).\n\n\n\\bibitem{Ballagh} R. J. Ballagh, K. Burnett and T. F. Scott, Phys. Rev.\nLett. {\\bf 78}, 3276-3279 (1997).\n\n\n\\bibitem{Wallis} H. Wallis, A. Rohrl, M. Naraschewski and A. Schenzle,\nPhys. Rev. {\\bf A55}, 2109 (1997).\n\n\n\\bibitem{Trip1} M. Trippenbach and Y.\\ B.\\ Band, Phys. Rev. {\\bf A56},\n4242 (1997)\n\n\n\\bibitem{Trip2} M. Trippenbach and Y.\\ B.\\ Band, Phys. Rev. {\\bf A57},\n4791 (1998).\n\n\n\\bibitem{BTBJ}\nY.\\ B.\\ Band, M.\\ Trippenbach, J.\\ P.\\ Burke, and P.\\ S.\\ Julienne, \n``Elastic scattering loss of atoms from colliding Bose-Einstein condensate \nwavepackets\", Phys. Rev. Lett. (submitted).\n\n\n\\bibitem{SK-K}\nA.P. Chikkatur, A. Goerlitz, D.M. Stamper-Kurn, S. Gupta, S. Inouye,\nD.E. Pritchard, and W. Ketterle, to be published.\n\n\n\\bibitem{Tiesinga96}\nE. Tiesinga, C. J. Williams, P. S. Julienne, K. M Jones, P. D. Lett,\nand W. D. Phillips, J. Res. Natl. Inst. Stand. Technol. {\\bf 101},\n505 (1996).\n\n\n\\bibitem{SplitFFT} \nJ. A. Fleck, J. R. Morris and M. D. Feit, Appl. Opt. \n{\\bf 10}, 129 (1976); M. D. Feit and J. A. Fleck, Appl. Opt.{\\bf 17}, \n3390 (1978); Appl. Opt. {\\bf 18}, 2843 (1979).\n\n\n\\bibitem{CD} Y. Castin and R. Dum,Phys. Rev. Lett {\\bf 77}, 5315 (1996).\n\n\n\\bibitem{Band90}\nY. B. Band, Phys. Rev. {\\bf A42}, 5530 (1990).\n\n\n\\bibitem{BTiesBJ}\nY.\\ B.\\ Band, E.\\ Tiesinga, J.\\ P.\\ Burke, and P.\\ S.\\ Julienne, \nunpublished (2000).\n\n"
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cond-mat0002120
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Energy level dynamics in systems with weakly multifractal eigenstates: equivalence to 1D correlated fermions at low temperatures
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"author": "V.E.Kravtsov"
}
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It is shown that the parametric spectral statistics in the critical random matrix ensemble with multifractal eigenvector statistics are identical to the statistics of correlated 1D fermions at finite temperatures. For weak multifractality the effective temperature of fictitious 1D fermions is proportional to $T_{eff}\propto (1-d_{n})/n \ll 1$, where $d_{n}$ is the fractal dimension found from the $n$-th moment of inverse participation ratio. For large energy and parameter separations the fictitious fermions are described by the Luttinger liquid model which follows from the Calogero-Sutherland model. The low-temperature asymptotic form of the two-point equal-parameter spectral correlation function is found for all energy separations and its relevance for the low temperature equal-time density correlations in the Calogero-Sutherland model is conjectured.
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[
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"name": "cond-mat0002120.tex",
"string": "\\documentstyle[prl,aps,multicol]{revtex}\n\\renewcommand{\\narrowtext}{\\begin{multicols}{2}\n\\global\\columnwidth20.5pc}\n\\renewcommand{\\widetext}{\\end{multicols} \\global\\columnwidth42.5pc}\n\\newcommand{\\Rrule}{\\vspace{-0.1in}\\hfill\\vrule depth1em height0pt \\vrule\nwidth3.5in height.2pt depth.2pt\\vspace*{-0.125in}}\n\\begin{document}\n\\draft\n\\title{Energy level dynamics in systems with weakly multifractal\neigenstates: equivalence to 1D correlated fermions at low\ntemperatures}\n\\author{V.E.Kravtsov}\n\\address{The Abdus Salam International Centre for Theoretical Physics,\nP.O.B.\n586, 34100 Trieste, Italy, \\\\ Landau Institute for Theoretical\nPhysics, 2 Kosygina st., 117940 Moscow, Russia}\n\\author{A.M.Tsvelik}\n\\address{Department of Physics, University of Oxford\n1 Keble Road, Oxford, OX1 3NP, United Kingdom}\n%\\date{Month DD, Year}\n\\maketitle\n\\begin{abstract}\nIt is shown that the parametric spectral statistics in the critical random\nmatrix\nensemble with multifractal eigenvector statistics are identical to the\nstatistics of correlated 1D fermions at finite temperatures. \nFor weak\nmultifractality the effective temperature of fictitious 1D\nfermions is proportional to $T_{eff}\\propto (1-d_{n})/n \\ll 1$, where\n$d_{n}$ is the\nfractal\ndimension found from the $n$-th moment of inverse participation ratio.\nFor large energy and parameter separations the fictitious fermions are\ndescribed by\nthe Luttinger liquid model which follows from the Calogero-Sutherland\nmodel. The low-temperature asymptotic form of the two-point\nequal-parameter spectral \ncorrelation\nfunction is found for all energy separations and its relevance for the low\ntemperature equal-time density correlations in the Calogero-Sutherland\nmodel is conjectured. \n\n\\end{abstract}\n\n\\pacs{PACS number(s): 72.15.Rn, 72.70.+m, 72.20.Ht, 73.23.-b}\n\\narrowtext\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe spectral statistics in complex quantum systems are signatures of the\nunderlying dynamics of the corresponding classical counterpart. The\nspectral statistics in chaotic and disordered systems in the limit of\ninfinite dimensionless\nconductance $g$ is described by the\nclassical random matrix theory of Wigner and Dyson\n\\cite{RMT} (WD statistics). The WD statistics possess a remarkable\nproperty of universality: it depends only on the symmetry class with\nrespect to the time-reversal transformation ${\\cal T}$. The three symmetry\nclasses correspond to the lack of ${\\cal T}$-invariance (the unitary\nensemble, $\\beta=2$); the ${\\cal T}$-invariant systems with ${\\cal T}^2\n=1$ (the orthogonal ensemble, $\\beta=1$), and the ${\\cal T}$-invariant\nsystems with ${\\cal T}^2 \n=-1$ (the symplectic ensemble, $\\beta=4$), respectively. The physical\nground behind this universality is the structureless eigenfunctions in the\nergodic regime which implies the invariance of the eigenfunction\nstatistics with respect to a unitary transformation of the basis.\n\nIn real disordered metals the eigenfunctions are not basis-invariant. The\nbasis-preference reaches its extreme form for the strongly impure metals\nwhere all eigenfunctions are localized in the coordinate space but\ndelocalized in the momentum space. In this case the spectral statistics is\nPoissonian in the thermodynamic (TD) limit.\n\nFor low-dimensional systems $d=1,2$, where all states are\nlocalized in the TD limit, one can observe the smooth crossover\nfrom the WD to the Poisson spectral statistics as a function of the\nparameter $\\xi/L$, where $\\xi$ is the localization radius and $L$ is the\nsystem size. The dependence of the spectral correlation functions on the\nenergy variable $s=E/\\Delta$ ($\\Delta$ is the mean level separation) \nis non-universal for\nfinite\n$L/\\xi$ but all of them tend to the Poisson\nlimit\nas $L/\\xi\\rightarrow\\infty$.\n\nIn systems of higher dimensionality $d>2$ the situation is different\nbecause of the presence of the Anderson localization transition at a\ncritical disorder $W=W_{c}$. In the metal phase $W<W_{c}$ the dimensional\nconductance $g(L)\\rightarrow \\infty$ as $L\\rightarrow\\infty$\nand one obtains the WD spectral statistics in the TD limit. In the\ninsulator state $g(L)\\rightarrow 0$ at $L\\rightarrow\\infty$ and the\nlimiting statistics is Poissonian. However, there is a fixed point\n$W=W_{c}$ in which the spectral statistics are nearly independent of $L$.\nThus at the critical point there exist universal spectral\nstatistics which are neither WD\nnor Poissonian but rather a hybrid of \nboth \\cite{Shkl}. However, the universality of the critical spectral\nstatistics (CSS) is somewhat limited, since it depends not only on the\nDyson symmetry parameter $\\beta$ but also on the critical value of the\ndimensional conductance $g^{*}$ which in turn depends on the \ndimensionality $d$ of the system \\cite{Zhar1}. Thus for each\nuniversality class there is a {\\it\nfamily} of critical spectral statistics parametrized by the critical\ndimensionless conductance $g^{*}$.\n\nThe very existence of the subject of critical level statistics imposes a\nconstraint on the possible values of the localization length exponent\n$\\nu=[d\\beta(g)/d\\ln g]^{-1}|_{g=g^{*}}$, where $\\beta(g)\\equiv d\\ln\ng/d\\ln L$ is the scaling \nfunction. Indeed, for the spectral statistics to be meaningful\nthe width\nof\nthe critical energy window $\\delta E$ must be much larger than the mean\nlevel separation $\\Delta \\propto 1/L^{d}$. The quantity $\\delta E$ is\ndefined as the\ndistance from the mobility edge $E=E_{c}$ at which the\nlocalization or correlation radius\n$\\xi(\\delta E)\\propto |\\delta E|^{-\\nu}$ is equal to the system size $L$.\nThe number of critical eigenstates ${\\cal\nN}=\\delta E/\\Delta$ is proportional to $L^{d-\\frac{1}{\\nu}}$. \nFor \n$\\nu d >1$\nthis number tends to infinity in the\nTD limit $L\\rightarrow\\infty$ despite the width of the critical energy\nwindow shrinks to zero. This necessary condition for the existence of the\ncritical statistics is secured by the famous Harris criterion $\\nu d\n> 2$.\n\nHowever, the critical exponent $\\nu$ enters not only in the necessary\ncondition for the CSS but also in the correlation functions of the\ndensity of energy levels $\\rho(E)$.\nIt has been shown in \\cite{KLAA,AKL} that there is a power-law tail in the\ncritical \ntwo-level correlation function (TLCF)\n$R(\\omega)=\\langle\\langle\\rho(E)\\rho(E+\\omega)\n\\rangle\\rangle$ that arises because of the\nfinite-size\ncorrection to the dimensionless conductance\n$g(L_{\\omega})/g^{*}-1\\propto\n(L_{\\omega}/L)^{1/\\nu}= s ^{-\\frac{1}{\\nu d}}$, where\n$L_{\\omega}\\propto \\omega^{-1/d}\\ll L$ is\nthe length scale set\nby the\nenergy difference $\\omega=E-E'\\equiv s \\Delta$ between two levels.\nThe sign of this tail depends on whether the critical energy levels are\non the metal ($E<E_{c}$) or on the insulator ($E>E_{c}$) side of the\nmobility\nedge, in the same way as for the 2D systems in the weak delocalization\n($\\beta=4$) or weak localization ($\\beta=1$) regimes \\cite{KL2}. \nClearly, this power-law tail does not reflect properties of the\ncritical\neigenstates but rather the behavior $\\xi(|E-E_{c}|)$ of the size of the\nspace region\nwhere eigenstates show the critical\nspace correlations. \n\nIn order to study the relationship between the properties of the critical\neigenstates and the CSS in its pure form one should consider a system with\na continuous line of critical points where $\\beta(g)=0$. This case\nformally corresponds to $\\nu=\\infty$ and the finite-size effects are\nabsent. \n\nAnother complication which makes ambiguous the definition of CSS \nis the fact that being independent of the system size, the spectral \ncorrelation\nfunctions depend on\nthe boundary conditions \\cite{Mon,PSch,KY} and topology of a system. \nTherefore we will consider the system of the torus topology where CSS\ntakes its `canonical' form. In particular, the TLCF decays exponentially\nin this case. \n\nAs has been already mentioned the universality of the WD statistics\nis based on the ergodic, basis-invariant statistics of\neigenfunctions which one may encounter in different physical situations.\nThe characteristic feature of all critical quantum systems is the\nmultifractal statistics of the critical eigenfunctions \\cite{W,CCP,AlKL}.\nThe simplest two-point correlations of the critical wave functions\ncan be obtained from the renormalization group result\n\\cite{W2} for $l<r<\\xi<L$:\n\\begin{equation}\n\\label{Wegn}\n\\langle |\\Psi_{E}(0)|^{2n}|\\Psi_{E}(r)|^{2n}\\rangle\n=p\\,\n<|\\Psi_{E}(0)|^{2n}>^2 \\,(\\xi/r)^{\\alpha_{n}},\n\\end{equation}\nwhere $l$ is the short-distance cut-off of the order of the elastic\nscattering length, $p=(\\xi/L)^{d}$ is the probability for a reference\npoint to be inside a localization region. The exponent\n$\\alpha_{n}=2n(d_{n}-d_{2n})+d_{2n} -2d_{n}+d$\nis expressed through the fractal dimensions\n$d_{n}$ defined by the $L$-dependence of the moments of inverse\nparticipation ratio:\n\\begin{equation}\n\\label{mf} \nL^{d} \\langle|\\Psi_{E}(r)|^{2n} \\rangle \\propto\nL^{-d_{n}\\,(n-1)},\\;\\;\\;\\;\\;n\\geq 2.\n\\end{equation}\nAt the critical point $p=1$ and the correlation radius\n$\\xi\\rightarrow\\infty$ in\nEq.(\\ref{Wegn}) must be\nreplaced by the sample size $L$. Spectral statistics are\nrelated with eigenfunction correlations at \ndifferent energies $E$ and $E'=E+\\omega$. If $\\omega\\gg \\Delta$ and\nthus $L_{\\omega}\\ll L$ one should substitute $L_{\\omega}$ for $L$ in the\n$r$-dependent term of Eq.(\\ref{Wegn}). In this way the multifractality\nexponents $d_{n}$ enter the spectral $\\omega$-dependences\n\\cite{Ch}. \n\nFor weak\nmultifractality one can expect the fractal dimensions to be a linear\nfunction of $n$ which is controlled by only one\nparameter $a$:\n\\begin{equation}\n\\label{weak}\nd_{n}/d=1-a n,\\;\\;\\;\\;\\;\\;\\;\\;a\\sim 1/g^{*}\\ll 1.\n\\end{equation}\nThis relationship holds approximately for the Anderson transition in\n$2+\\epsilon$\ndimensions \\cite{W,CCP,AlKL} and is fulfilled exactly for the critical\neigenstate of the Dirac equation in the random vector-potential\n\\cite{Mud}. \n\nThus for critical quantum systems with weak multifractality it is\nnatural to\nexpect that the spectral statistics depends \n on only one system-specific parameter - the critical\nconductance $g^{*}$. \n\nIn view of the expected universality, it is useful to find a simple\none-parameter random matrix ensemble with the multifractal eigenfunction\nstatistics which would play the same role for the critical systems as the\nclassical RMT does for the ergodic systems. \nAs a matter of fact there are few candidates \\cite{KMut}. However, here\nwe focus only on one of them \\cite{MirFyod}, since for this ensemble \nthe multifractality of\neigenstates has been rigorously proven \\cite{MirFyod,AlLev}.\n\nConsider a Hermitean $N\\times N$ matrix with the real ($\\beta=1$), complex\n($\\beta=2$)\nor quaternionic ($\\beta=4$) entries $H_{ij}$ $(i\\geq j)$ which are\nindependent\nGaussian random numbers with zero mean and the variance:\n\\begin{equation}\n\\label{MFmat}\n\\langle|H_{ij}|^{2} \\rangle =\n\\frac{1}{1+\\frac{(i-j)^2}{B^2}}\n\\left\\{\n\\begin{array}{cc}\n1/\\beta, & i=j, \\\\\n1/2 , & i\\neq j\n\\end{array}\n\\right.\\equiv J(i-j)\n\\end{equation}\nThis model has been shown to be critical both for large \\cite{MirFyod}\nand for small \\cite{AlLev} values of $B$ with the fractal dimensionality\n$d_{2}$ at the center of the spectral band being:\n\\begin{equation}\n\\label{eta}\nd_{2}=\\left\\{\n\\begin{array}{cc}\n1-\\frac{1}{\\pi\\beta B} , & B\\gg 1, \\\\\n2 B , & B\\ll 1\n\\end{array}\n\\right.\n\\end{equation}\nThus the 1D system with long-range hopping described by the matrix\nHamiltonian Eq.(\\ref{MFmat})\npossesses the line of critical points $B\\in (0,\\infty)$, the fractal\ndimensionality $d_{2}$ changing from 1 to 0 with decreasing $B$.\n\nOne can extend this matrix 1D model by closing it into a ring and applying\na `flux' $\\varphi\\in [0,1]$. In this case\n\\begin{equation}\n\\label{fl}\nH_{ij}(\\varphi)=H_{ij}+H_{ij}^{(1)}\\,e^{2\\pi i\\varphi\\, sgn(i-j)} \n\\end{equation}\nis a\nsum of\ntwo independent\nGaussian\nrandom numbers with the variance of $H_{ij}^{(1)}$ given by:\n\\begin{equation}\n\\label{ring}\n\\langle|H_{ij}^{(1)}|^{2} \\rangle = J(N-|i-j|).\n\\end{equation}\n\nFor large values of $B$ which correspond to weak multifractality\none can derive \\cite{MirFyod} an effective field theory -- the\nsupersymmetric nonlinear sigma-model \\cite{Ef} -- which describes the\nspectral and eigenfunction correlations of the critical random matrix\nensemble Eq.(\\ref{MFmat}):\n\\begin{equation}\n\\label{sigma}\nF[{\\bf Q}]= -\\frac{ g^{*}}{16}\\,\\sum_{i,j=1}^{N}\\,Str\\,\n[{\\bf Q}_{i}\\,U_{|i-j|}\\,{\\bf Q}_{j}]+\\frac{i\\pi\ns}{4N}\\,\\sum_{i=1}^{N}\\,Str[\\sigma_{z}\n{\\bf Q}_{i}],\n\\end{equation}\nwhere ${\\bf Q}$ is the supermatrix with ${\\bf Q}_{i}^{2}={\\bf 1}$ and\n\\begin{equation}\n\\label{g}\ng^{*}= 4 \\beta B.\n\\end{equation}\nThe symmetry with respect to time reversal is encoded in the symmetry of\n${\\bf Q}_{i}$ in exactly the same\nway as for the\ndiffusive sigma-model \\cite{Ef}. The only difference is the long-range\nkernel $U_{|i-j|}$ with the Fourier-transform $\\tilde{U}_{k}=|k|$.\nFor a torus geometry $k=2\\pi m/N$, where $m$ is an {\\it arbitrary}\ninteger.\n \nOne can explicitly resolve the constraint ${\\bf Q}^{2}=1$ by switching to\nthe\nintegration over the `angles' $W$. Then the Gaussian fluctuations of\n`angles' recover the spectrum of `quasi-diffusion' modes: \n\\begin{equation}\n\\label{qdif}\n\\varepsilon_{m}=g^{*}\\,|m|,\\;\\;\\;\\;\\;m=0,\\pm \n1,\\pm 2,...\n\\end{equation}\nThe problem of spectral statistics can be generalized to include the\ndependence of spectrum on the flux $\\varphi$ introduced by Eq.(\\ref{fl}).\nOne can define \\cite{SA} the\n{\\it parametric} two-level correlation function\n$R(s,\\varphi)=\\langle\\langle \n\\rho(E,0)\\rho(E+s\\Delta,\\varphi)\\rangle\\rangle$ which can be treated in the\nframework of the same nonlinear sigma-model but with the \nphase-dependent $\\varepsilon_{m}$: \n\\begin{equation}\n\\label{phase}\n\\varepsilon_{m}(\\varphi)=g^{*}\\,|m-\\varphi|. \n\\end{equation}\nFollowing the work by Andreev and Altshuler Ref.\\cite{AA} we introduce the\nspectral determinant:\n\\begin{equation}\n\\label{sd}\nD^{-1}(s,\\varphi)=\\prod_{m\\neq 0}\n\\frac{\\varepsilon_{m}^{2}(\\varphi)+s^2}{\\varepsilon_{m}^{2}(0)}\n\\end{equation}\nThen it can be shown in the same way as in Ref.\\cite{AA} that the\nparametric TLCF for $s\\gg 1$ and\n$g^{*}\\varphi\\gg 1$ can be expressed\nin terms of the spectral determinant as follows:\n\\begin{eqnarray}\n\\label{AAu}\nR^{u}(s,\\varphi)&=&-\\frac{1}{4\\pi^2}\\frac{\\partial^2\nG(s,\\varphi)}{\\partial\ns^2}+\n\\cos(2\\pi s)\\,e^{G(s,\\varphi)}\\\\ \\label{AAo}\nR^{o}(s,\\varphi)&=&-\\frac{1}{2\\pi^2}\\frac{\\partial^2\nG(s,\\varphi)}{\\partial\ns^2}+2\n\\cos(2\\pi s)\\,e^{2G(s,\\varphi)}\\\\ \\nonumber\nR^{s}(s,\\varphi)&=&-\\frac{1}{8\\pi^2}\\frac{\\partial^2\nG(s,\\varphi)}{\\partial\ns^2}+\\frac{\\pi}{\\sqrt{8}}\\,\n\\cos(2\\pi s)\\,e^{G(s,\\varphi)/2}+\\\\ \\label{AAs} &+& \\frac{1}{8}\\,\\cos(4\\pi\ns)\\,e^{2G(s,\\varphi)},\n\\end{eqnarray}\nwhere $G(s,\\varphi)=G(s,\\varphi+1)$ is a periodic in $\\varphi$ function:\n\\begin{equation}\n\\label{G}\ne^{G(s,\\varphi)}=\\frac{D(s,\\varphi)}{2\\pi^2\\, (s^2 +\n\\varepsilon_{0}^{2}(\\varphi))}.\n\\end{equation}\nEqs.(\\ref{AAu}-\\ref{AAs}) coincide with the corresponding formulae in\nRef.\\cite{AA}\nfor the unitary, orthogonal and symplectic ensembles \nafter some\nmisprints are corrected as in\nRef.\\cite{KamMez} and $s\\rightarrow 2s$ for the symplectic ensemble \nto take account of the Kramers degeneracy. The only difference is in the\nform of the spectral determinant Eq.(\\ref{sd}) due to the specific\nspectrum $\\varepsilon_{m}(\\varphi)$ of the quasi-diffusion modes.\n\nUsing the functional representation Eq.(\\ref{sigma}) one can also find the\nleading term in the deviation from the WD statistics\nat $s\\ll g^{*}$ and $\\varphi=0$ using the results of Ref.\\cite{KrMir,MirFyod}: \n\\begin{equation}\n\\label{small}\n\\delta R(s)= \\frac{1}{2\\pi^2 \\beta}\\,\\left(\\sum_{m\\neq\n0}\\frac{1}{\\varepsilon_{m}^{2}}\\right)\\,\\frac{d^2}{ds^2}\\,\\left[s^2\\,R_{WD}(s)\n\\right],\n\\end{equation}\nwhere $R_{WD}(s)$ is the Wigner-Dyson TLCF. \n\nA remarkable property of the function $G(s,\\varphi)$ for\nthe\ncritical\nsigma-model Eq.(\\ref{sigma}) on a torus is that\nit\ncan be decomposed into\nthe sum\n$G(s,\\varphi)=F(z)+F(\\bar{z})$ of\nanalytic\nfunctions $F(z)$ and $F(\\bar{z})$ where $\\tau=g^{*}\\varphi$,\n$z=\\tau+is$,\n$\\bar{z}=\\tau-is$ and\n\\begin{equation}\n\\label{F}\nF(z)= -\\ln[\\sqrt{2}g^{*}\\,\\sin(\\pi z/g^{*})].\n\\end{equation}\nEq.(\\ref{F}) results from a straightforward evaluation \\cite{KMut} of the\nproduct in\nEq.(\\ref{sd}).\n\nOn the other hand, it can be easily verified that $G(s,\\varphi)$\ngiven by Eq.(\\ref{F}) is proportional to the Green's function of the\nfree-boson field $\\Phi(z)$\non\na torus in the $(1+1)$ $z$-space:\n$0<\\Re z<g^{*}$, $-\\infty<\\Im z<+\\infty$:\n\\begin{equation}\n\\label{gf}\n\\langle\\Phi(s,g^{*}\\varphi)\\,\\Phi(0,0) \\rangle_{S} -\n\\langle\\Phi(0,0)\\,\\Phi(0,0) \\rangle_{S} =K\\, G(s,\\varphi),\n\\end{equation}\nwhere $\\langle ... \\rangle_{S}$ denotes the functional average with the\nfree-boson action:\n\\begin{equation}\n\\label{bos}\nS[\\Phi]=\\frac{1}{8\\pi K}\\int_{0}^{g^{*}}d\\tau\\int_{-\\infty}^{+\\infty}\nds\\;\\left[(\\partial_{s}\\Phi)^2 + (\\partial_{\\tau}\\Phi)^2 \\right].\n\\end{equation}\nNow we are in a position to make a crucial step and suggest\nthat for the critical RMT described by Eq.(\\ref{MFmat}), the\nAndreev-Altshuler\nequations\n(\\ref{AAu}-\\ref{AAs}) are nothing but density-density correlations\nin the Luttinger liquid of fictitious 1D fermions at a\nfinite temperature $T=1/g^{*}$:\n\\begin{equation}\n\\label{dd}\nR(s,\\varphi)=\\bar{n}^{-2}\\,\\langle\nn(s,\\tau)\\,n(0,0)\\rangle_{S}\\,-1,\\;\\;\\;\\; \\tau=g^{*}\\varphi.\n\\end{equation}\nIndeed, the density operator $n(s,\\tau)$ ($s$- is space and $\\tau\\in\n(0, 1/T)$ is imaginary time coordinate) for 1D interacting \nfermions with \nthe\nFermi-momentum $k_{F}=\\pi$ can be expressed through\nthe free boson field $\\Phi(s,\\tau)$ as follows \\cite{shura}: \n\\begin{eqnarray}\n\\label{dens}\nn(s,\\tau)&=&\\frac{1}{2\\pi}\\,\\partial_{s}\\Phi(s,\\tau)+\nA_{K}\\,\\cos[2\\pi s +\\Phi(s,\\tau)]+\\\\ \\nonumber &+& B_{K}\\,\\cos[4\\pi s +\n2\\Phi(s,\\tau)].\n\\end{eqnarray}\nThe constants $A_{K}$ and $B_{K}$ are independent of `temperature'\n$1/g^{*}$ but\ndepend on the interaction constant $K$. They can be uniquely determined\nfrom the WD limit $g^{*}\\rightarrow\\infty$.\n\nUsing Eqs.(\\ref{dd},\\ref{dens}) and the well known result for the Gaussian\naverage\nof the\nexponent:\n\\begin{equation}\n\\label{Ga}\n\\langle e^{ip \\Phi(s,\\tau)}\\,e^{-ip\n\\Phi(0,0)}\\rangle=e^{K p^2\\,[G(s,\\tau)-G(0,0)]},\n\\end{equation}\none can verify that for the choice: \n\\begin{equation}\n\\label{K}\nK=\\frac{2}{\\beta},\\;\\;\\;\\;\\;\\beta=1,2,4\n\\end{equation}\nthe Andreev-Altshuler formulae Eqs.(\\ref{AAu}-\\ref{AAs})\nare reproduced exactly for the orthogonal, unitary and symplectic\nensembles, respectively.\nNow we remind on a known result that the parametric\nspectral \nstatistics in the WD limit $g^{*}=\\infty$ is equivalent to the \nTomonaga-Luttinger\nliquid at zero temperature. It follows directly from \nRef.\\cite{SLA}\nand the equivalence of the Calogero-Sutherland model and the Tomonaga-Luttinger\nliquid for large distances $|z|\\gg 1$. The critical random matrix ensemble\nEq.(\\ref{MFmat},\\ref{fl},\\ref{ring}) and the critical 1D sigma-model\nEq.(\\ref{sigma}) turns out to be the simplest {\\it generalization} of the\nWD\ntheory that\nretains the Tomonaga-\nLuttinger liquid analogy extended for finite `temperatures'\n$T=1/g^{*}$ which are related with the spectrum of fractal dimensions\nEq.(\\ref{weak}).\n\nThe Wigner-Dyson two-level statistics for all three symmetry classes can\nbe expressed through the single kernel $K(s)=\\sin(\\pi s)/(\\pi s)$ in the\nfollowing way \\cite{RMT}:\n\\begin{eqnarray}\n\\label{ALLu}\nR^{u}(s)=-K^{2}(s),\\\\ \\label{ALLo}\nR^{o}(s)=-K^{2}(s)-\\frac{dK(s)}{ds}\\,\\int_{s}^{\\infty}K(x)\\,dx,\\\\\n\\label{ALLs}\nR^{s}(s)=-K^{2}(s)+\\frac{dK(s)}{ds}\\,\\int_{0}^{s}K(x)\\,dx. \n\\end{eqnarray}\nIt turns out that such a representation is also valid for the critical\nTLCF at $g^{*}\\gg 1$ if the kernel is replaced by:\n\\begin{equation}\n\\label{kern}\nK(s)=\\frac{T}{\\alpha}\\,\\frac{\\sin(\\pi\\alpha s)}{\\sinh(\\pi T\ns)},\\;\\;\\;\\;T=\\frac{1}{g^{*}}. \n\\end{equation}\nwhere $\\alpha=1$ for the orthogonal and the unitary ensemble and\n$\\alpha=2$ for the symplectic ensemble. The form of the kernel Eq.(28)\ncan be guessed from the well known density correlation function for the\ncase of free fermions in one dimension at a finite temperature that\ncorresponds to the unitary ensemble \\cite{KMut,SLA,MNS}.\nIn order to prove this statement we note that for $s\\gg 1$ and $g^{*}\\gg\n1$\nEqs.(\\ref{ALLu}-\\ref{ALLs}) with the kernel Eq.(\\ref{kern}) give the same\nleading terms as Eqs.(\\ref{AAu}-\\ref{AAs}) with $G(s,0)$ given by\nEq.(\\ref{F}). For $s\\ll g^{*}$ Eqs.(\\ref{ALLu}-\\ref{ALLs}) give \ncorrections to the WD statistics that coincide with Eq.(\\ref{small}).\nThus the representation Eqs.(\\ref{ALLu}-\\ref{ALLs}) with the kernel\nEq.(\\ref{kern}) are the correct asymptotic expressions for {\\it both}\n$s\\gg 1$ and $s\\ll g^{*}$. At $g^{*}\\gg 1$ these regions have a\nparametrically large overlap so that Eqs.(\\ref{ALLu}-\\ref{ALLs}) \nare valid for all values of $s$. Given that for all $s$ at $T=0$\nand for large energy separations $s\\gg 1$ at $T\\ll 1$\nexpressions\nEqs.(\\ref{ALLu}-\\ref{ALLs})\ncorrespond to equal-time correlations in the Calogero-Sutherland\nmodel, one can expect Eqs.(\\ref{ALLu}-\\ref{ALLs}) with the\nkernel\nEq.(\\ref{kern}) to describe the low-temperature equal-time correlations in\nthe\nCalogero-Sutherland \nmodel \\cite{CSuth} for for all $s$ and the interaction constant\n$K=2/\\beta$.\n\nOne can use the kernel Eq.(28) to compute the spacing distribution\nfunction $P(s,g^{*})$ which can be compared with the corresponding\ndistribution function $P_{c}(s)$ obtained by the numerical diagonalization\nof the three-dimensional Anderson model at the critical point. Such a\ncomparison is done in Ref.\\cite{N} for $\\beta=1,2,4$. It turns out that\nidentifying the parameter $g^{*}$ from the fitting $P(s,g^{*})\\sim\ne^{-\\kappa(g^{*})s}$ with the far exponential tail of $P_{c}(s)\\sim\ne^{-\\kappa s}$ one reproduces the entire distribution function $P_{c}(s)$\nextremely well.\n\n A. M. T. acknowledges a kind hospitality of Abdus Salam \nICTP where this work was\n performed. \n\\begin{references}\n\\bibitem{RMT} M. L. Mehta, {\\it Random Matrices},\nAcademic Press, Boston 1991\n\\bibitem{Shkl} B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianidis,\nand H .B. Shore, Phys. Rev. B {\\bf 47} (1993) 11487\n\\bibitem{Zhar1} I. Kh. Zharekeshev and B. Kramer, Ann. Phys. (Leipzig)\n{\\bf 7} (1998) 442 \n\\bibitem{KLAA} V. E. Kravtsov, I. V. Lerner, B. L. Altshuler and A. G.\nAronov, Phys. Rev. Lett. {\\bf 72} (1994) 888\n\\bibitem{AKL} A. G. Aronov, V. E. Kravtsov and I. V. Lerner, Phys. Rev.\nLett. {\\bf 74} (1995) 1174\n\\bibitem{W} F. Wegner, Z. Phys. B {\\bf 36} (1980) 209\n\\bibitem{CCP} C. Castellani and L. Peliti, J. Phys. A {\\bf 19} (1986)\nL429\n\\bibitem{AlKL} B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, Sov.\nPhys. JETP {\\bf 64} (1986) 1352\n\\bibitem{Ch} J. T. Chalker, Physica A {\\bf 167} (1990) 253\n\\bibitem{W2} F. Wegner in: {\\it Localization and metal-insulator\ntransitions} edited by H. Fritsche and D. Adler, Plenum, New York 1985,\np. 337.\n\\bibitem{Mud} I. I. Kogan, C. Mudry, A. M. Tsvelik,\nPhys. Rev. Lett. {\\bf 77}, 707 (1996) ; J.-S. Caux,\nPhys. Rev. Lett. {\\bf 81 }, 4196 (1998). \n\\bibitem{KL2} V. E. Kravtsov and I. V. Lerner, Phys. Rev. Lett. {\\bf 74}\n(1995) 2563. \n\\bibitem{Mon} D. Braun, G. Montambaux and M. Pascaud, Phys. Rev. Lett.\n{\\bf 81} (1998) 1062\n\\bibitem{PSch} H. Potempa and L. Schweitzer, J. Phys. C {\\bf 10}\n(1998) L431\n\\bibitem{KY} V. E. Kravtsov and V. I. Yudson, Phys. Rev. Lett. {\\bf 82}\n(1999) 157\n\\bibitem{KMut} V. E. Kravtsov and K. A. Muttalib, Phys. Rev. Lett.\n{\\bf 79} (1997) 1913\n\\bibitem{MirFyod} A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada\nand T. H. Seligman, Phys. Rev. E\n{\\bf 54} (1996) 3221\n\\bibitem{AlLev} B. L. Altshuler and L. S. Levitov, Phys.Rep. {\\bf 288}\n(1997) 487;\nL. S. Levitov, Phys.Rev.Lett. {\\bf 64}(1990) 547; \nF. Evers and A. D. Mirlin, cond-mat/0001083.\n\\bibitem{Ef} K. B. Efetov, Adv. Phys. {\\bf 32} (1983) 53\n\\bibitem{SA} B. D. Simons and B. L. Altshuler, Phys. Rev. Lett. \n{\\bf 70} (1993) 4063\n\\bibitem{AA} A. V. Andreev and B. L. Altshuler, Phys. Rev. Lett. {\\bf 75}\n(1995) 902\n\\bibitem{KamMez} A. Kamenev and M. Mezard, cond-mat/9903001\n\\bibitem{KrMir} V. E. Kravtsov and A. D. Mirlin, JETP Letters {\\bf 60}\n(1994) 656\n\\bibitem{shura} see, for example, \nA. O. Gogolin, A. M. Tsvelik and A. A. Nersesyan,\n{\\it The Bosonization Approach to Strongly Correlated Systems},\nCambridge University Press 1998.\n\\bibitem{SLA} B. D. Simons, P. A. Lee and B. L. Altshuler, Phys. Rev.\nLett. {\\bf 70} (1993) 4122\n\\bibitem{CSuth} B. Sutherland in: {\\it Lecture notes in physics}, {\\bf\n242}, Springer\nVerlag, Berlin 1985\n\\bibitem{MNS} M.MOshe, H.Neuberger, and B.Shapiro, Phys.Rev.Lett., {\\bf\n73}, 1497 (1994).\n\\bibitem{N} S.M.Nishigaki, Phys. Rev. E {\\bf 59}, 2853 (1999). \n\\end{references}\n\\widetext\n\\end{document}\n\n\n\n"
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"name": "cond-mat0002120.extracted_bib",
"string": "\\bibitem{RMT} M. L. Mehta, {\\it Random Matrices},\nAcademic Press, Boston 1991\n\n\\bibitem{Shkl} B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianidis,\nand H .B. Shore, Phys. Rev. B {\\bf 47} (1993) 11487\n\n\\bibitem{Zhar1} I. Kh. Zharekeshev and B. Kramer, Ann. Phys. (Leipzig)\n{\\bf 7} (1998) 442 \n\n\\bibitem{KLAA} V. E. Kravtsov, I. V. Lerner, B. L. Altshuler and A. G.\nAronov, Phys. Rev. Lett. {\\bf 72} (1994) 888\n\n\\bibitem{AKL} A. G. Aronov, V. E. Kravtsov and I. V. Lerner, Phys. Rev.\nLett. {\\bf 74} (1995) 1174\n\n\\bibitem{W} F. Wegner, Z. Phys. B {\\bf 36} (1980) 209\n\n\\bibitem{CCP} C. Castellani and L. Peliti, J. Phys. A {\\bf 19} (1986)\nL429\n\n\\bibitem{AlKL} B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, Sov.\nPhys. JETP {\\bf 64} (1986) 1352\n\n\\bibitem{Ch} J. T. Chalker, Physica A {\\bf 167} (1990) 253\n\n\\bibitem{W2} F. Wegner in: {\\it Localization and metal-insulator\ntransitions} edited by H. Fritsche and D. Adler, Plenum, New York 1985,\np. 337.\n\n\\bibitem{Mud} I. I. Kogan, C. Mudry, A. M. Tsvelik,\nPhys. Rev. Lett. {\\bf 77}, 707 (1996) ; J.-S. Caux,\nPhys. Rev. Lett. {\\bf 81 }, 4196 (1998). \n\n\\bibitem{KL2} V. E. Kravtsov and I. V. Lerner, Phys. Rev. Lett. {\\bf 74}\n(1995) 2563. \n\n\\bibitem{Mon} D. Braun, G. Montambaux and M. Pascaud, Phys. Rev. Lett.\n{\\bf 81} (1998) 1062\n\n\\bibitem{PSch} H. Potempa and L. Schweitzer, J. Phys. C {\\bf 10}\n(1998) L431\n\n\\bibitem{KY} V. E. Kravtsov and V. I. Yudson, Phys. Rev. Lett. {\\bf 82}\n(1999) 157\n\n\\bibitem{KMut} V. E. Kravtsov and K. A. Muttalib, Phys. Rev. Lett.\n{\\bf 79} (1997) 1913\n\n\\bibitem{MirFyod} A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada\nand T. H. Seligman, Phys. Rev. E\n{\\bf 54} (1996) 3221\n\n\\bibitem{AlLev} B. L. Altshuler and L. S. Levitov, Phys.Rep. {\\bf 288}\n(1997) 487;\nL. S. Levitov, Phys.Rev.Lett. {\\bf 64}(1990) 547; \nF. Evers and A. D. Mirlin, cond-mat/0001083.\n\n\\bibitem{Ef} K. B. Efetov, Adv. Phys. {\\bf 32} (1983) 53\n\n\\bibitem{SA} B. D. Simons and B. L. Altshuler, Phys. Rev. Lett. \n{\\bf 70} (1993) 4063\n\n\\bibitem{AA} A. V. Andreev and B. L. Altshuler, Phys. Rev. Lett. {\\bf 75}\n(1995) 902\n\n\\bibitem{KamMez} A. Kamenev and M. Mezard, cond-mat/9903001\n\n\\bibitem{KrMir} V. E. Kravtsov and A. D. Mirlin, JETP Letters {\\bf 60}\n(1994) 656\n\n\\bibitem{shura} see, for example, \nA. O. Gogolin, A. M. Tsvelik and A. A. Nersesyan,\n{\\it The Bosonization Approach to Strongly Correlated Systems},\nCambridge University Press 1998.\n\n\\bibitem{SLA} B. D. Simons, P. A. Lee and B. L. Altshuler, Phys. Rev.\nLett. {\\bf 70} (1993) 4122\n\n\\bibitem{CSuth} B. Sutherland in: {\\it Lecture notes in physics}, {\\bf\n242}, Springer\nVerlag, Berlin 1985\n\n\\bibitem{MNS} M.MOshe, H.Neuberger, and B.Shapiro, Phys.Rev.Lett., {\\bf\n73}, 1497 (1994).\n\n\\bibitem{N} S.M.Nishigaki, Phys. Rev. E {\\bf 59}, 2853 (1999). \n"
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cond-mat0002121
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Comment on: "Auger decay, Spin-exchange, and their connection to Bose-Einstein condensation of excitons in $Cu_2O$ " \\ by G.M. Kavoulakis, A. Mysyrowicz ( cond-mat/0001438 )
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"name": "cond-mat0002121.tex",
"string": "\\documentstyle[aps,epsf]{revtex}\n\\begin{document}\n\n\\twocolumn\n\n\\title{Comment on: \"Auger decay, Spin-exchange, and their\nconnection to Bose-Einstein condensation \nof excitons in $Cu_2O$ \" \\\\\nby \nG.M. Kavoulakis, A. Mysyrowicz ( cond-mat/0001438 )}\n\n\\author{A. B. Kuklov}\n\\address{ Dept. of Eng. and Physics,\nCollege of Staten Island, CUNY\n }\n\n\\maketitle\n\n\\vskip0.5 cm\n\nIn the very recent work \\cite{KAV} a new mechanism \nof the interconversion of the triplet excitons\ninto singlet excitons in $Cu_2O$ has been suggested. In\naccordance with it, two triplet excitons\nwith the opposite (internal)\n angular momenta may collide and interconvert into a pair \nof the singlet excitons. \n Estimates presented in\n\\cite{KAV} show that such an interconversion\nis the most effective channel for the\ndecrease of the triplet exciton density. This\nquestions the commonly accepted view \nthat the Auger decay \nis the primary channel\nfor the decay (see in \\cite{OHARA}). \nFurthermore, it has been pointed out in \\cite{KAV}\nthat \nthe actual rate of the Auger decay must be\nseveral orders of magnitude less than \nit was previously calculated. \n\nIn this comment, it is suggested that \nthe mechanism \\cite{KAV} leads to a verifiable\nprediction about the rate of the decrease of the\ntriplet exciton density as function of the\npolarization of the incident laser light\ninducing the two-photon creation of the triplet excitons.\nAs a matter of fact, this rate can be greatly reduced\nif the polarization of the laser light is properly\nchosen.\n\nFirst, it is worth noting that\na necessary condition \nfor the triplet-singlet\ninterconversion \\cite{KAV} is that\nbefore the collision a pair of the triplet excitons\nhas total angular momentum $J=0$, that is, they form\na singlet state (invariant under the point\ngroup rotations). Thus, the rate of the \ninterconversion should be extremely sensitive\nto the initial state of the excitonic\nensemble. If this state is a thermal mixture\nwith random orientation of the excitonic\nspins, then each exciton can easily\nfind another one with the opposite $J_z$,\nso that the collision between them would result\nin a pair of the singlet excitons \\cite{KAV}.\nOn the contrary, if initially excitonic spins \nwere aligned, the collision induced\ninterconversion will be completely suppressed\nbecause of the conservation of the angular\nmomentum of the colliding pairs. Thus,\naligning angular momenta of the triplet \nexcitons in one way or another should\nprevent the triplet excitons from \ntransforming into the singlet excitons\nin accordance with the mechanism \\cite{KAV}.\nThis property can be used as a test for\nthe mechanism \\cite{KAV}.\n\nOne way for preparing\ntriplet excitons with preferential orientation\nof their spins is a direct creation\nof the coherent triplet excitons employed in\n\\cite{GOTO}. In this work, the triplet\nexcitons have been created by the two-photon \ndirect transitions: The incoming\nlaser field was tuned to the half of the\nexcitonic frequency, so that the two-photon\ntransition was in exact resonance with\nthe triplet excitons. Such a method\nallows to create a dense and coherent \ncloud of the triplet excitons. This\nis practically a direct mean of creating\na condensate of the triplet excitons.\nHowever,\nfast collision induced \ndecay of the triplet excitons may destroy\nthis coherence on the scale of few\nnanoseconds. On one hand, \nthis is the case if\nthe Auger decay is responsible for \nthe triplet exciton depletion because\nthis channel is not sensitive to the \norientation of the angular momenta of\nthe colliding pairs. On the other hand,\nif the primary \nchannel for the decay is\nthe mechanism \\cite{KAV}, it should be\npossible to use such a polarization\nof the cloud that the created \ncondensate of the triplet excitons\nis stable on much longer time scale.\n\nIt is possible to employ general\nsymmetry considerations, and make \na suggestion for the choice of \nthe orientation of the incoming laser\nfields in the geometry of the\nexperiment \\cite{GOTO}. Indeed,\nthe two-photon process \nof creation of the triplet\nexciton corresponds\nto the interaction term\nin the energy density \n\n\\begin{eqnarray}\nH_{le}=\\sum_{a=\\pm 1,0;i,j}\n\\psi^{\\dagger}_{(a)}Q^{(a)}_{ij}E_iE_j +H.c.\n\\label{1}\n\\end{eqnarray}\n\\noindent\nwhere $\\psi_{(a)}$ stands for the \ntriplet exciton Bose field which\nhas three projections $a=\\pm 1$\nand $a=0$ of the (internal)\nangular momenta; $E_j$ denotes\nthree space components of the incoming\nlaser field $\\sim E_j\\exp (-i\\omega t)$\nwhich is taken in the\nrotating wave approximation;\n$ Q^{(a)}_{ij}$ are corresponding matrices\nrepresenting the point symmetry\n(including spins) of $Cu_2O$ in such a way\nthat (\\ref{1}) is invariant under\nthis symmetry. The interaction\nterm responsible for the decay \\cite{KAV}\ncan be represented in the contact \nform (S-wave channel) as\n\n\\begin{eqnarray}\nH_{op}=g_{op}\\psi^{\\dagger}\\psi^{\\dagger}\n(\\psi_{(+1)} \\psi_{(-1)}+ \\psi_{(-1)} \n\\psi_{(+1)} +\n\\label{2} \\\\\n+ \\psi_{(0)} \\psi_{(0)}) + H.c.\n\\nonumber\n\\end{eqnarray}\n\\noindent\nwhere $\\psi$ is the field \nof the singlet excitons, and\n$g_{op}$ is the interaction constant\nsuch that the rate estimated\nin \\cite{KAV} is $\\sim g_{op}^2$.\nIt is worth noting that the term\nin the brackets in (\\ref{2})\nis invariant under\nthe symmetry group (including spins)\nof $Cu_2O$,\nwhere the excitonic states are formed\non the total angular momenta states\nof $Cu$ (see in \\cite{CUO}). If the\ntriplet excitons are created in such\na manner that this invariant is zero,\nthe interconversion process will be \n suppressed. \n\nThe induced fields $\\psi_{(a)}$ are\ngiven from (\\ref{1}) as\n$ \\psi_{(a)}\\sim \\sum_{i,j}\nQ^{(a)}_{ij}E_iE_j $. If \nsubstituted into\n(\\ref{2}), this will result in the\nterm describing four-photon\nproduction of the singlet\nexcitons which in general should be significant\nas long as\nthe mechanism \\cite{KAV}\nis dominant, provided the \ndensity $|\\psi_{(a)}|^2\n\\sim |\\sum_{i,j}\nQ^{(a)}_{ij}E_iE_j |^2$\n of the induced triplet excitons is\nlarge enough. In fact, the\nsymmetry of $Q^{(a)}_{ij}$ is the\nsame as that of the tensors of the\ndirect quadrupole transitions for the\ntriplet excitons. Using this, it is possible\nto find the energy density (\\ref{2}) as \n\n\\begin{eqnarray}\nH_{op}\\sim g_{op}\\psi^{\\dagger}\\psi^{\\dagger}\n(E_x^2E_y^2 + E_x^2E_z^2+ E_y^2E_z^2) + H.c.\n\\label{3}\n\\end{eqnarray}\n\\noindent\nwhere $E_x,\\, E_y,\\, E_z$ refer to the\ncomponents\nof the laser field with respect to\nthe principal cubic axes of $Cu_2O$.\nAccordingly,\nthe requirement \n\n\\begin{eqnarray}\nE_x^{-2}+E_y^{-2} + E_z^{-2}=0\n\\label{4}\n\\end{eqnarray}\n\\noindent\ninsures that the interconversion\nprocess \\cite{KAV} described\nby (\\ref{2}), (\\ref{3}) is zero in the\ndominant s-wave channel as long as no\nthermalization of the created triplet\nexcitons occurs.\n\nA solution of (\\ref{4}) for the \nsix components of $E_j=E'_j + iE''_j$,\nwhere $ E'_j $ and $E''_j$ stand for\nthe real and imaginary parts of $E_j$,\nrespectively, can be represented as \nfollows\n\n\\begin{eqnarray}\nE_j={1\\over \\epsilon'_j +i\\epsilon''_j}\n\\label{5}\n\\end{eqnarray}\n\\noindent\nin terms of the two auxiliary\nreal vectors $\\epsilon'_j,\\, \\epsilon''_j$\nwhich are arbitrary except for the\nconditions\n\n\\begin{eqnarray}\n\\sum_j \\epsilon'^2_j=\\sum_j\\epsilon''^2_j, \\quad \n\\sum_j \\epsilon'_j\\epsilon''_j=0.\n\\label{6}\n\\end{eqnarray}\n\\noindent\nThe interpretation of these conditions is\nstraightforward: the complex vector $E_j$\nrepresented by (\\ref{5}) \nshould be chosen in such a way that\nthe two auxiliary vectors $\\epsilon'_j$\nand $\\epsilon''_j$ are equal in magnitude\nto each other and are mutually orthogonal.\nFor the case of the incidence of the light \nalong the direction (1,1,1), the solution\nof (\\ref{6}), (\\ref{5}) gives \nthat $\\sum_j E'_j E''_j=0$ and\n$\\sum_j E'^2_j=\\sum_j E''^2$.\nNote that, given this, the\ninterconversion rate $k(T=0)=0$, as opposed\nto the rate of the \nAuger decay which is not sensitive to\ntemperature and the mutual orientation\nof the excitonic\nangular momenta of the colliding pairs.\n\nAt finite temperatures $T\\neq 0$,\nthe normal component - thermal triplet excitons -\nis present in addition to the \ncondensate. This component\nshould be characterized by zero net spin\npolarization due to the interaction with phonons.\nThus, the interconversion process \\cite{KAV}\nwill take place. Its rate $k(T)$ is proportional\nto the normal density $n'$. Thus, it \nmust be strongly temperature dependent. \nIt is straightforward to find an estimate for $k(T)$ \nin the temperature range \n$\\mu(0)<T \\ll T_0$,\nwhere $\\mu(0)=4\\pi an_0$ and $a,\\, n_0$\nstand\nfor the excitonic scattering length\nand the exciton condensate (spin-polarized)\ndensity, respectively; and $T_0$ denotes\nthe temperature of the excitonic Bose-Einstein\ncondensation. Indeed, in this range \nthe normal component\nbehaves almost as an ideal gas. Thus, the\nestimate follows from Eqs.(8-10) of Ref.\n\\cite{KAV} where the total density is replaced\nby the density of the normal component\n$n'=n_0(T/T_0)^{3/2}$ \\cite{BEC}. Accordingly,\nthe ratio of the rate of the interconversion\n$k(T)$ at $T\\neq 0$ for the spin-polarized excitonic\ncondensate to the rate $1/\\tau_{o,p}$ \\cite{KAV}\nestimated for the case of the non-polarized\ncloud is\n\n\\begin{eqnarray}\nk(T)\\tau_{o,p}\\approx {n^{\\prime}\\over n_0}=\n\\left({T\\over T_0}\\right)^{3/2} \\ll 1\n\\end{eqnarray}\n\\noindent\nfor the temperatures under consideration.\nAt temperatures $T<\\mu(0)$, further significant\nreduction\nof the rate should occur due to the interaction\nbetween the triplet exciton condensate and the normal component.\n\n\\begin{references}\n\\bibitem{KAV}\nG.M. Kavoulakis, A. Mysyrowicz,\ncond-mat/0001438.\n\\bibitem{OHARA}\nK.E. O'Hara, L. O. Suilleabhain, J.P. Wolfe,\n Phys. Rev. B {\\bf 60}, 10565 (1999);\nK.E. O'Hara, J. R. Gullingsrud, J.P. Wolfe,\n Phys. Rev. B {\\bf 60}, 10872 (1999).\n\\bibitem{GOTO}\nT. Goto, M. Y. Shen, S. Koyama, T. Yokouchi,\n Phys. Rev. B {\\bf 55}, 7609 (1997).\n\\bibitem{CUO}\nG.M. Kavoulakis, Yia-Chung Chang, G.Baym,\nPhys.Rev. B {\\bf 55}, 7593 (1997).\n\\bibitem{BEC}\nK. Huang, {\\it Statistical Mechanics}, John Wiley \\& Sons,\nNY, 1963.\n\\end{references}\n\\end{document}\n\n"
}
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[
{
"name": "cond-mat0002121.extracted_bib",
"string": "\\bibitem{KAV}\nG.M. Kavoulakis, A. Mysyrowicz,\ncond-mat/0001438.\n\n\\bibitem{OHARA}\nK.E. O'Hara, L. O. Suilleabhain, J.P. Wolfe,\n Phys. Rev. B {\\bf 60}, 10565 (1999);\nK.E. O'Hara, J. R. Gullingsrud, J.P. Wolfe,\n Phys. Rev. B {\\bf 60}, 10872 (1999).\n\n\\bibitem{GOTO}\nT. Goto, M. Y. Shen, S. Koyama, T. Yokouchi,\n Phys. Rev. B {\\bf 55}, 7609 (1997).\n\n\\bibitem{CUO}\nG.M. Kavoulakis, Yia-Chung Chang, G.Baym,\nPhys.Rev. B {\\bf 55}, 7593 (1997).\n\n\\bibitem{BEC}\nK. Huang, {\\it Statistical Mechanics}, John Wiley \\& Sons,\nNY, 1963.\n"
}
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cond-mat0002122
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Spin-fermion model near the quantum critical point: one-loop renormalization group results
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[
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"author": "Ar. Abanov and Andrey V. Chubukov"
}
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We consider spin and electronic properties of itinerant electron systems, described by the spin-fermion model, near the antiferromagnetic critical point. We expand in the inverse number of hot spots in the Brillouin zone, $N$ and present the results beyond previously studied $N = \infty $ limit. We found two new effects: (i) Fermi surface becomes nested at hot spots, and (ii) vertex corrections give rise to anomalous spin dynamics and change the dynamical critical exponent from $z=2$ to $z>2$. To first order in $1/N$ we found $z = 2N/(N-2)$ which for a physical $N=8$ yields $z\approx 2.67$.
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[
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"name": "renormalization_2.3.tex",
"string": "%\\documentstyle[preprint,prb,aps,refcheck]{revtex}\n\\documentstyle[prl,aps,twocolumn,epsf,floats]{revtex}\n%\\documentstyle[prb,aps]{revtex}\n%\\documentstyle[preprint,aps]{revtex}\n%\\documentstyle[version2,preprint,prb,aps]{revtex}\n\n%\\documentstyle{article}\n%\\textwidth\t6in\n%\\textheight\t7.7in\n%\\oddsidemargin\t0.2in\n%\\evensidemargin\t0.2in\n\n\\newcommand{\\di}{\\mbox{d}}\n\n\\begin{document}\n\n\\draft \n\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname %\n@twocolumnfalse\\endcsname\n\n%\\showrefnames\n\n\\date{\\today}\n\n\\title{ Spin-fermion model near the quantum critical point: \none-loop renormalization group results}\n \n\\author{Ar. Abanov and Andrey V. Chubukov}\n\n\\address\n{Department of Physics, University of Wisconsin, Madison, WI 53706}\n\n\n%\\address{........}\n\n\\maketitle\n\n\\begin{abstract}\nWe consider spin and electronic \n properties of itinerant electron systems, described by \nthe spin-fermion model, near \nthe antiferromagnetic critical point.\nWe expand in the inverse number of hot spots in the Brillouin zone, $N$ \nand present the results beyond previously studied $N = \\infty $ limit.\nWe found two new effects: \n(i) Fermi surface becomes nested at hot spots, \nand (ii) vertex corrections give rise to\n anomalous spin dynamics and change the dynamical critical exponent from $z=2$ to $z>2$.\n To first order in $1/N$ we found $z = 2N/(N-2)$ which for a physical $N=8$ \nyields $z\\approx 2.67$.\n\\end{abstract}\n\n\\pacs{PACS numbers: 74.20.Fg, 75.20Hr}\n]\n\n\\narrowtext\n\nThe problem of fermions interacting with critical antiferromagnetic spin fluctuations attracts a lot of attention at the moment due to its relevance to both\nhigh temperature superconductors and heavy-fermion materials~\\cite{review}.\n The key interest\nof the current studies is to understand the system \nbehavior near the quantum critical point (QCP) where the magnetic \ncorrelation length diverges at $T=0$~\\cite{scs}.\nAlthough in reality the QCP is almost always\n masked by either superconductivity or \nprecursor effects to superconductivity, \nthe vicinity of the QCP can be reached by \n varying external parameter such as pressure in\n heavy fermion compounds, or doping concentration in cuprates.\n \nIn this paper, we study the properties of the QCP without taking pairing fluctuations into account. We assume that the singularities\nassociated with the closeness to the QCP extent up to energies which \n exceed typical energies associated with the pairing.\nThis assumption is consistent with the recent calculations of the pairing instability temperature in cuprates~\\cite{acf}. \nFrom this perspective, the understanding\n of the properties of the QCP without pairing correlations \nis a necessary preliminary step for subsequent studies of the pairing problem.\n\nA detailed study of the antiferromagnetic QCP was performed by \nHertz~\\cite{hertz} and later by Millis~\\cite{millis} who chiefly \nfocused on finite $T$ properties near the QCP.\nThey both argued that if the Fermi surface contains hot spots (points \nseparated by antiferromagnetic momentum $Q$, see Fig.~\\ref{fig1}),\nthen spin excitations possess \npurely relaxational dynamics with $z=2$. They further argued that in\n $d=2$, $d+z=4$, i.e., the critical theory is at marginal dimension,\n in which case one should expect that spin-spin interaction yields\nat maximum logarithmical corrections to the relaxational dynamics. \nMillis argued~\\cite{millis} that this is true provided that \nthe effective Ginsburg-Landau functional for spins (obtained by integrating out the fermions) is an analytic function of the spin ordering field.\nThis is a'priori unclear as the expansion coefficients\nin the Ginsburg-Landau functional are made out of particle-hole bubbles and\ngenerally are sensitive to the closeness to quantum criticality due to \n feedback effect from near critical spin fluctuations \non the electronic subsystem. Millis however\ndemonstrated that the quartic term in the \nGinsburg-Landau functional is governed by high energy fermions and is \nfree from singularities. \n\nIn this communication, we, however, argue that the regular Ginsburg-Landau \nexpansion is not possible in 2D by the reasons different from those displayed in ~\\cite{hertz,millis}. Specifically, we argue that the damping term in the\nspin propagator (assumed to be linear in $\\omega$ in ~\\cite{hertz,millis}) is\nby itself made out of a particle hole bubble, and, contrary to $\\phi^4$ coefficient, is governed by low-energy fermions. We demonstrate that due to singular vertex corrections, the \nfrequency dependence of the spin damping term at the QCP is actually\n $\\omega^{1-\\alpha}$. In the one loop approximation, we find \n$\\alpha \\approx 0.25$. \n\nAnother issue which we study is the form of the renormalized \nquasiparticle Fermi surface near the \nmagnetic instability. In a mean-field SDW theory, the Fermi surface \nin a paramagnetic phase is not affected by the closeness to the QCP. \nBelow the instability, the doubling of the unit cell induces a \nshadow Fermi surface at $k_F +Q$,\n with the residue proportional to the deviation from criticality. This\ngives rise to the opening of the SDW gap near hot spots and eventually (for\na perfect antiferromagnetic long range order) yields a Fermi surface in the \nform of small pockets around $(\\pi/2,\\pi/2)$ and symmetry related \npoints (see Fig.~\\ref{fig1}a).\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n%\\epsfxsize=7.5cm\n\\epsfxsize=3.0in \n\\epsfysize=1.1in \n\\epsffile{artem5fig1n.eps}\n\\end{center}\n\\caption{\nThe Fermi surface with hot spots and the directions of Fermi velocities\nat hot spots separated by $Q$, and the evolution of the\n Fermi surface evolution for \n(a) mean-field ($N=\\infty)$ SDW theory,\n and (b) finite $N$. In both cases, the doubling of the unit cell due to antiferromagnetic SDW ordering introduces shadow Fermi surface and yields a gap opening near hot spots. At finite $N$, however, the Fermi surface at the quantum critical point \nbecomes nested at hot spots due to vanishing of renormalized $v_y$.}\n\\label{fig1} \n\\end{figure} \nSeveral groups argued~\\cite{ks} \nthat this mean-field scenario is modified by fluctuations, \nand the Fermi surface evolution towards hole pockets begins\nalready within the paramagnetic phase. \nWe show that the Fermi surface near hot spots does\nevolve as $\\xi \\rightarrow \\infty$, but due to strong fermionic damping (not \nconsidered in~\\cite{ks}), this evolution is a minor effect\nwhich at $\\xi = \\infty$ only gives rise to a nesting at the hot spots (see Fig. ~\\ref{fig1}b). \n\nThe point of departure for our analysis is the spin-fermion model \nwhich describes low-energy fermions \ninteracting with their own collective spin degrees of freedom. \nThe model is described by \n\\begin{eqnarray}\n{\\cal H} &=&\n \\sum_{{\\bf k},\\alpha} {\\bf v}_F ({\\bf k}-{\\bf k}_F) \n c^{\\dagger}_{{\\bf k},\\alpha} c_{{\\bf k},\\alpha}\n+ \\sum_q \\chi_0^{-1} ({\\bf q}) {\\bf S}_{\\bf q} {\\bf S}_{-{\\bf q}} +\\nonumber \\\\\n&&g \\sum_{{\\bf q,k},\\alpha,\\beta}~\nc^{\\dagger}_{{\\bf k+ q}, \\alpha}\\,\n{\\bf \\sigma}_{\\alpha,\\beta}\\, c_{{\\bf k},\\beta} \\cdot {\\bf S}_{\\bf -q}\\, .\n\\label{intham}\n\\end{eqnarray}\nHere $c^{\\dagger}_{{\\bf k}, \\alpha} $ is the fermionic creation operator\nfor an electron with momentum ${\\bf k}$ and spin projection $\\alpha$,\n$\\sigma_i$ are the Pauli matrices, and \n $g$ measures the strength of the \ninteraction between fermions and their collective bosonic spin\ndegrees of freedom. The latter are described by \n ${\\bf S}_{\\bf q}$ and are characterized by a bare spin susceptibility\nwhich is obtained by integrating out high-energy fermions.\n\nThis spin-fermion model can be viewed as \nthe appropriate low-energy theory for\nHubbard-type lattice fermion models provided that spin fluctuations are\nthe only low-energy degrees of freedom. This model explains a number \nof measured features of cuprates both in the normal and the \nsuperconducting states~\\cite{ac}. Its application to heavy-fermion \nmaterials is more problematic as in these compounds conduction \nelectrons and spins are independent degrees of freedom, and the \ndynamics of spin fluctuations may \nbe dominated by local Kondo physics rather than the \ninteraction with fermions~\\cite{piers}. \n\nThe form of the bare susceptibility $\\chi_0 (q)$ is an input \nfor the low-energy theory. We \nassume that $\\chi_0 (q)$ is non-singular and peaked at ${\\bf Q}$, i.e., \n$\\chi_0 ({\\bf q}) = \\chi_0/(\\xi ^{-2} + ({\\bf q}-{\\bf Q})^2)$, \nwhere $\\xi$ is the magnetic \ncorrelation length. \nIn principle, $\\chi_0$ can \nalso contain a nonuniversal frequency dependent term in the form \n$(\\omega/W)^2$ \nwhere $W$ is of order of fermionic bandwidth. We, however, will see that \nfor a Fermi surface with hot spots which we consider here, \nthis term will be overshadowed by a universal $\\omega^{1-\\alpha}$ \nterm produced by low-energy fermions.\n\nThe earlier studies of the spin-fermion model have demonstrated that the\nperturbative expansion for both fermionic and bosonic self-energies \nholds in power of $\\lambda = 3g^2 \\chi_0/(4\\pi v_F \\xi^{-1})$ where \n$v_F$ is the Fermi velocity at a hot spot. \nThis perturbation theory\n obviously does not converge when $\\xi \\rightarrow \\infty$. \nAs an alternative to a conventional perturbation theory, we suggested the\nexpansion in inverse number of hot spots in the Brillouin zone $N$ \n($=8$ in actual case)~\\cite{acf,ac}.\nPhysically, large $N$ implies that a spin fluctuation has many channels \nto decay into a particle-hole pair, \nwhich gives rise to a strong ($\\sim N$) spin damping rate. \nAt the same time, a fermion near a hot spot can only scatter into a single \nhot spot separated by ${\\bf Q}$. Power counting arguments than show that a \nlarge damping rate appears \nin the denominators of the fermionic self-energy and vertex corrections \nand makes them small to the extent of $1/N$. \nThe only exception from this rule is the fermionic self-energy due to a \nsingle spin fluctuation exchange, which contains\n a frequency dependent piece\n without $1/N$ prefactor due to an infrared singularity \nwhich has to be properly regularized~\\cite{chubukov}. \n\nThe set of coupled equations\nfor fermionic and bosonic self-energies at $N=\\infty$ has \nbeen solved in ~\\cite{chubukov}, and we merely quote the result. \nNear hot spots, we have\n\\begin{eqnarray}\nG_{k}^{-1}(\\omega)&=&\\omega -\\epsilon_k\n+\\Sigma (\\omega ), \n%%%%%G_{k+Q}^{-1}(\\omega _{m}) = i\\omega _{m} - \\epsilon_{k+Q}\n%+(v_x {\\tilde k}_x - v_y {\\tilde k}_y)\n%+\\Sigma _{k+Q}(\\omega _{m}),\n\\nonumber \\\\\n\\chi (q,\\Omega _{m})&=&\\chi _{0}\\xi\n^{2}/(1+({\\bf q}-{\\bf Q})^{2}\\xi ^{2} -i \\Pi _\\Omega). \n\\label{def}\n\\end{eqnarray}\nHere $\\epsilon_k = v_x {\\tilde k}_x + v_y {\\tilde k}_y$, \nwhere ${\\tilde k} = k - k_{hs}$, and $v_x$, $v_y$, which we set to be positive, are the components of the \nFermi velocity at a hot spot ($v^2_F = v^2_x + v^2_y$). \nThe fermionic self-energy $\\Sigma_k (\\omega)$ and the spin \npolarization operator $\\Pi_{\\Omega}$ are given by \n\\begin{equation}\n\\Sigma (\\omega)=2~\\lambda~\\frac{\\omega}{1+%\n\\sqrt{1 -\\frac{i|\\omega|}{\\omega _{sf}}}};~\\Pi _\\Omega=\\frac{%\n|\\Omega|}{\\omega _{sf}} \\label{input}\n%\\Sigma (\\omega)=2~\\lambda~\\omega/(1+\n%(1 -\\frac{i|\\omega|}{\\omega _{sf}})^{1/2});~\\Pi _\\Omega=\\frac{%\n%|\\Omega|}{\\omega _{sf}} \\label{input}\n\\end{equation}\nand $\\omega _{sf}=(4\\pi/N)~v_x v_y/(g^2 \\chi_0 \\xi ^{2})$. \n\nWe see from Eq.(\\ref{input}) that for $\\omega \\leq \\omega _{sf}$, $%\nG(k_{hs},\\omega )=Z/(\\omega +i\\omega |\\omega |/(4\\omega _{sf}))$, i.e., as\nlong as $\\xi $ is finite, the system preserves the Fermi-liquid behavior at\nthe lowest frequencies. The quasiparticle residue $Z$ however depends on \nthe interaction strength, $Z=(1+ \\lambda)^{-1}$, and \nprogressively goes down when the spin-fermion coupling increases. At\nlarger frequencies $\\omega \\geq \\omega _{sf}$, the system crosses over to a\nregion, which is in the basin of attraction of the quantum critical point, $%\n\\xi =\\infty $. In this region, $G^{-1}(k_{F},\\omega ) \\approx \n3 g~(v_x v_y \\chi_0 /\\pi Nv^2_F)^{1/2}~(i|\\omega|)^{1/2}\n{\\rm sgn}(\\omega )$~\\cite{chubukov,Mi}.\nAt the same time, spin propagator has a simple $z=2$ relaxational dynamics unperturbed by strong frequency dependence of the fermionic self-energy~\\cite{kadanof}.\n\nOur present goal is to go beyond $N=\\infty$ limit \nand analyze the role of $1/N$ corrections. \nThe $1/N$ terms give rise to two new features:\n vertex corrections which renormalize both fermionic and bosonic self-energies, and static \nfermionic self-energy $\\Sigma_k$. \nThe corresponding diagrams are presented in\n Fig~\\ref{fig2}.\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n%\\epsfxsize=7.5cm\n\\epsfxsize=2.8in \n\\epsfysize=1.1in \n\\epsffile{artem5fig3.eps}\n\\end{center}\n\\caption{ \nThe one-loop RG diagrams for the fermionic self-energy and vertex renormalization.\nSolid lines are full fermionic propagators, wavy lines are full spin susceptibilities, and black triangles are full vertices. The lowest order diagrams are\nobtained by replacing full internal lines and vertices by their $N=\\infty$ forms}\n\\label{fig2} \n\\end{figure} \n The lowest-order $1/N$ corrections have been calculated \nbefore~\\cite{chubukov,ch-morr}. Both vertex correction and the static\n self-energy are logarithmical in $\\xi$:\n\\begin{eqnarray}\n\\frac{\\Delta g}{g} &=& \\frac{Q(v)}{N}~ \\log {\\xi}, \n\\label{vert}\\\\\n\\Delta \\epsilon_k &=& - \\epsilon_{k+Q}~\n\\frac{12}{\\pi N}~ \\frac{v_x v_y}{v^2_F}~ \\log{\\xi}\n\\label{se}\n\\end{eqnarray}\nwhere $\\epsilon_{k+Q} = -v_x {\\tilde k}_x + v_y {\\tilde k}_y$, and\n%\\begin{equation}\n$Q(v) =(4/\\pi) \\arctan (v_{x}/v_{y})$\n%. This function is positive and\n%-\\frac{4}{\\pi^2} Re \\int_0^\\pi d\\phi~\n%\\frac{v_x v_y~\\log{\\sin[\\phi/2]}}{v^2_y\\cos^2 \\phi/2 - v^2_x \\sin^2 \\phi/2}.\n%\\label{q}\n%\\end{equation}\n%The analysis of (\\ref{q}) shows that\n%Clearly, $Q(v)$ is positive and non-singular for arbitrary \n%ratio of $v_y/v_x$, \ninterpolates between \n$Q=1$ for $v_x = v_y$, and $Q=2$ for $v_y \\rightarrow 0$. \n\nBesides, the $1/N$ corrections also contribute \n$(1/N) \\omega \\log \\xi$ to $G^{-1}_k (\\omega)$, but \nthis term is negligible compared to $\\Sigma (\\omega)$ and we neglect it.\n\n%Two comments are in order at this point. First, we neglected in \n%(\\ref{vert},\\ref{se}) regular $\\xi$-independent terms, which also \n%turn out to be universal, i.e., independent on the upper \n%cutoff~\\cite{ch-morr}. In practice, these terms can be neglected \n%only for sufficiently large $\\xi$, otherwise they reduce the strength\n%of the $1/N$ corrections. Thus for\n%$\\xi =2$ and $v_x \\approx v_y$ (as in optimally doped cuprates), \n%the total $1/N$ correction to $g$ is only $3\\%$ while (\\ref{vert}) yields a\n% $9\\%$ correction. \n%Second, w\n\nWe see from (\\ref{vert},\\ref{se}) that the $1/N$ corrections \nto the vertex and to the velocity of the excitations\nare almost decoupled from each other: the velocity renormalization does \nnot depend on the coupling strength at all, \nwhile the renormalization of the vertex depends on the ratio of velocities \nonly through a non-singular $Q(v)$.\nThis is a direct consequence of the fact that the dynamical part of the \nspin propagator is \n obtained self-consistently within the model. Indeed, the\noverall factors in $\\Delta\\epsilon_k$ and \n$\\Delta g/g$ are $g^2 (\\omega_{sf} \\xi^2)$ where \n$\\omega_{sf} \\xi^2$ comes from the dynamical part of the spin susceptibility.\nSince the fermionic damping is produced by\n the same spin-fermion interaction as the fermionic self-energy, \n$\\omega_{sf}$ scales as $1/g^2$, \nand the coupling constant disappears from the r.h.s. of (\\ref{vert},\\ref{se}). \n\nThe logarithmical dependence on $\\xi$ implies that $1/N$ expansion \nbreaks down near the QCP, and one has to sum up the series of the \nlogarithmical corrections. We will do this in a standard one-loop \napproximation by summing up the series in $(1/N) \\log \\xi$ but \nneglecting regular \n$1/N$ corrections to each term in the series. We verified that in \nthis approximation, the cancellation of the coupling constant holds \neven when $g$ is a running, scale dependent \n coupling. This in turn implies that one can separate the velocity \nrenormalization \nfrom the renormalization of the vertex to all orders in $1/N$.\n\nSeparating the corrections to $v_x$ and $v_y$ and \nperforming standard RG manipulations, we obtain a set of two RG equations\nfor the running $v^R_x$ and $v^R_y$\n\\begin{eqnarray}\n\\frac{\\di v^R_{x}}{\\di L}&=&\\frac{12}{\\pi N} \n\\frac{ (v^R_x)^2 v^R_y}{ (v^R_{x})^{2}+ (v^R_{y})^{2}} \\nonumber \\\\\n\\frac{\\di v^R_{y}}{\\di L}&=&-\\frac{12}{\\pi N} \n\\frac{ (v^R_{y})^2 v^R_x}{ (v^R_{x})^{2}+ (v^R_{y})^{2}}\\label{Renorm-velocity}\n\\end{eqnarray}\nwhere $L = \\log \\xi$. \nThe solution of these equations is straightforward, and yields\n\\begin{equation}\nv^R_x = v_x Z; v^R_y = v_y Z^{-1}; \nZ=\\left(1 + \\frac{24 L}{\\pi N} \\frac{v_y}{v_x}\\right)^{1/2}\n% v^R_x = v_x \\left(1 + \\frac{24 L}{\\pi N} \\frac{v_y}{v_x}\\right)^{1/2};\n% v^R_y = v_y \\left(1 + \\frac{24 L}{\\pi N} \\frac{v_y}{v_x}\\right)^{-1/2}\n\\label{sol}\n\\end{equation}\nwhere, we remind, $v_x$ and $v_y$ are the bare values of the \nvelocities (the ones which appear in the Hamiltonian). \n\nWe see that $v^R_y$ vanishes logarithmically at\n$\\xi \\rightarrow \\infty$. This implies that right at the QCP, the\nrenormalized velocities at $k_{hs}$ and $k_{hs}+Q$ are antiparallel \nto each other, i.e. the Fermi surface becomes nested at hot spots \n(see Fig~\\ref{fig1}b).\nThis nesting \n%is the first step in the\n%evolution of the Fermi surface towards hole pockets~\\cite{ch-morr}. If the \n%nesting occurred at some finite $\\xi$, as the lowest \n%order result (\\ref{se}) might have indicated,\n% then the evolution could proceed\n% further along the same path as in ~\\cite{ch-morr}, and the system \n%might develop strong SDW precursors already in a paramagnetic phase.\n% It turns out, however, that this process is \n%precluded because nesting at finite $\\xi$ (i.e., $v^R_y =0$ at\n%finite $v^R_x$)\n%would imply the vanishing of $\\omega_{sf}$ which in turn\n% appears as the overall factor in the velocity renormalization.\n%Earlier, one of us~\\cite{cms} considered a toy model in which spin damping was completely neglected. In this model, the feedback effect on the velocity renormalization is absent, and the \n%Fermi surface evolves towards hole pockets already in a paramagnetic phase. \n%\n%The vanishing of $v^R_y$ at $\\xi =\\infty$ \ncreates a ``bottle neck effect'' immediately below the criticality as the original and the \nshadow Fermi surfaces approach hot spots with equal derivatives (see Fig. ~\\ref{fig1}b). This obviously helps developing a SDW gap at $k_{hs}$ below the \nmagnetic instability. However, above the transition, no SDW precursors \nappear at $T=0$.\n\nAnother feature of the RG equations (\\ref{Renorm-velocity}) is\n that they leave the product $v_x v_y$ unchanged. This is a combination in which velocities appear in $\\omega_{sf}$. The fact that $v_x v_y$ is not renormalized implies that, without vertex renormalization, $\\omega_{sf} \\xi^2$ remains finite at $\\xi = \\infty$, i.e., spin fluctuations preserve a simple $z=2$ relaxational dynamics. \n%This result agrees with ~\\cite{hertz,millis}.\n\nWe now consider vertex renormalization. Using again the fact that $g^2 \\omega_{sf}$ does not depend on the running \ncoupling constant,\n one can straightforwardly extent the second-order result for the vertex renormalization, Eqn (\\ref{vert}), \nto the one-loop RG equation\n\\begin{equation}\n\\frac{\\di g^R}{\\di L} = \\frac{Q(v)}{N} g^R \n\\label{rgg}\n\\end{equation}\nwhere $g^R$ is a running coupling constant, and \n$Q(v)$ is the same as in (\\ref{vert}) but contain renormalized velocities \n$v^R_x$ and $v^R_y$. \n%Observe that the coupling constant {\\it increases} as $\\xi \\rightarrow \\infty$.\nAt the QCP, the dependence on $\\xi$ obviously transforms into the dependence\non frequency ($L = \\log \\xi \\rightarrow \n(1/2) \\log |\\omega_0/\\omega|$, where $\\omega_0$ is the upper cutoff). \nUsing the fact that for $\\xi \\rightarrow \\infty$,\n% $v_y$ vanishes logarithmically with decreasing frequency as\n $v^R_y/v^R_x \\approx N\\pi/24 L$ and expanding $Q(v)$ \nnear $v^R_y =0$, we find\n $Q(v) \\approx 2 (1 - (2/\\pi) v^R_y/v^R_x) = 2 - N/3 L$.\nSubstituting this result into \n(\\ref{rgg}) and solving the differential equation we obtain (${\\bar \\omega} = \\omega/\\omega_0$)\n\\begin{equation}\ng^R = g~ |\\bar{\\omega}|^{-1/N}~|\\log {\\bar\\omega}|^{-1/6} \n\\label{gr}\n\\end{equation}\n%Observe that the power of logarithm does not depend on $N$.\nWe see that at the QCP, running coupling constant diverges as $\\omega \\rightarrow 0$ roughly as $|\\omega|^{-1/N}$. Substituting this result into \nthe spin polarization operator and using the fact that \n$\\omega_{sf} \\propto (g^R)^{-2}$ we find that at the QCP, \n\\begin{equation}\n\\Pi_\\Omega \\propto |\\omega|^{\\frac{N-2}{N}}~ |\\log \\omega|^{-\\frac{1}{3}}\n\\label{chan}\n\\end{equation}\nThis result implies that vertex corrections change the\n dynamical exponent $z$ from its mean-field value $z=2$\n to $z = 2N/(N-2)$.\nFor $N=8$, this yields $z \\approx 2.67$ and $\\chi (Q,\\omega) \\propto \n|\\omega|^{1-\\alpha}$ where $\\alpha =0.75$. \n\nSingular vertex corrections also renormalize the fermionic self-energy\nas $\\Sigma (\\omega) \\propto g^R \\sqrt{|\\omega|}/v_F$. \nUsing the results for $g^R$ and $v_F \\approx v_x$\nwe obtain at criticality\n\\begin{equation}\n\\Sigma(\\omega) \\propto |\\omega|^{\\frac{N-2}{2N}}~ \n|\\log \\omega|^{-\\frac{2}{3}}\n\\label{sian}\n\\end{equation}\nEqs. (\\ref{sol}), (\\ref{chan}) and (\\ref{sian}) are the central results of the paper.\nWe see that the singular corrections to the Fermi velocity cause nesting but\ndo not affect the spin dynamics.\n The corrections to the vertex on the other hand do not affect velocities, but change the dynamical critical exponent for spin fluctuations. \n\n%So far, our analysis was restricted to $T=0$. \n%We now briefly discuss the form of the susceptibility at finite $T$.\n%Previous studies demonstrated~\\cite{scs,millis} that in the absence of\n%vertex corrections, the magnetic correlation length in $2D$ is \n%$\\xi^{-2} \\propto u T$ up to logarithmical prefactors, i.e., at the QCP,\n%$\\chi (Q,\\omega) \\propto T -i |\\omega|$, consistent with $\\omega/T$ scaling.\n%This linear dependence of $\\xi^{-2}$ is due to the scattering of a given \n%spin fluctuation by thermal spin excitations, and $u$ is the overall \n%factor in a $\\phi^2$ term in the Ginsburg-Landau potential.\n\n%So far, our analysis was restricted to $T=0$. \nWe now briefly discuss the form of the susceptibility at finite $T$.\nPrevious studies have demonstrated~\\cite{scs,millis} that the \nscattering of a given spin fluctuation by classical, thermal spin \nfluctuations yields, up to logarithmical prefactors,\n$\\xi^{-2} \\propto u T$, where \n$u$ is the coefficient \n in the $\\phi^4$ term in the Ginsburg-Landau potential. This implies that \nat the QCP, $\\chi (Q,\\omega) \\propto T -i |\\omega|$.\n\nWe, however, argue that the linear in $T$ and the \nlinear in $\\omega$ terms have completely different origin: the linear in\n$\\omega$ term comes from low-energies and is universal, while the linear \nin $T$ term comes from high energies and is model dependent.\nThis can be understood by \nanalyzing the particle-hole bubble at finite $T$. \nWe found that as long as one restricts with the linear expansion \nnear the Fermi surface, $\\Pi_\\Omega$ preserves exactly the same form \nas at $T=0$, to all orders in the perturbation theory. \n The temperature dependence of $\\Pi$ appears only due to a\nnonzero curvature of the electronic dispersion and is obviously sensitive\n to the details of the dispersion at energies comparable to the bandwidth. Similarly, the derivation of the Landau-Ginsburg potential from \n(\\ref{intham}) shows~\\cite{millis} that \n$u$ vanishes for linearized $\\epsilon_k$, \nand is finite only due to a nonzero curvature of the fermionic dispersion. \n\nThe different origins of $T$ and $\\omega$ dependences in $\\chi \n(Q,\\omega)$ imply that the anomalous $\\omega^{1-\\alpha}$ frequency \ndependence of \n$\\chi (Q,\\Omega)$ is not accompanied by the \nanomalous temperature dependence of $\\chi (Q,0)$ simply because\nfor high energy fermions, vertex corrections are non-singular. \nThis result \nimplies, in particular, that our theory does not explain \nanomalous spin dynamics observed in heavy fermion \n\\cite{l} despite the similarity in the\nexponent for the frequency dependence of $\\Pi_\\Omega$, because the \nexperimental data imply the existence of the $\\Omega/T$ scaling in \n$CeCu_{6-x}Au_x$~. More likely, the explanation should involve the \nlocal Kondo physics~\\cite{piers}. \n%This result implies, in particular, that although \n%our results yield almost the same anomalous despite the similarity in the\n%exponent for the frequency dependence of $\\Pi_\\Omega$,\n%our theory does not explain \n%anomalous spin dynamics observed in heavy fermion \n%$CeCu_{6-x}Au_x$~\\cite{l}. Although the \n%experimental data for those materials are consistent with the \n%$\\Omega/T$ scaling. \n\nFinally, we consider how anomalous vertex corrections affect the \nsuperconducting problem. We and Finkel'stein argued recently~\\cite{acf} \nthat at $\\xi = \\infty$, the \n kernel $K(\\omega, \\Omega)$ of the \nEliashberg-type gap equation for the $d-$wave anomalous vertex \n$F(\\Omega) = (\\pi T/2) \\sum_{\\omega} K(\\omega, \\Omega) F(\\omega)$ \nbehaves as $K (\\omega,\\Omega)\\propto g^2/(v^2_F \\Sigma^2 (\\omega) \n\\Pi_{\\Omega -\\omega})^{1/2}$\nAt $N=\\infty$, this yields (including prefactor)\n$K (\\omega,\\Omega)= |\\omega (\\Omega -\\omega)|^{-1/2}$. Although\nthis kernel is qualitatively different from the one in the BCS theory \nbecause it depends on both frequencies, it still scales as \ninverse frequency due to an interplay between a non-Fermi liquid \nform of the fermionic self-energy and\nthe absence of the gap in the spin susceptibility which mediates pairing.\nWe demonstrated in ~\\cite{acf} that \nthis inverse frequency dependence \ngives rise to a finite pairing instability temperature \neven when $\\xi = \\infty$.\n\nTo check how the kernel is affected by vertex corrections, we substitute\nthe results for $g^R$, $v_F$, $\\Sigma (\\omega)$ and $\\Pi_\\Omega$ \ninto $K(\\omega,\\Omega)$. We find after simple manipulations that \n%$K^{-1}(\\omega,\\Omega) \\propto |\\omega|^{(N+2)/(2N)} \n%|\\Omega - \\omega|^{(N-2)/(2N)}$. Simple power counting then shows that \n{\\it despite\nsingular vertex corrections, the kernel in the gap equation still scales \ninversely proportional to frequency}. A simple extension of the \nanalysis in ~\\cite{acf} then shows that the system still possesses a \npairing instability at $\\xi = \\infty$ at a temperature which differs \nfrom that without vertex \nrenormalization only by $1/N$ corrections. \n\nTo summarize, in this paper we considered the properties of the \nantiferromagnetic quantum critical point for itinerant electrons \nby expanding in the inverse number of hot spots in the Brillouin zone $N=8$. \nWe went beyond a self-consistent $N=\\infty$ theory and found two new effects:\n(i) Fermi surface becomes nested at hot spots which is a weak\n SDW precursor effect, and (ii) vertex corrections account for anomalous \nspin dynamics and change the dynamical critical exponent from \n$z=2$ to $z>2$. To first order in $1/N$ we found \n$z = 2N/(N-2)\\approx 2.67$. We argued that anomalous\n frequency dependence is not accompanied by anomalous $T$ dependence.\n\nIt is our pleasure to thank G. Blumberg, P. Coleman, \nM. Grilli, A. Finkel'stein, D. Khveshchenko, \nA. Millis, H. von L\\\"{o}hneysen, J. Schmalian, Q. Si, and A. Tsvelik \n for useful conversations. \nThe research was supported by NSF DMR-9979749.\n\\begin{references} \n\n\\bibitem{review} for a review, see e,g., \nN.D. Mathur et al, Nature, {\\bf 394}, 39 (1998); \nD.J. Scalapino, Phys. Rep. {\\bf 250}, 329 (1995);\nP. Monthoux and D. Pines, Phys. Rev. B {\\bf 47}, 6069 (1993).\n\n\\bibitem{scs} S. Sachdev, A. Chubukov, and A. Sokol, \nPhys Rev B {\\bf 51}, 14874 (1995); \nA. Gamba, M. Grilli, and C. Castellani, Nucl. Phys. B {\\bf 556}, 463 (1999); \nM. Lavagna and C. P\\'epin, cond-mat/0001259.\n\n\\bibitem{acf} Ar. Abanov, A. Chubukov, and A. M. Finkel'stein,\ncond-mat/9911445.\n%``{\\it $T_{c}$ and the angular\\dots}''\n\n\\bibitem{hertz} J. A. Hertz, Phys. Rev. B {\\bf 14}, 1165 (1976).\n\n\\bibitem{millis} A. J. Millis, Phys. Rev. B {\\bf 48}, 7183 (1993).\n\n\\bibitem{ks} A. Kampf and J.R. Schrieffer, \nJ. Phys. Chem. Solids {\\bf 56}, 1673 (1995);\nA. Chubukov, D. Morr, and K. Shakhnovich,\nPhilos. Mag. B {\\bf 74}, 563 (1996); \nJ. Schmalian, D. Pines and B. Stojkovic, Phys. Rev. B {\\bf 60}, 667 (1999).\n\n\\bibitem{ac} Ar. Abanov and A. Chubukov, \nPhys. Rev. Lett. {\\bf 83}, 1652 (1999). In this paper\nwe assumed for simplicity that the Fermi velocities at hot spots separated \nby $Q$ are almost orthogonal to each other, i.e.,\n $v_x \\approx v_y \\approx v_F/\\sqrt{2}$.\n% This is what one obtains for $t-t^\\prime$ form of fermionic dispersion with \n%$|t^\\prime| \\ll t$.\n\n\\bibitem{piers} P. Coleman, Physica B {\\bf 259-261}, 353 (1999) and references therein; Q. Si et al, Int. J. Mod. Phys. B {\\bf 13}, 2331 (1999).\n\n\\bibitem{chubukov} A. Chubukov, Europhys. Lett. {\\bf 44}, 655 (1997).\n\n\\bibitem{Mi} A.J. Millis, Phys. Rev. B {\\bf 45}, 13047 (1992).\n\n\\bibitem{kadanof} L. Kadanoff, Phys. Rev. {\\bf 132}, 2073 (1963).\n\n\\bibitem{ch-morr} A. Chubukov and D. Morr, Phys. Rep. {\\bf 288}, 355\n (1997).\n\n\\bibitem{l} O. Stockert et al, Phys. Rev. Lett. {\\bf 80}, 5627 (1998); \nsee also A. Schr\\\"{o}der et al, ibid, {\\bf 80}, 5623 (1988).\n%\\bibitem{pines} David + ...\n\n\n%%%%%\\bibitem{norman} M. R. Norman {\\it et al.}, Phys. Rev. Lett. {\\bf 79}, 3506 (1997). M.R. Norman and H. Ding, Phys. Rev. B {\\bf 57} R11089 (1998).\n\n%\\bibitem{shennat} Z-X. Shen et al, Science {\\bf 280}, 259 (1998).\n\n%\\bibitem{kotliar} A. Georges et al, Rev. Mod. Phys., {\\bf 68}, 13 (1996).\n\n%%%%%\\bibitem{solution} ``{\\it Some paper where it was solved for the normal state.}''\n\n%\\bibitem{kadanoff} L. Kadanoff, Phys. Rev. {\\bf 132}, 2073 (1963).\n\n%\\bibitem{renormalization} ``{\\it Something about renormalization technique.Zinn-Justin???}''\n\\end{references}\n\n\\end{document}\n\n\nThe absence of the linear in $T$ term in $\\chi (Q,0)$ also implies \nthat as $T$ is increased from the QCP,\n the spin correlation length scales as $\\xi^{-1} \\propto T$, just as in transitions with no spin damping.\nThis result \ndiffers from the one obtained in the studies of the phenomenological \n$\\sigma$ model with the damping term~\\cite{scs}. In the latter case, \nthe constraint\non the local spin value yielded $\\Omega/T$ scaling in $\\chi (Q,\\Omega)$ and \n$\\xi^{-2} \\propto T$ at the lowest temperatures, and a crossover to\n$\\xi^{-1} \\propto T$ at higher temperatures comparable to $J$. \nWe believe that the difference in the results is related to the fact that\nthe use of the normal state susceptibility for the sum rule is justified\nonly above the pairing instability temperature. This pairing instability \nis mediated by the dynamical spin susceptibility\nand is therefore intrinsic for electronic \nsubsystem interacting with spin fluctuations.\n The instability \ntemperature $T_{ins}$ (which actually signals the onset of \nthe pseudogap behavior~\\cite{acf}) \nis rather large -- \nof the order of $J$ at strong coupling. Therefore, the $\\sigma$ model \nresults (which ignore pairing) are actually valid only at high $T \\geq J$,\n where they are consistent with the results based on the spin-fermion model. \nBelow $T_{ins}$,\n the form of the spin propagator is modified by feedback effects from susceptibility, and the spin excitations gradually \nrecover the form of spin waves, i.e. the dynamical exponent evolves to $z=1$.\nThis effect has to be phenomenologically \nincluded into the $\\sigma-$model description. On the other hand,\n in the spin-fermion approach, the feedback effects \nnaturally emerge in the solution of the full set of \nEliashberg equations. \nThese effects change the frequency dependence of $\\Pi_\\Omega$, \nbut do not affect the static susceptibility, i.e., $\\xi^{-1}$ remains proportional to $T$ both above and below $T_{ins}$.\n \n\n\nunusual features of cuprates continue to attract a lot of attention \nmotivated by a search for a new physics in strongly correlated fermionic\nsystems. A lot of papers dealing with optimally doped and slightly \nunderdoped cuprates have been published recently~\\cite{alotofpapers}. \nThe intriguing physics of the highly underdoped systems is still \nattracting a lot of attention of the researches in the field.\n\nAn unusual feature of those materials is the proximity to two\ndifferent phases -- superconducting and antiferromagnetic. It means,\nthat one has to consider magnetic fluctuation dynamics alongside with \nelectron dynamics.\n\nAn ``obvious'' input parameter in such systems is the spin correlation\nlength $\\xi $. It varies with doping. The limit of highly underdoped\ncuprates corresponds to the limit $\\xi \\rightarrow \\infty $. It was\nargued in~\\cite{Tc} that in this limit a new ``universal'' physics\nemerges. In the present paper we are reporting the results of the\nrenormalization group (RG) analysis in this limit.\n%[{\\it Something about new, universal physics in the limit \n%$\\xi \\rightarrow \\infty $}~\\cite{Tc}]\n\nOne way of approaching the ``universal'' physics of such systems is\nby means of the spin-fermion model. According to this model the whole\n2D electronic system can be divided on two subsystems; the spin fluctuation\nwith the propagator peaked at the antiferromagnetic vector $Q$, and\nthe fermions. \n\nIt was argued~\\cite{chubukov} that this model is the \n\n\nEq. (\\ref{intham}) gives rise to fermionic and bosonic self-energies and \nis particularly relevant for fermions near\nhot spots -- the \npoints at the Fermi surface separated by $Q$. In cuprates, the hot spots are\nlocated near $(0,\\pi)$ and symmetry related points. \nThe presence of hot spots \n\n\nThe normal state properties of the spin-fermion model have recently \nbeen analyzed and\ncompared with the experiments~\\cite{chubukov,chub-morr}. \nIt was argued that the experimental situation in cuprates\ncorresponds to a strong coupling limit $R = {\\bar g}/v_F \\xi^{-1} \n\\gg 1$, where ${\\bar g} = g^2 \\chi_0$ is the measurable effective\n coupling constant.\nThe clearest experimental indication for this is the absence of the \nsharp quasiparticle peak in the normal state ARPES data for optimally doped\nand underdoped cuprates~\\cite{norman,shennat}.\nAt strong coupling, a conventional perturbation theory does not work, but it\nxturns out that \na variant of perturbative expansion is still possible.\nThe point is that at large $R$,\nthere is a single self-energy diagram which depends only on frequency and\nscales as $R$, and \ninfinite set of self-energy and \nvertex corrections diagrams which scale as powers of \n$\\log(R)/N$~\\cite{chubukov}, where $N=8$ is the number of hot spots. \nOne can then incorporate the $O(R)$ term into\nthe new zero-order theory and treat $\\log(R)/N$ terms perturbatively,\nin the RG formalism. \n%In practice, however,\n%he prefactors for the $\\log R$ vertex corrections are \n% very small such that one\n% can safely neglect these corrections except very near the\n%antiferromagnetic transition. \n%Below we just neglect vertex corrections and solve \n%the problem in the self-consistent Born (i.e. FLEX) approximation. \n\n\nThis approximation is already highly nontrivial \nand \nhas clear similarities with mean-field $d=\\infty$ theories\n~\\cite{kotliar}: it incorporates the\ndominant ($\\sim R$) self-energy correction which depends only on\nfrequency, \nand also includes the dominant bosonic self-energy which \ngives rise to a fermionic damping.\nThe corresponding set of\nself-consistent \nequations is presented in Eq.~(\\ref{set})\n\\begin{eqnarray}\n\\Sigma_{\\omega} &=&3 i g^2 \\int \\frac{d^2 q d\\Omega}{(2\\pi)^3}~ \nG(k +q, \\omega + \\Omega)~\\chi (q,\\Omega) \\nonumber \\\\\n\\Pi_\\Omega &=&-2Ni{\\bar g} \\int \\frac{d^2 k d\\omega}{(2\\pi)^3}~ \nG(k,\\omega)~G(k+Q, \\omega + \\Omega) \\label{set} \n\\end{eqnarray}\nwhere the spin-fluctuation Green's function is:\n\\begin{equation}\\label{spin-Green}\n\\chi ({\\bf q},\\omega )=\\frac{\\chi _{0}}{\\xi ^{-2}+(q-Q)^{2}-\\Pi (\\omega )}.\n\\end{equation}\nThe electron Green's function has the usual form:\n\\begin{equation}\\label{electron-green}\nG({\\bf q}, \\omega )=\\frac{1}{\\omega -v_{x}g_{x}-v_{y}q_{y}+\\Sigma (\\omega )},\n\\end{equation}\nwhere $v_{x}$ and $v_{y}$ are the $x$ and $y$ projections of the fermi\nvelocity at hot spots (see Fig~\\ref{fig1}). \n\nThe set of equations (\\ref{set}) has been solved before~\\cite{solution} \nand we just quote here the result. \n\nIn underdoped cuprates the spin correlation length $\\xi $ is large\nand we neglect $\\xi ^{-2}$ term in $\\chi$. It is clear that any finite\n$\\xi $ will stop renormalization at scales $\\omega \\sim \\xi ^{-1}$.\n\nAt the given order the electron self-energy \ndoes not depend on momentum. As was shown in~\\cite{kadanoff} in this case\nthe polarization operator also depends on frequency only\nand has a pure relaxational form. The strait-forward calculation gives\n\\begin{equation}\\label{gamma}\n\\Pi(\\omega ) =\\frac{i|\\omega |}{\\Gamma} \\mbox{, where }\n\\frac{1}{\\Gamma }=\\frac{N}{4\\pi }\\frac{\\bar{g}}{v_{x}v_{y}}\n\\end{equation}\nThe electron self-energy can now also be calculated:\n\\begin{equation}\\label{sigma}\n\\Sigma (\\omega )=\\sqrt{i\\gamma |\\omega |}\\mbox{sign}(\\omega )\n\\mbox{, where } \\gamma =\\frac{9\\bar{g}^{2}}{(2\\pi )^{2}}\n\\frac{\\Gamma }{v_{x}^{2}+v_{y}^{2}}.\n\\end{equation}\n\nThe equations (\\ref{gamma},\\ref{sigma}) show that at small frequencies\nthe dominant $\\omega $ dependence in the spin-fluctuation and electron\nGreen's functions comes from the polarization operator and the electron\nself-energy. It completely overshadows the bare $\\omega $ dependence of the\nGreen's functions. Both electrons and spin fluctuations have an overdumped\ndynamics. \n\nThis is the starting point for the RG analysis. Simple power counting\nshows, that the corrections to the fermi velocity ${\\bf v}$, electronic and\nspin fluctuation dumping constants $\\gamma $ and $\\Gamma $, \nand coupling constant $\\bar{g}$ scale as logarithms of upper cutoff. \n\nFirst we note, that in our approach$\\Gamma $ is found self-consistently. \nIt is not an independent parameter in the problem. All the dynamics in \nthe system comes from the electron-spin interaction. As a result\nthere is no small parameter in the problem. Nevertheless, in order\nto organize the perturbative series we employ a formal $1/N$ expansion\nand keep only the terms of the first order in this parameter.\n\nIn the spirit of RG analysis~\\cite{renormalization} \nwe first calculate the corrections to the \nfermi velocity using (\\ref{spin-Green}) and (\\ref{electron-green}) with\ndynamics given by (\\ref{gamma}) and (\\ref{sigma}). Then introducing\n$L=\\int _{\\lambda }^{\\Lambda }\\di \\omega /\\omega $, where $\\lambda $ and\n$\\Lambda $ are lower and upper cutoffs respectively we get:\n\n\nFor large $L$ the solution of the equations (\\ref{Renorm-velocity}) gives\n$v_{x}\\sim (L/N)^{1/2}$ and $v_{y}\\sim (L/N)^{-1/2}$. It shows that \nunder renormalization the velocities in the corresponding hot spots\nbecome antiparallel.\n\nWe note that those results did not require the vertex renormalization,\nbecause the coupling constant $\\bar{g}$ always comes in the combination\n$\\Gamma \\bar{g}$, which drops out of the equation according to (\\ref{gamma}).\n\nWe have already pointed out that the main contribution into system dynamics\ncomes from the processes that transfer electrons from one hot spot\nto another. It means that the most interesting vertex corrections are the ones\nto the vertex at a transfer momentum $q$ equal to antiferromagnetic\nmomentum $Q$. \n\nAt small energy scales $v_{x}\\gg v_{y}$ (see (\\ref{Renorm-velocity})).\nIt allows us to drop\n$k_{x}^{2}$ term in the spin-fluctuation propagator.\nThen taking into account the fact that the two electrons are in \nthe different hot spots the calculation of the logarithmical \nvertex correction gives:\n\\begin{equation}\\label{vertex}\n\\Delta g = g\\frac{\\bar{g}\\Gamma }{4\\pi v_{x}v_{y}}L\n\\end{equation}\nAgain, using (\\ref{gamma}), and rewriting (\\ref{vertex}) in differential form\nwe get the Gell-Mann-Low equation:\n\\begin{equation}\\label{vertex-Gell}\n\\frac{\\di \\bar{g}}{\\di L}=\\frac{2}{N}\\bar{g}\n\\end{equation}\nThis shows, that the coupling constant scales as $\\bar{g}\\sim \\omega ^{-2/N}$\nfor small $\\omega $.\n\nA quick estimation of the higher order corrections to the coupling constant\nshows that they will have a higher powers of $\\bar{g}\\Gamma /v_{x}v_{y}$.\nAccording to (\\ref{gamma}) those terms give additional powers \nof $1/N$ and can be neglected.\n\nOne can introduce a $d$-wave superconducting vertex into the \nHamiltonian (\\ref{intham}).\n\\begin{equation}\\label{SC-ham}\n\\Delta {\\cal H}_{sc}=\\sum_{{\\bf k},\\alpha}~\n\\hat{g}_{sc}({\\bf k})c^{\\dagger}_{{\\bf -k }, \\alpha}\\,\n c^{\\dagger}_{{\\bf k},\\alpha }+\\mbox{H.S.},\n\\end{equation}\nwhere $g_{sc}({\\bf k})$ has $d$-wave symmetry.\n\nAgain, the most interesting regions of the fermi surface are \nthose close to the \nhot spots~\\cite{Tc}. Denoting $g_{sc}=\\hat{g}_{sc}({\\bf k}_{h.s.})$ for\nsome hot spot $h.s.$ ($\\hat{g}_{sc}({\\bf k}_{h.s.}+{\\bf Q})=-g_{sc}$ \ndue to $d$-symmetry) we\nask about the renormalization of the $g_{sc}$ vertex.\n\\begin{equation}\\label{SC-vertex}\n\\Delta g_{sc}=g_{sc}\\frac{3}{8\\pi }\n\\frac{\\bar{g}\\Gamma }{v\\gamma \\sqrt{\\Gamma }}L\n\\end{equation}\n\nIn this situation unlike (\\ref{vertex}) both electrons are\nin the same hot spot. As a result instead of the combination\n$\\bar{g}\\Gamma /v_{x}v_{y}$ we get $\\bar{g}\\sqrt{\\Gamma }/v\\sqrt{\\gamma }$.\nOr using (\\ref{sigma}), in differential form we obtain \n\\begin{equation}\\label{SC-vertex-Gell}\n\\frac{\\di g_{sc}}{\\di L}=\\frac{1}{2}g_{sc}\n\\end{equation}\nOne can see that the superconducting vertex scales as $\\omega ^{-1/2}$ at\nsmall $\\omega $. We note that there is no $1/N$ in the answer.\nWe note that although 2D is the marginal dimensionality of the\nsystem, the Gell-Mann-Low function starts with the first power in $g$\ndue to the self-consistent solution (\\ref{gamma}).\n\nIn conclusion, we have considered the spin-fluctuation model of the High $Tc$\nmaterials. We argued that the dominant dynamics comes from the processes\nof scattering the electrons by the antiferromagnetic fluctuations between\nthe corresponding hot spots. The dissipative dynamics which comes from such\nprocesses completely overshadows the bare dynamics and should be found\nself consistently. \n\nUnder the renormalization the fermi velocities at the hot spots are tend\nto become antiparallel. Such that the angle $\\phi \\sim v_{y}/v_{x}\\sim\n|\\log \\max(\\xi ^{-1},\\omega)/N |^{-1}$. \n%One can as well see from \n%(\\ref{Renorm-velocity}) that for large $N$ the \n\nThe equation (\\ref{vertex-Gell}) shows, that the vertex grows under \nrenormalization $g\\sim \\max(\\xi ^{-1},\\omega )^{-2/N}$. Nevertheless,\nin the limit of large $N$ this change is negligible. \n\nThe situation with the superconducting vertex is completely different.\nDue to the fact that interacting electrons in this case are both at the\nsame hot spot the $1/N$ factor does not enter the final result\n$g_{sc}\\sim \\max(\\xi ^{-1},\\omega )^{-1/2}$.\n\n\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002122.extracted_bib",
"string": "\\bibitem{review} for a review, see e,g., \nN.D. Mathur et al, Nature, {\\bf 394}, 39 (1998); \nD.J. Scalapino, Phys. Rep. {\\bf 250}, 329 (1995);\nP. Monthoux and D. Pines, Phys. Rev. B {\\bf 47}, 6069 (1993).\n\n\n\\bibitem{scs} S. Sachdev, A. Chubukov, and A. Sokol, \nPhys Rev B {\\bf 51}, 14874 (1995); \nA. Gamba, M. Grilli, and C. Castellani, Nucl. Phys. B {\\bf 556}, 463 (1999); \nM. Lavagna and C. P\\'epin, cond-mat/0001259.\n\n\n\\bibitem{acf} Ar. Abanov, A. Chubukov, and A. M. Finkel'stein,\ncond-mat/9911445.\n%``{\\it $T_{c}$ and the angular\\dots}''\n\n\n\\bibitem{hertz} J. A. Hertz, Phys. Rev. B {\\bf 14}, 1165 (1976).\n\n\n\\bibitem{millis} A. J. Millis, Phys. Rev. B {\\bf 48}, 7183 (1993).\n\n\n\\bibitem{ks} A. Kampf and J.R. Schrieffer, \nJ. Phys. Chem. Solids {\\bf 56}, 1673 (1995);\nA. Chubukov, D. Morr, and K. Shakhnovich,\nPhilos. Mag. B {\\bf 74}, 563 (1996); \nJ. Schmalian, D. Pines and B. Stojkovic, Phys. Rev. B {\\bf 60}, 667 (1999).\n\n\n\\bibitem{ac} Ar. Abanov and A. Chubukov, \nPhys. Rev. Lett. {\\bf 83}, 1652 (1999). In this paper\nwe assumed for simplicity that the Fermi velocities at hot spots separated \nby $Q$ are almost orthogonal to each other, i.e.,\n $v_x \\approx v_y \\approx v_F/\\sqrt{2}$.\n% This is what one obtains for $t-t^\\prime$ form of fermionic dispersion with \n%$|t^\\prime| \\ll t$.\n\n\n\\bibitem{piers} P. Coleman, Physica B {\\bf 259-261}, 353 (1999) and references therein; Q. Si et al, Int. J. Mod. Phys. B {\\bf 13}, 2331 (1999).\n\n\n\\bibitem{chubukov} A. Chubukov, Europhys. Lett. {\\bf 44}, 655 (1997).\n\n\n\\bibitem{Mi} A.J. Millis, Phys. Rev. B {\\bf 45}, 13047 (1992).\n\n\n\\bibitem{kadanof} L. Kadanoff, Phys. Rev. {\\bf 132}, 2073 (1963).\n\n\n\\bibitem{ch-morr} A. Chubukov and D. Morr, Phys. Rep. {\\bf 288}, 355\n (1997).\n\n\n\\bibitem{l} O. Stockert et al, Phys. Rev. Lett. {\\bf 80}, 5627 (1998); \nsee also A. Schr\\\"{o}der et al, ibid, {\\bf 80}, 5623 (1988).\n%\n\\bibitem{pines} David + ...\n\n\n%%%%%\n\\bibitem{norman} M. R. Norman {\\it et al.}, Phys. Rev. Lett. {\\bf 79}, 3506 (1997). M.R. Norman and H. Ding, Phys. Rev. B {\\bf 57} R11089 (1998).\n\n%\n\\bibitem{shennat} Z-X. Shen et al, Science {\\bf 280}, 259 (1998).\n\n%\n\\bibitem{kotliar} A. Georges et al, Rev. Mod. Phys., {\\bf 68}, 13 (1996).\n\n%%%%%\n\\bibitem{solution} ``{\\it Some paper where it was solved for the normal state.}''\n\n%\n\\bibitem{kadanoff} L. Kadanoff, Phys. Rev. {\\bf 132}, 2073 (1963).\n\n%\n\\bibitem{renormalization} ``{\\it Something about renormalization technique.Zinn-Justin???}''\n"
}
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cond-mat0002123
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Vortex Pinning and Dynamics in Layered Superconductors with Periodic Pinning Arrays
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[
{
"author": "Charles Reichhardt $^{a}$"
}
] |
We examine vortex dynamics and pinning in layered superconductors using three-dimensional molecular dynamics simulations of magnetically interacting pancake vortices. Our model treats the magnetic interactions of the pancakes exactly, with long-range logarithmic interactions both within and between planes. At the matching field the vortices are aligned with the pinning array. As a function of tilt angle for the pinning arrays a series of commensuration effects occur, seen as peaks in the critical current, due to pancakes finding a favorable alignment. \vspace{1pc}
|
[
{
"name": "houston.tex",
"string": "\\documentstyle[twoside,fleqn,espcrc2,epsf]{article}\n\\input{epsf}\n\n\\title{Vortex Pinning and Dynamics in Layered Superconductors with Periodic\nPinning Arrays} \n\\author{Charles Reichhardt $^{\\rm a}$, Cynthia J. Olson \n\\address{Department of Physics, University of California, Davis, California\n95616}\nand\nNiels 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 examine vortex dynamics and pinning in layered superconductors\nusing three-dimensional molecular dynamics simulations of magnetically\ninteracting pancake vortices. \nOur model treats the\nmagnetic interactions of the pancakes exactly, with long-range logarithmic\ninteractions both within and between planes. \nAt the matching field the vortices are aligned with the pinning array. \nAs a function of tilt angle for the pinning arrays a series of commensuration\neffects occur, seen as peaks in the critical current, due to\npancakes finding a favorable alignment.\n\\vspace{1pc}\n\\end{abstract}\n\n% typeset front matter (including abstract)\n\\maketitle\n\nIn superconductors with periodic pinning arrays interesting commensurability\neffects occur when\nthe periodicity of the vortex lattice matches the \nperiodicity of the pinning lattice. \nExperiments \\cite{Baert,Schuller} \nand simulations \\cite{Reichhardt} \nso far have been done with\nthin film superconductors where the vortex lattice and pinning \ncan be considered two-dimensional. The case of vortex lattices interacting\nwith a periodic pinning array in a layered 3D superconductor has not been\nstudied. Such a system would correspond to an anisotropic superconductor\nsuch as BSCCO \nwith a periodic arrangement of columnar defects. In this system \nthe $z$-direction\nbecomes important as the applied field or the pinning array is tilted. \nThe dynamical effects of\nvortices moving in periodic pinning arrays \nin such a system have not been examined,\nin particular\nhow the vortex lattice structure of the moving state differs from that\nof the pinned state. \nTo study vortex pinning and \ndynamics in layered superconductors,\nwe 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 using a rapidly converging summation method \\cite{ngj}. \n\nThe overdamped equation of motion, for $T=0$, 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 and $\\rho$ and $z$ are the distance\nbetween pancakes in cylindrical coordinates.\n%, and we take $\\eta=1$.\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)=-\\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$E_{1}(x) = \n\\int^{\\infty}_{\\rho}\\exp(-x/\\lambda)/\\rho^{\\prime}d\\rho^{\\prime}$ and\n$\\epsilon_{0} = \\Phi_{0}^{2}/(4\\pi\\xi)^{2}$.\nThe pinning is placed in a square array of parabolic traps with \na radius $r_{p}$ much smaller than the distance between pins. \nThe location of the pinning sites is the same in every layer corresponding\nto correlated defects. \nA driving force $f_{d}$ is slowly increased and the vortex velocities are \nmeasured. \nHere we consider the \nfirst matching field\ncase where the\nnumber of vortices $N_{v}$ equals the number of pinning sites $N_{p}$. \nWe conduct a series of \nsimulations in which the pinning sites are tilted \nat an increasing angle\nwith respect to the \n$z$-axis. We will only consider driving \nthat produces vortex motion\ntransverse to the direction of the tilt angle. \nWe examine systems with 8 layers \ncontaining 64 vortices and pins in each layer. \nWork for larger systems, varied \nfields and coupling strength will be presented elsewhere \\cite{toappear}.\n\nIn Fig. 1(a) we present \nthe critical depinning force $f_{dp}^{c}$ \nas a function of tilt angle $ \\theta$. \nHere $f_{dp}^{c}$ peaks \nat $\\theta = 0^{\\circ}$ when the \npancakes are aligned with pins on all layers. \nAs $ \\theta$ is increased\n$f_{dp}^{c}$ drops. \nFor small tilt angles $ \\theta < 5^{\\circ}$ \nthe vortex lines tilt with the pins. For larger angles the vortex lines\nrealign in the $z$ direction. \nThe depinning force $f_{dp}^{c}$ \nwill then remain low \nas only one pancake in the straight vortex line will \nbe sitting at a pinning site. At \n$\\theta = 45^{\\circ}$ $f_{dp}^{c}$ shows a peak of the same magnitude \nas the peak at \n$\\theta = 0$. At this tilt angle, and also for any angle\nsatisfying \n$ \\theta = \\tan^{-1}(n)$ where $n$ is an \ninteger, the pinning sites are again\naligned in the $z$-direction so that a vortex line can be formed that\nis also aligned in the $z$-direction with \nall the pancakes in a single vortex being able to \nsit in a pinning site. \nThere are also peaks in $f_{dp}^{c}$ at \n$\\theta = 26.6^{\\circ}$ and $56.3^{\\circ}$.\nAt these angles the pancakes again sit on all the pinning sites.\nThe individual vortex lines now consists of half the number of pancakes\nas at $\\theta = 0.0^{\\circ}$; however, there are now twice as many vortex\nlines with the pancakes from an individual vortex line being coupled in\nevery other layer. \nThe view from the\n$z$-direction as shown in Fig.~1 for these angles \nindicates that the vortex lattice is now rectangular with\ntwice as many vortex lines as at the other angles. \nAt $\\theta = 36.9^{\\circ}$ a smaller peak is observed. The vortex structure\nat this angle will be presented elsewhere \\cite{toappear}. \n \nIn (b) and (c) we show the vortex structures for the pinned phase and\nmoving phase for $\\theta = 1.5^{\\circ}$ as seen from the $z$-direction. \nIn (b) the vortices can be seen to stay aligned with the pins. In (c)\nfor $f_{d} > f_{dp}^{c}$ the vortices realign with the z-direction. \nSuch a transition from a tilted to straight vortex lattice as a function\nof drive may be visible with neutron scattering experiments.\n\nWe acknowledge helpful discussions with L. N. Bulaevskii, A. Kolton,\nR.T. Scalettar, and G. T. Zim{\\' a}nyi. \nThis work was supported by CLC and CULAR (LANL/UC) and by the Director,\nOffice of Adv. Scientific Comp. Res., Div. of Math., Information and\nComp. Sciences, U.S.~DoE contract DE-AC03-76SF00098. \n\n\\begin{thebibliography}{9}\n\n\\bibitem{Baert} M.~Baert {\\it et al.}, Phys.~Rev.~Lett.~{\\bf 74}, 3269 \n(1995); K.~Harada {\\it et al.}, Science {\\bf 271}, 1393 (1996). \n\n\\bibitem{Schuller} \nJ.I.~Mart\\'{\\i}n {\\it et al.}, Phys.~Rev.~Lett.~{\\bf 79}, 1929 (1997);\nY.~Fasano {\\it et al.}, Phys.~Rev.~B {\\bf 60}, R15047 (1999). \n\n\\bibitem{Reichhardt}\nC.~Reichhardt, C.J.~Olson and F.~Nori, Phys.~Rev.~B {\\bf 57}, 7937 (1998).\n\n\\bibitem{clem} J.R. Clem, Phys. Rev. B {\\bf 43}, 7837 (1990).\n\n\\bibitem{ngj} N. Gr{\\o}nbech-Jensen, Comp. Phys. Comm. {\\bf 119}, 115 (1999).\n\n\\bibitem{toappear} C.~Reichhardt, \nC.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%\\epsfxsize=3.5in \n\\epsfbox{Fig.2.ps}}\n\\caption{The critical depinning force $f_{dp}^{c}$ versus the tilt angle\n$\\theta$ of the pinning sites. The vortex arrangements as seen from the\n$z$-direction are outlined for different tilt angles $\\theta = 0$ left and\n$\\theta = 26.6$ right.\n(b) shows the pinned vortex arrangement for $\\theta = 1.5^{\\circ}$ where\nthe vortices stay aligned with the pins. (c) shows the moving vortex \nstate for $\\theta = 1.5^{\\circ}$ \nwhere the vortices have realigned with the $z$ direction.}\n\\label{fig:fig1}\n\\end{figure}\n\n\\end{document}\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002123.extracted_bib",
"string": "\\begin{thebibliography}{9}\n\n\\bibitem{Baert} M.~Baert {\\it et al.}, Phys.~Rev.~Lett.~{\\bf 74}, 3269 \n(1995); K.~Harada {\\it et al.}, Science {\\bf 271}, 1393 (1996). \n\n\\bibitem{Schuller} \nJ.I.~Mart\\'{\\i}n {\\it et al.}, Phys.~Rev.~Lett.~{\\bf 79}, 1929 (1997);\nY.~Fasano {\\it et al.}, Phys.~Rev.~B {\\bf 60}, R15047 (1999). \n\n\\bibitem{Reichhardt}\nC.~Reichhardt, C.J.~Olson and F.~Nori, Phys.~Rev.~B {\\bf 57}, 7937 (1998).\n\n\\bibitem{clem} J.R. Clem, Phys. Rev. B {\\bf 43}, 7837 (1990).\n\n\\bibitem{ngj} N. Gr{\\o}nbech-Jensen, Comp. Phys. Comm. {\\bf 119}, 115 (1999).\n\n\\bibitem{toappear} C.~Reichhardt, \nC.J. Olson and N. Gr{\\o}nbech-Jensen, to be published.\n\n\\end{thebibliography}"
}
] |
cond-mat0002124
|
Ab initio treatment of electron correlations in polymers: lithium hydride chain and beryllium hydride polymer
|
[
{
"author": "Ayjamal Abdurahman"
},
{
"author": "$^{1}$~\\cite{email} Alok Shukla"
},
{
"author": "$^{2}$~\\cite{add1} and Michael Dolg$^{1}$\\cite{add2}"
}
] |
Correlated {\em ab initio\/} electronic structure calculations are reported for the polymers lithium hydride chain $[LiH]_{\infty}$ and beryllium hydride $[Be_{2}H_{4}]_{\infty}$. First, employing a Wannier-function-based approach, the systems are studied at the Hartree-Fock level, by considering chains, simulating the infinite polymers. Subsequently, for the model system $[LiH]_{\infty}$, the correlation effects are computed by considering virtual excitations from the occupied Hartree-Fock Wannier functions of the infinite chain into the complementary space of localized unoccupied orbitals, employing a full-configuration-interaction scheme. For $[Be_{2}H_{4}]_{\infty}$, however, the electron correlation contributions to its ground state energy are calculated by considering finite clusters of increasing size modelling the system. Methods such as M$\o$ller--Plesset second--order perturbation theory and coupled--cluster singles, doubles and triples level of theory were employed. Equilibrium geometry, cohesive energy and polymerization energy are presented for both polymers, and the rapid convergence of electron correlation effects, when based upon a localized orbital scheme, is demonstrated.
|
[
{
"name": "cond-mat0002124.tex",
"string": "\\documentstyle[prb,aps,preprint]{revtex} \n%\\documentstyle[prb,twocolumn,aps,floats,tighten]{revtex}\n%\\sloppy\n\\begin{document}\n\\tighten\n\\draft \n\\title{Ab initio treatment of electron correlations in polymers: \nlithium hydride chain and beryllium hydride polymer}\n\\author{Ayjamal Abdurahman,$^{1}$~\\cite{email} \nAlok Shukla,$^{2}$~\\cite{add1} and Michael Dolg$^{1}$\\cite{add2}}\n\\address{$^1$ Max-Planck-Institut f\\\"ur Physik komplexer Systeme,\nN\\\"othnitzer Str. 38, D-01187 Dresden, Germany}\n\\address{$^2$Department of Physics and The Optical Sciences Center, \nUniversity of Arizona, Tucson, AZ 85721}\n\\maketitle\n\n%\\begin{center}\n%\\today\n%\\end{center}\n\n%\\newpage\n\n\\begin{abstract}\nCorrelated {\\em ab initio\\/} electronic structure calculations are \nreported for the polymers lithium hydride chain $[LiH]_{\\infty}$ and \nberyllium hydride $[Be_{2}H_{4}]_{\\infty}$. First, employing \na Wannier-function-based approach, the systems are studied\nat the Hartree-Fock level, by considering chains, simulating the infinite \npolymers. Subsequently, for the model system $[LiH]_{\\infty}$, \nthe correlation effects are computed by considering virtual \nexcitations from the occupied Hartree-Fock Wannier functions \nof the infinite chain into the complementary space of localized unoccupied \norbitals, employing a full-configuration-interaction scheme.\nFor $[Be_{2}H_{4}]_{\\infty}$, however, the electron correlation\ncontributions to its ground state energy are calculated by considering finite \nclusters of increasing size modelling the system. Methods such as\nM$\\o$ller--Plesset second--order perturbation theory and coupled--cluster singles, \ndoubles and triples level of theory were employed. Equilibrium geometry, cohesive \nenergy and polymerization energy are presented for both polymers, and \nthe rapid convergence of electron correlation effects, when based upon a \nlocalized orbital scheme, is demonstrated.\n\\end{abstract}\n\n%\\pacs{}\n%{\\bf Keywords:} polymers, lithium hydride, beryllium hydride, \n%electron correlations, Wannier orbitals. \\\\\n\n\\section{Introduction}\n\\label{intro}\nPolymers represent a class of one--dimensional infinite crystalline systems \nwhere {\\em ab initio\\/} Hartree--Fock (HF) self--consistent field (SCF) \nmethods are well developed~\\cite{ladik}. An available program \npackage is CRYSTAL~\\cite{crystal}. However, in order to be able to \ncalculate the structural and electronic properties of polymers with \nan accuracy that allows a meaningful comparison with experiment, \nit is usually necessary to include the effects of electron correlations \ninto the theory. The most widely used approach here is density--functional \ntheory (DFT). Despite its indisputable success in solid state physics \nand computational chemistry as a computationally cheap routine tool \nfor large-scale investigations, DFT has the drawback that results depend \nhighly on the chosen functional, and cannot be improved in a systematic way.\nWave-function--based quantum chemical {\\em ab initio } techniques on the\nother hand are free from this flaw, and provide a large array of \nmethods of different accuracy and computational cost. Thus it is \ndesirable to extend their applicability to infinite systems such as polymers.\n\nElectron correlations are mostly a local effect and therefore \nlocalized molecular orbitals are preferable to the canonical HF solutions \nfor the treatment of large molecules.~\\cite{hampel} Similarly, in infinite\nsystems (localized) Wannier functions provide a better starting point \nfor an {\\em ab initio\\/} treatment of electronic correlations than the\n(canonical) Bloch functions. Previous studies of polymers obtained the \nWannier orbitals from an {\\em a posteriori\\/} localization of the Bloch \nfunctions according to a given prescription.~\\cite{ladik}\nDuring the last years, in our group a HF approach was developed \nwhich allows the direct determination of Wannier orbitals within the \nSCF process.~\\cite{shukla1}\nVarious applications to one- and three-dimensional infinite systems \nproved the numerical equivalence of our Wannier--function--based HF approach \nto the conventional Bloch--function--based \ncounterpart.~\\cite{shukla2,shukla3}\n\nIn this paper HF--SCF calculations and subsequent correlation energy calculations \nare presented for the lithium hydride chain $[LiH]_{\\infty}$ and the beryllium \nhydride polymer $[Be_{2}H_{4}]_{\\infty}$. As a simple, but due to its ionic \ncharacter, nontrivial model polymer, the lithium hydride chain system has \nbeen previously dealt with in a number of studies.~\\cite{shukla3,teramae,tunega}. \nIn the present contribution we extend our previous calculation ~\\cite{shukla3} to\na wave-function-based {\\em ab initio\\/} study of electron correlation effects\nusing a combination of the full configuration interaction (FCI) method and the\nthe so-called incremental scheme.~\\cite{stoll1,stoll2,stoll3} \nThe latter approach consists basically in an expansion of the total correlation \nenergy per unit cell in terms of interactions of increasing complexity among the \nelectrons assigned to localized orbitals (Wannier functions) comprising the \npolymer under consideration.\nThe electron correlation energy increments needed to establish the total\nenergy per unit cell are evaluated by considering virtual excitations from a \nsmall region of space in and around the reference cell, keeping the \nelectrons of the rest of the crystal frozen at the Hartree--Fock (HF) level. \nThe fast convergence of the incremental expansion allows to truncate it at\nrelatively low order and thus to calculate the correlation energy of an\ninfinite system without modelling it as a finite cluster.\nHowever, neither the FCI method nor the incremental approach based on\npolymer Wannier orbitals can at present be used for systems with a more\ncomplicated unit cell. Therefore, the second system investigated by us, the \nberyllium hydride polymer, was treated at the coupled-cluster (CC) and\nM$\\o$ller--Plesset second--order perturbation (MP2) level of theory. Starting \nfrom the Wannier HF data the correlation corrections to the total energy \nper unit cell were derived from quantum chemical calculations of \nfinite model systems using the MOLPRO molecular orbital \n{\\em ab initio\\/} program package.~\\cite{molpro} \nTo our knowledge, this system was studied at the HF level two decades ago\nby Karpfen using the crystal orbital method, i.e., without including\ncorrelation effects.~\\cite{karpfen} Recently, its monomer beryllium\ndihydride $BeH_{2}$ has been well characterized theoretically using reliable \nab initio and density functional theory methods.~\\cite{hinze,jursic} \n\nThe remainder of the paper is organized as follows. In section \\ref{methods} \nthe applied methods are briefly described. The calculations and results are \nthen presented in section \\ref{results}. Finally, a summary is given in \nsection \\ref{summary}. \n\n\\section{Applied methods}\n\\label{methods}\nSection \\ref{wanhf} gives brief outline for the theory within a \nrestricted HF (RHF) framework. Sections \\ref{inc}\nand \\ref{sa}, respectively, describe the incremental \nscheme and a simple approach, to compute electron correlation effects in \npolymers. \n\\subsection{Wannier--orbital--based Hartree--Fock approach}\n\\label{wanhf}\nOur approach, described in more detail in previous publications.~\\cite\n{shukla1,shukla2,shukla3} is based upon the direct determination\nof the orthonormal Wannier--type (localized) \norbitals for the polymer. Denoting by $\\mid\\alpha(\\mathbf{R}_{j})\\rangle$ \nthe Wannier orbitals of a unit cell located at lattice \nvector $\\mathbf{R}_{j}$, the set $\\{ |\\alpha({\\bf R}_{i})\\rangle;\n{\\alpha} = 1, n_c; j = 1, N \\}$ \nspans the occupied HF space. \n Here, $n_{c}$ is the number of orbitals per unit cell, \nand $N (\\to \\infty)$ is the total number of unit cells in the system.\nIn our previous work we showed that one\ncan obtain $n_c$ RHF Wannier functions, $\\{|\\alpha \\rangle, \\; \\alpha =1,n_c\\}$ occupied by $2n_c$ electrons \nlocalized in the reference unit cell (denoted ${\\cal C}$) by solving the \nequations~\\cite{shukla1,shukla2,shukla3} \n\\begin{equation}\n( T + U\n + \\sum_{\\beta} (2 J_{\\beta}- K_{\\beta}) \n+\\sum_{k \\in{\\cal N}} \\sum_{\\gamma} \\lambda_{\\gamma}^{k} \n|\\gamma({\\bf R}_{k})\\rangle\n\\langle\\gamma({\\bf R}_{k})| ) |\\alpha\\rangle \n = \\epsilon_{\\alpha} |\\alpha\\rangle\n\\mbox{,}\n\\label{eq-rhf} \n\\end{equation} \nwhere $T$ represents the kinetic-energy operator, $U$ represents\nthe interaction of the electrons of ${\\cal C}$ with the nuclei\nof the whole of the crystal, while $J_{\\beta}$, $K_{\\beta}$, \nrespectively, represent the Coulomb and exchange interactions felt\nby the electrons occupying the $\\beta$-th Wannier function \nof ${\\cal C}$, due to the rest of the electrons of the infinite system.\nThe first three terms of Eq.(\\ref{eq-rhf}) constitute the canonical \nHartree-Fock operator, while the last term is a projection\noperator which makes the orbitals localized in ${\\cal C}$ orthogonal to those \nlocalized in the unit cells in the immediate neighborhood of ${\\cal C}$\nby means of infinitely high shift parameters $\\lambda_{\\gamma}^{k}$'s. These\nneighborhood unit cells, whose origins are labeled by lattice vectors\n${\\bf R}_{k}$, are collectively referred to as ${\\cal N}$. The \nprojection operators along with the shift\nparameters play the role of a localizing potential in the Fock matrix, and \nonce self-consistency has been achieved, the occupied eigenvectors of \nEq.(\\ref{eq-rhf}) are localized in ${\\cal C}$, and are orthogonal to the \norbitals of ${\\cal N}$---thus making them Wannier \nfunctions~\\cite{shukla1,shukla2,shukla3}. As far as the\northogonality of the orbitals of ${\\cal C}$ to those contained in unit cells\nbeyond ${\\cal N}$ is concerned, it should be automatic for systems with\na band gap once ${\\cal N}$ has been chosen to be large enough. Based upon\nour past experience regarding a suitable\nchoice of ${\\cal N}$,~\\cite{shukla1,shukla2,shukla3} in the \npresent calculation we\nincluded up to the third nearest-neighbor unit cells in ${\\cal N}$.\nFor the details concerning the computation\nof various terms involving lattice sums ($U$, $J$, and $K$) involved in\nEq. (\\ref{eq-rhf}) for the\ncase of polymers, we refer the reader to \nreference~{\\cite{shukla3}}.\n\n\\subsection{Incremental method}\n\\label{inc}\nElectron correlation effects in the ground states of a large number of \nthree-dimensional ionic and covalent solids,~\\cite{inc-cal} as well as \npolymers~\\cite{yu} have been studied with the incremental scheme. \nAll these calculations used localized orbitals of finite clusters as a\nbasis set for the correlation treatment. In the present work on the \nlithium hydride chain we use directly the Wannier--functions of the infinite\nsystem. A related study of the three--dimensional lithium hydride solid \nhas been published elsewhere~\\cite{lih}.\n\nThe correlation energy per unit cell is expanded as\n\\begin{equation}\nE_{corr} = \\sum_{i} \\varepsilon_{i}\n+ \\sum_{<ij>} \\Delta\\varepsilon_{ij}\n+ \\sum_{<ijk>} \\Delta\\varepsilon_{ijk}\n+ ... \\label{eq-inc}\n\\end{equation}\nwhere the summation over $i$ involves Wannier functions located the \nreference cell, while those over $j$ and $k$ include all the Wannier\nfunctions of the crystal. \nThe ``one--body\" increments $\\varepsilon_{i}$ = $\\Delta\\varepsilon_{i}$\nare computed by considering virtual excitations only from the $i$-th \nWannier function, freezing the rest of the polymer at the HF level.\nThe ``two--body\" increments $\\Delta\\varepsilon_{ij}$ are defined as \n$\\Delta\\varepsilon_{ij}$ = \n$\\varepsilon_{ij}-(\\Delta\\varepsilon_{i}+\\Delta\\varepsilon_{j})$ \nwhere $\\varepsilon_{ij}$ is the correlation energy of the system obtained \nby correlating two distinct Wannier functions ${i}$ and ${j}$. Thus \n$\\Delta\\varepsilon_{ij}$ represents the correlation contribution \nof electrons localized on two ``bodies\" ${i}$ and ${j}$. \nHigher--order increments are defined in an analogous way. \nFinally, summing up all increments, with the proper\nweight factors (according to their occurrence in the unit cell \nof the polymer), one obtains the exact correlation energy per \nunit cell of the infinite system.\nIn order to get reliable results a size--extensive correlation \nmethod should be used, although non size--extensive schemes \nalso may provide reasonable estimates if the\nincremental expansion is truncated at low order.\nIn the present work for the lithium hydride chain we choose the strictly\nsize-extensive full configuration interaction (FCI) method. \nAs mentioned earlier, when computing the correlation contributions via\nEq. (\\ref{eq-inc}), except for the orbitals involved (say orbitals $i$ and $j$\nfor the two-body increment $\\Delta \\epsilon_{ij}$), the rest of the occupied \nWannier orbitals of the\ninfinite solid are held frozen at the HF level. The region\ncontaining these frozen orbitals plays the role of the ``environment''\nfor the electrons involved in the correlated calculations, and its\ncontribution can be absorbed in the so-called ``environment potential''\n$U^{\\mbox{env}}$ defined as \n\\begin{equation}\n U^{\\mbox{env}}_{pq}= \\sum_{\\alpha({\\bf R}_{j}) \\in {\\cal E} } \n( 2 \\langle p \\alpha({\\bf R}_{j})|\\frac{1}{r_{12}}|q \\alpha({\\bf R}_{j}) \\rangle\n - \\langle p \\alpha({\\bf R}_{j})|\\frac{1}{r_{12}}|\\alpha({\\bf R}_{j}) q \\rangle\n ) \\; \\mbox{,} \\label{eq-uenv}\n\\end{equation}\nwhere ${\\cal E}$ represents the unit cells of the environment, $p$ and $q$ \nare two arbitrary basis functions, and the factor of\ntwo in the first term is due to the spin summation.\nThe sum of Eq.(\\ref{eq-uenv}) involves infinite lattice sum over the\nenvironment unit cells, and is computed\nby simply subtracting from the lattice summed $J$ and $K$ integrals (cf.\nEq. (\\ref{eq-rhf})) obtained at the\nend of the HF iterations, the contributions corresponding to the orbitals\nbeing correlated. Once \n$U^{\\mbox{env}}_{pq}$ has been computed, one is left with an effective\nHamiltonian involving a finite number of electrons located in the\nregion whose Wannier orbitals are being correlated. Physically speaking \n$U^{\\mbox{env}}_{pq}$ represents the influence of the environment electrons \non the electrons being correlated, explicitly.\nIn the present calculations the Li $1s^2$ core shell was also kept frozen, and its\ncontribution was also included in $U^{\\mbox{env}}_{pq}$. The basis functions\n$p$ and $q$ were restricted to those of the reference cell and the adjacent cells \nup to the third-nearest neighbors.\n\nThe virtual orbitals used for computing the correlation effects were also\nlocalized. They were obtained by first orthogonalizing the basis set\nto the occupied space by using corresponding projection operators, as \nsuggested by Pulay.~\\cite{pulay} Subsequently the basis functions\nare orthogonalized to each other using the symmetric-orthogonalization\nprocedure, yielding a localized and orthonormal virtual orbital set.~\\cite{lih} \nThe number of virtual orbitals per unit cell considered for a specific\nincrement corresponds to the number of basis functions per unit cell\nminus the number of occupied orbitals per unit cell. The virtual orbitals have\nbeen expanded in the same basis set as described above for $U^{\\mbox{env}}_{pq}$.\n\\subsection{A simple approach}\n\\label{sa}\nIn principle the total energy $E_{tot}$ per $[Be_{2}H_{4}]$ unit cell of \nberyllium hydride may be obtained as the limit\n%\n%\n\\begin{equation}\nE=\\lim_{n \\to \\infty}{E(Be_{2n+1}H_{4n+2})\\over n} ,\n\\end{equation} \ni.e., by performing calculations for increasingly long oligomers \n$H(BeH_{2})_{2n}BeH$. In order to reduce finite-size effects due to the \ntermination of the oligomers by one beryllium and two hydrogen atoms \nsaturating the dangling bonds of ${\\cdot } (BeH_{2})_{2n} {\\cdot }$ , one \nmay consider instead\n%\n%\n\\begin{equation}\nE=\\lim_{n \\to \\infty}\\bigtriangleup E_{n}=\\lim_{n \\to\n\\infty}\\biggl[E(Be_{2n+3}H_{4n+6})-E(Be_{2n+1}H_{4n+2})\\biggr],\n\\label{eq-sa}\n\\end{equation}\ni.e., the energy change between subsequent oligomers differing by \na single unit cell. Therefore, identical unit cells were used as building\nblocks for both oligomers, i.e., the geometrical optimization was \nrestricted only to parameters relevant for the polymer beryllium hydride.\\\\ \nSince the convergence of $\\bigtriangleup E_{n}$ with respect to n is much\nfaster for the correlation contributions than for the HF energy, and HF\nprograms treating the infinite system are at hand (CRYSTAL, WANNIER), we use\nEq. (\\ref{eq-sa}) only for the correlation energy per unit cell. \nThis approach has previously been used successfully in calculations \nfor trans-polyacetylene,~\\cite{yu} and some boron-nitrogen \npolymers.~\\cite{ayjamal} \n\\section{Calculations and Results}\n\\label{results}\n\\subsection{$[LiH]_{\\infty}$} \nHF ground state calculations are a necessary prerequisite for the application \nof the incremental approach to electron correlation. \nWe performed such calculations for a lithium hydride chain oriented \nalong the x-axis using the WANNIER code \\cite{wannier}.\nThe reference cell contained hydrogen at the (0,0,0) and \nlithium at the $({\\it a}/2,0,0)$, where ${\\it a}$ is lattice constant.\nWe adopted the extended basis set optimized by Dovesi \n{\\em et al.}~\\cite{dovesi} \nFirst, all--electron Wannier HF calculations were performed \nat the different\nlattice constants in the range 2.8--4.0 (\\AA) and the total HF energy \nper unit cell for various lattice constants \nnear the equilibrium was fitted to a cubic polynomial in \norder to derive the ground state HF equilibrium lattice constant \nand total energy.\nAfter determining the Wannier orbitals for each value of the lattice \nconstant, the corresponding \nFCI calculations were performed by means of the incremental scheme. \nThe expansion of the correlation energy per unit cell\nwas restricted to one-- and two--body increments, and included \ninteractions up to third--nearest neighbor unit cells. Contributions \nfrom higher order increments as well as from interactions\nbetween more distant cells proved to be negligible.\nThe equilibrium values for the FCI energy per unit cell \nand the lattice constant were determined as described for \nthe HF results.\nThe main contribution of 98.8 \\% to the correlation energy per unit cell \nat the equilibrium geometry ($E$=$-0.0307 a.u.$) comes from the one-body term.\nTwo-body terms for first--, second-- and third--nearest neighbors contribute \nwith 1.15, 0.01 and 0.001 \\%, respectively. Our results are summarized\nin table \\ref{t1}. It is quite obvious from table \\ref{t1} that, as a function\nof distance, the two-body correlation effects converge very rapidly.\n\nSince the Li basis set used here is suitable only for the ionic LiH molecule, \nwe cannot get \na good result for the atomic reference energy of the neutral Li atom \n(which is needed to determine the cohesive energy).\n Therefore, for this almost ideally ionic chain the cohesive energies both at the HF and the\ncorrelated level are obtained \nby subtracting the electron affinities (EA) and ionization potential \n(IP) from the dissociation energy calculated with respect to the ions \n$Li^{+}$ and $H^{-}$. The HF values of EA and IP are determined using \nthe finite--difference atomic HF program MCHF \\cite{mchf}. The experimental \nvalues of EA and IP were taken as the CI limit, i.e., disregarding the very\nsmall relativistic effects. For the \npolymerization energy we optimized the Li--H distance for the\n$^1\\Sigma^+$ ground state of the monomer at the HF and CI level.\nOur results are summarized in table \\ref{t2}. It is clear from table\n\\ref{t2} that, as expected, correlation effects contribute significantly\nto the cohesive energy. However, they do not make any significant contribution\nto the lattice constant of the system.\n\\subsection{$[Be_{2}H_{4}]_{\\infty}$}\nBeryllium hydride has attracted considerable interest as a rocket fuel on\naccount of its high heat of combustion. It has also been considered as a\nmoderator for nuclear reactors. From the previous studies \\cite{bery} \nwe also know that it is poisonous and difficult to prepare for experiment. \nEven though there is no or very little experimental information about \nthe polymer, it has been studied theoretically using reliable \n{\\em ab initio\\/} methods at the HF level by Karpfen \\cite{karpfen}. \nIn the present work we have studied this polymer at the HF and the correlated \nlevel. The Wannier--orbital--based HF--SCF approach, \ncoupled-cluster (CC), and M$\\o$ller--Plesset \nsecond--order perturbation (MP2) theory were employed to \ndetermine the equilibrium structures and total energies per unit cell.\n In our calculations the unit cell included two beryllium and four\nhydrogen atoms and has a perfect tetrahedral structure with all four Be-H bond \ndistances equal, i.e., there are two HBeHBeH planes that are perpendicular to\neach other, with the beryllium atoms in their crossings. \nIn the cluster approximation the unit cell is terminated by one beryllium and two\nhydrogen atoms. In this structure the terminal beryllium atoms have \ntrigonal coordination while all others are distorted tetrahedrons.\n First we optimized the structure \nof this polymer at the HF--SCF level using the CRYSTAL \\cite{crystal} program. \nThe total HF energies obtained with the CRYSTAL program were then taken as an \ninput for a re-optimization at the MP2, CCSD (CC singles and \ndoubles) and CCSD(T) (CCSD with a perturbative estimate of triples) level. \nThe correlation energy contributions at each geometry have been calculated \nwith the MOLPRO molecular orbital {\\em ab initio\\/} program \npackage \\cite{molpro} \nby using the simplified finite--cluster approach in which we \nput {\\em n}=3 in Eq. (\\ref{eq-sa}). In this system the correlation\nenergy converges rapidly with respect to cluster size, i.e., for {\\em n}=3, \none finds $\\bigtriangleup E_{4}-\\bigtriangleup E_{3}{\\approx}10^{-6}$ a.u..\nWe have optimized the beryllium--hydride bond length $(r_{BeH})$ and \nthe lattice constant (a). We adopted polarized valence \ndouble--zeta ( 6--31G$^{**}$) basis sets for beryllium and for hydrogen. \nThe polarization functions \nconsisted of a single p--type exponent of $0.75$ Bohr$^{-2}$ on hydrogen and \nsingle d--type exponents of $0.4$ Bohr$^{-2}$ on beryllium. \nIn our HF calculations for polymers we optimized the most diffuse s--type \nexponent, which is less than $0.1$ in the original 6--31G$^{**}$ basis set,\nand obtained $0.15$. A smaller value causes linear dependencies in the basis \nset when applied in the infinite system.\\\\ \nWe have also calculated the cohesive energy per unit cell at the HF \nand correlated level. The atomic HF--SCF, MP2, CCSD and CCSD(T) reference \nenergies (Be: $-14.5668$ a.u., $-14.5928$ a.u., $-14.6131$ a.u. \nand $-14.6131$ a.u.; H: $-0.4982$ a.u.) were obtained with the original \n6--31G$^{**}$ basis sets. In addition to the cohesive energy, we \nhave also calculated the polymerization energy. The geometry of \nthe monomers was optimized at \nthe SCF, MP2, CCSD, and CCSD(T) \nlevels of theory employing the MOLPRO program \\cite{molpro}. Our final results are\nsummarized in table \\ref{t3}. Due to the absence of experimental data \nor theoretical results at the correlated level, we compare our \nresult only at the HF level. To the best of our \nknowledge, only Karpfen \\cite{karpfen} has performed a geometry \noptimization for this polymer within an ab initio crystal \nHartree--Fock approach and \nhis results are also given in table \\ref{t3}. Our beryllium--hydrogen \nbond length is in good agreement with the one obtained by Karpfen,\nbut our HF energy is lower 0.05 a.u. than the value of\nKarpfen \\cite{karpfen}. A possible reason is the use of d functions\nin our basis sets. \n\\section{Summary}\n\\label{summary}\nIn conclusion, given a well-localized basis set of Wannier orbitals\nsize-extensive standard quantum chemical methods such as \nfull configuration interaction, coupled-cluster or many-body perturbation \ntheory can be applied to evaluate ground state properties of polymers.\nRapid convergence of the incremental expansion of the correlation energy\nis obtained for ionic systems, e.g., the simple model of the lithium\nhydride chain. In beryllium hydride polymer electron correlation \naccounts for 12--14{\\%} of the cohesive energy and 22--24{\\%} of the \npolymerization energy at all three levels \nof theory and reduces the lattice constant.\nIn all the cases it was demonstrated that the use of localized orbitals\nleads to a rapid convergence of electron correlation effects, thus making it \npossible for one to compute the electron correlation effects of \ninfinite systems.\n\n\\begin{thebibliography}{99}\n%\n\\bibitem[\\ast]{email}{email: ayjamal@mpipks-dresden.mpg.de}\n\\bibitem[\\dagger]{add1}{Present address: Department of Physics, Indian \nInstitute of Technology, Powai, Mumbai 400 076, India}\n\\bibitem[\\ddagger]{add2}{Permanent address: Institut f\\\"ur Physikalische und Theoretische\nChemie, Universit\\\"at Bonn, Wegeler Str. 12, 53115 Bonn, Germany}\n\\bibitem{ladik} See, e.g., J. J. Ladik, Quantum Theory of Polymers as Solids. \nPlenum press, New York, NY (1988).\n%\n\\bibitem{crystal} R. Dovesi, V.R. Saunders, C. Roetti, M. Causa,\nN.M. Harrison, R. Orlando, E. Apra CRYSTAL95 user's manual. University of\nTurin, Italy (1996).\n%\n\\bibitem{hampel} C. Hampel and H. -J. Werner, J. Chem. Phys. {\\bf 104}, 6286\n(1996), and references cited therein.\n%\n\\bibitem{shukla1}A. Shukla, M. Dolg, H.Stoll and P. Fulde, Chem. Phys. Lett.\n{\\bf 262}, 213 (1996).\n%\n\\bibitem{shukla2} A. Shukla, M. Dolg, P. Fulde, and H. Stoll, Phys. Rev. B, {\\bf 57}, 1471 (1998).\n%\n\\bibitem{shukla3} A. Shukla, M. Dolg, and H. Stoll, Phys. Rev. B {\\bf 58}, 4325 (1998).\n%\n\\bibitem{teramae} H. Teramae, Theor. Chim. Acta {\\bf 94}, 311 (1996).\n%\n\\bibitem{tunega} D. Tunega, J. Noga, Theor. Chim. Acta {\\bf 100}, 78 (1998). %\n%\n\\bibitem{stoll1} H. Stoll, Phys. Rev. B {\\bf 46}, 6700 (1991).\n%\n\\bibitem{stoll2} H. Stoll, J. Chem. Phys. {\\bf 97}, 8449 (1992).\n%\n\\bibitem{stoll3} H. Stoll, Chem. Phys. Lett. {\\bf 191}, 548 (1992).\n%\n\\bibitem{molpro} H.-J. Werner and P. Knowles, MOLPRO, 1994, is a package \nof {\\em ab initio\\/} programs written by H.-J. Werner and P.J. Knowles, \nwith contributions from J.Alml{\\\"o}f, R. D. Amos, A. Berning, C. Hampel, \nR. Lindh, W. Meyer, A. Nicklass, P. Palmieri, K.A. Peterson, \nR.M. Pitzer, H. Stoll, A.J. Stone, P.R. Taylor.\n%\n\\bibitem{karpfen} A. Karpfen, Theor. Chim. Acta {\\bf 50}, 49 (1978). \n%\n\\bibitem{hinze} J. Hinze, O. Friedrich, and A. Sundermann, Mol. Phys, {\\bf 96}, 711 (1999). \n%\n\\bibitem{jursic} B. S. Jursic J. Mol. Struct. (Theochem) {\\bf 467}, 7 (1999).\n%\n\\bibitem{inc-cal}{See, e.g., B. Paulus, P. Fulde and H. Stoll, Phys. Rev. B \n{\\bf 54}, 2556 (1996); K. Doll, M. Dolg, P. Fulde, and H. Stoll, Phys. Rev. B\n{\\bf 55}, 10282 (1997).}\n\\bibitem{yu} M. Yu, S. Kalvoda, and M. Dolg, Chem, Phys {\\bf 224}, 121 (1997).\n\\bibitem{ayjamal}{A. Abdurahman, M. Albrecht, A. Shukla, and M. Dolg,\nJ. Chem. Phys. {\\bf 110}, 8819 (1999).}\n%\n\\bibitem{lih} A. Shukla, M. Dolg, P. Fulde, and H. Stoll, Phys. Rev. B {\\bf\n60}, 5211 (1999).\n%\n\\bibitem{pulay}{P. Pulay, Chem. Phys. Lett. {\\bf 100}, 151 (1983).}\n%\n\\bibitem{wannier} Computer program WANNIER, A. Shukla, M. Dolg, H. Stoll, and \nP. Fulde (unpublished).\n%\n\\bibitem{dovesi} R. Dovesi, C. Ermondi, E. Ferrero, C. Pisani, and C. Roetti \nPhys. Rev. B {\\bf 29}, 3591 (1984).\n%\n\\bibitem{mchf} MCHF atomic electronic structure code, C. Froese-Fischer, \nThe Hartree-Fock Method for Atoms -- A Numerical Approach, \nWiley, New York, 1976. \n%\n\\bibitem{bery} Ullmann's Encyclopedia of Industrial Chemistry, Fifth,\nCompletely Revised Edition {\\bf A13}, 205 (1989). \n%\n\\end{thebibliography}\n\n\\clearpage\n\\newpage\n\n\\begin{table}\n\\caption{Various increments to the correlation energy (in\nHartrees) computed by the Wannier-function-based approach\npresented in this work. The results refer to\nlattice constant of 3.30 (\\AA). NN stands for nearest\nneighbors.}\\label{t1}\n\\begin{tabular}{lll}\n\\hline\nCorrelation & & Energy \\\\\nIncrement & & \\\\\n\\hline\n\\hline\none-body & &-0.0303345\\\\ \ntwo-body (1NN)& &-0.0003538\\\\ \ntwo-body (2NN)& &-0.0000035\\\\ \ntwo-body (3NN)& &-0.0000003\\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Total energy E$_{tot}$ (Hartree),\ncohesive energy $\\bigtriangleup$E$_{coh}$\n(eV), polymerization energy $\\bigtriangleup$E$_{pol}$ (eV) per unit cell and\nlattice constant a (\\AA) of the lithium hydride chain.}\n\\label{t2}\n\\begin{tabular}{lllll}\n\\hline\nMethod & E$_{tot}$ & $\\bigtriangleup$E$_{coh}$ & $\\bigtriangleup$E$_{pol}$&a \\\\\n\\hline\n\\hline\nWANNIER SCF&-8.038047&3.8760&1.8067 &3.3273\\\\ \nCRYSTAL SCF&-8.038031&3.8759&1.8063 &3.3274\\\\ \nFCI &-8.068744&4.6545&1.4854 &3.3300\\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Total energy E$_{tot}$ (Hartree), cohesive energy $\\bigtriangleup$E$_{coh}$ (eV), \npolymerization energy $\\bigtriangleup$E$_{pol}$ \n(eV) per unit $Be_{2}H_{4}$ and lattice constant a (\\AA), Be--H distance h (\\AA) of beryllium hydride.}\n\\label{t3}\n\\begin{tabular}{llllll}\n\\hline\nMethod & E$_{tot}$ & $\\bigtriangleup$E$_{coh}$ & $\\bigtriangleup$E$_{pol}$&a&h \\\\\n\\hline\n\\hline\nCRYSTAL SCF &-31.6300 &13.70&2.645&3.958&1.467 \\\\\nMP2$^{a}$ &-31.7608&15.85&3.445 &3.958&1.456 \\\\ \nCCSD$^{a}$ &-31.7908 &15.56&3.402&3.969&1.457 \\\\ \nCCSD(T)$^{a}$ &-31.7944&15.66&3.478&3.968 &1.458 \\\\\nKarpfen$^{b}$&-31.5780&--&--&4.024&1.470\\\\\n\\hline\n\\hline\n\\end{tabular}\n\n$^{a}$ correlation contributions added to CRYSTAL SCF energies.\\\\\n$^{b}$ performed with $7, 1/4$ basis sets considering third neighbor's \ninteractions.\\\\\n\\end{table}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002124.extracted_bib",
"string": "\\begin{thebibliography}{99}\n%\n\\bibitem[\\ast]{email}{email: ayjamal@mpipks-dresden.mpg.de}\n\\bibitem[\\dagger]{add1}{Present address: Department of Physics, Indian \nInstitute of Technology, Powai, Mumbai 400 076, India}\n\\bibitem[\\ddagger]{add2}{Permanent address: Institut f\\\"ur Physikalische und Theoretische\nChemie, Universit\\\"at Bonn, Wegeler Str. 12, 53115 Bonn, Germany}\n\\bibitem{ladik} See, e.g., J. J. Ladik, Quantum Theory of Polymers as Solids. \nPlenum press, New York, NY (1988).\n%\n\\bibitem{crystal} R. Dovesi, V.R. Saunders, C. Roetti, M. Causa,\nN.M. Harrison, R. Orlando, E. Apra CRYSTAL95 user's manual. University of\nTurin, Italy (1996).\n%\n\\bibitem{hampel} C. Hampel and H. -J. Werner, J. Chem. Phys. {\\bf 104}, 6286\n(1996), and references cited therein.\n%\n\\bibitem{shukla1}A. Shukla, M. Dolg, H.Stoll and P. Fulde, Chem. Phys. Lett.\n{\\bf 262}, 213 (1996).\n%\n\\bibitem{shukla2} A. Shukla, M. Dolg, P. Fulde, and H. Stoll, Phys. Rev. B, {\\bf 57}, 1471 (1998).\n%\n\\bibitem{shukla3} A. Shukla, M. Dolg, and H. Stoll, Phys. Rev. B {\\bf 58}, 4325 (1998).\n%\n\\bibitem{teramae} H. Teramae, Theor. Chim. Acta {\\bf 94}, 311 (1996).\n%\n\\bibitem{tunega} D. Tunega, J. Noga, Theor. Chim. Acta {\\bf 100}, 78 (1998). %\n%\n\\bibitem{stoll1} H. Stoll, Phys. Rev. B {\\bf 46}, 6700 (1991).\n%\n\\bibitem{stoll2} H. Stoll, J. Chem. Phys. {\\bf 97}, 8449 (1992).\n%\n\\bibitem{stoll3} H. Stoll, Chem. Phys. Lett. {\\bf 191}, 548 (1992).\n%\n\\bibitem{molpro} H.-J. Werner and P. Knowles, MOLPRO, 1994, is a package \nof {\\em ab initio\\/} programs written by H.-J. Werner and P.J. Knowles, \nwith contributions from J.Alml{\\\"o}f, R. D. Amos, A. Berning, C. Hampel, \nR. Lindh, W. Meyer, A. Nicklass, P. Palmieri, K.A. Peterson, \nR.M. Pitzer, H. Stoll, A.J. Stone, P.R. Taylor.\n%\n\\bibitem{karpfen} A. Karpfen, Theor. Chim. Acta {\\bf 50}, 49 (1978). \n%\n\\bibitem{hinze} J. Hinze, O. Friedrich, and A. Sundermann, Mol. Phys, {\\bf 96}, 711 (1999). \n%\n\\bibitem{jursic} B. S. Jursic J. Mol. Struct. (Theochem) {\\bf 467}, 7 (1999).\n%\n\\bibitem{inc-cal}{See, e.g., B. Paulus, P. Fulde and H. Stoll, Phys. Rev. B \n{\\bf 54}, 2556 (1996); K. Doll, M. Dolg, P. Fulde, and H. Stoll, Phys. Rev. B\n{\\bf 55}, 10282 (1997).}\n\\bibitem{yu} M. Yu, S. Kalvoda, and M. Dolg, Chem, Phys {\\bf 224}, 121 (1997).\n\\bibitem{ayjamal}{A. Abdurahman, M. Albrecht, A. Shukla, and M. Dolg,\nJ. Chem. Phys. {\\bf 110}, 8819 (1999).}\n%\n\\bibitem{lih} A. Shukla, M. Dolg, P. Fulde, and H. Stoll, Phys. Rev. B {\\bf\n60}, 5211 (1999).\n%\n\\bibitem{pulay}{P. Pulay, Chem. Phys. Lett. {\\bf 100}, 151 (1983).}\n%\n\\bibitem{wannier} Computer program WANNIER, A. Shukla, M. Dolg, H. Stoll, and \nP. Fulde (unpublished).\n%\n\\bibitem{dovesi} R. Dovesi, C. Ermondi, E. Ferrero, C. Pisani, and C. Roetti \nPhys. Rev. B {\\bf 29}, 3591 (1984).\n%\n\\bibitem{mchf} MCHF atomic electronic structure code, C. Froese-Fischer, \nThe Hartree-Fock Method for Atoms -- A Numerical Approach, \nWiley, New York, 1976. \n%\n\\bibitem{bery} Ullmann's Encyclopedia of Industrial Chemistry, Fifth,\nCompletely Revised Edition {\\bf A13}, 205 (1989). \n%\n\\end{thebibliography}"
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cond-mat0002125
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[
{
"name": "paper.tex",
"string": "\\NeedsTeXFormat{LaTeX2e}\n\\documentclass[12pt,twoside]{report}\n\\usepackage{english,a4}\n\\usepackage{epsf}\n\\usepackage{epsfig}\n\\usepackage{times}\n\n\\setlength{\\textheight} {260mm}\n\\setlength{\\textwidth} {180mm}\n\\setlength{\\topmargin} {-40mm}\n\\setlength{\\evensidemargin} {-10mm}\n\\setlength{\\oddsidemargin} {-10mm}\n\n\\setlength{\\textfloatsep} {-2mm}\n\n\\pagestyle{empty}\n\n\n\\begin{document}\n\n\\vspace*{0.6cm}\n\n\\begin{quote}\n\n\\begin{center}\n{\\Large \nStatic and dynamical properties of a supercooled liquid\nconfined in a pore\\footnote{Talk presented in the workshop {\\it Dynamics in\nConfinement} Grenoble, 26 - 29 January, 2000}}\n\nPeter Scheidler, Walter Kob, and Kurt Binder\n\n{\\small \\it \nInstitut f\\\"ur Physik, Johannes Gutenberg-Universit\\\"at Mainz,\nStaudinger Weg 7, D-55099 Mainz, Germany }\n\\end{center}\n\\end{quote}\n\n\\vspace{0.7cm}\n\n\\hspace*{1mm} \\hfill\n\\begin{minipage}[t]{165mm}\n{\\footnotesize\n%\\baselineskip10pt\n{\\bf Abstract}. We present the results of a Molecular Dynamics computer\nsimulation of a binary Lennard-Jones liquid confined in a narrow pore.\nThe surface of the pore has an amorphous structure similar to that of\nthe confined liquid. We find that the static properties of the liquid are\nnot affected by the confinement, while the dynamics changes dramatically.\nBy investigating the time and temperature dependence of the intermediate\nscattering function we show that the dynamics of the particles close\nto the center of the tube is similar to the one in the bulk, whereas\nthe characteristic relaxation time $\\tau_q(T,\\rho)$ of the intermediate\nscattering function at wavevector $q$ and distance $\\rho$ from the axis\nof the pore increases continuously when approaching the wall, leading\nto an apparent divergence in the vicinity of the wall. This effect is\nseen for intermediate temperatures down to temperatures close to the\nglass transition. The $\\rho$-dependence of $\\tau_q(T,\\rho)$ can be\ndescribed by an empirical law of the form $\\tau_q(T,\\rho)=f_q(T) \\exp\n[\\Delta_q/(\\rho_p-\\rho)]$, where $\\Delta_q$ and $\\rho_q$ are constants,\nand $f_q(T)$ is the only parameter which shows a significant temperature\ndependence. }\n\\end{minipage}\n\n\\vspace{0.75cm}\n\n\\noindent {\\bf 1. INTRODUCTION}\n\\vspace{12pt}\n\n\\noindent\nThe dynamics of a bulk liquid in its supercooled state has been\ninvestigated extensively in experiments and computer simulations\nand is understood reasonably well [1,2]. Much less is known about\nthe influence of a spatial confinement on the dynamic properties of\na liquid. The growing interest in this topic in recent years is based\non the fact that new nanoscale materials with adjustable pore size, such\nas Vycor glass, have been developed, which allow to study the influence\nof the confinement. Concerning the details of the glass transition, experiments\nhave so far given controversial results. Depending on the nature of the\npores and the contained liquid, some authors report an increase in the\nglass temperature~[3,4], while others find a decrease~[5,6]. In this\npaper we report the results of computer simulations which were done to\ninvestigate this phenomenon. Within such simulations (see also~[7,8]) it\nis possible to control the nature of the wall (roughness and wall-liquid\ninteraction), and to do a local analysis of the dynamics of the particles.\nTherefore this method is well suited to increase our understanding of\nthe effects of the confinement.\n\\vspace{0.75cm}\n\n\\noindent {\\bf 2. MODEL AND DETAILS OF THE SIMULATION}\n\\vspace{12pt}\n\n\\noindent \nTo mimic the experimental setup of a fluid confined in porous materials\nwe take as the spatial confinement a cylindrical tube. The contained\nliquid is chosen to be a simple Lennard-Jones fluid. To prevent\ncrystallization at low temperatures we take a binary mixture of 80\\% A\nand 20\\% B particles with the same mass and interacting via a Lennard-Jones\npotential of the form $V_{\\alpha\\beta}(r)=4\\epsilon_{\\alpha\\beta}\n[(\\sigma_{\\alpha\\beta}/r)^{12}-(\\sigma_{\\alpha\\beta} /r)^{6}]$ with\n$\\alpha,\\beta \\in\\{{\\rm A,B}\\}$ and cut-off radii $r_{\\alpha,\\beta}^C$=$2.5\n\\cdot \\sigma_{\\alpha\\beta}$. The parameters were chosen as\n$\\epsilon_{\\rm AA}$=1.0, $\\sigma_{\\rm AA}$=1.0, $\\epsilon_{\\rm AB}$=1.5,\n$\\sigma_{\\rm AB}$=0.8, $\\epsilon_{\\rm BB}$=0.5, and $\\sigma_{\\rm BB}$=0.88. \nThe bulk\nproperties of this system have been investigated in the past~[9]. In the\nfollowing, all results will be given in reduced units, i.e. length in\nunits of $\\sigma_{\\rm AA}$, energy in units of $\\epsilon_{\\rm AA}$ and time\nin units of $(m\\sigma_{\\rm AA}^2/48\\epsilon_{\\rm AA})^{1/2}$. For Argon these\nunits correspond to a length of 3.4\\AA, an energy of 120K$k_B$ and a\ntime of $3\\cdot10^{-13}$s.\n\nTo minimize the influence of the changes in {\\it static} properties\ndue to the confinement, such as layering or a change in the static\nstructure factor, on changes in its {\\it dynamic} properties, we chose\nthe wall of the pore to have an amorphous structure similar to the one\nof the confined liquid. For this purpose, we equilibrated a large\nbulk system at an intermediate temperature, $T$=0.8, and extracted\na cylinder with radius $\\rho_T+r_{AA}^C$ with a tube radius $\\rho_T$\nof 5.0. During the simulation of the tube the particles in the outer\nring, $\\rho$$\\ge$$\\rho_T$, remained fixed while the inner particles,\n$\\rho$$<$$\\rho_T$, interact with each other and the wall particles and\nwere allowed to move. (Here $\\rho$ is the distance from the center of\nthe cylinder.)\n\nThe time evolution of the system was calculated by solving the equations\nof motion with the velocity form of the Verlet algorithm with a time\nstep of 0.01 at high ($T$$\\ge$1.0) and 0.02 at low ($T$$\\le$0.8)\ntemperatures. To improve the statistics we simulated between 8 and\n16 independent systems, each containing 1905 fluid particles and about\n2300 wall particles. The tube length of 20.137 was chosen such that the\naverage particles density is 1.2, the same value as used in the earlier\nsimulations of the bulk. The temperatures investigated were $T$=2.0,\n1.0, 0.8, 0.7, 0.6, and 0.55. The equilibration was done by periodically\ncoupling the liquid to a stochastic heat bath. All data presented here\nwas produced during a microcanonical run at constant energy and volume.\n\n\\vspace{0.75cm}\n\n\n\\noindent {\\bf 3. RESULTS}\n\\vspace{12pt}\n\n\n%\n%\n\\begin{figure}[t]\n\\begin{center}\\begin{picture}(2000,286)\n \\epsfxsize=180mm\\put(0,0){\\epsfbox{grenoble_fig12.eps}}\n\\end{picture}\n\\parbox{170mm}{\n{\\label{fig12}\n\\parbox[t]{7.8cm}{\n%\\begin{flushleft}\n\\footnotesize {\\bf Figure~1.~}Density profiles for A and B particles at $T$=2.0\nand $T$=0.55.\n%\\end{flushleft}\n}\n\\hspace{4mm}\n\\parbox[t]{8.4cm}{\n%\\begin{flushleft}\n\\footnotesize {\\bf Figure~2.~}Radial distribution function $g^z_{\\rm AA}(r)$ \nplus vertical offset $x(T)$ in the outer region for $T$=2.0 ($x$=0.0),\n$T$=1.0 ($x$=0.75), $T$=0.8 ($x$=0.95), $T$=0.7 ($x$=1.15), \n$T$=0.6 ($x$=1.35), and $T$=0.55 ($x$=2.1); \ncomparison with bulk curves for $T$=2.0 and $T$=0.55. \\vspace{4mm}\n%\\end{flushleft}\n}\n}}\n\\end{center}\n\\end{figure}\n%\n\\noindent \nThe analysis of the static properties of the confined system gave the \nexpected results. \nLooking at the density profile in a plane perpendicular to the \naxis of the tube, normalized to its average value,\n%\n\\begin{equation}\nd_{\\alpha}(\\rho)=\\left \\langle \\int_{interior}{\\sum_{i=1}^{N_{\\alpha}}\n{\\delta \\left(\\sqrt{x_i^2+y_i^2} - \\rho \\right)}} \\right \\rangle \\cdot \n\\left[ \\frac{N_{\\alpha}}{V} \\right]^{-1} \\mbox{, where } \\alpha \\in \n\\{\\rm A,B\\},\n\\label{eq_profile}\n\\end{equation}\n%\nwe observe only a small dependence of $d$ on the distance $\\rho$ from the\ncenter (Fig.~1). Due to the roughness of the surface the fluid particles\ncan penetrate slightly into the wall. We can define a penetration radius\n$\\rho_p$ as the distance from the center of the tube at which there\nis almost no chance of finding a particle, namely the value of $\\rho$\nwere the density profile has decreased to $10^{-4}$ from its bulk value.\nFor A particles we find $\\rho_A$$\\approx$5.5$\\pm$0.2, and for the smaller\nB particles $\\rho_A$$\\approx$6.1$\\pm$0.2, values that depend only weakly\non temperature below $T$=0.7. From the figure we see that the fluid has\na tendency to form concentric layers, especially at low temperatures,\nbut that this effect is only weak.\n\nThe radial distribution function in $z$-direction,\n%\n\\begin{equation}\ng_{\\alpha \\beta}^z(r) \\propto \\left \\langle {\\sum_{i=1}^{N_{\\alpha}}\n\\sum_{j=1 \\atop x_{ij}^2+y_{ij}^2 < \\mu^2}^{N_{\\beta}} \n{\\delta \\left( r - \\left| \\vec r_i - \\vec r_j \\right| \\right)}\n} \\right \\rangle , \\,\\, \\alpha \\in \\{\\rm A,B\\} \\,\\, , \\,\\, \\mu^2=0.5^2 \n\\label{eq_raddis}\n\\end{equation}\n%\nshows no strong deviation from its bulk behavior (Fig.~2). Even at the lowest \ntemperature, $T$=0.55, there is only a small difference in \npeak positions and amplitudes between the bulk curve and the one for particles in\nthe outer region, $\\rho$$\\ge$4.0, of the tube. At high $T$ this difference can \nhardly be seen. Similar results are obtained for AB- and BB-correlations \nand also the static structure factor is hardly affected by the confinement.\n\nIn contrast to this, the dynamical properties of the system change\ndramatically due to the confinement. All investigated dynamic quantities\n(mean squared displacement, intermediate scattering function, and van\nHove correlation function) show a strong $\\rho$-dependence. The following\nresults were obtained by labeling a particle with its distance from\nthe $z$-axis at time $t$=0, and analysing its dynamics as a function of\nits position. Fig.~3 shows the $\\rho$-dependence of the self part of the\nintermediate scattering function, \n\\begin{equation} F_s(q,\\rho,t)=\\langle\n\\exp \\left[ i {\\vec q} \\cdot \\left({\\vec r}(t)-{\\vec r} \\right) \\right]\n\\cdot \\delta \\left( x^2(0)+y^2(0) -\\rho^2 \\right)\n\\rangle , \\label{eq_fq} \\end{equation} \nfor A particles and $\\vec q$ along the $z$-axis with modulus $q$ corresponding\nto the maximum of the static structure factor for AA correlations at the lowest\ntemperature investigated ($T$=0.55).\n%\n%\n\\begin{figure}[t]\n\\begin{center}\\begin{picture}(2000,279)\n \\epsfxsize=180mm\\put(0,0){\\epsfbox{grenoble_fig34.eps}}\n\\end{picture}\n\\parbox{170mm}{\n{\\label{fig34}\n\\parbox[t]{9cm}{\n%\\begin{flushleft}\n\\footnotesize {\\bf Figure~3.~}Time dependence of the self part of the \nintermediate scattering function for different values of $\\rho$ for A \nparticles at $T$=0.55.\n%\\end{flushleft}\n}\n\\hspace{5mm}\n\\parbox[t]{7cm}{\n%\\begin{flushleft}\n\\footnotesize {\\bf Figure~4.~}$\\rho$-dependence of the relaxation times \n$\\tau_q(T,\\rho)$ of $F_s(q,\\rho,t)$, compared with the corresponding bulk \nvalues.\\vspace{4mm}\n%\\end{flushleft}\n}\n}}\n\\end{center}\n\\end{figure}\n%\nWhile the relaxation of $F_s(q,\\rho,t)$ in the center is similar to the\none in the bulk, it becomes much slower with increasing $\\rho$, i.e.~on\napproach to the wall. The slowing down of the dynamics when approaching\nthe wall exceeds more than three orders of magnitude. Since any particle\ntagged at time zero moves within a range of radii with different intrinsic\nrelaxation times during the run, the averaged relaxation is more stretched\nthan in the bulk, especially close to the wall, where the differences\nbetween relaxation times is large.\n\nIn order to investigate the $\\rho$-dependence of the dynamics we define\na characteristic $\\rho$-dependent relaxation time $\\tau_q(T,\\rho)$\nof the intermediate scattering function as the time at which it has\ndecayed to $e^{-1}$ of its initial value and compare these times.\nA quantitative analysis of the $\\rho$-dependence of this relaxation\ntimes for the investigated temperatures below $T$=1.0 give the following\nresults (Fig.~4). At higher temperatures particles in the inner region,\n$\\rho$$\\le$2.0, show almost bulk behavior before a strong increase in\n$\\tau_q(T,\\rho)$ becomes apparent for higher $\\rho$. At low temperatures\nthe presence of the wall affects the dynamics of the particles even in\nthe center of the tube. The curves for all temperatures show an apparent\ndivergence in the vicinity of the wall. The given statements above hold \nfor A and B particles and also for different wave vectors.\n\nBased on these observations we found an empirical law which is able to describe\nthe $\\rho$-dependence of the relaxation times of the intermediate scattering \nfunction. The data from Fig.~4 is described well by the functional form \n\\begin{equation}\n\\tau_q(T,\\rho)=f_q(T) \\exp \\left[ \\Delta_q/(\\rho_p - \\rho) \\right] ,\n\\label{eq_law}\n\\end{equation}\nat least in the vicinity of \nthe wall, i.e. for $\\rho$$\\ge$3.5, which corresponds to half of the particles.\nIn Eq.~(\\ref{eq_law}), the penetration radius $\\rho_p$ is determined as \nmentioned above from the static properties and depends only weakly on \ntemperature. The quantity $\\Delta_q$ depends on particle type and the value of $q$.\nThe only temperature dependent quantity in this fit is the amplitude $f_q(T)$,\nwhich also depends on particle type and $q$.\nIf we assume that in the supercooled state Eq.~(\\ref{eq_law}) holds for most \nof the particles, the slowing down of the system in the supercooled state is \nmainly characterized by the temperature \ndependence of the amplitude $f_q(T)$. More details on this will be discussed \nelsewhere~[10].\n\\vspace{0.75cm}\n\n\n\\noindent {\\bf 4. SUMMARY}\n\\vspace{12pt}\n\n\\noindent\nWe have presented the results of a computer simulation of a simple\nglass former in a narrow tube. We find the relaxation times of the\nintermediate scattering function to be strongly dependent on the distance\nfrom the wall. We are able to describe this behavior by an empirical law\npredicting a divergence at $\\rho$=$\\rho_p$, where $\\rho_p$ is determined\nfrom static quantities. \nWe see a gradual slowing down of the dynamics of the whole system, especially \nno immobile layer close to the wall, in agreement with experiments by \nRichert [11]. Note that we expect a similar slowing down of\nthe dynamics in the vicinity of a wall also for other tube radii and\neven in a slit geometry, and that in the case of a narrow confinement,\nsuch as the one we investigated here, this slowing down will dominate\nthe dynamics of the whole system.\n\nThis work was supported by {\\it Deutsche Forschungsgemeinschaft} under SFB 262/D1\nand the {\\it NIC} in J\\\"ulich.\\vspace{0.75cm}\n\n\\begin{flushleft}\n\\noindent {\\bf References}\n\\vspace{12pt}\n\n\\hspace{1.08mm}\n[1] See, e.g., {\\it Proceedings of Third Intern.~Discussion\nMeeting}, J.~Non-Crysl.~Solids {\\bf 235-237} (1998). \\\\\n%\n\\hspace{1.08mm}\n[2] W.~Kob, J.~Phys.: Condens. Matter {\\bf 11}, R85 (1999). \\\\\n%\n\\hspace{1.08mm}\n[3] P.~Pissis, A.~Kyritsis, D.~Daoukaki, G.~Barut, R.~Pelster, and G.~Nimtz,\nJ.~Phys.: Condens. Matter {\\bf 10},\\\\\n\\hspace{8mm} 6205 (1998).\\\\\n%\n\\hspace{1.08mm}\n[4] C.~L.~Jackson and G.~B.~McKenna, Chem.~Mater.~{\\bf 8}, 2128 (1996). \\\\\n%\n\\hspace{1.08mm}\n[5] J.~Sch\\\"uller, B.~Yu, B.~Mel'nichenko, R.~Richert, and E.~W.~Fischer, \nPhys.~ Rev.~Lett.~{\\bf 73}, 2224 (1994). \\\\\n%\n\\hspace{1.08mm}\n[6] W.~E.~Wallace, J.~H.~van Zanten, and W.~L.~Wu,~Phys.~Rev.~E, {\\bf 52}, R3329 (1995).\\\\\n%\n\\hspace{1.08mm}\n[7] J.~Baschnagel and K.~Binder, J.~Phys.~I France {\\bf 6}, 1271 (1996) \\\\\n%\n\\hspace{1.08mm}\n[8] Z.~T.~N\\'emeth and H.~L\\\"owen, J.~Phys.: Condens. Matter {\\bf 10}, 6189 (1998).\\\\\n%\n\\hspace{1.08mm}\n[9] W.~Kob and H.~C.~Andersen, Phys.~Rev.~E {\\bf 51}, 4626 (1995); \nPhys.~Rev.~E {\\bf 52}, 4134 (1995). \\\\\n%\n\\noindent\n[10] P.~Scheidler, W.~Kob, and K.~Binder, to be published \\\\\n%\n\\noindent\n[11] R.~Richert, Phys.~Rev.~E {\\bf 54}, 15762 (1996) \n\\end{flushleft}\n\n\\end{document}\n"
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cond-mat0002126
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Structure and Magnetism of well-defined cobalt nanoparticles embedded in a niobium matrix
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"author": "M. Jamet$^1$"
},
{
"author": "V. Dupuis$^1$"
},
{
"author": "P. M\\'elinon$^1$"
},
{
"author": "G. Guiraud$^1$"
},
{
"author": "A. P\\'erez$^1$"
},
{
"author": "W. Wernsdorfer$^2$"
},
{
"author": "A. Traverse$^3$"
},
{
"author": "B. Baguenard$^4$"
}
] |
Our recent studies on Co-clusters embedded in various matrices reveal that the co-deposition technique (simultaneous deposition of two beams : one for the pre-formed clusters and one for the matrix atoms) is a powerful tool to prepare magnetic nanostructures with any couple of materials even though they are miscible. We study, both sharply related, structure and magnetism of the Co/Nb system. Because such a heterogeneous system needs to be described at different scales, we used microscopic and macroscopic techniques but also local selective absorption ones. We conclude that our clusters are 3 nm diameter f.c.c truncated octahedrons with a pure cobalt core and a solid solution between Co and Nb located at the interface which could be responsible for the magnetically inactive monolayers we found. The use of a very diluted Co/Nb film, further lithographed, would allow us to achieve a pattern of microsquid devices in view to study the magnetic dynamics of a single-Co cluster.
|
[
{
"name": "PRBCoNb.tex",
"string": "\\documentstyle[prl,aps,epsfig,twocolumn]{revtex}\n\n\\begin{document}\n\n\\draft\n\n\\title{Structure and Magnetism of well-defined cobalt nanoparticles \nembedded in a niobium matrix}\n\n \\author{M. Jamet$^1$, V. Dupuis$^1$, P. M\\'elinon$^1$, G. Guiraud$^1$, A. P\\'erez$^1$, W. Wernsdorfer$^2$, A. Traverse$^3$, B. Baguenard$^4$}\n \n\\address{$^1$ D\\'epartement de Physique \ndes Mat\\'eriaux, Universit\\'e Claude Bernard-Lyon 1 et CNRS, 69622 \nVilleurbanne, FRANCE. \\\\\n $^2$ Laboratoire Louis N\\'eel, CNRS, 38042 \nGrenoble, FRANCE. \\\\\n$^3$ LURE, CNRS CEA MENJS, B\\^at. 209A, BP 34, 91898 Orsay, FRANCE. \\\\\n$^4$ Laboratoire de Spectrom\\'etrie ionique et mol\\'eculaire, Universit\\'e Claude Bernard Lyon 1 et CNRS, 69622 Villeurbanne, FRANCE}\n\n\n\\maketitle \n\n\n\\begin{abstract}\n\nOur recent studies on Co-clusters embedded in various matrices reveal that \nthe co-deposition technique (simultaneous deposition of two beams : one for \nthe pre-formed clusters and one for the matrix atoms) is a powerful tool \nto prepare magnetic nanostructures with any couple of materials even \nthough they are miscible. We study, both sharply related, structure and \nmagnetism of the Co/Nb system. Because such a heterogeneous system needs \nto be described at different scales, we used microscopic and macroscopic \ntechniques but also local selective absorption ones. We conclude that our clusters are 3 nm diameter f.c.c truncated octahedrons with a pure \ncobalt core and a solid solution between Co and Nb located at the \ninterface which could be responsible for the magnetically \ninactive monolayers we found.\nThe use of a very diluted Co/Nb film, further lithographed, would allow us \nto achieve a pattern of microsquid devices in view to study the magnetic \ndynamics of a single-Co cluster. \n\n\\end{abstract}\n\n\\pacs{61.10.Ht, 61.46.+w, 75.50.Tt}\n\n\\narrowtext\n\n\\section {Introduction}\n\nStructural and magnetic properties of clusters, $\\textit{i.e.}$ particles containing from\ntwo to a few thousand atoms, are of great interest nowadays. From a \ntechnological point of view, those systems are part of the development of \nhigh density magnetic storage media, and, from a fundamental point of \nview, the physics of magnetic clusters still needs to be investigated. \nIndeed, to perform stable magnetic storage with small clusters, one has to \ncontrol the magnetization reversal process (nucleation and dynamics), and \nthus make a close connection between structure and magnetic behavior. To \nreach the magnetic properties of small clusters, there are two \navailable approaches : \"macroscopic\" measurements (using a Vibrating \nSample Magnetometer (VSM) or a Super Quantum Interference Device (SQUID)) \non a cluster collection (10$^{9}$ particles) that implicate statistical \ntreatments of the data, and \"microscopic\" measurements on a single \nparticle. From now, micro-magnetometers (MFM,~\\cite{Chan93} Hall \nmicro-probe,\\cite{Geim97} or\nclassical micro-squid\\cite{Wern97}) were not sensitive enough to perform magnetic \nmeasurements on a single cluster. The present paper constitutes the \npreliminary study toward magnetic measurements on a small single cluster \nusing a new microsquid design. We focus on and try to connect structural \nand magnetic properties of a cluster collection. With a view to clear up \nstructural questions, we first study the structure of nanocrystalline \nCo-particles embedded in a niobium matrix by means of Transmission \nElectron Microscope (TEM) observations, X-ray diffraction and absorption \ntechniques. Then magnetization measurements are performed on the same \nparticles to deduce their magnetic size and their anisotropy terms.\n\n\\section {experimental devices }\n\n\tWe use the co-deposition technique recently developed in our laboratory \nto prepare the samples.\\cite{Pare97} It consists in two independent beams reaching at \nthe same time a silicon (100) substrate at room temperature : the \npre-formed cluster beam and the atomic beam used for the matrix. The \ndeposition is made in a Ultra High Vacuum (UHV) chamber \n(p=$5.10^{-10}$ Torr) \nto limit cluster and matrix oxidation.\n\tThe cluster source used for this experiment is a classical laser \nvaporization source improved according to Milani-de Heer \ndesign.\\cite{Mila90} It \nallows to work in the Low Energy Cluster Beam Deposition (LECBD) regime : clusters \ndo not fragment arriving on the substrate or in the \nmatrix.\\cite{Pere97} The \nvaporization Ti:Sapphire laser used provides output energies up to 300 mJ \nat 790 nm, in a pulse duration of 3 $\\mu$s and a 20 Hz repetition rate. It \npresents many advantages described elsewhere,\\cite{Pell94} such as adjustable high \ncluster flux. The matrix is evaporated thanks to a UHV electron gun in \ncommunication with the deposition chamber. By monitoring and controlling \nboth evaporation rates with quartz balances, we can continuously adjust \nthe cluster concentration in the matrix.\nWe previously show that this technique allows to prepare nanogranular films \nfrom any couple of materials, even two miscible ones forbidden by the phase diagram \nat equilibrium.\\cite{Negr99} We determine the crystalline structure and the morphology \nof cobalt clusters deposited onto copper grids and protected by a thin \ncarbon layer (100 $\\AA$). From earlier High Resolution Transmission Electron \nMicroscopy (HRTEM) observations, we found that cobalt clusters form \nquasi-spherical nanocrystallites with a f.c.c structure, and a sharp size \ndistribution.\\cite{Tuai97,Pare97}\n\tIn order to perform macroscopic measurements on a cluster collection \nusing surface sensitive techniques, we need films having a 5-25 nm \nequivalent thickness of cobalt clusters embedded in 500 nm thick niobium \nfilms. We chose a low cluster concentration (1-5 $\\%$) to make structural and \nmagnetic measurements on non-interacting particles. One has to mention that \nsuch concentration is still far from the expected percolation threshold (about \n20 $\\%$).\\cite{Pare97}\n\tFrom both X-ray reflectometry and grazing X-ray scattering measurements, \nwe measured the density of the Nb films: 92 $\\%$ of the bulk one, and a \nb.c.c polycrystalline structure as reported for common bulk.\nX-ray absorption spectroscopy (XAS) was performed on D42 at the LURE facility in \nOrsay using the X-ray beam delivered by the DCI storage \nring\\cite{Baud93} at the Co \nK-edge (7709 eV) by electron detection at low temperature (T=80 K)\\cite{Mima94}. The porosity of the matrix is low \nenough and avoids the oxidation of the reactive Co clusters as shown in \nX-ray Absorption Near Edge Structure (XANES) spectra at the Co-K \nedge where no fingerprint of oxide on cobalt clusters embedded in \nniobium films is observed. The results of the Extended X-ray Absorption Fine Structure \n(EXAFS) simulations reveal the local distances between first Co-neighbors \nand their number for each component. \nMagnetization measurements on diluted samples were performed using a \nVibrating Sample Magnetometer (VSM) at the Laboratoire Louis N\\'eel in \nGrenoble. Other low temperature magnetization curves of the same samples \nwere obtained from X-ray Magnetic Circular Dichroism (XMCD) signal. The \nmeasurement was conducted at the European Synchrotron Radiation Facility \nin Grenoble at the ID12B beamline. The degree of circular polarization was \nalmost 80 $\\%$, and the hysteresis measurements were performed using a \nhelium-cooled UHV electromagnet that provided magnetic fields up to 3 \nTesla.\n\n\\section {structure}\n\nThe origin of the EXAFS signal is well established as mentioned in various \nreferences.\\cite{Carr90} If multiple scattering effects are neglected on the first \nnearest neighbors, the EXAFS modulations are described in terms of \ninterferences between the outgoing and the backscattered photoelectron wave \nfunctions. We use Mc-Kale tabulated phase and amplitude shifts for all types \nof considered Co-neighbors.\\cite{Kale86} The EXAFS analysis is restricted to solely \nsimple diffusion paths from the standard fitting code developed in \nthe Michalowicz version\\cite{Mich97} where an amplitude reduction factor \nS$_{0}^{2}$ equal to 0.7\\cite{MRoy97} and an asymmetric distance distribution based on hard \nsphere model\\cite{Prou97} are introduced. The first consideration traduces the possibility of \nmultiple electron excitations contributing to the total absorption \ncoefficient reduction. The \nsecond one is needed to take into account the difference between the core \nand the interface Co-atom distances in the cluster.\\cite{Tuai97} So, in \nthe fit, R$_{j}$ and s$_{j}$ values, corresponding to the shortest distance and to \nthe asymmetry parameter of the j$^{th}$ atom from the excited one \nrespectively, replace the average distance in \nthe standard EXAFS formulation. We also define N$_{j}$ the coordination number, $\\sigma_{j}$ the Debye-Waller factor \nof the j$^{th}$ atom, k the photoelectron momentum and $\\Gamma$(k) its mean free path.\nStructural parameters ($N_{j}, R_{j}, \\sigma_{j}, s_{j}$) were determined from the \nsimulation of the EXAFS oscillations (Fig. 1). As for some systems with \ntwo components (for example in metallic superlattices previously \nstudied\\cite{Baud93,Vdup93}), the first Fourier transform peak of the EXAFS \nspectrum presents a shoulder which can be understood unambiguously in terms of \nphase-shift between Co and Nb backscatterers for k values around \n5$\\AA^{-1}$. \nThis splitting in the real space corresponds to a broadening of the second \noscillation in the momentum space\\cite{Baud93} (see Fig. 1). Thus, in the simulations, we first consider two kinds of Co-neighbors : cobalt and niobium. But, a preliminary study of cobalt and niobium core levels by X-ray photoelectron spectroscopy reveals a weak concentration of oxygen inside the sample owing to the UHV environment. The core level yielding provides an oxygen concentration of about 5 $\\%$. Such Co-O bonding is taken into account for the fit improvement. Moreover, from HRTEM and X-ray \ndiffraction patterns \\cite{Pare97,Tuai97}, we know : \nthe mean size of the clusters (3 nm), their inner f.c.c \nstructure with a lattice parameter close to the bulk one and their shape close to the Wulff\nequilibrium one (truncated octahedron). Finally, the Co/Nb system can \nbe usefully seen as a cobalt core with the bulk parameters and a more \nor less sharp Co/Nb interface. From these assumptions, we use the simulation of EXAFS oscillations to describe the Co/Nb interface and to verify it is relevant with an observed alloy in the phase diagram (tetragonal Co$_{6}$Nb$_{7}$).\nThe best fitted \nvalues of EXAFS oscillations are the following : \\\\ \n\t- 70 $\\%$ of Co atoms are surrounded with cobalt neighbors in the f.c.c phase \nwith the bulk-like distance (d$_{Co-Co}$=2.50 $\\AA$), corresponding \nto N$_{1}$=8.4, \nR$_{1}$=2.495 $\\AA$, $\\sigma_{1}$=0.1 $\\AA$, s$_{1}$=0.18 $\\AA$ in EXAFS \nsimulations. \\\\\n\t- 26 $\\%$ of Co atoms are surrounded with niobium neighbors in the tetragonal \nCo$_{6}$Nb$_{7}$ phase, corresponding to N$_{2}$=3.1, R$_{2}$=2.58 $\\AA$, \n$\\sigma_{2}$=0.16 $\\AA$, s$_{2}$=0.06 $\\AA$ \nin EXAFS simulations. \\\\\n\t- 4 $\\%$ of Co atoms are surrounded with oxygen neighbors with a distance \nequal to d$_{Co-O}$=2.0 $\\AA$ based on the typical oxygen atomic radii in chemisorption systems\\cite{Drey99} or transition metal oxides. This environment \ncorresponds to N$_{3}$=0.5, R$_{3}$=1.9 $\\AA$, $\\sigma_{3}$=0.04 \n$\\AA $, s$_{3}$=0.1 $\\AA$ in EXAFS \nsimulations. \\\\\nAccording to Ref.\\cite{Hard69}, a 3 nm-diameter f.c.c truncated octahedron consists of 35.6 $\\%$ core atoms (zone a), 27 $\\%$ atoms in the first sublayer (zone b) and 37.6 $\\%$ atoms in the surface layer (zone c). Let us propose the following compositions : a pure f.c.c Co phase in zone a, a Co$_{4}$Nb phase in zone b, and a Co$_{6}$Nb$_{7}$O$_{2}$ phase in zone c (i.e. : at the cluster-matrix interface). The corresponding coordination numbers : N$_{1}$(Co-Co)=8.5, N$_{2}$(Co-Nb)=3.1 and N$_{3}$(Co-O)=0.4 are in good quantitative agreement with the coordination numbers N$_{1}$, N$_{2}$ and N$_{3}$ we obtain from EXAFS simulations.\n\tConcerning the other fitting parameters, what is found is the high value for the mean free path of the photoelectron \n\t($\\Gamma =1.6$) \nand the Debye Waller factor for Co-metal environment ($\\sigma$$>$ \n0.1 $\\AA$). Notice \nthat because we did not dispose of experimental phase and amplitude, but \ncalculated ones, a large difference between sample and reference is \nexpected, so their absolute values do not represent physical reality but \nonly are necessary to attenuate the amplitude of oscillations. On the \ncontrary, the total number of neighbors is fixed by TEM experiments which \nreveal a f.c.c-phase for the Co-clusters (so N$_{1}$+N$_{2}$+N$_{3}$=11$\\pm$1). To follow the shape, position and relative amplitudes of the \noscillations, N$_{j}$ is a free parameter for each \ncomponent in the simulation and besides is related to the concentration \nof the j$^{th}$ atom from the Co-absorber one in the sample. This study \nfinally evidences a diffuse interface between cobalt and \nniobium mostly located on the first monolayer.\n\nIn summary, we made a consistent treatment of all the experimental results obtained from different techniques. \nWe notice that EXAFS spectra show unambiguously a smooth interface between \nmiscible elements as cobalt \nand niobium. This information will be of importance and is the key to \nunderstand the magnetic behavior discussed below. \n\n\\section{magnetism}\n\n\tHere, we present the magnetic properties of these nanometer sized \nclusters embedded in a metallic matrix. Furthermore such a system \nwill be used to perform microsquid devices in order to reach magnetization \nmeasurements on an isolated single domain cluster. The present study deals with \nmacroscopic measurements performed on a particle assembly (typically \n10$^{14}$) of cobalt clusters in a niobium matrix to describe the magnetic properties \nof the Co/Nb system. Because of the goal mentioned above, we focus on very \ndiluted samples (less than 2 $\\%$ volumic for Co concentrations). \nFor these low cluster concentrations, magnetic couplings between particles are negligible whereas \ndipolar and RKKY interactions in the case of metallic matrix, \nare considered. Nevertheless, both last contributions which vary as \n1/d$_{ij}^{3}$ (where d$_{ij}$ is the mean distance between particles) \nare expected to be weak compared \nto ferromagnetic order inside the cluster. In a first approximation, \nwe neglect any kind of surface disorder so that a single domain cluster can be seen as an \nisolated macrospin with uniform rotation of its magnetization. It means \nthat the atomic spins in the cluster remain parallel during the cluster magnetization rotation. In an external applied field, the magnetic energy of a nanoparticle \nis the sum of a Zeeman interaction (between \nthe cluster magnetization and the local field), and anisotropy terms (as shape, magnetocrystalline, surface \n(interface in our case) or strain anisotropy).\nAt high temperatures (T$>$100 K), anisotropy contributions of nanometric \nclusters can be neglected compared to the thermal activation \n($K_{eff}V/k_{B}\\approx 30 K$\\cite{Comm99}) \nand clusters act as superparamagnetic independent entities. A way to \nestimate the interparticle interactions is to plot 1/$\\chi$ vs. T in the \nsuperparamagnetic regime. 1/$\\chi$ follows a \nCurie-Weiss-like law :\n\\begin{equation}\n\\frac{1}{\\chi} =C(T-\\theta)\n\\end{equation}\nand $\\theta$ gives an order of magnitude of the particle \ninteractions.\\cite{Dorm97} From \nexperimental data, we give 1/$\\chi$ vs. T on Fig. 2, and find $\\theta$=1-2 K, which is \nnegligible compared with the other energies of the clusters. \nIn the superparamagnetic regime, we also estimate in Section A \nthe magnetic size distribution of the clusters.\nAt low temperatures, clusters have a ferromagnetic behavior due to the \nanisotropy terms. And, in Section B, we \nexperimentally estimate their mean anisotropy constant. \n\n\\subsection{Magnetic size measurement}\n\n In the following, we make the approximation that the atomic magnetic moment is equal to 1.7 $\\mu _{B}$ at any \ntemperature (or 1430 emu/cm$^{3}$ like in the bulk h.c.p cobalt). \nBesides, our synthesized cobalt clusters have approximately a 3 nm diameter and contain at least \n1000 atoms. According to references \\cite{Bill94,Resp98}, a magnetic moment \nenhancement only appears for particles containing less than 500 atoms. So \nin our size range we can assume that the atomic cobalt moment is close to \nthe bulk phase one (m$_{Co}$=1.7$\\mu _{B}$). \nWe consider a log-normal size distribution :\n\\begin{equation}\nf(D)=\\frac{1}{D\\sqrt{2\\pi\\sigma^{2}}}exp\\Biggl(-\\biggl(ln\\Bigl(\\frac{D}{D_{m}}\\Bigr)\\biggr)^{2}\n\\frac{1}{2\\sigma^{2}}\\Biggr) \n\\end{equation}\nwhere D$_{m}$ is the mean cluster diameter and $\\sigma$ the dispersion. In the superparamagnetic regime,\n we can use a classical Langevin function $L(x)$ and write :\n\\begin{equation}\n\\frac{m(H,T)}{m_{sat}}=\\frac{\\int _{0}^{\\infty}D^{3}L(x)f(D)\\,dD}{\\int \n_{0}^{\\infty}D^{3}f(D)\\,dD}, x=\\frac{\\mu_{0}H(\\pi D^{3}/6)M_{S}}{k_{B}T}\n\\end{equation}\nwhere H is the applied field ($\\mu_{0}$H in Tesla), T the temperature and \nm$_{sat}$ \nthe saturation magnetic moment of the sample estimated on magnetization \ncurves at low temperatures under a 2 Tesla field. First of all, on Fig. 3, \none can see that for T$>$100 K, m(H/T) curves superimpose according to Eq. \n(3) for a magnetic field being applied in the sample plane (we checked \nthat the results are the same for a perpendicular applied field). \nSecondly, one can notice that for T=30 K, the magnetization deviation to \nthe high temperature curves comes from the fact that the anisotropy is not \nnegligible anymore, and one has to use a modified Langevin function in the \nsimulation.\\cite{Mull73} In this equation, we also assume the particles to feel the \napplied field, actually, they feel the local field which is the sum of the \nexternal field and the mean field created by the surrounding particles in \nthe sample.\nFurthermore, in the superparamagnetic regime, we fit experimental m(H,T) \ncurves obtained from VSM measurements to find D$_{m}$ and $\\sigma$, the mean diameter \nand dispersion of the \"magnetic size\" distribution, respectively (see Fig. \n4). For those fits, we still use the M$_{S}$ bulk value (the use of other ones \ngiven in references \\cite{Brun89,Alde92,Dora97} leads quite to the same results (with an \nerror less than 5 $\\%$), the determining factors being D$_{m}$ and \n$\\sigma$). Figure (5) \ndisplays D$_{m}$ and $\\sigma$ for two niobium deposition rates (V$_{Nb}$=3 \n$\\AA$/s and V$_{Nb}$=5 $\\AA$/s, \nrespectively). Such results are compared with the real cluster sizes \ndeduced from TEM observations. The magnetic domain is always smaller than \nthe real diameter. Furthermore, the magnetic domain decreases as the \ndeposition rate increases. This indicates that the kinetics of the \ndeposition plays a crucial role for the nature of the interface. For \nexample, we found a magnetic domain size of 2.3 nm (resp. 1.8 nm) for a \n3 nm diameter cluster when V$_{Nb}$=3 $\\AA$/s (resp. \nV$_{Nb}$=5 $\\AA$/s), the dispersion $\\sigma$=0.24 remained the same.\n\n\\subsection{Anisotropy}\n\n The bulk value of the f.c.c cobalt cubic magnetocrystalline \n anisotropy constant is : K$_{MA}$=2.7.10$^{6}$ \nerg/cm$^{3}$\\cite{Chen94} less than the h.c.p bulk phase one\n(4.4.10$^{6}$ erg/cm$^{3}$). The shape anisotropy constant K$_{shape}$ can be \ncalculated from \nthe demagnetizing factors and the saturation magnetization. \n In case of weak distortions in the sphericity, the shape anisotropy for a \nprolate spheroid can be expressed as follows : \n\\begin{equation}\nE_{shape}=\\frac{1}{2}\\mu_{0}M_{S}^{2}(N_{z}-N_{x})\\cos^{2}(\\theta)=K_{shape}\\cos^{2}(\\theta)\n\\end{equation}\nM$_{S}$ is the saturation magnetization of the particle : M$_{S}$ =1430 \nemu/cm $^{3}$, $\\theta$ \nthe angle between the magnetization direction and the easy axis, and \nN$_{x}$, \nN$_{z}$ the demagnetizing factors along x-axis and z-axis respectively.\n We plot on Fig. 6, the constant anisotropy K$_{shape}$ as a function of the prolate \nspheroid deformation c/a \\cite{Ahar96} (with c and a representing the wide and \nsmall ellipsoid axis, respectively). For a truncated octahedron, the ratio \nc/a has been evaluated lower than 1.2 which restricts the K$_{shape}$ value of the \norder of 10$^{6}$ erg/cm$^{3}$. However, we have no information about \nthe magnitude of interface and strain anisotropies in our system. \\\\\n Let us now experimentally evaluate the anisotropy constant K$_{eff}$ of \ncobalt clusters from low temperature measurements. Hysteresis curves are obtained from VSM experiments, but at very low temperatures ($\\textit{i.e.}$ T$<$8 K), superconducting fluctuations appear due to the niobium matrix and prevent any magnetization measurements on the whole sample. So, we also use X-ray \nMagnetic Circular Dichroism as a local magnetometer by recording \nthe MCD signal at the cobalt L$_{3}$ white line as a function of the applied \nmagnetic field (for details on the method see Ref.\\cite{Chen93}). The angle of the \nincident beam is fixed at 55$^{\\circ}$ with respect to the surface normal and the magnetic field is parallel to the sample surface. The \nabsorption signal is recorded by monitoring the soft X-ray fluorescence \nyield chosen for its large probing depth (1000 $\\AA$). Finally, from hysteresis curves given by both VSM and XMCD techniques, we deduce m$_{r}$(T), the remanent magnetic moment vs. T down to 5.3 K, and we normalize it by \ntaking : m$_{r}$(8.1K)$_{VSM}$=m$_{r}$(8.1K)$_{XMCD}$, the curve \nm$_{r}$(T)/m$_{r}$(5.3K) is given on Fig. \n7. \nTo evaluate m$_{r}$(T), one can write :\n\\begin{equation}\nm_{r}(T)=\\frac{m_{sat}}{Cte}\\frac{\\int _{D_{B}(T)}^{\\infty} \nD^{3}f(D)\\,dD}{\\int _{0}^{\\infty} D^{3}f(D)\\,dD}\n\\end{equation}\nwhere D$_{B}$(T) is the particle blocking diameter at temperature T. Cte is a \nparameter independent of the particle size. Cte=$2$ if clusters have a \nuniaxial magnetic behavior and $3-\\sqrt{3}$ if they have a cubic magnetic \none. In order to rule out this Cte, we plot the ratio :\n\\begin{equation}\n\\frac{m_{r}(T)}{m_{r}(5.3K)}=\\frac{\\int _{D_{B}(T)}^{\\infty} \nD^{3}f(D)\\,dD}{\\int _{D_{B}(5.3K)}^{\\infty} D^{3}f(D)\\,dD}\n\\end{equation}\nOne finds D$_{B}$(T) when the relaxation time of the particle is equal to the \nmeasuring time : $\\tau=\\tau_{0}exp(K_{eff}V/k_{B}T)=\\tau_{mes}$.\n\\begin{equation}\nD_{B}^{3}(T)=aT, a=\\frac{6k_{B}}{\\pi \nK_{eff}}ln\\Bigl(\\frac{\\tau_{mes}}{\\tau_{0}}\\Bigr)\n\\end{equation} \n$\\tau_{0}$ is the microscopic relaxation time of the particle, taken independent \nof the temperature. The fit result is presented on Fig. 7. We find \na=3.5$\\pm$0.1 \nnm$^{3}$/K, and by taking $\\tau_{mes}$=10 s, and \n$\\tau_{0}$=10$^{-12}$-10$^{-9}$ s, we obtain K$_{eff}$=2.0 \n$\\pm$0.3.10$^{6}$ erg/cm$^{3}$.\nBy fitting Zero Field Cooled (ZFC) curves for different applied fields, we \ncan also evaluate K$_{eff}$. Besides, if we neglect the blocked particle \nsusceptibility, we have :\n\\begin{equation}\n\\frac{m_{ZFC}(H,T)}{m_{sat}}=\\frac{\\int_{0}^{D_{B}(H,T)}D^{3}L(x)f(D)\\,dD}{\\int_{0}^{\\infty}D^{3}f(D)\\,dD}\n\\end{equation}\nMoreover, for low field values compared with the anisotropy field of \ncobalt clusters (estimated to be $\\mu_{0}$H$_{a}$=0.4 T), we can make \nthe approximation :\n\\begin{equation}\nD_{B}^{3}(H,T)=af\\Bigl(\\frac{H}{H_{a}}\\Bigr)T\\approx a\\Bigl(1+\\alpha \\frac{H}{H_{a}}\\Bigr)T\n\\end{equation}\nwhere a is the coefficient of Eq. (7), $\\alpha$ a numerical constant. The ZFC \ncurve fits are presented on Fig. 8. A linear extrapolation to \n$\\mu_{0}$H=0 T \nalso gives a$\\approx$3.5 nm$^{3}$/K and an anisotropy constant of \n2.0$\\pm$0.3.10$^{6}$ erg/cm$^{3}$ \nfor the same numerical values as above. We found a similar result for the \nsecond sample with a niobium evaporation rate of 5 $\\AA$/s. Finally, \nwe experimentally found an anisotropy constant close to the one of \nquasi-spherical f.c.c cobalt clusters.\n\n\\section{discussion}\n\nThe \"magnetic size\" distribution is compared to the one obtained from \nTEM observations of pure Co-clusters prepared in the same experimental \nconditions (see Fig. 5(a)). For all the studied Co/Nb samples, we \nsystematically find a global size reduction which might be related to the \nformation of a non-magnetic alloy at the interface as suggested by EXAFS \nsimulations. The most significant parameter in the magnetically dead alloy \nthickness, seems to be the rate of deposition of the niobium matrix \n(V$_{Nb}$). \nAs an example, we mention that for V$_{Nb}$=5 $\\AA$/s, the reduction is twice the \none for V$_{Nb}$=3 $\\AA$/s (see Fig. 5(b)). That result suggests the model proposed \nin Fig. 9(a), 9(b).\nAs cobalt-niobium forms a miscible system\\cite{Mass73}, we show that the more V$_{Nb}$ \nincreases, the more the quantity of Nb-atoms introduced at the cobalt \ncluster surface increases. \\\\\nTo study the magnetism of the perturbed monolayers at the interface, we \nprepared a cobalt-niobium alloy using induction-heating under argon \natmosphere with 40 $\\%$-Co and 60 $\\%$-Nb atomic weights. From classical X-ray \n$\\theta$/2$\\theta$ diffraction ($\\lambda$=1.5406 $\\AA$), we identified \nthe $\\beta$-phase given by the \nbinary phase diagram : Co$_{6}$Nb$_{7}$. From VSM measurements on this sample, we \nfound a remaining paramagnetic susceptibility $\\chi$=10$^{-4}$ (for 2$<$T$<$300 K) \ncorresponding to the \"Pauli\" paramagnetism of the sample. This feature \ncould explain the \"dead\" layer at the cluster surface.\nObi and al.\\cite{Yobi99} obtained two \"magnetically dead\" cobalt monolayers on \ncobalt-niobium multilayers evaporated by a rf-dual type sputtering method \n(\"magnetically dead\" layers were also suggested by M\\\"uhge and \nal.\\cite{Muhg97} for Fe/Nb multilayers). \nFinally, we can underline the fact that the pre-formed cobalt clusters by \nLECBD technique are very compact nanocrystallites which conserve a \nmagnetic core even if embedded in a miscible matrix. \nThe existence of a \"magnetically dead\" layer at the cluster-matrix \ninterface may reduce surface effects compared with recent results obtained \non smaller cobalt particles (150-300 atoms) stabilized in polymers.\\cite{Resp98} The estimated mean\nanisotropy constant might correspond to cubic magnetocrystalline \nor shape effects. To confirm this assumption, works are in progress to investigate the magnetic properties of a single cluster \nin a niobium matrix using a new microsquid technique. \\\\\nOne can also mention that for XMCD signals detected from the total \nelectron yield method, the extraction of quantitative local magnetic \nvalues from the applicability of the individual orbital and spin sum rules \nis in progress.\\cite{Chen95} Nevertheless, one can mention a small enhancement of the \norbital/spin magnetic moment ratio.\\cite{Chen95} Such increase might come from the \norbital magnetic moment enhancement expected for small particles. \n\\cite{Dora97} Systematic XMCD studies on clusters assembled Co/X films \nshould be performed on Si-protected layers under synchrotron radiation to \nconfirm these results.\n\n\\section{conclusion}\n\nWe have shown that the magnetic properties of nanoparticles can be \nevaluated unambiguously if we know the size, the shape and the nature of \nthe interface. This latter is given by EXAFS spectroscopy. We summarize \nthe main results : \\\\\n\t- the mean like-bulk Co-Co distance (d$_{Co-Co}$=2.50 $\\AA$) concerns the 3/4 of \nthe atoms (namely : the core atoms) \\\\\n\t- Co-Nb bonds are located on roughly one monolayer at the surface of the \nCo-clusters embedded in the Nb-matrix. \\\\\n Even though this interface is \nrather sharp, it is of importance since the interface thickness is on the \nsame order of magnitude than the cluster radius.\nIn addition, some magnetic properties were approached by different \ncomplementary techniques as VSM magnetometry (at temperatures higher than \n8 K) and XMCD signal detected by the fluorescence yield method (at \ntemperatures from 5.3 K to 30 K) under a magnetic field. We show the good \nresult coherence on the superimposed range (8 K$<$T$<$30 K) for both techniques \nprobing the whole thickness of the sample. The main result is the \npossibility of a \"magnetically dead\" layer at the interface Co/Nb, to \nrelate to the alloyed interface (from EXAFS measurements) and to the \nmoderate anisotropy value (found around 2.10$^{6}$ erg/cm$^{3}$). To confirm this assumption and to \nunderstand the role of the interface on the anisotropy terms involved in \nso low dimension magnetic nanostructures, XMCD measurements at the \nCo-L$_{2,3}$ \nedge have to be provided on a Co/Nb bilayer stacking (alternating 2 \nmonolayers of Co and 2 monolayers of Nb) with the same Nb-deposition rates \nas in our systems.\n\n\\section{Aknowledgements}\n\nThe authors would like to thank M. NEGRIER and J. TUAILLON for fruitfull discussions, C. BINNS from the University of Leceister, \nUnited Kingdom and J.VOGEL from the Laboratoire Louis N\\'eel at Grenoble, \nFrance for their help during the first XMCD tests on the ID12B line of N. \nBROOKES at the ESRF in Grenoble.\n \n\n\\begin{thebibliography}{32}\n\n\\bibitem{Chan93}\nT. Chang, J. G. Zhu and J. H. Judy, J. Appl. Phys. $\\bf{73}$,6716 (1993)\n\n\\bibitem{Geim97}\nA. K. Geim, S. V. Dubonos, J. G. S. Lok, I. V. Grigorieva, J. C. Maan, \nL. Theil Hansen and P. E. Lindelof, Appl. Phys. Lett. $\\bf{71}$, 2379 \n(1997)\n\n\\bibitem{Wern97}\nW. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, \nN. Demoncy, A. Loiseau, H. Pascard and D. Mailly, Phys. Rev. Lett. \n$\\bf{78}$, 1791 (1997)\n\n\\bibitem{Pare97}\nF. Parent, J. Tuaillon, L. B. Steren, V. Dupuis, B. Pr\\'evel, P. \nM\\'elinon, A. P\\'erez, G. Guiraud, R. Morel, A. Barth\\'el\\'emy and A. \nFert, Phys. Rev. B $\\bf{55}$, 3683 (1997)\n\n\\bibitem{Mila90}\nP. Milani and W. A. de Heer, Rev. Sci. Instr. $\\bf{61}$, 1835 (1990)\n\n\\bibitem{Pere97}\nA. P\\'erez, P. M\\'elinon, V. Dupuis, P. Jensen, B. Pr\\'evel, M. Broyer, \nM. Pellarin, J. L. Vialle, B. Palpant, J. Phys. D $\\bf{30}$, 1 (1997)\n\n\\bibitem{Pell94}\nM. Pellarin, E. Cottancin, J. Lerm\\'e, J. L. Vialle, J. P. Wolf, M. \nBroyer, V. Paillard, V. Dupuis, A. P\\'erez, J. P. P\\'erez, J. \nTuaillon, P. M\\'elinon, Chem. Phys. Lett. $\\bf{224}$, 338 (1994)\n\n\\bibitem{Negr99}\nM. N\\'egrier, J. Tuaillon, V. Dupuis, P. M\\'elinon, A. P\\'erez, A. \nTraverse, submitted to Phil. Mag. A (1999)\n\n\\bibitem{Tuai97}\nJ. Tuaillon, V. Dupuis, P. M\\'elinon, B. Pr\\'evel, M. Treilleux, A. \nP\\'erez, M. Pellarin, J. L. Vialle, M. Broyer, Phil. Mag. A $\\bf{76}$, \n493 (1997)\n\n\\bibitem{Baud93}\nF. Baudelet, A. Fontaine, G. Tourillon, D. Gay, M. Maurer, M. Piecuch, \nM. F. Ravet, V. Dupuis, Phys. Rev. B $\\bf{47}$, 2344 (1993)\n\n\\bibitem{Mima94}\nJ. Mimault, J. J. Faix, T. Girardeau, M. Jaouen and G. Tourillon, Meas. Sci. Technol. $\\bf{5}$, 482 (1994)\n\n\\bibitem{Carr90}\nB. Carriere, G. Krill, Mat. Sci. Forum $\\bf{59}$, 221 (1990)\n\n\\bibitem{Kale86}\nA. G. Mc. Kale, G. Sknapp and S. K. Chan, Phys. Rev. B $\\bf{33}$, 841 \n(1986)\n\n\\bibitem{Mich97}\nA. Michalowicz, N. Allali, J. Phys. IV France $\\bf{7}$, C2-261 (1997)\n\n\\bibitem{MRoy97}\nM. Roy, J. Phys. IV France $\\bf{C2}$, 151 (1997)\n\n\\bibitem{Prou97}\nE. Prouzet, A. Michalowicz and N. Allali, J. Phys. IV France $\\bf{7}$, C2-261 (1997)\n\n\\bibitem{Vdup93}\nV. Dupuis, M. Maurer, M. Piecuch, M. F. Ravet, J. Dekoster, S. \nAndrieu, J. F. Bobo, F. Baudelet, P. Bauer, A. Fontaine Phys. Rev. B $\\bf{48}$, 5585 (1993)\n\n\\bibitem{Drey99}\n\\v S. Pick, H. Dreyss\\'e, J. Magn. Magn. Mater. $\\bf{198}$, \n312 (1999)\n\n\\bibitem{Hard69}\nR. V. Hardeveld and F. Hartog, Surf. Sci. $\\bf{15}$, 189 (1969)\n\n\\bibitem{Comm99}\nThe anisotropy constant used for this calculation is given in Section B.\n\n\\bibitem{Yobi99}\nY. Obi, M. Ikebe, T. Kubo, H. Fujimori, Physica C $\\bf{317-18}$, 149 \n(1999)\n\n\\bibitem{Bill94}\nI. M. L. Billas, A. Chatelain and W. A. de Heer, Science $\\bf{265}$, \n1682 (1994)\n\n\\bibitem{Resp98}\nM. Respaud, J. M. Broto, H. Rakoto, A. R. Fert, L. Thomas, B. \nBarbara, Phys. Rev. B $\\bf{57}$, 2925 (1998)\n\n\\bibitem{Chen94}\nJ. P. Chen, C. M. Sorensen, K. J. Klabunde and G. C. Hadjipanayis, J. \nAppl. Phys. $\\bf{76}$, 6676 (1994)\n\n\\bibitem{Ahar96}\nA. Aharoni, {\\it Introduction to the theory of ferromagnetism} (Oxford \nScience Publications) (1996)\n\n\\bibitem{Chen93}\nC. T. Chen, Y. U. Idzerda, H.-J. Lin, G. Meigs, A. Chaiken, G. A. \nPrinz, G. H. Ho, Phys. Rev. B $\\bf{48}$, 642 (1993)\n\n\\bibitem{Mull73}\nK. M\\\"uller and F. Thurley, Int. J. Magnetism $\\bf{5}$, 203 \n(1973)\n\n\\bibitem{Dorm97}\nJ. L. Dormann, D. Fiorani, E. Tronc, Adv. in Chem. Phys. Vol. \n$\\bf{XCVIII}$, Ed. by I. Prigogine and Stuart A. Rice (1997)\n\n\\bibitem{Brun89}\nP. Bruno, Ph. D. Thesis, Paris Orsay, France, 1989\n\n\\bibitem{Alde92}\nM. Alden, S. Mirbt, H. L. Skriver, N. M. Rosengaard, B. Johansson, Phys. \nRev. B $\\bf{46}$, 6303 (1992)\n\n\\bibitem{Dora97}\nJ. Dorantes-Davila, H. Dreyss\\'e, G. M. Pastor, Phys. Rev. B \n$\\bf{55}$, 15033 (1997)\n\n\\bibitem{Mass73}\nT. B. Massalski, J. L. Murray, L. H. Bennett, H. Baker, {\\it Binary \nPhase Diagrams}, American Society for Metals, Metals Park, Ohio 440 73\n\n\\bibitem{Muhg97}\nTh. M\\\"uhge, K. Westerholt, H. Zabel, N. N. Garifyanov, Yu. V. \nGoryunov, I. A. Garifullin and G. G. Khaliullin, Phys. Rev. B \n$\\bf{55}$, 8945 (1997)\n\n\\bibitem{Chen95}\nC. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs, E. \nChaban, G. H. Ho, E. Pellegrin and F. Sette, Phys. Rev. Lett. \n$\\bf{75}$, 152 (1995)\n\n\\end{thebibliography}\n\n\n\n%% Figures\n\n\n\\begin{figure}\n\n\\caption{EXAFS spectrum obtained on the sample containing 5 $\\%$ of cobalt \nclusters embedded in niobium ($\\circ$ : experimental data, continuous line : \nsimulation). }\n\n\\label{fig 1}\n\\end{figure}\n\n\\begin{figure}\n\n\\caption{Inverse of the sample susceptibility plotted versus the \ntemperature T, making a linear extrapolation of this curve for high \ntemperatures, one obtains, when 1/$\\chi \\rightarrow$ 0, an idea of the interaction \ntemperature : $\\theta$=1-2 K.}\n\n\\label{fig 2}\n\\end{figure}\n\n\n\n\\begin{figure}\n\n\\caption{In the superparamagnetic regime, m(H/T) curves superimpose for \nT$=$100, 200 and 300 K. ($+$ : 300 K, $\\times$ : 200 K, $\\diamond$ : 100 K). At \nT$=$30 K ($\\bullet$ : 30 K), \nthe anisotropy energy is no more negligible, and one has to use a modified \nLangevin function to fit the curve.}\n\n\\label{fig 3}\n\\end{figure}\n\n\\begin{figure}\n\n\\caption{ We use a classical Langevin function to fit experimental \nmagnetization curves m(H) in the superparamagnetic regime ($\\bullet$ : \nexperimental data, continuous lines : fits). This allows us to deduce the \nmean particle diameter D$_{m}$ and the dispersion $\\sigma$ of the \n\"magnetic\" size \ndistribution considering a log-normal distribution for cobalt clusters.}\n\n\\label{fig 4}\n\\end{figure}\n\n\\begin{figure}\n\n\\caption{(a) TEM size distribution : D$_{m}$=3.0$\\pm$0.1 nm, \n$\\sigma$=0.24$\\pm$0.01 and \n\"magnetic\" size distribution : D$_{m}$=2.3$\\pm$0.1 nm, \n$\\sigma$=0.24$\\pm$0.01 for a niobium \ndeposition rate V$_{Nb}$=3 $\\AA$/s and a log-normal distribution, (b) TEM size \ndistribution : D$_{m}$=3.2$\\pm$0.1 nm, $\\sigma$=0.25$\\pm$0.01 and \"magnetic\" size \ndistribution : D$_{m}$=1.8$\\pm$0.1 nm, $\\sigma$=0.25$\\pm$0.01 for a niobium deposition rate \nV$_{Nb}$=5 $\\AA$/s.}\n\n\\label{fig 5}\n\\end{figure}\n\n\\begin{figure}\n\n\\caption{The volumic shape anisotropy energy (K$_{shape}$ in erg/cm$^{3}$) is plotted \n(continuous line) vs. the particle deformation c/a assuming it is a \nprolate spheroid. We also report volumic magnetocristalline anisotropy \nenergy for f.c.c and h.c.p cobalt.}\n\n\\label{fig 6}\n\\end{figure}\n\n\\begin{figure}\n\n\\caption{ Remanent magnetic moment plotted vs. the temperature T. The signal \nis first normalized writting : m$_{r}$(8.1K)$_{VSM}$=m$_{r}$(8.1K)$_{XMCD}$, and then we take \nm$_{r}$(5.3K)=1. We see that the continuous line curve fits both VSM ($\\bullet$) and XMCD ($\\circ$) measurements. From this fit, we can deduce the anisotropy constant K$_{eff}$.}\n\n\\label{fig 7}\n\\end{figure}\n\n\\begin{figure}\n\n\\caption{(a) ZFC curves (black dots) taken for 6 different applied fields : (a) \n0.002 T, (b) 0.005 T, (c) 0.0075 T, (d) 0.01 T, (e) 0.015 T, (f) 0.02 T. \n(b) Fits (continuous lines) allow to deduce af(H/H$_{a}$) \n(D$_{B}^{3}$(H,T)=af(H/H$_{a}$)T) and then \nthe anisotropy constant K$_{eff}$.}\n\n\\label{fig 8}\n\\end{figure}\n\n\\begin{figure}\n\n\\caption{(a) Expected magnetic structure of cobalt clusters embedded in a \nniobium matrix, we find one \n\"magnetically dead\" cobalt monolayer (2$\\times$3.5 $\\AA$ in diameter) for a V$_{Nb}$=3 $\\AA$/s deposition rate, (b) two \"magnetically dead\" cobalt \nmonolayers (4$\\times$3.5 $\\AA$ in diameter) are found for V$_{Nb}$=5 $\\AA$/s.}\n\n\\label{fig 9}\n\\end{figure}\n\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-mat0002126.extracted_bib",
"string": "\\begin{thebibliography}{32}\n\n\\bibitem{Chan93}\nT. Chang, J. G. Zhu and J. H. Judy, J. Appl. Phys. $\\bf{73}$,6716 (1993)\n\n\\bibitem{Geim97}\nA. K. Geim, S. V. Dubonos, J. G. S. Lok, I. V. Grigorieva, J. C. Maan, \nL. Theil Hansen and P. E. Lindelof, Appl. Phys. Lett. $\\bf{71}$, 2379 \n(1997)\n\n\\bibitem{Wern97}\nW. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, \nN. Demoncy, A. Loiseau, H. Pascard and D. Mailly, Phys. Rev. Lett. \n$\\bf{78}$, 1791 (1997)\n\n\\bibitem{Pare97}\nF. Parent, J. Tuaillon, L. B. Steren, V. Dupuis, B. Pr\\'evel, P. \nM\\'elinon, A. P\\'erez, G. Guiraud, R. Morel, A. Barth\\'el\\'emy and A. \nFert, Phys. Rev. B $\\bf{55}$, 3683 (1997)\n\n\\bibitem{Mila90}\nP. Milani and W. A. de Heer, Rev. Sci. Instr. $\\bf{61}$, 1835 (1990)\n\n\\bibitem{Pere97}\nA. P\\'erez, P. M\\'elinon, V. Dupuis, P. Jensen, B. Pr\\'evel, M. Broyer, \nM. Pellarin, J. L. Vialle, B. Palpant, J. Phys. D $\\bf{30}$, 1 (1997)\n\n\\bibitem{Pell94}\nM. Pellarin, E. Cottancin, J. Lerm\\'e, J. L. Vialle, J. P. Wolf, M. \nBroyer, V. Paillard, V. Dupuis, A. P\\'erez, J. P. P\\'erez, J. \nTuaillon, P. M\\'elinon, Chem. Phys. Lett. $\\bf{224}$, 338 (1994)\n\n\\bibitem{Negr99}\nM. N\\'egrier, J. Tuaillon, V. Dupuis, P. M\\'elinon, A. P\\'erez, A. \nTraverse, submitted to Phil. Mag. A (1999)\n\n\\bibitem{Tuai97}\nJ. Tuaillon, V. Dupuis, P. M\\'elinon, B. Pr\\'evel, M. Treilleux, A. \nP\\'erez, M. Pellarin, J. L. Vialle, M. Broyer, Phil. Mag. A $\\bf{76}$, \n493 (1997)\n\n\\bibitem{Baud93}\nF. Baudelet, A. Fontaine, G. Tourillon, D. Gay, M. Maurer, M. Piecuch, \nM. F. Ravet, V. Dupuis, Phys. Rev. B $\\bf{47}$, 2344 (1993)\n\n\\bibitem{Mima94}\nJ. Mimault, J. J. Faix, T. Girardeau, M. Jaouen and G. Tourillon, Meas. Sci. Technol. $\\bf{5}$, 482 (1994)\n\n\\bibitem{Carr90}\nB. Carriere, G. Krill, Mat. Sci. Forum $\\bf{59}$, 221 (1990)\n\n\\bibitem{Kale86}\nA. G. Mc. Kale, G. Sknapp and S. K. Chan, Phys. Rev. B $\\bf{33}$, 841 \n(1986)\n\n\\bibitem{Mich97}\nA. Michalowicz, N. Allali, J. Phys. IV France $\\bf{7}$, C2-261 (1997)\n\n\\bibitem{MRoy97}\nM. Roy, J. Phys. IV France $\\bf{C2}$, 151 (1997)\n\n\\bibitem{Prou97}\nE. Prouzet, A. Michalowicz and N. Allali, J. Phys. IV France $\\bf{7}$, C2-261 (1997)\n\n\\bibitem{Vdup93}\nV. Dupuis, M. Maurer, M. Piecuch, M. F. Ravet, J. Dekoster, S. \nAndrieu, J. F. Bobo, F. Baudelet, P. Bauer, A. Fontaine Phys. Rev. B $\\bf{48}$, 5585 (1993)\n\n\\bibitem{Drey99}\n\\v S. Pick, H. Dreyss\\'e, J. Magn. Magn. Mater. $\\bf{198}$, \n312 (1999)\n\n\\bibitem{Hard69}\nR. V. Hardeveld and F. Hartog, Surf. Sci. $\\bf{15}$, 189 (1969)\n\n\\bibitem{Comm99}\nThe anisotropy constant used for this calculation is given in Section B.\n\n\\bibitem{Yobi99}\nY. Obi, M. Ikebe, T. Kubo, H. Fujimori, Physica C $\\bf{317-18}$, 149 \n(1999)\n\n\\bibitem{Bill94}\nI. M. L. Billas, A. Chatelain and W. A. de Heer, Science $\\bf{265}$, \n1682 (1994)\n\n\\bibitem{Resp98}\nM. Respaud, J. M. Broto, H. Rakoto, A. R. Fert, L. Thomas, B. \nBarbara, Phys. Rev. B $\\bf{57}$, 2925 (1998)\n\n\\bibitem{Chen94}\nJ. P. Chen, C. M. Sorensen, K. J. Klabunde and G. C. Hadjipanayis, J. \nAppl. Phys. $\\bf{76}$, 6676 (1994)\n\n\\bibitem{Ahar96}\nA. Aharoni, {\\it Introduction to the theory of ferromagnetism} (Oxford \nScience Publications) (1996)\n\n\\bibitem{Chen93}\nC. T. Chen, Y. U. Idzerda, H.-J. Lin, G. Meigs, A. Chaiken, G. A. \nPrinz, G. H. Ho, Phys. Rev. B $\\bf{48}$, 642 (1993)\n\n\\bibitem{Mull73}\nK. M\\\"uller and F. Thurley, Int. J. Magnetism $\\bf{5}$, 203 \n(1973)\n\n\\bibitem{Dorm97}\nJ. L. Dormann, D. Fiorani, E. Tronc, Adv. in Chem. Phys. Vol. \n$\\bf{XCVIII}$, Ed. by I. Prigogine and Stuart A. Rice (1997)\n\n\\bibitem{Brun89}\nP. Bruno, Ph. D. Thesis, Paris Orsay, France, 1989\n\n\\bibitem{Alde92}\nM. Alden, S. Mirbt, H. L. Skriver, N. M. Rosengaard, B. Johansson, Phys. \nRev. B $\\bf{46}$, 6303 (1992)\n\n\\bibitem{Dora97}\nJ. Dorantes-Davila, H. Dreyss\\'e, G. M. Pastor, Phys. Rev. B \n$\\bf{55}$, 15033 (1997)\n\n\\bibitem{Mass73}\nT. B. Massalski, J. L. Murray, L. H. Bennett, H. Baker, {\\it Binary \nPhase Diagrams}, American Society for Metals, Metals Park, Ohio 440 73\n\n\\bibitem{Muhg97}\nTh. M\\\"uhge, K. Westerholt, H. Zabel, N. N. Garifyanov, Yu. V. \nGoryunov, I. A. Garifullin and G. G. Khaliullin, Phys. Rev. B \n$\\bf{55}$, 8945 (1997)\n\n\\bibitem{Chen95}\nC. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs, E. \nChaban, G. H. Ho, E. Pellegrin and F. Sette, Phys. Rev. Lett. \n$\\bf{75}$, 152 (1995)\n\n\\end{thebibliography}"
}
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cond-mat0002127
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Magnetic free energy at elevated temperatures and hysteresis of magnetic particles %\vspace{-1mm}
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[
{
"author": "H. Kachkachi$^*$ and D. A. Garanin$^{\\dagger}$"
}
] |
We derive a free energy for weakly anisotropic ferromagnets which is valid in the whole range of temperature and interpolates between the micromagnetic energy at zero temperature and the Landau free energy near the Curie point $T_c$. % This free energy takes into account the change of the magnetization length due to thermal effects, in particular, in the inhomogeneous states. % As an illustration, we study the thermal effect on the Stoner-Wohlfarth curve and hysteresis loop of a ferromagnetic nanoparticle assuming that it is in a single-domain state. % Within this model, the saddle point of the particle's free energy, as well as the metastability boundary, are due to the change in the magnetization length sufficiently close to $T_c$, as opposed to the usual homogeneous rotation process at lower temperatures. %
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[
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"name": "tsw.tex",
"string": "\n\n%\\documentstyle[aps,prb,twocolumn,floats,psfig]{revtex}\n\\documentstyle[a4,12pt,psfig]{article}\n\n\n\\textheight= 240 truemm\n%\\textwidth=25\n%\\topmargin= -5 truemm % for Eugene\n\\topmargin= -15 truemm % the standard setting\n\n\n\n\\newcommand{\\gtrsim}{\n\\,\\raisebox{0.35ex}{$>$}\n\\hspace{-1.7ex}\\raisebox{-0.65ex}{$\\sim$}\\,\n}\n\n\\newcommand{\\lesssim}{\n\\,\\raisebox{0.35ex}{$<$}\n\\hspace{-1.7ex}\\raisebox{-0.65ex}{$\\sim$}\\,\n}\n\n\\newcommand{\\onehalf}{\\mbox{\\scriptsize \n\\raisebox{1.5mm}{1}\\hspace{-2.7mm}\n\\raisebox{0.mm}{$-$}\\hspace{-2.8mm}\n\\raisebox{-0.9mm}{2}\\hspace{-0.7mm}\n\\normalsize }}\n\n\\newcommand{\\erfc}{ {\\rm erfc} }\n\\newcommand{\\const}{ {\\rm const} }\n\\newcommand{\\arctanh}{ {\\rm arctanh} }\n\n\n\n\\begin{document}\n\n\n\\bibliographystyle{prsty}\n\n%\\wideabs{\n\n%\\begin{flushleft}\n%{\\small \\em submitted to}\\\\\n%{\\small \n%PHYSICAL REVIEW B \n%\\hfill\n%VOLUME {\\normalsize XX}, \n%NUMBER {\\normalsize XX} $\\qquad\\qquad$\n%\\hfill \n%MONTH {\\normalsize XX}\n%{\\normalsize 1} NOVEMBER {\\normalsize 1997-}I, {\\normalsize 11102$-$11118} \n%, 3250-3256}\n%}\n%\\end{flushleft} \n\n\n\\title{ \nMagnetic free energy at elevated temperatures and hysteresis of magnetic particles\n%\\vspace{-1mm}\n} \n\n\\author{\nH. Kachkachi$^*$ and D. A. Garanin$^{\\dagger}$ \n}\n\\date{}\n\n\\maketitle\n%\\address{ \n$^*$Laboratoire de Magn\\'{e}tisme et d'Optique, Univ. de Versailles St. Quentin, \n45 av. des Etats-Unis, 78035 Versailles, France \n%}\n\n%\\address{ \n$^{\\dagger}$Max-Planck-Institut f\\\"ur Physik komplexer Systeme, N\\\"othnitzer Strasse 38,\nD-01187 Dresden, Germany \n%}\n\n\\begin{abstract}\nWe derive a free energy for weakly anisotropic ferromagnets which is valid in the\nwhole range of temperature and interpolates between the micromagnetic energy at zero temperature \nand the Landau free energy near the Curie point $T_c$.\n% \nThis free energy takes into account the change of the magnetization length due to thermal\neffects, in particular, in the inhomogeneous states.\n%\nAs an illustration, we study the thermal effect on the Stoner-Wohlfarth curve and hysteresis \nloop of a ferromagnetic nanoparticle assuming that it is in a single-domain state.\n%\nWithin this model, the saddle point of the particle's free energy, as well as the metastability\nboundary, are due to the change in the magnetization length sufficiently close to $T_c$, as opposed to the usual homogeneous rotation process at lower temperatures. \n%\n\\end{abstract} \n\\smallskip\n\\begin{flushleft}\nPACS numbers: 75.10.-b, 75.50.Tt\\\\\nKeywords: Thermal effects, hysteresis loop, Fine-particle systems\n\\end{flushleft}\n%}\n\n\\section{Introduction}\n\\label{introduction}\n\n\nThe macroscopic energy of weakly anisotropic ferromagnets which first appeared in the seminal\npaper by Landau and Lifshitz \\cite{lanlif35}, has been an instrument for\ninnumerable investigations of domain walls and other inhomogeneous states of magnetic systems.\n%\nLater an approach based on this macroscopic energy has been called\n``micromagnetics'' \\cite{bro63mic}. \n%\nStrictly speaking, micromagnetics is an essentially zero-temperature theory for {\\em\nclassical} magnets, as it considers the magnetization as a vector of fixed length.\n%\nUnder these conditions, it can be easily obtained as a continuous limit of the classical\nHamiltonian on a lattice.\n%\nPractically, micromagnetics has been applied to nonzero temperatures as well, with\ntemperature-dependent equilibrium magnetization and anisotropy constants.\n\n \nOn the other hand, close to the Curie temperature $T_c$ a magnetic free energy of the\nLandau type can be considered.\n%\nThis free energy allows the magnetization to change both in direction and length, and it\nis, in fact, a continuous limit of the free energy following from the mean-field\napproximation (MFA).\n%\nUsing this approach, Bulaevskii and Ginzburg \\cite{bulgin63} predicted a phase transition between the\nBloch walls and the Ising-like walls at temperatures slightly below $T_c$. \n%\nMuch later this phase transition was observed experimentally\n\\cite{koegarharjah93,harkoegar95} using the theoretical results for the mobility of domain\nwalls in this regime \\cite{gar91llb,gar91edw}.\n\n\nAlthough both of these approaches have been formulated at the same mean-field level, they have\nbeen considered as unrelated for a long time.\n%\nOn the other hand, the MFA itself is an all-temperature approximation, and thus one can\nquestion its macroscopic limit in the whole temperature range.\n%\nThe corresponding macroscopic free energy should interpolate between the Landau-Lifshitz\nenergy, or micromagnetics at $T=0$ and the Landau free energy in the vicinity of $T_c$.\n%\nThe Euler equation for the magnetization which minimizes this generalized free energy\nappears in Refs.\\ \\cite{gar91llb} and \\cite{gar97prb}.\n%\nWhereas the condition for the Landau theory is $M \\ll M_s$, where $M_s$ is the saturation\nmagnetization at $T=0$, the new equations only require $|M-M_e| \\ll M_s$, where $M_e$ is\nthe equilibrium magnetization at zero field.\n%\nFor weakly anisotropic magnets (the anisotropy energy is much less than the homogeneous exchange\nenergy, and this is satisfied by most compounds) the magnetization magnitude $M$ is either\nsmall or only slightly deviates from $M_e$, thus the condition above is satisfied in the\nwhole temperature range.\n\n\nDerivation of the generalized macroscopic free energy from the MFA is much subtler\nthan that of the Euler equations.\n%\nThis free energy appears in Ref.\\ \\cite{gar97prb} where its form was guessed.\n%\nIn this paper, we will present this derivation and illustrate the resulting magnetic free\nenergy in the case of a single-domain magnetic particle with a uniaxial anisotropy.\n%\nFor the sake of transparency, we will consider the exchange anisotropy rather than the single-site\nanisotropy; this does not change the qualitative results.\n%\nIt will be shown that the free-energy landscape for the magnetic particle has different\nforms at low temperatures and near $T_c$.\n%\nAt low temperatures, the saddle point between the ``up'' and ``down'' minima is located\nnear the sphere $|{\\bf M}| = M_e$, the value of $M$ being somewhat reduced in comparison with\n$M_e$ due to thermal effects at $T>0$.\n%\nNear $T_c$, the value of $M$ strongly changes by going from one minimum to another\nthrough the saddle point; for zero fields the saddle point becomes $\\bf M =0$. \n%\nThese effects also modify the metastability boundary of the magnetic particle (the famous\nStoner-Wohlfarth curve \\cite{stowoh4891} which was experimentally observed in \nRef.\\ \\cite{weretal97}), and its hysteresis curves. \n\n\n\nThe main body of this paper is organized as follows.\n%\nIn Sec.\\ \\ref{secfenergy} we give the derivation of the magnetic free energy at all\ntemperatures.\n%\nIn Sec.\\ \\ref{seclandscape} the free-energy landscape of a single-domain magnetic\nparticle is analyzed.\n%\nIn Secs.\\ \\ref{secastroid} and \\ref{sechysteresis} we study the Stoner-Wohlfarth curves and\nhysteresis loops at different temperatures.\n%\nIn Sec.\\ \\ref{HomoTherm} we discuss the possibility of observing the\nnew thermal effects for single-domain magnetic particles.\n\n\n\n\\section{Free energy of a magnetic particle}\n\\label{secfenergy}\n\nLet us start with the {\\em biaxial} ferromagnetic model described by the \n classical anisotropic Hamiltonian of the type\n%\n\\begin{eqnarray}\\label{biaxham}\n{\\cal H} = - \\mu_0{\\bf H} \\sum_i {\\bf s}_i\n- \\frac{1}{2}\\sum_{ij}J_{ij}\n(s_{zi}s_{zj} + \\eta_y s_{yi}s_{yj} + \\eta_x s_{xi} s_{xj}) ,\n\\end{eqnarray}\n%\nwhere $\\mu_0$ is the magnetic moment of the atom, $i,j$ are lattice sites, \n${\\bf s}_i$ is the normalised vector, \n$|{\\bf s}_i|=1$, and the dimensionless ani\\-so\\-t\\-ro\\-py factors satisfy \n$\\eta_x \\leq \\eta_y \\leq 1$.\n\n%\nTo study the macroscopic properties of this system at nonzero\ntemperatures, it is convenient to use a macroscopic free energy. \n%\nIn the literature one can find two types of macroscopic free energies\nfor magnets.\n%\nOne of them is the so-called micromagnetic free energy which is valid at zero\ntemperature, and the other one is Landau's free energy which is\napplied near the Curie temperature $T_c$. \n%\nSince both free energies are based on the mean-field approximation\n(MFA), it is possible to derive a simple form of the MFA free\nenergy for weakly anisotropic ferromagnets which is valid in the whole\ntemperature range and bridges these two well-known forms.\n%\n\nThe free energy $F=-T\\ln {\\cal Z}$ of a spin system \ndescribed by the Hamiltonian in Eq.\\ (\\ref{biaxham}) \ncan be calculated in the mean-field approximation by \nconsidering each spin on a site $i$ as an isolated \nspin in the effective field containing contributions determined \nby the mean values of the neighboring ones. Namely, \n%\n\\begin{equation}\\label{hammfa}\n%\n{\\cal H} \\Rightarrow {\\cal H}^{\\rm MFA} \n= \n{\\cal H}_{00} - \\sum_i {\\bf H}_i^{\\rm MFA} {\\bf s}_i ,\n%\n\\end{equation}\n%\nwhere\n%\n\\begin{equation}\\label{ham00}\n%\n{\\cal H}_{00} \n=\n\\frac{1}{2}\\sum_{ij}J_{ij}\n\\left(\n\\sigma_{zi}\\sigma_{zj} \n+ \n\\eta_x \\sigma_{xi} \\sigma_{xj}\n+ \n\\eta_y \\sigma_{yi}\\sigma_{yj}\n\\right) , \n%\n\\end{equation}\n%\n$\\mbox{\\boldmath $\\sigma$}_i \\equiv \\langle{\\bf s}_i\\rangle$ is the spin polarization, \nand the molecular field ${\\bf H}_i^{\\rm MFA}$ is given by\n%\n\\begin{equation}\\label{fieldmfa}\n%\n{\\bf H}_i^{\\rm MFA} = \\mu_0 {\\bf H} \n+ \\sum_j J_{ij} \n\\left(\n\\sigma_{zj} {\\bf e}_z \n+ \\eta_x \\sigma_{xj} {\\bf e}_x \n+ \\eta_y \\sigma_{yj} {\\bf e}_y\n\\right) .\n%\n\\end{equation}\n%\nThen the solution of the one-spin problem in Eq.\\ (\\ref{hammfa}) leads to\n%\n\\begin{eqnarray}\\label{fenergymicro}\n%\n&&\nF = {\\cal H}_{00} - NT\\ln(4\\pi) - T \\sum_i \\Lambda(\\xi_i) \\nonumber \\\\\n&&\n \\Lambda(\\xi)\\equiv \\ln\\left(\\frac{\\sinh(\\xi)}{\\xi}\\right) ,\n%\n\\end{eqnarray}\n%\nwhere $N$ is the total number of spins,\n$\\xi_i \\equiv |\\mbox{\\boldmath $\\xi$}_i|$, and\n$\\mbox{\\boldmath $\\xi$}_i \\equiv \\beta {\\bf H}_i^{\\rm MFA}$.\n%\nThe MFA free energy determined by Eq.\\ (\\ref{fieldmfa}) and \nEq.\\ (\\ref{fenergymicro}) can be minimized with respect to the spin \naverages $\\mbox{\\boldmath $\\sigma$}_i$ to find the equilibrium solution \nin the general case where the anisotropy $1-\\eta_{x,y}$ is not \nnecessarily small.\n%\nThe minimum condition for the free energy, \n$\\partial F/\\partial\\mbox{\\boldmath $\\sigma$}_i=0$,\nleads to the Curie-Weiss equation\n%\n\\begin{equation}\\label{cweissgen}\n%\n\\mbox{\\boldmath $\\sigma$}_i \n=\nB(\\xi_i)\\frac{\\mbox{\\boldmath $\\xi$}_i}{\\xi_i},\n%\n\\end{equation}\n%\nwhere $B(\\xi)=\\coth(\\xi) - 1/\\xi$ is the Langevin function.\n\nFor small anisotropy, $1-\\eta_{x,y} \\ll 1$, one can go over to the \ncontinuum limit and write for the short-range interaction $J_{ij}$\n%\n\\begin{equation}\\label{spinlapl}\n%\n\\sum_j J_{ij}\\mbox{\\boldmath $\\sigma$}_j \n\\cong \nJ_0\\mbox{\\boldmath $\\sigma$}_i \n+ \nJ_0\\alpha\\Delta\\mbox{\\boldmath $\\sigma$}_i ,\n%\n\\end{equation}\n%\nwhere $\\Delta$ is the Laplace operator acting on the components\nof $\\mbox{\\boldmath $\\sigma$}({\\bf r})$, \n$J_0$ is the zero Fourier component (the zeroth moment), and \n$J_0\\alpha$ is the second moment of the exchange interaction $J_{ij}$.\n%\nFor the simple cubic lattice with nearest neighbor interactions $\\alpha=a_0^2/z$,\n$z=6$, and $a_0$ is the lattice spacing.\n%\nGoing from summation to integration in Eq.\\ (\\ref{fenergymicro}), one obtains\n%\n\\begin{eqnarray}\\label{fenergymacro1}\n\\frac{F}{J_0} = - \\frac{NT}{J_0}\\ln(4\\pi) + \\frac{1}{v_0} \\!\\!\\int\\!\\! d{\\bf r}\n\\left\\{\n\\frac{1}{2}\\sigma^2 \n+ \n\\frac{1}{2}( \\mbox{\\boldmath $\\sigma$},{\\bf h}_{\\rm eff}-{\\bf h} )\n- \n\\frac{1}{\\beta J_0} \\Lambda(\\xi)\n\\right\\} ,\n\\end{eqnarray}\n%\nwhere $v_0$ is the unit-cell volume,\n%\n%\\marginpar{defheff}\n%\n\\begin{eqnarray}\\label{defheff}\n%\n&&\n\\mbox{\\boldmath $\\xi$} \n= \\beta J_0 ( \\mbox{\\boldmath $\\sigma$} + {\\bf h}_{\\rm eff} )\n\\nonumber \\\\\n&&\n{\\bf h}_{\\rm eff} = {\\bf h} + \\alpha \\Delta \\mbox{\\boldmath $\\sigma$} \n- (1-\\eta_x) \\sigma_x {\\bf e}_x - (1-\\eta_y) \\sigma_y {\\bf e}_y ,\n%\n\\end{eqnarray}\n%\nand ${\\bf h} \\equiv \\mu_0 {\\bf H}/J_0$.\n%\nWe will consider the case of small fields, $h \\ll 1$.\n%\nSince in this case in Eq.\\ (\\ref{defheff}) \n$|{\\bf h}_{\\rm eff}| \\ll |\\mbox{\\boldmath $\\sigma$}|$ in the whole \nrange below $T_c$ (near $T_c$ the value of $\\sigma$ is small but the susceptibility is\nlarge), the last term of Eq.\\ (\\ref{fenergymacro1}) can be \nexpanded to first order in ${\\bf h}_{\\rm eff}$ using \n%\n%\\marginpar{deltaxi}\n%\n\\begin{equation}\\label{deltaxi}\n%\n\\xi = \\xi_0 + \\delta\\xi, \\qquad\n%\n\\xi_0 = \\beta J_0 \\sigma, \\qquad\n%\n\\delta\\xi \n\\cong \n\\beta J_0 \\frac{\\mbox{\\boldmath $\\sigma$}{\\bf h}_{\\rm eff}}{\\sigma}\n%\n\\end{equation}\n%\nand the first two terms of the expansion\n%\n%\\marginpar{devlam}\n%\n\\begin{equation}\\label{devlam}\n%\n\\Lambda(\\xi) \n\\cong \n\\Lambda(\\xi_0) \n+ \nB(\\xi_0)\\delta\\xi \n+ \n\\frac{1}{2}B'(\\xi_0)(\\delta\\xi)^2 ,\n%\n\\end{equation}\n%\nwhere $B'(\\xi)\\equiv dB(\\xi)/d\\xi$.\n%\nHence, \n%\n%\\marginpar{fenergymacro}\n%\n\\begin{eqnarray}\\label{fenergymacro}\n\\frac{F}{J_0} &=& \\frac{1}{v_0} \\!\\!\\int\\!\\! d{\\bf r}\n\\left\\{\\frac{1}{2}\\sigma^2 -\\frac{1}{\\beta J_0} \\Lambda(\\xi_0) -\n\\frac{B(\\xi_0)}{\\sigma}\\mbox{\\boldmath $\\sigma$}{\\bf h} \\right.\n\\left.-\n\\left(\\frac{B(\\xi_0)}{\\sigma}-\\frac{1}{2}\\right)\n( \\mbox{\\boldmath $\\sigma$},{\\bf h}_{\\rm eff}-{\\bf h} )\n\\right\\}\\nonumber \\\\ \n&-& \\frac{NT}{J_0}\\ln(4\\pi). \n\\end{eqnarray}\n%\nNear $T_c = T_c^{\\rm MFA}= J_0/3$ the order parameter $\\mbox{\\boldmath $\\sigma$}$ becomes small, \nand using $\\Lambda(\\xi) \\cong \\xi^2/6 - \\xi^4/180$ and\n$B(\\xi_0) \\cong \\xi_0/3 \\cong \\mbox{$\\beta_c J_0\\sigma/3$} = \\sigma$ \none straightforwardly arrives at the Landau free energy\n%\n%\\marginpar{fenergylan}\n%\n\\begin{eqnarray}\\label{fenergylan}\n%\n\\frac{F}{J_0} &=& \\frac{1}{v_0} \\!\\!\\int\\!\\! d{\\bf r} \\left\\{\n\\frac{1}{2}\\alpha (\\nabla \\mbox{\\boldmath $\\sigma$})^2 - \n\\mbox{\\boldmath $\\sigma$}{\\bf h} + \n\\frac{1}{2}(1-\\eta_x) \\sigma_x^2 \\right.\n\\left.+ \\frac{1}{2}(1-\\eta_y) \\sigma_y^2\n- \n\\frac{\\epsilon}{2} \\sigma^2 + \\frac{3}{20} \\sigma^4\n\\right\\} \\nonumber \\\\\n&-& \\frac{NT}{J_0}\\ln(4\\pi) ,\n\\end{eqnarray}\n%\nwhere $\\epsilon \\equiv (T_c^{\\rm MFA}-T)/T_c^{\\rm MFA}$ and \n$({\\bf \\nabla \\sigma})^{2}=({\\bf \\nabla }\\sigma_{x})^{2}+({\\bf \\nabla }\n\\sigma_{y})^{2}+({\\bf \\nabla }\\sigma_{z})^{2}$.\n%\nNote that Eq.\\ (\\ref{fenergylan}) formally yields unlimitedly increasing values of $\\sigma$ at equilibrium,\nas a function of the field $h$.\n%\nTo comply with the condition $\\sigma \\ll 1$ in Landau's formalism, $h$ should be kept\nsmall, as was required above.\n%\nOne could also work out the $h^2$ corrections to Eq.\\ (\\ref{fenergylan}).\n\n\n\nNow we consider the temperature region where the influence of anisotropy \nand field on the magnitude of the spin polarization $\\sigma$ can be studied perturbatively.\n%\nConcerning the influence of anisotropy, the applicability criterion can \nbe obtained from the requirement that the local shift of $T_c$ for spins \nforced perpendicularly to the easy axis should be smaller than the \ndistance from $T_c$, i.e., $\\Delta T_c/T_c \\sim 1-\\eta \\ll \\epsilon$. \n%\nThe macroscopic free energy in the perturbative region can then be combined with \nthe Landau free energy in their common applicability range $1-\\eta \\ll \\epsilon \\ll 1$. \n%\nThus, in the perturbative region we expand the first two terms of expression \nEq.\\ (\\ref{fenergymacro}) up to the second order in $\\delta\\sigma\\equiv\\sigma-\\sigma_e$ using\n%\n%\\marginpar{deltaxie}\n%\n\\begin{equation}\\label{deltaxie}\n%\n\\xi_0 = \\xi_e + \\delta\\xi, \\qquad\n%\n\\xi_e = \\beta J_0 \\sigma_e, \\qquad\n%\n\\delta\\xi \\cong \\beta J_0 \\delta\\sigma \n%\n\\end{equation}\n%\nand the formula analogous to Eq.\\ (\\ref{devlam}).\n%\nThis leads to\n%\n%\\marginpar{fenergymacro2}\n%\n\\begin{eqnarray}\\label{fenergymacro2}\n%\n\\frac{F}{J_0} = \\frac{F_e}{J_0} + \\frac{1}{v_0} \\!\\!\\int\\!\\! d{\\bf r}\n\\left\\{\n\\frac{1}{2}(1-B'\\beta J_0)(\\sigma-\\sigma_e)^2 \\right.\n\\left.-\\frac{B(\\xi_0)}{\\sigma}\\mbox{\\boldmath $\\sigma$}{\\bf h}\n-\\left(\\frac{B(\\xi_0)}{\\sigma}-\\frac{1}{2}\\right)\n( \\mbox{\\boldmath $\\sigma$},{\\bf h}_{\\rm eff}-{\\bf h} )\n\\right\\} ,\n%\n\\end{eqnarray}\n%\nwhere $B'=B'(\\xi_e)$, and\n%\n%\\marginpar{feequi}\n%\n\\begin{equation}\\label{feequi}\n%\n\\frac{F_e}{J_0} = - \\frac{NT}{J_0}\\ln(4\\pi) \n+ N\n\\left[\n\\frac{1}{2}\\sigma_e^2 - \\frac{1}{\\beta J_0}\\Lambda(\\xi_e) \n\\right], \n%\n\\end{equation}\n%\nis the equilibrium free energy in the absence of magnetic field and\nthe quantity $\\sigma_e$ is the spin polarisation at equilibrium satisfying the homogeneous\nCurie-Weiss equation\n%\n%\\marginpar{curieweisseq}\n%\n\\begin{equation}\\label{curieweisseq}\n%\n\\sigma_e=B(\\xi_e).\n%\n\\end{equation}\n%\nThe minimum condition for Eq.\\ (\\ref{fenergymacro2}),\n$\\delta F/\\delta \\mbox{\\boldmath $\\sigma$}=0$,\nafter an accurate calculation taking into account the dependence of \n$B(\\xi_0)$ on $\\sigma$ and neglecting the terms quadratic in \n${\\bf h}_{\\rm eff}$, results in an equation of the form\n%\n%\\marginpar{diffeqmag}\n%\n\\begin{equation}\\label{diffeqmag}\n%\n\\frac{1}{\\bar\\chi_\\|}\n(\\sigma-\\sigma_e)\\frac{\\mbox{\\boldmath $\\sigma$}}{\\sigma}\n- \n{\\bf h}_{\\rm eff}\n+\n\\frac{\n[\\mbox{\\boldmath $\\sigma$}\n\\times [\\mbox{\\boldmath $\\sigma$}\n\\times {\\bf h}_{\\rm eff}]]\n}{\\sigma^2\\bar\\chi_\\|} = 0 ,\n%\n\\end{equation}\n%\nwhere ${\\bf h}_{\\rm eff}$ is given by Eq.\\ (\\ref{defheff})\nand the dimensionless longitudinal susceptibility $\\bar\\chi_\\|$ is given in Eq.\\ (\\ref{barchi}) below.\n%\nThe solution of Eq.\\ (\\ref{diffeqmag}) satisfies \n$\\mbox{\\boldmath $\\sigma$}\\| {\\bf h}_{\\rm eff}$,\nand the term with the double vector product plays no role.\n%\nConsidering the response to small fields ${\\bf h}= h_z {\\bf e}_z$ and \n${\\bf h}= h_{x,y} {\\bf e}_{x,y}$\nin Eq.\\ (\\ref{diffeqmag}) in a homogeneous situation\n(in the transverse case ${\\bf h}_{\\rm eff}=0$ and $\\sigma=\\sigma_e$), \none can identify the reduced susceptibilities for the spin polarization as\n%\n%\\marginpar{barchi}\n%\n\\begin{eqnarray}\\label{barchi}\n%\n&&\n\\bar\\chi_\\| \\equiv \\frac{d\\sigma_z}{dh_z} = \\frac{B'\\beta J_0}{1-B'\\beta J_0},\n\\nonumber \\\\\n&&\n\\bar\\chi_x = \\frac{1}{1-\\eta_x},\n%\n\\qquad\n\\bar\\chi_y = \\frac{1}{1-\\eta_y} .\n%\n\\end{eqnarray}\n\n\nOur expression for the free energy, Eq.\\ (\\ref{fenergymacro2}), is still\ncumbersome, but can be simplified if we make the observation \nthat in the perturbative region the deviation \n$\\delta\\sigma\\equiv \\sigma-\\sigma_e$ \nis proportional to $h_{\\rm eff}$, and\nthe terms of the type \n$\\delta\\sigma \\cdot h_{\\rm eff}$ and $(\\delta\\sigma)^2$ \nin Eq.\\ (\\ref{fenergymacro2}) are thus quadratic in $h_{\\rm eff}$.\n%\nSuch terms are nonessential in the calculation of $F$ itself, they are \nonly needed for the proper writing of the equilibrium equation \nEq.\\ (\\ref{diffeqmag}).\n%\nNow we can replace Eq.\\ (\\ref{fenergymacro2}) by a simplified form \n%\n%\\marginpar{fenergymacro3}\n%\n\\begin{eqnarray}\\label{fenergymacro3}\n\\frac{F}{J_0} = \\frac{F_e}{J_0}\n+ \\frac{1}{v_0} \\!\\!\\int\\!\\! d{\\bf r}\n\\left\\{\n\\frac{1}{2}\\alpha (\\nabla \\mbox{\\boldmath $\\sigma$})^2 \n- \n\\mbox{\\boldmath $\\sigma$}{\\bf h} + \n\\frac{1}{2}(1-\\eta_x) \\sigma_x^2 \\right.\n\\left.+\\frac{1}{2}(1-\\eta_y) \\sigma_y^2 \n+ \n\\frac{1}{2\\bar\\chi_\\|}(\\sigma-\\sigma_e)^2\n\\right\\} .\n\\end{eqnarray}\n%\nThis form coincides with Eq.\\ (\\ref{fenergymacro2}), if we set \n$\\sigma=\\sigma_e$, i.e., \nit yields the same value of $F$ in the leading first order in \n$h_{\\rm eff}$.\n%\nOn the other hand, Eq.\\ (\\ref{fenergymacro3}) leads to the same equilibrium \nequation, Eq.\\ (\\ref{diffeqmag}), without the nonessential last term.\n%\nNow, at the last step of the derivation, one can combine \nEq.\\ (\\ref{fenergymacro3}) with the Landau free energy in Eq.\\ (\\ref{fenergylan}), \nleading to\n%\n%\\marginpar{fenergymacro4}\n%\n\\begin{eqnarray}\\label{fenergymacro4}\n\\frac{F}{J_0} = \\frac{F_e}{J_0}\n+ \\frac{1}{v_0} \\!\\!\\int\\!\\! d{\\bf r}\n\\left\\{\n\\frac{1}{2}\\alpha (\\nabla \\mbox{\\boldmath $\\sigma$})^2 \n- \n\\mbox{\\boldmath $\\sigma$}{\\bf h} \n+ \\frac{1}{2\\bar\\chi_x} \\sigma_x^2 \\right.\n\\left.+ \\frac{1}{2\\bar\\chi_y} \\sigma_y^2 \n+ \\frac{1}{8\\sigma_e^2\\bar\\chi_\\|}(\\sigma^2-\\sigma_e^2)^2\n\\right\\} .\n\\end{eqnarray}\n%\nIndeed, in the Landau region, $\\epsilon \\ll 1$, from \nEq.\\ (\\ref{curieweisseq}) and Eq.\\ (\\ref{barchi}) it follows $\\sigma_e^2\\cong (5/3)\\epsilon$ \nand $\\bar\\chi_\\|\\cong (2\\epsilon)^{-1}$, and Eq.\\ (\\ref{fenergymacro4}) simplifies \nto Eq.\\ (\\ref{fenergylan}). \n%\nIn terms of the magnetization ${\\bf M}$ defined by \n%\n\\begin{equation}\\label{defM}\n%\n{\\bf M}({\\bf r}) = \\mu_0 \\mbox{\\boldmath $\\sigma$}({\\bf r})/v_0,\n\\qquad \\mbox{\\boldmath $\\sigma$}({\\bf r}) \\equiv \\langle {\\bf s}_{\\bf\nr}\\rangle,\n%\n\\end{equation}\n%\nand other dimensional quantities, the free energy Eq.\\ (\\ref{fenergymacro4}) takes on the form \n\\cite{harkoegar95,gar97prb}\n%\n\\begin{eqnarray}\\label{fenergymacro5}\nF = F_e + \\!\\!\\int\\!\\! d{\\bf r}\n\\left\\{\n\\frac{1}{2 q_d^2}(\\nabla {\\bf M})^2 - {\\bf M} \\mbox{\\boldmath $\\cdot$} {\\bf H} \n+ \\frac{1}{2\\chi_x} M_x^2 \\right.\n\\left.+ \\frac{1}{2\\chi_y} M_y^2 \n+ \\frac{1}{8 M_e^2\\chi_\\|}(M^2-M_e^2)^2\n\\right\\},\n\\end{eqnarray}\n%\nwith\n%\n%\\marginpar{fenergyident}\n%\n\\begin{equation}\\label{fenergyident}\n%\nq_d^2 = \\frac{W_D}{\\alpha J_0},\n%\n\\qquad\n\\chi_\\alpha = \\frac{W_D}{J_0}\\bar\\chi_\\alpha,\n%\n\\qquad\nW_D \\equiv \\frac{(\\mu_0)^2}{v_0},\n%\n\\end{equation}\n%\nwhere $q_d$ is the so-called dipolar wave number, $W_D$ is the characteristic energy\nof the dipole-dipole interaction, and $\\chi_\\alpha \\equiv dM_\\alpha/dH_\\alpha$ with\n$\\alpha=x,y,z$ are the susceptibilities [cf. Eq.\\ (\\ref{barchi})].\n%\nThe applicability of Eq.\\ (\\ref{fenergymacro5}) requires that the deviation\n$M-M_e$ from the equilibrium magnetization $M_e$ in the absence of anisotropy and field, \nis small in comparison with the saturation value $M_s=\\mu_0/v_0$.\n%\nThis is satisfied in the whole range of temperature if the anisotropy and field are small, i.e., \n$1-\\eta_{x,y} \\ll 1$ and $\\mu_0H \\ll J_0$.\n%\nOn the other hand, the free energy in Eq.\\ (\\ref{fenergymacro5}) can be transformed into the ``micromagnetic'' \nform by introducing the magnetization direction vector\n$\\mbox{\\boldmath$\\nu$} \\equiv {\\bf M}/M$.\n%\nOne can then write\n%\n%\\marginpar{identmicromag}\n%\n\\begin{equation}\\label{identmicromag}\n%\n\\frac{1}{2\\chi_{x,y}} M_{x,y}^2 = K_{x,y} \\nu_{x,y}^2,\n%\n\\qquad\nK_{x,y} = \\frac{M^2}{2\\chi_{x,y}},\n%\n\\end{equation}\n%\nwhere $K_{x,y}$ are the anisotropy constants.\n%\nIn particular, for the uniaxial model one can rewrite\n%\n%\\marginpar{identuniax}\n%\n\\begin{equation}\\label{identuniax}\n%\n\\frac{1}{2\\chi_\\perp} (M_x^2 + M_y^2) = - K \\nu_z^2 + K,\n%\n\\end{equation}\n%\nwhere $K$ is the uniaxial anisotropy constant.\n%\nNote that at nonzero temperatures $M$ and thus the anisotropy constants can be spatially\ninhomogeneous.\n%\nIn this case Eq.\\ (\\ref{fenergymacro5}) is more useful than its micromagnetic form.\n\n\nIt should be stressed that the traditional way of writing the magnetic free energy in\nterms of the magnetization is, at least from the theoretical point of view, somewhat\nartificial.\n%\nAll terms in Eq.\\ (\\ref{fenergymacro5}) apart from the Zeeman term, are non-magnetic\nand in fact independent of the atomic magnetic moment $\\mu_0$.\n%\nThe latter cancels out of the resulting formulae, as soon as they are reexpressed in terms\nof the original Hamiltonian (\\ref{biaxham}).\n%\nOn the other hand, Eq.\\ (\\ref{fenergymacro5}) is convenient if the parameters are taken\nfrom experiments. \n\n\nExamination of the formalism above shows that it can be easily generalized to\nquantum systems, leading to the same form as in Eq.\\ (\\ref{fenergymacro5}). \n%\nIn the derivation, the classical Langevin function $B(x)$ is replaced by the quantum\nBrillouin function $B_S(x)$.\n\n\n\n\\section{The free-energy landscape for a uniaxial magnetic particle}\n\\label{seclandscape}\n \nHenceforth, we will consider the uniaxial anisotropy, $\\eta_x=\\eta_y\\equiv\n\\eta_\\perp$.\n%\nFor single-domain magnetic particles in a homogeneous state, the gradient terms in the free energy\ncan be dropped and the free energy of Eq.\\ (\\ref{fenergymacro5}) can be\npresented in the form \n%\n\\begin{eqnarray}\\label{freduced}\n%\n&&\n F = F_e + (VM_e^2/\\chi_\\perp) f \\nonumber \\\\\n&&\nf=-{{\\bf n} \\cdot {\\bf\nh}}+\\frac{1}{2}(n_{x}^{2}+n_{y}^{2})+\\frac{1}{4a}(n^{2}-1)^{2},\n%\n\\end{eqnarray}\n%\nwhere $V$ is the particle's volume and $f$ the reduced free energy written in\nterms of the reduced variables\n%\n%\\marginpar{defnha}\n%\n\\begin{equation}\\label{defnha}\n%\n{\\bf n} \\equiv {\\bf M}/M_e, \n%\n\\qquad {\\bf h} \\equiv {\\bf H}\\chi_\\perp/M_e,\n%\n\\qquad a \\equiv 2\\chi_\\|/\\chi_\\perp.\n%\n\\end{equation}\n%\nOne can see that the parameter $a$ here controls the rigidity of the magnetization vector; \nit goes to zero in the zero-temperature limit (the fixed magnetization length) and\ndiverges at $T_c$ as $a\\cong (1-\\eta)/\\epsilon$ within the MFA.\n%\nAs the MFA is not quantitatively accurate, it is better to consider the susceptibilities\nand hence $a$ as taken from experiments.\n%\nAlthough this procedure is not rigorously justified, it can improve the results.\n%\nNote that the reduced free energy $f$ in Eq.\\ (\\ref{freduced}) is only defined\nfor $T<T_c$ since the reduced magnetization ${\\bf n}$ is normalized by $M_e$.\n\n\nFor fields ${\\bf h}$ inside the Stoner-Wohlfarth astroid, which will be\ngeneralized here to nonzero temperatures, $f$ has two minima separated by a barrier.\n%\nOwing to the axial symmetry, one can set $n_y=0$ for the investigation of the free \nenergy landscape.\n%\nThe minima, saddle points, and the maximum can be found from the equations \n$\\partial f/\\partial n_{x}=\\partial f/\\partial n_{z}=0$, or, explicitly\n%\n\\begin{eqnarray}\\label{eqextrema}\n% \n&&\nn_{z}(n^2-1) = ah_z \\nonumber \\\\ \n&&\nn_{x}(n^2-1+a)=ah_x.\n% \n\\end{eqnarray}\n% \nThese equations can be rewritten as \n%\n%\\marginpar{eqextrema2}\n%\n\\begin{equation}\\label{eqextrema2}\n%\n1-n^2 = -ah_z/n_z = a(1 - h_x/n_x),\n%\n\\end{equation}\n%\nwhereupon the important relation follows\n%\n%\\marginpar{xzrel}\n%\n\\begin{equation}\\label{xzrel}\n%\n\\frac{h_x}{n_x} - \\frac{h_z}{n_z} = 1 \\qquad {\\rm or} \\qquad\nn_x = \\frac{h_x n_z}{h_z + n_z}.\n%\n\\end{equation}\n%\nUsing the latter, one can solve for $n_x$ and obtain a 5th-order equation for\n$n_z$\n%\n%\\marginpar{nzeq}\n%\n\\begin{equation}\\label{nzeq}\n%\nh_x^2 n_z^3 = (h_z+n_z)^2(ah_z + n_z-n_z^3),\n%\n\\end{equation}\n%\nfrom which the parameters of the potential landscape such as energy minima, saddle\npoint, and energy barriers can be found.\n\n\nIn zero field, the characteristic points of the energy landscape can be simply\nfound from Eqs.\\ (\\ref{eqextrema}).\n%\nOne of these points is $n_x=n_z=0$, which is a local maximum for $a<1$ and a saddle point\nfor $a>1$. \n%\nThe minima are given by $n_x=0$, $n_z=\\pm 1$.\n%\nThe saddle points correspond to $n_z=0$, while from the second of Eqs.\\\n(\\ref{eqextrema}) one finds\n%\n\\begin{eqnarray}\\label{nxsaddle}\nn_x = \n\\left\\{\n \\begin{array}{ll}\n \\pm\\sqrt{1-a}, & a \\leq 1 \\\\\n 0, & a \\geq 1.\n \\end{array}\n\\right.\n\\end{eqnarray}\n%\nIn fact, due to the axial symmetry, for $a<1$ one has a saddle circle\n$n_x^2+n_y^2=1-a$ rather than two saddle points.\n%\nThe free-energy barrier following from this solution is given by\n%\n%\\marginpar{febarrier}\n%\n\\begin{equation}\\label{febarrier}\n%\n\\Delta f \\equiv f_{\\rm sad} - f_{\\rm min} = \n\\left\\{\n%\n\\begin{array}{ll}\n%\n(2-a)/4, & a \\leq 1 \\\\\n%\n1/(4a), & a \\geq 1.\n%\n\\end{array}\n%\n\\right.\n%\n\\end{equation}\n%\n\n%\n\\begin{figure}[t]\n\\unitlength1cm\n\\begin{picture}(11,6.5)\n%\n\\centerline{\\psfig{file=tsw_l05.eps,angle=-90,width=8cm}}\n%\n\\end{picture}\n%\n\\begin{picture}(11,6)\n%\n\\centerline{\\psfig{file=tsw_l2.eps,angle=-90,width=8cm}}\n%\n\\end{picture}\n%\n\\caption{ \\label{tsw_land}\nThe free energy of a ferromagnetic particle with uniaxial anisotropy\n[$f$ in Eq.\\ (\\protect\\ref{freduced}) in zero field] for $a\\equiv 2\\chi_\\|/\\chi_\\perp = 0.5$ (upper plot)\nand $a=2$ (lower plot) corresponding to lower and higher temperatures, respectively.\n}\n%\n\\end{figure}\n%\n\nThe free-energy landscape in zero field is shown in Fig.\\ \\ref{tsw_land}.\n%\nAt nonzero temperatures $a>0$, the magnitude of the magnetization at the saddle is smaller than\nunity since it is directed perpendicularly to the easy axis, and for this\norientation the ``equilibrium\" magnetization is smaller than in the direction\nalong the $z$ axis.\n%\nFor $a>1$, the two saddle points, or rather the saddle circle, degenerate into\na single saddle point at $n_x=n_z=0$, and the local maximum there disappears.\n%\nThat is, for the magnetization to overcome the barrier, it is easier to change\nits magnitude than its direction.\n%\nThis is a phenomenon of the same kind as the phase transition in ferromagnets between the\nIsing-like domain walls in the vicinity of $T_c$ (the magnetization changes its\nmagnitude and is everywhere directed along the $z$ axis) and the Bloch walls at\nlower temperatures \\cite{bulgin63,koegarharjah93}.\n\n\n\\section{The Stoner-Wohlfarth curve}\n\\label{secastroid}\n\n\nThe Stoner-Wohlfarth curve separates the regions where there are two minima and one\nminimum of the free energy.\n%\nOn this curve the metastable minimum merges with the saddle point and loses its\nlocal stability. The corresponding condition is\n%\n%\\marginpar{swcond}\n%\n\\begin{equation}\\label{swcond}\n%\n\\partial ^{2}f/\\partial n_{x}^{2}\\times \\partial ^{2}f/\\partial n_{z}^{2}\n-(\\partial ^{2}f/\\partial n_{x}\\partial n_{z})^{2} = 0,\n%\n\\end{equation}\n%\nor, explicitly \n%\n%\\marginpar{swcond2}\n% \n\\begin{equation}\\label{swcond2}\n%\n(1-n^{2})\\left( 1-3n^{2}-a\\right) +2an_{z}^{2}=0.\n% \n\\end{equation}\n%\nUsing Eq.\\ (\\ref{eqextrema2}), one can transform the equation above to the\nquartic equation for $n_z$\n%\n%\\marginpar{swcondnz}\n% \n\\begin{equation}\\label{swcondnz}\n%\nh_z[(2+a)n_z + 3ah_z] + 2n_z^4=0.\n% \n\\end{equation}\n%\nBefore considering the general case, let us analyze the limiting cases $a\\ll 1$\nand $a\\gg 1$.\n\n\nAt low temperatures, i.e., $a\\ll 1$, the magnetization only slightly deviates from its\nequilibrium value, and from Eq.\\ (\\ref{swcond2}), to first order in $a$, one obtains\n%\n%\\marginpar{nrel}\n%\n\\begin{equation}\\label{nrel}\n%\nn^2 \\cong 1 - an_z^2 \\qquad {\\rm or} \\qquad\nn_x^2 + (1+a) n_z^2 \\cong 1.\n%\n\\end{equation}\n%\nFrom Eq.\\ (\\ref{swcondnz}) and the analogous equation for $n_x$ and\nusing Eq.\\ (\\ref{eqextrema2}), one can derive\nthe field dependence of $n_z$ and $n_x$ on the Stoner-Wohlfarth curve\n%\n\\begin{eqnarray}\\label{nznxonsw}\n%\n&&\nn_z \\cong -h_z^{1/3} [1 - (a/2)(h_z^{2/3}-1/3)] \\nonumber \\\\\n&&\nn_x \\cong h_x^{1/3} [1 + (a/2)(h_x^{2/3}-1)].\n%\n\\end{eqnarray}\n%\nInserting these results in Eq.\\ (\\ref{nrel}), one arrives at the equation for the \nStoner-Wohlfarth astroid\n%\n%\\marginpar{swasrtsmall}\n%\n\\begin{equation}\\label{swasrtsmall}\n%\nh_x^{2/3} + [(1+a/2)h_z]^{2/3} \\cong 1, \\qquad a\\ll 1,\n%\n\\end{equation}\n%\nwhere $a$ is given in Eq.\\ (\\ref{defnha}).\n%\nOne can see that, in comparison with the standard zero-temperature Stoner-Wohlfarth astroid,\ni.e. at $a=0$, $h_z$ is rescaled.\n%\nThe critical field in the $z$ direction decreases because of the field dependence of the \nmagnetization magnitude at nonzero temperatures.\n \n\nIn the case $a\\gg 1$, i.e. near $T_c$, Eq.\\ (\\ref{swcond2}) relating $n_x$ and\n$n_z$ in the equation for Stoner-Wohlfarth curve simplifies to\n%\n%\\marginpar{nrel1}\n%\n\\begin{equation}\\label{nrel1}\n%\nn_x^2 + 3 n_z^2 \\cong 1.\n%\n\\end{equation}\n%\nUsing this equation together with Eqs.\\ (\\ref{eqextrema2}), one obtains\n%\n%\\marginpar{nznxonsw1}\n%\n\\begin{equation}\\label{nznxonsw1}\n%\nn_z \\cong -(ah_z/2)^{1/3}, \\qquad n_x \\cong h_x.\n%\n\\end{equation}\n%\nMaking use of this result in Eq.\\ (\\ref{nrel1}), one obtains another limiting case\nof the Stoner-Wohlfarth curve\n%\n%\\marginpar{swalarge}\n%\n\\begin{equation}\\label{swalarge}\n%\nh_x^2 + 3(ah_z/2)^{2/3} \\cong 1, \\qquad a\\gg 1.\n%\n\\end{equation}\n%\nIn this case, the critical field (i.e., the field on the Stoner-Wohlfarth curve) \nin the $z$ direction is strongly reduced, and\nthere is no singularity in the dependence $h_{cz}(h_x)$ at $h_x=0$.\n\n\n%\n\\begin{figure}[t]\n\\unitlength1cm\n\\begin{picture}(11,6)\n%\n\\centerline{\\psfig{file=tsw_hz.eps,angle=-90,width=8cm}}\n%\n\\end{picture}\n%\n\\caption{ \\label{tsw_hz}\n%\nDependence $h_z(a)$ at $h_x=0$ on the Stoner-Wohlfarth curve.\n}\n%\n\\end{figure}\n%\n\n%\n\\begin{figure}[t]\n\\unitlength1cm\n\\begin{picture}(11,7)\n%\n\\centerline{\\psfig{file=tsw_tsw.eps,angle=-90,width=8cm}}\n%\n\\end{picture}\n%\n\\caption{ \\label{tsw_tsw}\nThe Stoner-Wohlfarth curves at different temperatures, $a\\equiv 2\\chi_\\|/\\chi_\\perp=0$\n($T=0$), 0.3, 0.5, 2/3, 0.835, 1, 2.\n}\n%\n\\end{figure}\n%\n\nThe qualitatively different character of the Stoner-Wohlfarth curves in these two\ncases is due to the different mechanisms pertaining to the loss of the local stability for\nthe field applied along the $z$ axis.\n%\nFor $h_x=0$ the mixed derivative $(\\partial ^{2}f/\\partial n_{x}\\partial n_{z})$\nin Eq.\\ (\\ref{swcond}) vanishes, and Eq.\\ (\\ref{swcond}) factorizes.\n%\nExplicitly, we have\n%\n%\\marginpar{swcondfact}\n%\n\\begin{equation}\\label{swcondfact}\n%\n(a-1+n_z^2)(-1+3n_z^2)=0.\n%\n\\end{equation}\n%\nVanishing of the first factor in this equation corresponds to the loss of\nstability with respect to the rotation of the magnetization, \n$\\partial ^{2}f/\\partial n_{x}^{2}=0$.\n%\nVanishing of the second factor, $\\partial ^{2}f/\\partial n_{z}^{2}=0$, \nimplies the loss of stability with respect to the change of the magnetization length.\n%\nUsing $n_x=0$, with the help of the first equality in Eqs.\\ (\\ref{eqextrema2}) one\nobtains in the two cases\n%\n%\\marginpar{hz}\n%\n\\begin{equation}\\label{hzc}\n%\nh_{cz} = \n\\left\\{\n%\n\\begin{array}{ll}\n%\nh_{z\\|} \\equiv \\sqrt{1-a}, & a \\leq 2/3 \\\\\n%\nh_{z\\perp} \\equiv 2/(3^{3/2}a), & a \\geq 2/3.\n%\n\\end{array}\n%\n\\right.\n%\n\\end{equation}\n%\nNote that the transition between the two regimes occurs here at a different\nvalue of $a$ than in Eq.\\ (\\ref{nxsaddle}). \n%\nThe dependence $h_z(a)$ at $h_x=0$ is shown in Fig.\\ \\ref{tsw_hz}.\n\n\nIn the general case, it is easier to find the Sto\\-ner-\\-Wohlfarth curve numerically\nfrom Eqs.\\ (\\ref{eqextrema}) and (\\ref{swcond2}).\n%\nThe results in the whole range of $a$ are shown in Fig.\\ \\ref{tsw_tsw}.\n\n\n\n%\n\\begin{figure}[t]\n\\unitlength1cm\n\\begin{picture}(11,6)\n%\n\\centerline{\\psfig{file=tsw_hy0.eps,angle=-90,width=8cm}}\n%\n\\end{picture}\n%\n\\begin{picture}(11,6)\n%\n\\centerline{\\psfig{file=tsw_hy1.eps,angle=-90,width=8cm}}\n%\n\\end{picture}\n%\n\\caption{ \\label{tsw_hy}\n%\nThermal effect on the static hysteresis loops [$a(0)=0$, $a(T_c)=\\infty$]. \n%\nUpper plot: $\\psi=0$ (${\\bf h}\\| {\\bf e}_z$), lower plot: $\\psi=\\pi/4$.\n% \n}\n%\n\\end{figure}\n%\n\n\n\n\\section{Hysteresis loops}\n\\label{sechysteresis}\n\n\nIn this section we use our model to study the hysteresis loop, i.e., the dependence \n${\\bf n(h)}$ at the stable or metastable free-energy minimum.\n%\nAt first we consider the case $h_x=0$ in which the problem can be solved analytically.\n%\nSetting $n_x=0$ in the first of Eqs.\\ (\\ref{eqextrema}) one obtains the cubic equation\n$n_z^3-n_z-ah_z=0$, the solution of which, for the positive branch of the hysteresis curve,\nreads\n%\n%\\marginpar{hyst}\n%\n\\begin{equation}\\label{hyst}\n%\nn_z = \n\\left\\{\n%\n\\begin{array}{ll}\n%\n\\displaystyle\n\\frac{2}{\\sqrt{3}} \\cos \\left(\\frac{\\phi}{3}\\right), & \\left| h_z\\right| \\leq h_{z\\|} \\\\\n%\n\\displaystyle\n\\left[\\frac{ah_z}{2}+\\sqrt{D}\\right]^{1/3} +\n\\left[\\frac{ah_z}{2}-\\sqrt{D}\\right]^{1/3}, & \\left| h_z\\right| \\geq h_{z\\|},\n%\n\\end{array}\n%\n\\right.\n%\n\\end{equation}\n%\nwhere $h_{z\\|}$ is given by Eq.\\ (\\ref{hzc}) and \n%\n%\\marginpar{defDphi}\n%\n\\begin{equation}\\label{defDphi}\n%\n\\phi \\equiv \\arccos(h_z/h_{z\\|}), \\qquad D \\equiv (a/2)^2 (h_z^2 - h_{z\\|}^2).\n%\n\\end{equation}\n%\nThe negative branch of the hysteresis curve can be obtained by the reflection\n$h_z\\Rightarrow -h_z$ and $n_z\\Rightarrow -n_z$.\n%\nEq.\\ (\\ref{hyst}) is written in the form which is explicitly real.\n%\nIn fact, both forms hold in the whole range $h_z \\geq -h_{z\\|}$, and there is no change of\nbehavior at $h_z = h_{z\\|}$, as expected on physical grounds.\n%\nThe characteristic values of the magnetization on the positive branch are \n$n_z(-h_{z\\|}) = 1/\\sqrt{3}$ and $n_z(h_{z\\|})= 2/\\sqrt{3}$.\n%\nThe solution above was obtained by setting $n_x=0$ and thus ignoring the transverse\ninstability which can occur before the longitudinal instability at $h_z = -h_{z\\|}$ for\nthe positive branch.\n% \nThis competition of instabilities has been studied in the previous section.\n%\nIt was found that for $a<2/3$ the transverse instability occurs at the fields\n$|h_z|= |h_{z\\|}|=\\sqrt{1-a} < |h_{z\\perp}|$.\n%\nThus in this case the branches of the hysteresis curves should be cut at $\\pm h_{z\\|}$;\nat these fields the system jumps to the other branch.\n\n\nThe analytical results obtained above for $h_x=0$ and the numerical ones for the case of ${\\bf h}$\ndirected at the angle $\\psi = \\pi/4$ to the easy axis are shown in Fig.\\ \\ref{tsw_hy}.\n%\nFor $h_x=0$, the derivative $dn_z/dh_z$ diverges at the metastability\nboundary for $a\\geq 2/3$ (longitudinal instability).\n\n\n\\section{The homogeneity criterion, thermal activation, spin-waves}\n\\label{HomoTherm}\n\n\nIn the preceding sections we illustrated how the general magnetic free energy of \nEq.\\ (\\ref{fenergymacro5}) works for the simplest model of a uniaxial magnetic particle\nin a single-domain state at elevated temperatures.\n%\nHere we discuss the possibility of observing the new types of behavior found above.\n%\nThis requires satisfying two rather restrictive conditions.\n%\nFirst, the particle spends in the metastable minimum a time that is long enough for\nmeasuring only if the barrier height energy is much larger than thermal energy.\n%\nAt any $T>0$, the particle will escape from the metastable state via\nthermal activation with a rate exponentially small at low temperatures.\n%\nFor this reason, strictly speaking, the static hysteresis at nonzero temperatures does not\nexist and dynamic measurements are needed.\n%\nAt elevated temperatures, the required frequency of these measurements can become too\nlarge. \n%\nSecond, the free energy barrier in the single-domain state should be smaller than the\nbarrier energy related with the formation of a domain wall which would travel through the particle and\nswitch the magnetization from one state to the other. \n%\nThe two criteria can be combined as follows\n%\n%\\marginpar{Criteria}\n%\n\\begin{equation}\\label{Criteria}\n%\nT \\ll \\Delta F_{SD} < \\Delta F_{DW}.\n%\n\\end{equation}\n%\nThe first criterion here requires that the particle's volume is high enough whereas the\nsecond criterion requires that it does not exceed some maximal value.\n%\nLet us consider, for instance, the zero field case, in which $F_{SD}$ is the \nsingle-domain free-energy barrier of Eqs.\\ (\\ref{febarrier}) and (\\ref{freduced}). \n%\nAt not too high temperatures $a \\lesssim 1$, one has $\\Delta f \\sim 1$, and the domain\nwalls are usual Bloch walls with the energy per unit area\n%\n%\\marginpar{DWEnergy}\n%\n\\begin{equation}\\label{DWEnergy}\n%\nw = 2M_e^2/(q_d^2 \\delta), \\qquad \\delta = \\sqrt{\\chi_\\perp}/q_d,\n%\n\\end{equation}\n%\nwhere $\\delta$ is the domain-wall width. \n%\nFor a spherical particle of radius $R$, the saddle point of the energy\ncorresponds to the domain wall through the center of the particle.\n%\nUsing the definitions of parameters introduced in Sec.\\ \\ref{secfenergy}, one obtains the\nformula \n%\n%\\marginpar{EnergyRatio}\n%\n\\begin{equation}\\label{EnergyRatio}\n%\n\\frac{ \\Delta F_{SD} }{ \\Delta F_{DW} } = \\frac{ Rq_d }{ 12 \\chi_\\perp^{1/2} } \n\\sim \\sqrt{1-\\eta} \\frac{ R }{ a_0 },\n%\n\\end{equation}\n% \nwhere $a_0$ is the lattice spacing.\n%\nIt is seen that the single-domain behavior of particles with $R \\gg a_0$ requires small\nvalues of the anisotropy $1-\\eta$.\n%\nWorking out the first inequality in Eq.\\ (\\ref{Criteria}), one can rewrite these equations in the\nform\n%\n%\\marginpar{Criteria1}\n%\n\\begin{equation}\\label{Criteria1}\n%\n\\left( \\frac{ \\theta }{ (1-\\eta) \\sigma_e^2 } \\right)^{1/3} \n\\ll \\frac{ R }{ a_0 } \\lesssim \\frac 1 {\\sqrt{1-\\eta}} ,\n%\n\\end{equation}\n%\nwhere $\\theta \\equiv T/T_c^{\\rm MFA}$ and $\\sigma_e$ is the spin polarization at\nequilibrium.\n%\nClearly, for $\\theta \\ll 1$ these conditions can be satisfied.\n%\nObserving the new effects exhibited by the hysteresis suggested above requires \n$a \\equiv 2\\chi_\\|/\\chi_\\perp \\equiv 2\\bar\\chi_\\|/\\bar\\chi_\\perp \\sim 1$, which for\nsmall anisotropy, i.e. $1-\\eta \\ll 1$, requires approaching $T_c$. \n%\nIndeed, near $T_c$ the first of Eqs.\\ (\\ref{barchi}) yields\n$\\bar\\chi_\\| \\cong (2\\epsilon)^{-1}$, thus $a \\sim 1$ implying $\\epsilon \\sim 1-\\eta$.\n%\nIn this region in Eq.\\ (\\ref{Criteria1}) one has $\\theta \\sim 1$ and\n$\\sigma_e^2\\cong (5/3)\\epsilon$. \n%\nThen the existence of the interval for $R/a_0$ in Eq.\\ (\\ref{Criteria1}) requires \n$(1-\\eta)/\\epsilon \\ll \\epsilon$ which is impossible since in this region \n$(1-\\eta)/\\epsilon \\sim 1$ and $\\epsilon \\ll 1$.\n%\nA similar analysis shows that also in the region where $a > 1$ Eqs.\\ (\\ref{Criteria})\ncannot be satisfied.\n\n\n\nThus we are led to the conclusion that the qualitatively different types\nof behavior of single-domain magnetic particles for $a\\gtrsim 1$ cannot be observed with the standard\ntechniques.\n%\nIf the particle's size is small enough, the single-domain criterion is satisfied, but\nincreasing temperature to $a\\sim 1$ causes strong thermal fluctuations.\n%\nThe potential landscape shown in Fig.\\ \\ref{tsw_land} is still valid but for studying the dynamics\nof the magnetic particle in this range we need a special kind of Fokker-Planck\nequation for {\\em non-rigid} magnetic moments. Such an equation has not been considered yet. \n\n\nOn the other hand, for large particle sizes the energy barriers are high and thus the process\nof thermal activation is suppressed, which is favorable for the observation of hysteresis\nloops.\n%\nIn this case, however, the barrier states are those with a domain wall across the\nparticle.\n%\nAnalyzing these states goes beyond the scope of this article.\n%\nWe only mention that for $a> 1/2$ the structure of domain walls in a ferromagnet is\ncompletely different from that of a Bloch wall: The transverse magnetization component in\nthe wall is zero everywhere whereas the longitudinal component changes its magnitude and goes through\nzero in the center of the wall \\cite{bulgin63}.\n%\nFor the observation of thermal effects in the hysteresis via inhomogeneous states, higher\nvalues of the anisotropy $1-\\eta$ are needed.\n% \nIn this case, thermal effects manifest themselves starting from low temperatures.\n%\nAn example of a strongly anisotropic material is Co, for which one obtains $1-\\eta \\simeq 0.02$.\n%\nThis value is still much smaller than one, so that the validity condition for the magnetic\nfree energy of Eq.\\ (\\ref{fenergymacro5}) is satisfied. \n\n\nReturning to the results of this paper, we can say that one can only observe corrections\nto the well-known results, such as that of Eq.\\ (\\ref{swasrtsmall}), in the range \n$a \\ll 1$, i.e., not close to $T_c$. \n%\nOn the other hand, one should not forget that the mean-field approximation used in this paper, \nwhile leading to qualitatively correct predictions, fails to account for some subtler\neffects that \ncan be responsible for important modifications of the results.\n%\nOne of these effects is the influence of spin waves on the longitudinal susceptibility\nwhich enters the definition $a \\equiv 2\\chi_\\|/\\chi_\\perp$. \n%\nWhereas within the MFA $\\chi_\\|$ rapidly decreases with decreasing temperature below\n$T_c$ and is independent of the anisotropy, it is {\\em infinite} in the whole range below $T_c$ for \nisotropic ferromagnets because of spin waves.\n%\nThe square-root singularity in the dependence $M(H)$ at zero field has been\nexperimentally observed and reported on in Ref.\\ \\cite{koegoedompie94}.\n%\nIn uniaxial ferromagnets, $\\chi_\\|$ becomes large for small anisotropies. \n%\nThis means that if the mean-field expression for $a$ is replaced by its value taken from\nexperiments, which is much larger due to spin-wave effects, the thermal effects\ndiscussed in this paper will considerably increase in intensity.\n% \nSuch a redefinition of $a$ is, of course, not rigorously justified, although it captures\nthe essential physics.\n%\nA more involved approach taking into account spin-wave effects for an exactly solvable model confirms the\nconcomitant increase of thermal effects on the variation of the magnetization length, as\nwas explicitly shown for domain walls in Ref.\\ \\cite{gar96jpa}.\n\nFinally, we would like to mention that quite recently, experimental\nresults have been obtained by Wernsdorfer \net al. \\cite{WW00} on 3 nm cobalt nanoparticles which clearly show the\ndisappearance of the singularity near $H_x = 0$ at a temperature circa\n8 K (the blocking temperature being 14 K). The height of the\nexperimental astroid decreases nearly as its width with increasing\ntemperature, but it does not become flat as predicted by our\ncalculations, which is not surprising considering the fact that $T\\ll T_c$, but the\ndisappearance of the singularity is definitive. \n\n\n\n\\section{Conclusion}\n\\label{secconclusion}\n\n\nIn this paper we have derived a macroscopic free energy for\nweakly anisotropic ferromagnets which is based on the mean-field approximation and is valid\nin the whole range of temperature interpolating between the micromagnetic energy at\n$T=0$ and the Landau free energy near $T_c$.\n%\nAs an illustration, we have considered single-domain magnetic particles with uniaxial\nanisotropy and we have shown that thermal effects qualitatively change the free-energy\nlandscape at sufficiently high temperatures, so that the passage from one free-energy minimum to the\nother is realized by the {\\em uniform change of the magnetization length} rather than the {\\em uniform\nrotation}. \n%\nThis also qualitatively changes the character of the Stoner-Wohlfarth curve and hysteresis\nloops.\n%\nThe latter effects cannot be observed with standard methods, however, because keeping the height of the\nfree-energy barrier much larger than thermal energy requires so large particle sizes that the\nsingle-domain criterion is no longer satisfied.\n%\nFor the uniform states, the theory is valid at low temperatures, but then the thermal effects\nconsidered in the paper are small corrections to the zero-temperature results.\n%\nLarge thermal effects on hysteresis should be searched for at temperatures close to $T_c$\nin particles\nof larger sizes, where the saddle point of the free energy is an {\\em inhomogeneous}\nstate.\n%\nInvestigation of the corresponding more complicated processes is beyond the scope of this\npaper. \n\n\n\n\\section*{Acknowledgements}\nD. A. Garanin is indebted to Laboratoire de Magn\\'etisme et d'Optique\nfor the warm hospitality extended to him during his stay in Versailles in January 2000. \n\n\n%\\bibliography{gar}\n\nElectronic addresses: \\\\\n$^*$kachkach@physique.uvsq.fr\\\\\n$^{\\dagger}$garanin@mpipks-dresden.mpg.de; http://www.mpipks-dresden.mpg.de/$\\sim$garanin/\n\n\\begin{thebibliography}{10}\n\n\\bibitem{lanlif35}\n{L. D. Landau and E. M. Lifshitz}, Phys. Z. Sowjetunion {\\bf 8}, 153 (1935).\n\n\\bibitem{bro63mic}\n{W. F. Brown, Jr.}, {\\em Micromagnetics} (Interscience, New York, 1963).\n\n\\bibitem{bulgin63}\n{L. N. Bulaevskii and V. L. Ginzburg}, Zh. Eksp. Teor. Fiz. {\\bf 45}, 772\n (1963) [JETP {\\bf 18}, 530 (1964)].\n\n\\bibitem{koegarharjah93}\n{J. K\\\"otzler, D. A. Garanin, M. Hartl, and L. Jahn}, Phys. Rev. Lett. {\\bf\n 71}, 177 (1993).\n\n\\bibitem{harkoegar95}\n{M. Hartl-Malang, J. K\\\"otzler, and D. A. Garanin}, Phys. Rev. B {\\bf 51},\n 8974 (1995).\n\n\\bibitem{gar91llb}\n{D. A. Garanin}, Physica A {\\bf 172}, 470 (1991).\n\n\\bibitem{gar91edw}\n{D. A. Garanin}, Physica A {\\bf 178}, 467 (1991).\n\n\\bibitem{gar97prb}\n{D. A. Garanin}, Phys. Rev. B {\\bf 55}, 3050 (1997).\n\n\\bibitem{stowoh4891}\n{E. C. Stoner and E. P. Wohlfarth}, Philos. Trans. R. Soc. London, Ser. A {\\bf\n 240}, 599 (1948);\n IEEE Trans. Magn. {\\bf MAG-27}, 3475\n (1991).\n\n\\bibitem{weretal97}\n{W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, N.\n Demoncy, A. Loiseau, and D. Mailly}, Phys. Rev. Lett. {\\bf 78}, 1791\n (1997).\n\n\\bibitem{koegoedompie94}\n{J. K\\\"otzler, D. G\\\"orlitz, R. Dombrowski, and M. Pieper}, Z. Phys. B {\\bf\n 94}, 9 (1994).\n\n\\bibitem{gar96jpa}\n{D. A. Garanin}, J. Phys. A {\\bf 29}, 2349 (1996).\n\n\\bibitem{WW00}\n{W. Wernsdorfer et al.}, preprint January 2000.\n\n\\end{thebibliography}\n\n\n\\end{document}\n\n\ntar -cvzf tsw.tar.gz tsw.tex tsw_l05.eps tsw_l2.eps tsw_hz.eps tsw_tsw.eps tsw_hy0.eps tsw_hy1.eps \n\nuntar: tar xvf [filename]\n\ngunzip [filename]\n"
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"name": "cond-mat0002127.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{lanlif35}\n{L. D. Landau and E. M. Lifshitz}, Phys. Z. Sowjetunion {\\bf 8}, 153 (1935).\n\n\\bibitem{bro63mic}\n{W. F. Brown, Jr.}, {\\em Micromagnetics} (Interscience, New York, 1963).\n\n\\bibitem{bulgin63}\n{L. N. Bulaevskii and V. L. Ginzburg}, Zh. Eksp. Teor. Fiz. {\\bf 45}, 772\n (1963) [JETP {\\bf 18}, 530 (1964)].\n\n\\bibitem{koegarharjah93}\n{J. K\\\"otzler, D. A. Garanin, M. Hartl, and L. Jahn}, Phys. Rev. Lett. {\\bf\n 71}, 177 (1993).\n\n\\bibitem{harkoegar95}\n{M. Hartl-Malang, J. K\\\"otzler, and D. A. Garanin}, Phys. Rev. B {\\bf 51},\n 8974 (1995).\n\n\\bibitem{gar91llb}\n{D. A. Garanin}, Physica A {\\bf 172}, 470 (1991).\n\n\\bibitem{gar91edw}\n{D. A. Garanin}, Physica A {\\bf 178}, 467 (1991).\n\n\\bibitem{gar97prb}\n{D. A. Garanin}, Phys. Rev. B {\\bf 55}, 3050 (1997).\n\n\\bibitem{stowoh4891}\n{E. C. Stoner and E. P. Wohlfarth}, Philos. Trans. R. Soc. London, Ser. A {\\bf\n 240}, 599 (1948);\n IEEE Trans. Magn. {\\bf MAG-27}, 3475\n (1991).\n\n\\bibitem{weretal97}\n{W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, N.\n Demoncy, A. Loiseau, and D. Mailly}, Phys. Rev. Lett. {\\bf 78}, 1791\n (1997).\n\n\\bibitem{koegoedompie94}\n{J. K\\\"otzler, D. G\\\"orlitz, R. Dombrowski, and M. Pieper}, Z. Phys. B {\\bf\n 94}, 9 (1994).\n\n\\bibitem{gar96jpa}\n{D. A. Garanin}, J. Phys. A {\\bf 29}, 2349 (1996).\n\n\\bibitem{WW00}\n{W. Wernsdorfer et al.}, preprint January 2000.\n\n\\end{thebibliography}"
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cond-mat0002128
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Statistical Physics of Structural Glasses
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"author": "Marc M\\'ezard"
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%Paper presented at the ICTP conference on `Unifying Concepts in Glass Physics, %september 1999. This paper gives an introduction and brief overview of some of our recent work on the equilibrium thermodynamics of glasses. We have focused onto first principle computations in simple fragile glasses, starting from the two body interatomic potential. A replica formulation translates this problem into that of a gas of interacting molecules, each molecule being built of $m$ atoms, and having a gyration radius (related to the cage size) which vanishes at zero temperature. We use a small cage expansion, valid at low temperatures, which allows to compute the cage size, the specific heat (which follows the Dulong and Petit law), and the configurational entropy. The no-replica interpretation of the computations is also briefly described. The results, particularly those concerning the Kauzmann tempaerature and the configurational entropy, are compared to recent numerical simulations.
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"string": "\n\\documentclass[12pt]{iopart}\n\n\\input{epsf} \n\n\\usepackage{graphicx}\n\n\\newcommand {\\spa} {\\vskip 1cm}\n\\newcommand {\\via} {\\end{center} \\end{slide*} \\begin{slide*} \\begin{center}}\n\\newcommand {\\bc} {\\begin{center}}\n\\newcommand {\\ec} {\\end{center}}\n\\newcommand {\\bd}{\\begin{displaymath}}\n\\newcommand {\\ed}{\\end{displaymath}}\n \\newcommand {\\be} {\\begin{equation}}\n\\newcommand {\\bea} {\\begin{eqnarray} \\nonumber }\n\\newcommand {\\ee} {\\end{equation}}\n\\newcommand {\\eea} {\\end{eqnarray}}\n \\newcommand {\\eps} {\\epsilon}\n \\newcommand {\\si} {\\sigma}\n\\newcommand {\\de} {\\delta}\n\\newcommand {\\De} {\\Delta}\n\\newcommand {\\ga} {\\gamma}\n\\newcommand {\\la} {\\lambda}\n\\newcommand {\\La} {\\Lambda}\n\\newcommand {\\Si} {\\Sigma}\n \\newcommand {\\al} {\\alpha}\n \\newcommand {\\NE }{\\not=}\n \\newcommand {\\N} {{\\cal N}}\n\\newcommand {\\R} {{\\cal R}}\n\\newcommand {\\LL}{{\\cal L}}\n\\newcommand {\\ba} {\\overline}\n\\newcommand {\\lan} {\\langle}\n\\newcommand {\\ran} {\\rangle}\n \n\\newcommand {\\cA} {{\\cal A}}\n\\newcommand {\\cC} {{\\cal C}}\n\\newcommand {\\cD} {{\\cal D}}\n\\newcommand {\\cH} {{\\cal H}}\n\\newcommand {\\cL} {{\\cal L}}\n\\newcommand {\\cN} {{\\cal N}}\n\\newcommand {\\cP} {{\\cal P}}\n\\newcommand {\\cR} {{\\cal R}}\n\\newcommand {\\cS} {{\\cal S}}\n\\newcommand {\\cX} {{\\cal X}}\n\\newcommand {\\cE} {{\\cal E}}\n\\def\\eps{\\epsilon}\n\\def\\al{\\alpha}\n\\def\\mb{\\mathbf}\n\\def\\la{\\langle}\n\\def\\ra{\\rangle}\n\\def\\epp{\\epsilon '}\n\\def\\s{\\sigma}\n \\def\\(({\\left(}\n \\def\\)){\\right)}\n\\def\\[[{\\left[}\n\\def\\]]{\\right]}\n\\def\\bi{\\bibitem}\n\\newcommand {\\tC} { {\\tilde{C}} }\n\\newcommand {\\for} {\\ \\ \\ \\mbox{for}\\ \\ }\n\\newcommand {\\bra }{\\langle 0 |}\n\\newcommand {\\ket} {| 0 \\rangle }\n\\newcommand {\\by} {{\\bf y}}\n\n\\newcommand {\\bx} {{\\bf x}}\n\n\\newcommand {\\ato} {\\left(}\n\\newcommand {\\sign} {\\mbox{sign}}\n\\newcommand {\\cto} {\\right)}\n\n\\def \\form#1 {eq. (\\ref{#1}) }\n\\def \\parziale#1#2 {{\\partial {#1} \\over \\partial {#2}}}\n\n\n \\begin{document}\n\\title{Statistical Physics of Structural Glasses}\n\\author{Marc M\\'ezard}\n\\address{Laboratoire de Physique Th\\'eorique de l'Ecole\nNormale Sup\\'{e}rieure \\footnote{UMR 8548: Unit\\'e Mixte du Centre National de la Recherche\nScientifique, et de\nl'\\'Ecole Normale Sup\\'erieure}\\\\\n24 rue\n Lhomond, F-75231 Paris Cedex 05, (France)\\\\\nmezard@physique.ens.fr}\n\\author{Giorgio Parisi}\n\\address{Dipartimento di Fisica and Sezione INFN,\\\\\nUniversit\\`a di Roma ``La Sapienza'',\nPiazzale Aldo Moro 2,\nI-00185 Rome (Italy)\\\\\ngiorgio.parisi@roma1.infn.it}\n\n\\date{\\today}\n\n\\maketitle\n\n\n\\begin{abstract}\n%Paper presented at the ICTP conference on `Unifying Concepts in Glass Physics,\n%september 1999.\nThis paper gives an introduction and brief overview of\nsome of our recent work on the \n equilibrium \nthermodynamics of glasses. We have focused onto first principle computations in \nsimple fragile glasses, starting\nfrom the two body interatomic potential. A replica formulation translates \nthis problem into\nthat of a gas of interacting molecules, each molecule being built of $m$ \natoms, and having a \ngyration radius\n(related to the cage size) which vanishes at zero temperature. We use a \nsmall cage expansion, valid\nat low temperatures, which allows to compute the cage size, the specific \nheat (which\nfollows the Dulong and Petit law), and the configurational entropy. The\nno-replica interpretation of the computations is also briefly described.\nThe results, particularly those concerning the Kauzmann\ntempaerature and the configurational entropy, are compared to \nrecent numerical simulations.\n\\end{abstract}\n\\pacs{05.20, 75.10N}\n\n\n\\section{Introduction}\nWhile the experimental and phenomenological knowledge on\nglasses has improved a lot in the last decades\\cite{glass_revue}, \nthe progress on a first\nprinciple, statistical mechanical study of the glass phase has turned out\nto be much more difficult. \n\nTake any elementary textbook on solid state physics.\nIt deals with a special class of solid state, the crystalline state, and \nusually\navoids to elaborate on the possibility of amorphous solid states. \nThe reason is very simple:\nthere is no theory of amorphous solid states. \nSchematically, the first elementary steps of the theory of crystals are \nthe following.\n One computes the ground\nstate energy of all the crystalline structures. The small vibrations \naround these\nstructures are easily handled, either using the simple Einstein \napproximation\nof independent atoms in harmonic traps, or computing the phonon\ndispersion relations and going to the Debye\ntheory. Then one can study the one electron problem and compute the band \nstructure.\nThe basic thermodynamic properties are already well reproduced by these \nelementary \ncomputations.\nAnharmonic vibrations, electron-phonon and electron-electron interactions \ncan then\nbe added to these basic building blocks.\n\nUntil very recently, none of the above computations, even in\nthe simplest-minded approximation,\ncould be done in the case of the glass state. The reason is obvious: all \nof them are\nmade possible in crystals by the existence of the symmetry group. The\nabsence of such a symmetry, which is a defining property of the glass \nstate, \nforbids the use of all the solid state techniques. If one takes a \nsnapshot of a glass\nstate, an instantaneous configuration of atoms, it looks more like a \nliquid\nconfiguration. In fact the techniques which we shall use are often \nborrowed\nfrom the theory of the liquid state. But while the liquid phase is \nergodic (which\nmeans that the probability distribution of positions is translationally \ninvariant),\nthe glass phase is not. The problem is to describe a non-ergodic phase \nwithout a symmetry:\nan amorphous solid state.\n\nThe work which we report on here has been elaborated during the last year \nand aims\nat building the first steps of a first principle theory of glasses. The \nfact that this\nis being made possible now is not fortuitous, but rather results from\na conjunction of several sets of ideas, and the general progress of the \nlast two\ndecades on the theory of amorphous systems. \n\nThe oldest ingredients are the\nphenomenological ideas, originating in the work of Kauzmann \\cite{kauzmann}, and \ndeveloped among others by\nby Adam, Gibbs and Di-Marzio \\cite{AdGibbs}, which identify the glass transition \nas a `bona fide'\nthermodynamic transition blurred by some dynamical effects.\nAs we shall discuss below, in this scenario the transition is associated \nwith an `entropy crisis', namely the\nvanishing of the configurational entropy of the thermodynamically \nrelevant glass\nstates.\n\n A very different, and more indirect, route, was the study of\nspin glasses. These are also systems which freeze into amorphous solid \nstates,\nbut one of their constitutive properties is very different from the \nglasses\nwe are interested in here: there exists in spin glasses some `quenched \ndisorder':\n the exchange-interaction coupling constants between the spin\ndegrees of freedom are quenched (i.e. time independent on all \nexperimental time scales)\n random variables\\cite{rubber}.\n Anyhow, a few years after the replica symmetry breaking (RSB) \nsolution of the mean field theory of spin glasses \\cite{MPV}, it was \nrealized \nthat there exists another category of mean-field spin\nglasses where the transition is due to an entropy crisis \\cite{REM}. These\nare now called discontinuous spin glasses because their phase transition, \nalthough\nit is of second order in the Ehrenfest sense, has a discontinuous order\nparameter, as first shown in \\cite{GrossMez}.\n Another name often found in the literature is \n `one step RSB' spin glasses, because of the\nspecial pattern of symmetry breaking involved in their solution. \nThe simplest example of these is the Random Energy Model \\cite{REM},\nbut many other such discontinuous spin glasses were found subsequently,\ninvolving multispin interactions \\cite{GrossMez,KiThWo,crisanti}.\n\nThe analogy\nbetween the phase transition of discontinuous spin glasses and the \nthermodynamic\nglass transition was first noticed by Kirkpatrick, Thirumalai and Wolynes\nin a series of inspired papers of the mid-eighties \\cite{KiThWo}. While some of \nthe basic ideas\nof the present development were around at that time, there still missed a\nfew crucial ingredients. On one hand one needed to get more confidence \nthat\n this analogy was not just fortuitous. \nThe big obstacle was the existence (in spin glasses) versus\nthe absence (in structural glasses) of quenched disorder. The \ndiscovery of discontinuous spin glasses without any \nquenched disorder\n\\cite{nodis1,nodis2,nodis3}\nprovided an important new piece of information: contrarily to what had \nbeen\nbelieved for long, quenched disorder is not necessary for the existence of\na spin glass phase (but frustration is). A second confirmation came very \nrecently from the developments on out of equilibrium dynamics of the \nglass phase.\nInitiated by the exact solution of the dynamics in a discontinuous spin \nglass\nby Cugliandolo and Kurchan \\cite{cuku}, this line of research has made a lot \nof progress\nin the last few years. It has become clear that, in realistic systems \nwith short\nrange interactions, the pattern of replica symmetry breaking can be \ndeduced\nfrom the measurements of the violation of the fluctuation dissipation \ntheorem \\cite{fdr}.\nAlthough these difficult measurements are not yet available, numerical\nsimulations performed on different types of glass forming\nsystems have provided an independent and spectacular confirmation of their\n`one step rsb' structure \\cite{gpglass,bk1,bk2} on the (short) time\nscales which are accessible. The theory was then facing the big \nchallenge:\nunderstanding what this replica symmetry breaking could mean, in systems \nvoid of quenched disorder, in which there is thus no a priori\nreason to introduce replicas. The recent progress has brought the answer \nto\nthis question and turned it into a computational method, allowing for\na first principle computation of the equilibrium thermodynamics of \nglasses \\cite{MePa1,Me,MePa2,sferesoft,LJ,LJ2}.\n\nIn the context of glasses,\nthe words `equilibrium thermodynamics' call for some comments. First, \nit is\nnot obvious whether the glass phase is an equilibrium phase of matter. It \nmight be\na metastable phase, reachable only by some fast enough quench, while the \n`true\nequilibrium' phase would always be crystalline. The answer depends on the \ninteraction potential. Numerically it is known that the frustration\ninduced by considering for instance binary mixtures of soft spheres\nof different radii strongly inhibits crystallisation. But what is the\ntrue equilibrium state is unknown,\nand not very relevant. One can study crystals without having proven that \nthey\nare stable phases of matter (by the way, simply proving that the fcc-hcp \nis\nthe densest packing of hard spheres in 3 dimensions, a simple zero \ntemperature\nstatement, has resisted the efforts of scientists for centuries\n\\cite{kepler}),\nand one can study the properties of diamond, even though it is notoriously\nunstable. The point is to have reproducible properties, which is \ncertainly\nthe case. Letting aside the crystal, a more interesting question is how\nto reach equilibrium glass states. Experimentally nobody knows how to \nachieve\nthis. In a ferromagnet, one can reach an equilibrium state and eliminate \ndomain walls\nby using an external magnetic field. In a glass there is no such field \nconjugate\nto the order parameter, and the fate is an out of equilibrium situation. \nThe same\nis true in spin glasses, and in fact in all kind of glass phases. Why \nstudy the\nequilibrium thermodynamics then? The answer is twofold. First\nprinciple computations are certainly much easier as far as the \nequilibrium is concerned,\ntherefore it is natural to start with these in order to first get some\ndetailed understanding of the free energy landscape, which will be useful \nin the\nmore realistic dynamical studies. Secondly, we have strong indications, \nand some\ngeneral arguments, in favour of a close relationship between the \nequilibrium\nproperties and the observable out of equilibrium dynamical observations \n\\cite{fdr}.\nLet us also mention here the recent developments of some phenomenological\ntheory of the out of equilibrium theory of glasses \\cite{theo}.\n\nIn this paper we shall introduce the main ideas of the recent elaboration \nof the\nequilibrium theory of glasses. We shall not present the details which \ncan be found in the literature. \nThe general replica strategy can be found in \\cite{remi,Me}. The\nexplicit computations have been done first for soft spheres in\n\\cite{MePa1,MePa2}, and then generalized to binary mixtures of \nsoft spheres \\cite{sferesoft} or Lennard Jones particles \\cite{LJ,LJ2}.\n\n\\section{Hypotheses on the glass phase}\nThe general framework of our approach is a familiar one in physics: we \nshall\nstart from a number of basic hypotheses on the glass phase, derive some \nquantitative\nproperties starting from these hypotheses, and then compare them with \nnumerical,\nand hopefully, in the future, experimental results. We work with a simple \nglass\nformer, $N$ undistinguishable particles move in a volume $V$ of a \nd-dimensional space, and we take the thermodynamic limit $N,V \\to \\infty$ \nat fixed density \n$\\rho=N/V$. The interaction potential is a two body one,\ndefined by a short range function $v(x)$ (for instance one may\nconsider a soft spheres system where $v(x)=1/x^{12}$).\n\nLet us introduce a free energy functional \n$F(\\rho)$ \nwhich depends on the density $\\rho(x)$ and on the temperature. We \nsuppose that at sufficiently low \ntemperature this functional has many minima (i.e. the number of minima \ngoes to infinity with the \nnumber $N$ of particles). Exactly at zero temperature these minima, \nlabelled by\nan index $\\alpha$, coincide with the mimima of \nthe potential energy as function of the coordinates of the particles.\nA more detailed discussion of the valleys and their relationship to the\ninherent structures \\cite{inherent} will be given in sect. \\ref{entrop}. \n To each valley we can associate a free energy $F_\\al$ and a free energy \ndensity \n$f_\\al= F_\\al/N$. The number of free energy minima with \nfree energy density $f$ is supposed to be exponentially large:\n\\be\n{\\cal N}(f,T,N) \\approx \\exp(N\\Sigma(f,T)),\\label{CON}\n\\ee\nwhere the function $\\Sigma$ is called the complexity or the \nconfigurational entropy (it is the \ncontribution to the entropy coming from the existence of an exponentially \nlarge number of locally \nstable configurations). This function is not defined in the regions \n$f>f_{max}(T)$ or $f<f_{min}(T)$, where ${\\cal N}(f,T,N)=0$, it is convex \nand \nit is supposed to go to zero continuously\nat $f_{min}(T)$, as found in all existing models so far (see \nfig.\\ref{sigma_qualit}). \nIn \nthe low temperature region the total free energy of the system, $\\Phi$, \ncan be well approximated by:\n\\be\n e^{-\\beta N \\Phi} \\simeq \\sum_\\al e^{-\\beta N f_\\al(T)} =\n\\int_{f_{min}}^{f_{max}} df \\ \\exp\\((N[ \\Sigma(f,T)-\\beta f]\\)) \\ ,\n\\label{SUM}\n\\ee\nwhere $\\beta=1/T$.\nThe minima which dominate the\nsum are those with a free energy density $f^*$\n which minimizes the quantity $\\Phi(f)=f-T\\Sigma(f,T)$.\nAt large enough temperatures the saddle point is at $f>f_{min}(T)$. When \none\ndecreases $T$ the saddle point free energy decreases.\nThe Kauzman temperature $T_K$ is that below which the saddle point sticks\nto the minimum: $f^*=f_{min}(T)$. It is a genuine phase transition, the\n`ideal glass transition'.\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{Scqualit.ps}\n\\caption{Qualitative shape of the configurational entropy versus free energy.\nThe\nwhole curve depends on the temperature. The\nsaddle point which dominates the partition function,\nfor $m$ constrained replicas, is the point $f^*$ such\nthat the slope of the curve equals $m/T$ (for the usual unreplicated system,\n$m=1$).\nIf the temperature is small enough the saddle point sticks\nto the minimum $f=f_{min}$ and the system is in its glass phase. }\n\\label{sigma_qualit}\n\\end{figure}\n\n\nThis scenario for the glass transition is precisely the one which is at \nwork in discontinuous\nspin glasses, and can be studied there in full details. The transition is \nof a rather special\ntype. It is of second order because the entropy and internal energy are \ncontinuous. \nWhen decreasing the temperature through $T_K$ there is a discontinuous \ndecrease of\nspecific heat, as seen experimentally. On the other hand the order \nparameter is\ndiscontinuous at the transition, as in first order transitions. To show \nthis we have to provide\na definition of the order parameter in our framework of equilibrium \nstatistical mechanics.\nThis is not totally trivial because of the lack of knowledge on the \nvalleys themselves.\nThe best way is to introduce two identical copies of the system. We have \none system\nof undistinguishable `red' particles, \ninteracting between themselves through $v(x)$, another\nsystem of undistinguishable `blue' particles, interacting between \nthemselves through $v(x)$, and we turn\non a small interaction between the blue and red particles, which is short \nrange.\nWe take the thermodynamic limit first, and then send this red-blue \ncoupling to zero.\nIf the position correlations between the red and blue particles disappear \nin\nthis double limit, the system is in a liquid phase, otherwise it is in a \nsolid phase.\nClearly, the order parameter, which is the red-blue pair correlation\nfunction, is discontinuous at the transition: there is no correlation\nin the liquid phase, while in the solid phase one gets an oscillating \npair correlation,\nsimilar to that of a dense liquid, but with an extra peak at the origin.\n In some sense, in this framework, the role of the unknown\n conjugate field, needed in order to polarize the system into one state,\nis played by the coupling to the second copy of the system.\nThe small red-blue coupling is here to insure that the two systems will \nfall into\nthe same glass state.\n\nThe above scenario, relating the glass transition to the vanishing\nof the configurational entropy, is the main hypothesis of our work. \nClearly\nit is in agreement with the phenomenology\nof the glass transition, and with the old ideas of Kauzman, Gibbs and \nDi-Marzio. It is also very interesting from the point of view of the\ndynamical behaviour.\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{tau.ps}\n\\caption{Relaxation time versus temperature. The right hand curve is the prediction\nof mode-coupling theory without any activated processes: it is a mean field prediction,\nwhich is exact for instance in\n the discontinuous mean-field spin glasses. The left-hand curve is the observed\nrelaxation time in a glass. The mode coupling theory provides a quantitative prediction for the increase\nof the relaxation time when decreasing temperature, at high enough temperature (well above the\nmode coupling transition $T_c$). The departure from the mean field prediction at lower temperatures is \nusually attributed to 'hopping' or 'activated' processes, in which the system is trapped\nfor a long time in some valleys, but can eventually jump out of it. The ideal\nglass transition, which takes place at $T_s$, \ncannot be observed directly since the system \nbecomes out of equilibrium on laboratory time scales at the `glass temperature' $T_g$.\nBecause of the special scenario of the static transition in mean field\nspin glasses, due to some entropy crisis, the transition temperature\n $T_s$ should be identified with the Kauzman temperature $T_K$. }\n\\label{figtau}\n\\end{figure}\n In discontinuous mean field spin glasses, the slowing down \nof the dynamics takes a very special form.\nThere exist a dynamical transition temperature $T_c>T_K$. When T decreases and gets near to $T_c$,\nthe correlation function relaxes with a characteristic two step forms: a\nfast $\\beta$ relaxation leading to a plateau takes place on a characteristic time\nwhich does not grow, while the $\\alpha$ relaxation from the\nplateau takes place on a time scale which diverges when $T \\to T_c$. This dynamic\ntransition is exactly described by the schematic mode coupling equations. The\nexistence of a dynamic relaxation at a temperature above the true thermodynamic one\nis possible only in mean field, and the conjecture\\cite{KiThWo}\n is that in a realistic system like a glass, \nthe region between $T_K$ and $T_c$ will have instead a finite, but very rapidly \nincreasing, relaxation time, as shown in fig. \\ref{figtau}.\n\nOn this figure we see the existence of several temperature regimes:\n\n-a relatively high temperature regime where mode coupling theory applies\n\n- an intermediate region, extending from $T_k$ up to the temperature\nabove $T_c$ where mode coupling predictions start to be correct. This is the region\nof activated processes, where one can identify some traps in phase space in which the system\nstays for a long time, and then jumps.\n\n-the low temperature, glass phase $T<T_K$. \n\nThe dynamics of the glass is expected to show aging effects in the glass region, but also in\nthe intermediate region provided the laboratory time is smaller than the relaxation time.\n\nHere we shall focus onto the equilibrium study of the low temperature phase. One main reason is\nthat the direct study of out of equilibrium dynamics is more difficult, and that one might be able\nto make progress by a careful analysis of the landscape \\cite{angelani}. Another motivation\nis to go into a more quantitative test of the basic scenario: while it agrees qualitatively\nwith several observations, as we just discussed, it should also be able to help\nmake more quantitative predictions.\n\n Our strategy\nwill be to start from this set of hypotheses and derive the quantitative\npredictions which can be checked independently.\nWe shall be\nable to compute for instance the configurational entropy versus free\nenergy within some well controlled approximations, and compare it to\nthe results of some numerical simulations.\n\n\\section{Replicas}\nIn order to cope with the degeneracy of glass states and the \nexistence of a configurational entropy, a choice method is the\nreplica method. Initially replicas were introduced in order to\nstudy systems with quenched disorder, in which one needs to compute\nthe disorder average of the logarithm of the partition function \\cite{MPV}.\nIt took a few years to realize that a large amount of information is\nencoded in the distribution of distances between replicas. This is\n true again in structural glasses. The simplest example was given above\nwhen we explained the use of two replicas in order to define\nthe order parameter. A much more detailed information can be gained if\none studies in general a set of $m$ replicas, sometimes named `clones' in\nthis context, coupled through a small \nextensive attraction which \n will eventually go to zero \\cite{remi,Me}. In the glass phase, the \nattraction will force all $m$ systems\n to fall into the same glass state, so that\nthe partition function is:\n\\be\nZ_{m} = \\sum_\\al e^{-\\beta Nm f_\\al(T)}= \\int_{f_{min}}^{f_{max}} df\n \\ \\exp\\((N [ \\Sigma(f,T)-m \\beta f]\\))\n\\label{zm}\n\\ee\nIn the limit where $m \\to 1$ the corresponding partition function \n$Z_m$ is dominated by the correct saddle point $f^*$ for $T>T_K$. \nThe interesting regime is when the temperature is $T<T_K$, \nand the number $m$ is allowed to become smaller than one. The saddle \npoint $f^*(m,T)$ in the expression (\\ref{zm}) is the solution\nof $\\partial \\Sigma(f,T) / \\partial f=m/T$. Because of the\nconvexity of $\\Sigma$ as function of $f$, the saddle point is\nat $f>f_{min}(T)$ when $m$ is small enough, and it sticks at \n$f^*=f_{min}(T)$\nwhen $m$ becomes larger than a certain value $m=m^{*}(T)$,\na value which is smaller than one when $T<T_K$. The free energy\nin the glass phase, $F(m=1,T)$, is equal to $ F(m^*(T),T)$. As the free \nenergy\nis continuous along the transition line $m=m^*(T)$, one can compute \n$F(m^*(T),T)$ from the region $m \\le m^*(T)$, which is a region where the\nreplicated system is in the liquid phase. This is the clue to\nthe explicit computation of the free energy in the glass phase. \nIt may sound a bit strange because one is tempted to think of $m$ as an \ninteger\nnumber. However the computation is much clearer if one sees $m$ as \na real parameter in (\\ref{zm}). As one considers low temperatures $T<T_K$ \nthe\n$m$ coupled replicas fall into the same glass state and thus they build\nsome molecules of $m$ atoms, each molecule being built from one atom of \neach \n'colour'. Now the interaction strength of one such molecule with another \none\nis basically rescaled by a factor $m$ (this \nstatement becomes exact in the limit of zero temperature\nwhere the molecules become point like). If $m$ is small enough this \ninteraction is small\n and the system of molecules is liquid. When $m$ increases, the molecular \nfluid\nfreezes into a glass state at the value $m=m^*(T)$.\nSo our method requires to estimate the \n replicated free energy, \n$\nF(m,T)=-{\\log(Z_m) /( \\beta m N )}\n$,\n in a molecular\nliquid phase, where the molecules consist of $m$ atoms and\n$m$ is smaller than one. For $T<T_K$, $F(m,T)$ is maximum at\nthe value of $m=m^{*}$ smaller than one,\nwhile for $T>T_K$ the maximum is reached at a value $m^*$ is larger than one.\n The knowledge of $F_m$ as a \nfunction\nof $m$ allows to reconstruct the configurational entropy\nfunction $Sc(f)$ at a given temperature $T$\nthrough a Legendre transform, using the parametric representation (easily\ndeduced from a saddle point \nevaluation of (\\ref{zm})):\n\\be\nf={\\partial \\[[m F(m,T)\\]] \\over \\partial m} \\ \\ \\ ; \\ \\ \\Sigma(f)={m^2 \\over T}\n{\\partial F(m,T)\\over \\partial m} \\ .\n\\label{legend}\n\\ee\n\n The Kauzmann temperature ('ideal \nglass\ntemperature') is the one such that $m^*(T_K)=1$. For $T<T_K$ the equilibrium\nconfigurational entropy vanishes. Above $T_K$ one obtains the equilibrium\nconfigurational entropy $\\Sigma(T)$ by solving (\\ref{legend}) at $m=1$. \n\nMore explicitly, one must thus \nintroduce $m$ clones of \neach particle, with positions $x_i^a, a\\in{1,...,m}$.\nThe replicated partition function is:\n\\bea\nZ_m={1 \\over N!^m} \\int \\prod_{i=1}^N \\prod_{a=1}^m d x_i^a \\ \n \\exp\\((-\\beta \\sum_{1 \\le i < j \\le N} \\ \\sum_{a=1}^m v(x_i^a-x_j^a)\n\\right. \\\\ \\left.\n-\\beta \\epsilon \\sum_{i,j=1}^N \\ \\sum_{1 \\le a < b \\le m}\nw(x_i^a-x_j^b) \\)) \\ ,\n\\label{Z1}\n\\eea \nwhere $v$ is the original interparticle potential and $w$ is an\n attractive potential.\n This attractive potential must be of short range\n(the range should be less than the\ntypical interparticle distance in the solid phase),\n but its precise form is irrelevant. Assuming\nthat $w$ is equal to $-1$ at very small distances,\nand zero at large distances (notice\nthat the scale of the inter-replica interaction is fixed by the \n parameter $\\epsilon$), the coupling $w$ can be used to define an overlap between two \nconfigurations,\nin a way similar to the crucial concept of overlaps in spin glasses. \nTaking\ntwo configurations $x_i$ and $y_i$ of the $N$ particles, one defines the \noverlap \nbetween the configurations as\n$q(x,y) = -1/N \\sum_{i,k=1,N} w(x_{i}-y_{k})$, or the distance as $1-q$. \nThe replicated\npartition function with $m$ clones is thus (in more compact notations \nwhere \n$dx=\\prod_{i=1}^N dx_i/N!$ and $H(x)\\equiv \\sum_{i<j}v(x_i-x_j)$ is \nthe total energy of the system):\n\\be\nZ_m=\\int \\prod_a dx^a \\exp\\(( -\\beta \\sum_{a}H(x_{a})+ \\beta \\eps N \n\\sum_{a,b}q(x_{a},x_b)\\)) \\ .\n\\label{zm_ov}\n\\ee\nThis can be defined also for non integer $m$ using an analytic \ncontinuation\n(if our hypothesis of the glass transition being of the same nature as \nthe one\nstep rsb in spin glasses is correct, there is no replica symmetry \nbreaking between\nthe clones\\cite{remi,Me}, and the continuation is straightforward). \nAlternatively, one can\ndefine it through the formula\n\\be\nZ_m \\propto \\int d\\mu(\\phi) Z(\\phi)^m\n\\ee\nwhere $\\phi$ is a quenched random potential defined in the full space,\nwhich has a Gaussian distribution with moments:\n\\be \n\\int d\\mu(\\phi)=1 \\ \\ ,\\ \\int d\\mu(\\phi) \\ \\phi(x)=0\n \\ \\ ,\\ \\int d\\mu(\\phi)\\ \\phi(x) \\phi(y) =c^t- w(x-y) \\ ,\n\\ee\nand $Z(\\phi)$ is the partition function of one system in the external\npotential $\\phi$:\n\\be\nZ(\\phi)= \\int dx \\exp\\((-\\beta H(x) -\\sqrt{\\beta \\epsilon} \\sum_{i=1}^N\\phi(x_{i})\\)) \\ .\n\\ee\n\n\\section{The molecular liquid}\nThe explicit computation of $Z_m$ in the regime $m<m^*(T)$ \nis a complicated problem of dense molecular \nliquids,\nwhich requires some approximate treatments. Several types\nof approximations have been developed recently, leading\nto fully consistent results.\nFocusing onto the low temperature regime, where the molecules have a\nsmall radius, it is natural to write the partition function in terms\nof the center of mass and relative\ncoordinates $\\{ r_i, u_i^a \\}$, with $x_i^a=r_i+u_i^a$ and $\\sum_a u_i^a=0$, \nand to\nexpand the interaction in powers of the relative displacements $u$. After\na proper renumbering of the particles, so that particles in the\nsame molecule have the same $i$ index, one gets:\n\\bea\nZ_m &= &{1 \\over N!} \\int dr \\prod _{a=1}^m du^a \\prod_{i=1}^N \n\\((m^3 \\delta(\\sum_{a=1}^m u_i^a) \\)) \n \\exp\\((-\\beta \\sum_{i<j,a} \\[[v(r_i-r_j) \\right. \\right. \\\\\n&& \\left. \\left. +\\sum_{p=2}^\\infty \n(u_i^a-u_j^a)^p\n{v^{(p)}(r_i-r_j) \\over p!}\\]]\n-{\\eps \\over 4} \\sum_{i,a,b} (u_i^a-u_i^b)^2 \\))\n\\ .\n\\label{zexpanded}\n\\eea\nThe last term is the small inter-replica coupling ($\\eps$ will be\nsent to zero in the end), which we have approximated for\nconvenience by its quadratic approximation. \nThe expression (\\ref{zexpanded}) can be expanded, at low \ntemperatures,\nin the following ways:\n\\begin{itemize}\n\\item\n`Harmonic resummation': One keeps only the $p=2$ term. The action is \nquadratic\nin $u$, and after performing the exact $u$ integral one obtains\nan effective interaction for the center of mass degrees of freedom,\nwhich we shall detail below. The parameter $m$ appears as a coupling \nconstant,\nthe analytic continuation in $m$ is thus trivial, \nand the whole problem reduces to treating the liquid\nof center of masses, interacting through the effective interaction.\n\\item\n`Small cage expansion': One expands the exponential in powers of the \nrelative \nvariables $u$, keeping only the $\\eps$ term in the exponent. Again, the\n$u$ integrals can be done exactly to each order\nof the approximation. In this way one generates\nan expansion of the free energy in powers of $1/\\eps$. This function can \nbe\nLegendre transformed with respect to $\\eps$, leading to a generalized free\nenergy expressed as a series in terms of the `cage radius', $ \nA=2/(3m(m-1)) \\sum_{a,b} \n<(u_i^a-u_i^b)^2>$. Notice that the $1/\\eps$ expansion is just an \nintermediate step\nin order to generate the small $A$ expansion of the potential (the same \ncan be done\nfor instance when computing the Gibbs potential of an Ising model in \nterms of the\nmagnetization $M$ at low temperatures: even if one is interested in the \nzero magnetic\nfield case, one can introduce the field as an intermediate device and \nfirst\nexpand in powers of $\\exp(-\\beta h)$, before turning\nthe result into\nan expansion in $1-M$).\n\\end{itemize}\nThe two methods are complementary. They both lead to the study of a \nliquid of\ncenter of mass positions. The\nsmall cage expansion is simpler because the result is expressed in terms \nof various\ncorrelation functions\nof the pure liquid of center of masses at the effective\ntemperature $T/m$, which can be handled using traditional\nliquid state techniques. On the other hand the\nleading ($p=2$) term at low temperatures is not treated exactly.\nIn the harmonic resummation scheme the interaction\npotential of the center of masses is modified: one gets\n\\be\nZ_m= Z_m^0 \\int dr\n\\exp\\((-\\beta m H(r) -{m-1 \\over 2} Tr \\log M \\)) \n\\label{Zharmo}\n\\ee\nwhere $ Z_m^0 ={m^{Nd/2} \\sqrt{2 \\pi T}^{N d (m-1)} / N!}$, and\nthe matrix $M$, of dimension $dN \\times dN$, is given by:\n\\be\nM_{(i \\mu) (j \\nu)}= {\\partial^2 H(r) \\over \\partial r_i^\\mu \\partial r_j ^\\nu}\n= \\delta_{ij} \\sum_k v_{\\mu\\nu}(r_i-r_k)- \nv_{\\mu\\nu}(r_i-r_j)\n\\ee\nand $v_{\\mu\\nu}(r) =\\partial^2 v /\\partial r_\\mu \\partial r_\\nu$ \n(the indices $\\mu$ and $\\nu$ denote space directions). The effective \ninteraction\ncontains the complicated `$Tr \\log M$' piece\nwhich is not a pair potential. Because of this term, in the whole glass \nphase where\none is interested in the $m<1$ regime, the partition function receives some \ncontributions only from\nthose configurations $r_i$ such that all eigenvalues\nof $M$ are positive: these are locally stable glass configurations. In order\nto handle this additional constraint, we used so far the following\n (rather crude) approximate treatment, which consists of two steps. \n First, a 'quenched approximation', which amounts to\n neglecting the feedback of\nvibration modes onto the centers of masses, substitutes\n$\\la \\exp\\((-{m-1 \\over 2} Tr \\log M \\)) \\ra$ by \n$ \\exp\\((-{m-1 \\over 2} \\la Tr \\log M \\ra\\)) $, where $\\la . \\ra$ \nis the Boltzmann expectation value at\nthe effective temperature $T/m$.\nOne is then left with the computation of the spectrum of $M$ in a liquid. \nThis is an\ninteresting problem in itself. The treatment done in \\cite{MePa1,MePa2} corresponds \nto keeping the\nleading term in a high density limit. Further recent progress \\cite{INM,MEPAZEE,CAGIAPA} \nshould allow\nfor a better controlled approximation of the spectrum.\n\n\nWe shall not review here the details of these computations, which can be \nfound\nin \\cite{MePa2} as far as the simple glass former with the\n`soft sphere' $1/x^{12}$ potential is \nconcerned, in \\cite{sferesoft} for the mixtures of soft spheres and\nin \\cite{LJ,LJ2} for mixtures of Lennard-Jones \nparticles.\nOnce one has derived an expression for the replicated free energy, one can\ndeduce from it the whole thermodynamics, as described above.\nIn all three cases, one finds an estimate of the Kauzman temperature which is in \nreasonable agreement with simulations, with a jump in specific\nheat, from a liquid value at $T>T_K$ to the Dulong-Petit value\n$C=3/2$ (we have included only positional\ndegrees of freedom) below $T_K$.\nThis is similar to the experimental result, where the glass\nspecific heat jumps down to the crystal value when one decreases the temperature\n(Our approximations so far are similar to the Einstein approximation of\nindependent vibrations of atoms, in which case the contribution\nof positional degrees of freedom to the crystal specific heat is $C=3/2$).\nThe parameter $m^*(T)$ and the cages sizes\nare nearly linear with temperature in the whole glass phase. \nThis means, in particular, that the effective temperature $T/ m$ is always \nclose to $T_K$, so in our theoretical computation we need\nonly to evaluate the expectation values of observables in the liquid phase, \nat temperatures where the HNC approximation for the liquid still works quite well. \n\nA more detailed numerical checks of these analytical predictions involves\nthe measurement of the configurational entropy. We shall review these checks in\nsect. \\ref{entrop}, but we first wish to present some\nalternative derivation of the low temperature results.\n\n\n\\section{Without replicas}\nFor those who do not appreciate the beauty and efficacy of the replica approach,\nit may be useful to derive some of the above results without resorting to the\nreplica method \\cite{pedago}. Specifically, we shall study the simplest case of\nthe zero temperature limit in the harmonic approximation through\n a direct approach, and reinterpret the above results.\nAt low temperatures, the critical value $m^*$ of the parameter $m$ goes to zero linearly with $T$.\nWe thus write $\\gamma=\\beta m$ and take the $T,m \\to 0$ limit of (\\ref{Zharmo})\nat fixed $\\gamma$. This gives:\n\\be\nZ_m \\simeq \\(( {\\gamma \\over 2 \\pi}\\))^{Nd/2} \\int_C dr \\ \n \\sqrt{\\det M(r)} \\exp\\((-\\gamma H(r)\\))\\ ,\n\\label{Zm_lowT}\n\\ee\nwhere $\\int_C$ is restricted to configurations in which all eigenvalues\nof $M$ are positive. \nA direct derivation of this formula, making all hypotheses\nexplicit, is the following. At zero temperature one \nis interested in configurations where every particle is in equilibrium:\n$\\forall i,\\mu, \\ {\\partial H / \\partial x_i^\\mu} =0$. The number of such configurations\nat energy $NE$,\n\\be\n\\mu(E)= \\int dx \\ | det M(x)|\\ \\delta(NE-H(x)) \\prod_{i,\\mu}\n \\delta\\(({\\partial H \\over \\partial x_i^\\mu}\\)) \\ ,\n\\ee\ncan be approximated at low enough energy,\nwhere most extrema are minima \\cite{CAGIAPA}, by the expression\n\\be\n\\nu(E)= \\int_C dx \\ det M(x) \\ \\delta(NE-H(x)) \\prod_{i,\\mu}\n \\delta\\(({\\partial H \\over \\partial x_i^\\mu}\\)) \\ .\n\\ee\nWithin this approximation $\\nu(E)$ is related to the configurational entropy \nthrough $\\nu(E)=\\exp(N\\Sigma(E))$,\nand one can compute its Laplace transform:\n\\be\n\\zeta(\\gamma) \\equiv \\int dE \\ \\nu(E) \\exp\\((-\\gamma N E\\))\n= \\int dE \\exp\\((N\\[[\\Sigma(E)-\\gamma E\\]]\\)) \\ .\n\\ee\nUsing an exponential representation of the ground state constraints,\nthis effective partition function is:\n\\be\n\\zeta(\\gamma)= \\(( { \\gamma \\over 2 \\pi} \\))^{Nd} \n\\int dx \\prod_k d \\lambda_k^\\mu \\ det M(x) \\ \\exp\\((-\\gamma H(x)+\ni \\gamma \\sum_{k,\\mu} \\lambda_k^\\mu {\\partial H \\over \\partial x_k^\\mu} \\)) \n\\label{zeta}\n\\ee\nOne can change variables from $x_k$ to $y_k=x_k-i\\lambda_k$. At low temperatures it is reasonable to assume that the only configurations which contribute\nare those in the neighborhood of the minima. Expanding in powers of $\\lambda$,\nand neglecting anharmonic terms,\none writes:\n\\bea\nH(x)&\\simeq&H(y) +i \\sum_{k,\\mu} \\lambda_k^\\mu {\\partial H(y) \\over \\partial y_k^\\mu}\n-{1 \\over 2} \\sum_{k,\\mu,l,\\nu} \\lambda_k^\\mu \\lambda_l^\\nu {\\partial^2 H(y) \\over \\partial y_k^\\mu\n\\partial y_l^\\nu}\\\\\n{\\partial H(x) \\over \\partial x_k^\\mu} &\\simeq& {\\partial H(y) \\over \\partial y_k^\\mu}\n+ \\sum_{l,\\nu} \\lambda_l^\\nu {\\partial^2 H(y) \\over \\partial y_k^\\mu\n\\partial y_l^\\nu} \\ .\n\\eea\nThe $\\lambda$ integral in (\\ref{zeta}) is then quadratic, and one gets:\n\\be\n\\zeta(\\gamma)= \\(( { \\gamma \\over 2 \\pi} \\))^{Nd/2} \n\\int_C dy\\ \\sqrt{ det M(y)} \\ \\exp(-\\gamma H(y)) \\ ,\n\\ee\na result identical to the low $T$ limit (\\ref{Zm_lowT}) of the replica approach within the\nharmonic approximation.\n\n\n\\section{Configurational entropy: theory and simulations}\n\\label{entrop}\nThe configurational entropy (sometimes called also complexity) is a key concept\n in the theory of \nglasses.\nThere is no difficulty of principle in defining a valley and its entropy in\nthe low temperature phase $T<T_K$. As we have seen, we can take a \nthermalized configuration as a reference system, add a small attraction \nto this configuration, and take the thermodynamic limit before the limit of \na vanishing attraction. This procedure defines the restricted partition function\nin the valley containing the reference configuration $y$,\nand therefore the free energy of the valley. Computing $S_c(f,T)$ is thus in principle\ndoable, but it is still a formidable challenge to get equilibrated configurations $y$\nin this temperature range.\n\nOn the other hand in the intermediate temperature regime $T_K<T<T_c$, the valleys and\nthe configurational entropy remain well defined in the mean field theory. The existence\nof a decoupling of time scales\n points\nto the possibility of defining metastable valleys in the whole region where \nactivated ('hopping') processes are found. This region is particularly interesting,\nboth because of the rapid change of relaxation times,\nand because part of this region can be studied experimentally or numerically. \nIt often happens that different authors use different \ndefinitions of the configurational entropy, which should be hopefully be \nequivalent at low temperature \nbut behave rather differently at high temperatures.\nTherefore\nit seems to us appropriate to start\nthis section with a comparison of the various definitions of configurational entropies\nwhich have been introduced and studied so far.\n\nIf we consider the configurational entropy versus temperature, which is non-zero \nfor $T>T_K$, in a first approximation we can distinguish three different types of \ndefinitions:\n\\begin{itemize}\n \\item A first definition is based on the presence of many minima of the Hamiltonian, i.e. \n inherent structures.\n \\item A second definition is based on the fact that the phase space at \nsufficient low energy \n may be decomposed in many disconnected region (let us call it the microcanonical one).\n \\item A third definition is based on the thermodynamics. One starts from the definition \n \\be\n S(T)=\\Sigma(T)+S_{valley}(T)\n \\ee\n where $S(T)$ is the total entropy and $S_{valley}$ is the entropy of the generic valley at \n temperature $T$. In this case the problem consists in finding a precise definition of $S_{valley}$.\n\\end{itemize}\nIn this paper we have used the third definition, however we think useful to recall the other two \ndefinitions in order to avoid possible misunderstanding.\n\\subsection{The inherent structure entropy}\n Given the Hamiltonian $H(x)$ of a system with $N$ particles, we can \nconsider the solution $x(t)$ of the equation\n\\be\n{dx \\over dt}=- {\\partial H \\over \\partial x}\n\\ee\nas function of the initial conditions $x(0)$. At large time $x(t)$ will go to one of the minima of the \nHamiltonian, called an inherent structure. \nWe label by $a$ each coherent structure and we call $\\cD_{a}$ the set of those \nconfigurations which for large times go to the coherent structure labeled by $a$.\nThe union of all the sets $\\cD_{a}$ is the whole phase space.\nThe probability of finding the system at a temperature $T$ inside a given inherent structure is \nproportional to \n\\be\nP(a)=Z(a)/\\sum_b Z(b) \\ \\ \\ ; \\ \\ \\ Z(a)\\equiv \\int_{x \\in \\cD_{a} }dx \\exp (-\\beta H(x)) \\ .\n\\label{za}\n\\ee\n\nThe configurational entropy density, $\\Si_{is}$, is defined by\n\\be\nN \\Si_{is}(T)= -\\sum_{a} P(a)\\ln(P(a)).\n\\label{Sis}\n\\ee\nThis definition makes sense at all temperatures. In the limit of large $T$ one finds\n\\be\n\\lim_{T\\to\\infty} \\Si_{is}(T) =-\\sum_{a} V(a)\\ln(V(a)),\n\\ee\nwhere $V(a)$ is proportional to the volume in phase space of the region $\\cD_{a}$, normalized in \nsuch a way that $\\sum_{a}V(a)=1$.\nIt is reasonable to expect that this inherent-structures configurational\n entropy starts to decrease when the temperature is decreased around $T=T_{c}$ and \nvanishes at $T=T_{K}$.\n\n\\subsection{Microcanonical entropy}\n We consider the hypersurface of constant energy density, \n$\nH(x)=EN\n$,\nand decompose this energy surface in connected components which we label by $a$. The number of \nconnected components clearly depends on $E$. \n\nCalling $V_{a}$ the normalized phase space volume of each connected component, we define the\nmicrocanonical configurational entropy density as\n\\be\nN \\hat\\Si_{m}(E)=-\\sum_{a} V(a)\\ln(V(a))\n\\ee\nThe microcanonical configurational entropy density as function of the temperature is naturally \ndefined as\n\\be\n \\Si_{m}(T) =\\hat\\Si_{m}(E(T))\n \\ee\nwhere $E(T)$ is the internal energy density as function of the temperature.\nIt is clear that at high energies the configuration space contains only one connected component and \ntherefore \n\\be\n\\lim_{T\\to\\infty} \\Si_{m}(T)=0\n\\ee\n\nThe two configurational entropies introduced so far,\n$\\Si_{is}(T)$ and $\\Si_m(T)$ certainly differ at \nhigh temperature and many hands must be waved in \norder to argue that both entropies behave in a similar way at low temperature and vanish together at \n$T_{K}$.\n\n\\subsection{The thermodynamic configurational entropy}\nAs we have already stated the thermodynamic configurational entropy can be defined by the \nrelation\n\\be\n\\Sigma_{t}=S(T)-S_{valley}(T)\n\\label{Stdef}\n\\ee\nThe main difficulty is the precise definition of the valleys, and of $S_{valley}(T)$,\nin the regime $T>T_K$ where the system is still ergodic. The basic idea \\cite{SPEEDY} is \nto take a generic equilibrium configuration ($y$) at temperature $T$ and to define $S_{valley}(T)$ \nas the thermodynamic entropy of the system constrained to stay at a distance not too large \n from the \nequilibrium configuration $y$.\nIf we impose a strong constraint (i.e. $x$ too near to $y$) the entropy will depend on the \nconstraint, but the constraint cannot be taken vanishingly small\nbecause the system is ergodic.\n\nOne may be worried that this method contains an unavoidable ambiguity. It turns out that\nthere exists a way to modify this method slightly in order to get rid of this ambiguity.\nThe modified method was introduced in \\cite{pot} and called the potential method. Let\nus summarize it here briefly.\nGiven two configurations $x$ and $y$ we define their overlap as before as\n$\nq(x,y) = -1/N \\sum_{i,k=1,N} w(x_{i}-y_{k}),\n$\nwhere $w(x)=-1 \\for x$ small, $w(x)=0 \\for x$ larger than the typical interatomic distance.\nInstead of adding a strict constraint we add an extra term to the Hamiltonian:\nwe define\n\\bea\n\\exp (-N \\beta F(y,\\eps))=\\int dx \\exp( -H(x)+ \\beta \\eps N q(x,y)),\\\\\nF(\\eps)=\\lan F(y,\\eps) \\ran ,\n\\eea\nwhere $\\lan f(y) \\ran$ denotes the average value of $f$ over equilibrium configurations $y$\nthermalized at temperature $\\beta^{-1}$.\n\nWe introduce the Legendre transform $W(q)$ of the free energy $F(\\eps)$:\n\\be\nW(q)=F(\\eps)+\\eps q \\ \\ \\ ; \\ \\ \\ q={-\\partial F \\over \\partial \\eps} \\ .\n\\ee\nAnalytic computation in mean field models \\cite{pot}, as well as in glass forming liquids \nusing the replicated HNC \napproximation \\cite{card}, show that the \nbehaviour of $W(q)$ is qualitatively given by the graphs of fig. \\ref{figW}.\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{TgtT_c.eps}\n\\includegraphics[width=0.45\\textwidth]{TeqT_c.eps}\n\n\\includegraphics[width=0.45\\textwidth]{TltT_c.eps}\n\\includegraphics[width=0.45\\textwidth]{TeqT_k.eps}\n\\caption{Qualitative behaviour of the potential $W(q)$, in the four\n regions\n$T>T_{c}$, $T=T_{c}$, $T_{K}<T<T_{c}$ and $T=T_{K}$. In these graphs the metastable part can be \neasily identified by remembering that $W(q)$ must be a convex function of $q$.}\n\\label{figW}\n\\end{figure}\nFig. \\ref{figqeps} shows the expectation value of $q$ as function of $\\eps$\nin the corresponding four temperature ranges.\nThe results for the potential $W(q)$ in the unstable region where its second derivative is \nnegative and $q$ is a decreasing function of $\\eps$\nare a clear artefact of the mean field approximation, while the results in the metastable region \ncorrespond to phenomena that can be observed \non time scales shorter than the lifetime of the metastable state.\n\nThe thermodynamic configurational entropy is the value of the potential\n$W(q)$ at the secondary minimum with $q \\ne 0$ \\cite{pot}, and it\ncan be defined only if the minimum do exist (i.e. for $T<T_{c}$). It is evident that the secondary \nminimum for $T>T_{k}$ is always in the metastable region. \n However if one would start from a large value of $\\eps$ \nand would decrease $\\eps$ to zero not too slowly,\nthe system would not escape from the metastable region and one \nobtains a proper definition of the thermodynamic configurational entropy in\nthis region $T>T_K$. In a similar way \none could compute $q(\\eps)$ in the region ($\\eps > \\eps_{c}$) where the high $q$ phase is \nthermodynamically stable\nand extrapolate it to $\\eps \\to 0$.\nThe ambiguity in the definition of the thermodynamic configurational entropy \nat temperatures above $T_k$ becomes larger and \nlarger when the temperature increases. It cannot be defined for $T>T_{c}$.\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{eqth.eps}\n\\includegraphics[width=0.45\\textwidth]{eqtc.eps}\n\n\\includegraphics[width=0.45\\textwidth]{eqti.eps}\n\\includegraphics[width=0.45\\textwidth]{eqtk.eps}\n\\caption{Qualitative behaviour of the order parameter $q$, measuring the\ntypical distance to\nthe reference configuration, versus the strength $\\eps$ of the coupling\nto this reference configuration,\n in the four regions\n$T>T_{c}$, $T=T_{c}$, $T_{K}<T<T_{c}$ and $T=T_{K}$. The dashed line \nshows the true thermodynamically stable curve, where the full line \nis the metastable and the unstable part of the curve.\n}\n\\label{figqeps}\n\\end{figure}\n\n\\subsection{Numerical estimates of the configurational entropy}\nMost attempts at estimating numerically the thermodynamic configurational\nentropy start from the decomposition (\\ref{Stdef}). The liquid entropy is estimated \nby a thermodynamic integration of the specific heat from the very dilute \n(ideal gas) limit. It turns out that in the deeply supercooled region\nthe temperature dependence of the liquid entropy is well fitted by the\nlaw predicted in \\cite{taraz}: $S_{liq}(T)=a T^{-2/5}+b$, which presumably allows\nfor a good extrapolation at temperatures $T$ which cannot be simulated.\nAs for the 'valley' entropy, it can be estimated as that of an harmonic\nsolid. One needs however the vibration frequencies of the solid. These have been \napproximated by several methods, which are all based on some\nevaluation of the Instantaneous Normal Modes (INM) \\cite{INM} in\nthe liquid phase,\nand the\nassumption that the spectrum of frequencies does not depend much on\ntemperature below $T_K$. Starting from a typical configuration\nof the liquid, one \ncan look at the INM around it. In general there exist some negative eigenvalues\n(the liquid is not a local minimum of the energy) which one must take care of. \nSeveral methods have been tried: either \nkeep only the positive eigenvalues, or one considers the absolute values of the eigenvalues\n\\cite{sferesoft,LJ,LJ2}.\nAlternatively one can also consider the INM around the nearest inherent structure\nwhich has by definition a positive spectrum \\cite{sferesoft,LJ,LJ2,SKT}. The\ncomputation of the thermodynamic entropy, using its definition as\na system coupled to a reference\nthermalized configuration, has also been studied in \\cite{sferesoft}.\n\nThe results for the configurational entropy as a function of temperature are\nshown in fig. \\ref{figSc}, for binary mixtures of soft spheres and of Lennard-Jones particles. \nThe agreement with the analytical result obtained from the replicated fluid system\nis rather satisfactory, considering the various approximations\ninvolved both in the analytical estimate and in the numerical ones.\n\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth,angle=270]{complex_soft.ps}\n\\includegraphics[width=0.45\\textwidth,angle=270]{ScdeTLJcomp.eps}\n\\caption{The configurational entropy versus temperature in \nbinary mixtures of soft-spheres and of Lennard-Jones particles.\nThe soft sphere result (left curve), from \\cite{sferesoft}, compares the analytical prediction\nobtained within the harmonic resummation scheme (full line), to simulation\nestimates of $S_{liq}-S_{valley}$, where the valley entropy is \nthat of a harmonic solid with INM eigenvalues projected onto positive\neigenvalues (+), taken in absolute values ($\\times$), or taken around\nthe nearest inherent structure ($\\ast$). The squares\ncorrespond to\nthe numerical estimate of the\nthermodynamic configurational entropy obtained by studying the system coupled \nto a reference configuration (see text, and \\cite{sferesoft} for details). \nThe Lennard-Jones result (right curve), shows as a \nfull (black) curve the theoretical prediction obtained from the\ncloned molecular liquid approach\\cite{LJ,LJ2}. The dotted (green) curve is the result from\nthe simulations of \\cite{LJ,LJ2} and the dashed (red) curve is the result\nfrom the simulations of \\cite{SKT}. Both simulations use the $S_{liq}-S_{valley}$\nestimate where the harmonic solid vibration modes are approximated by the ones\nof the nearest inherent structure.}\n\\label{figSc}\n\\end{figure}\n\nIn a recent work, Sciortino Kob and Tartaglia \\cite{SKT} have computed the configurational\nentropy of inherent structures, $\\Si_{is}(T)$, defined in (\\ref{Sis}),\nin binary Lennard-Jones system. \nAssuming that the free energy $-T \\log Z(a)$ of an inherent structure $a$ \n($Z(a)$ is defined in (\\ref{za}))\ncan be approximated by $E_a+\\delta F(T)$, with a correction $\\delta F$ which is\nnearly independent of $E_a$, then the logarithm of the\nprobability of finding an inherent structure with a given energy $E_{IS}$ is\ngiven by $-\\beta E_{IS}+\\Si_{is}(E_{IS}) +c^t$. One \ncan thus deduce the $E_{IS}$ dependence of $\\Si_{IS}$. Shifting the curves vertically\nin order to try to superimpose them with the thermodynamic configurational entropy,\nthey have checked that all these curves coincide in the region of small enough\nenergy, confirming thus that these two definitions\nof the configurational entropy agree at low enough energy or temperature.\nIn fig. \\ref{scdee} we compare their result for the configurational entropy\nof inherent structures to the one obtained analytically, using\nthe description of the molecular fluid of binary Lennard-Jones particles\nof \\cite{LJ,LJ2}. Apart from a small shift in the ground state energy\nwhich may have several origins (finite size effects, small uncertainties in the \ndescription of the correlation in the molecular fluid),\nthe figures are in rather good agreement.\n\n\n\\begin{figure}\n\\includegraphics[width=0.65\\textwidth,angle=270]{ScdeE_comp.eps}\n\\caption{\nThe left (red) curve is the configurational entropy of inherent structures\nversus energy for a binary Lennard-Jones fluid, computed numerically\nin \\cite{SKT} (with respect to the curve plotted in \\cite{SKT}, the\nenergies have been shifted in order to take into account the truncation of \nthe Lennard-Jones potential used in the simulations of \\cite{SKT}). The right\ncurve is the analytic prediction, using\nthe description of the molecular fluid of binary Lennard-Jones particles\nof \\cite{LJ,LJ2}. There is a small shift in energy between the two curves, \nbut the overall agreement is satisfactory.\n}\n\\label{scdee}\n\\end{figure}\n\n\\section{Remarks}\nWe believe that we have now a consistent scheme for computing the \nthermodynamic properties of glasses at equilibrium. What is needed is on the one hand\nsome better approximations of the molecular liquid state, on the other hand some\nprecise numerical results in the glass phase at equilibrium, as \nwell as measurements of the fluctuation dissipation ratio\nin the out of equilibrium dynamics (which should give the value of $m$ \\cite{fdr}).\n Another\nobvious direction is to study, with the present\nmethods, various types of interaction potentials, including some which are characteristic\nof strong glasses. Eventually, one would like to proceed to a first \nprinciple study of the out of equilibrium dynamics.\n\\section{Acknowledgments}\n\\label{acknowledgements}\nWe wish to thank W. Kob for providing the data discussed in the last section, and\nfor giving us the energy shift of the truncated Lennard-Jones problem, used\nin the comparison of fig. \\ref{scdee}. We wish to thank P. Verrocchio for\nproviding the analytic prediction shown in fig. \\ref{scdee}. \n\n\\section{References}\n\\begin{thebibliography}{999}\n\n\\bi{glass_revue}\nRecent reviews can be found in: C.A. Angell, Science, \n{\\bf 267}, 1924 (1995) and P.De Benedetti, `Metastable liquids', Princeton \nUniversity\nPress (1997). An introduction to the theory is: J.J\\\"ackle, Rep.Prog.\nPhys. {\\bf 49} (1986) 171.\n\\bi{kauzmann}\n A.W. Kauzman, Chem.Rev. {\\bf 43} (1948) 219.\n A nice recent discussion can be found \nin R. Richert and C.A. Angell, J.Chem.Phys. {\\bf 108} (1999) 9016.\n\n\\bi{AdGibbs}\nG. Adams and J.H. Gibbs J.Chem.Phys {\\bf 43} (1965) 139; J.H. Gibbs and E.A.\nDi \nMarzio, \nJ.Chem.Phys. {\\bf 28} (1958) 373.\n\n\\bi{rubber}\nIn this sense, the rubber is a structural glass which is much closer to\nspin glasses, because of the quenched random links between\nthe macromolecules. Theoretical studies of rubber\nare reviewed in P.M. Goldbart, H.E. Castillo and A. Zippelius\nAdv. Phys. {\\bf 45} (1996) 393.\n\n\\bi{MPV} For a review, see M. M\\'ezard, G. Parisi and M.A. Virasoro, {\\sl Spin glass theory \nand \nbeyond}, World Scientific (Singapore 1987)\n\n\\bi{REM}\nB. Derrida, {\\em Phys. Rev.} {\\bf B24}, 2613 (1981)\n\n\\bi{GrossMez}\nD.J. Gross and M. M\\'ezard, Nucl. Phys. {\\bf B240} (1984) 431.\n\n\n\\bi{KiThWo}\n T.R. Kirkpatrick and P.G. Wolynes, Phys. Rev. {\\bf\nA34}, 1045 (1986); T.R. Kirkpatrick and D. Thirumalai, Phys. Rev. Lett. {\\bf \n58},\n2091 (1987); T.R. Kirkpatrick and D. Thirumalai, Phys. Rev. {\\bf B36}, 5388 \n(1987); \nT.R. Kirkpatrick, D. Thirumalai and P.G. Wolynes, Phys. Rev. {\\bf\nA40}, 1045 (1989).\n\n\\bi{crisanti} A. Crisanti, H. Horner and H.J. Sommers,\nZ. Physik B {\\bf 92}, 257 (1993).\n\n\n\\bi{nodis1}\n J.-P. Bouchaud and M. M\\'ezard; J. Physique I (France) {\\bf \n4} (1994) 1109.\nE. Marinari, G. Parisi and F. Ritort; J. Phys. {\\bf A27} (1994) 7615; J. \nPhys. {\\bf A27} (1994) 7647.\n\n\\bi{nodis2}\nP.Chandra, L.B.Ioffe and D.Sherrington, Phys. Rev. lett. {\\bf 75} (1995) 713,\nand cond-mat/9809417.\nP.Chandra, M.V. Feigelman and L.B.Ioffe, Phys. Rev. lett. {\\bf 76} (1996) 4805.\n\n\\bi{nodis3}\n E. Marinari, G. Parisi and F. Ritort, cond-mat/9410089.\n S. Franz and J. Hertz, {\\it Phys. Rev. Lett.} {\\bf 74}, 2114 (1995).\n\n\\bi{cuku} L. F. Cugliandolo and J.Kurchan, Phys. Rev. Lett. {\\bf 71}, \n1 (1993).\n\n\\bibitem{fdr} S. Franz, M. M\\'ezard, G. Parisi and L. Peliti,\nPhys. Rev. Lett. {\\bf 81} 1758 (1998); {\\em The response of glassy systems\n to random perturbations: A bridge between equilibrium and \noff-equilibrium}, cond-mat/9903370, to appear in J.Stat.Phys.\n\n\\bi{gpglass}\n G. Parisi Phys.Rev.Lett. {\\bf 78}(1997)4581.\n\n\\bi{bk1}\n W. Kob and J.-L. Barrat, Phys.Rev.Lett. {\\bf 79} (1997) 3660.\n\n\\bi{bk2}\n J.-L. Barrat and W. Kob, cond-mat/9806027.\n\n\\bi{MePa1} M. M\\'ezard and G. Parisi, Phys. Rev. Lett. {\\bf 82}, 747 \n(1998).\n \n\\bi{Me} M. M\\'ezard, Physica A {\\bf 265}, 352 (1999).\n\n\\bi{MePa2} M. M\\'ezard and G. Parisi J. Chem. Phys. {\\bf 111}, 1076 (1999).\n\n\\bi{sferesoft} B. Coluzzi, M. M\\'ezard, G. Parisi and P. Verrocchio, {\\em\nThermodynamics of binary \nmixture glasses}, cond-mat/9903129.\n\n\\bi{LJ} B. Coluzzi, G. Parisi and P. Verrocchio, {\\em Lennard-Jones\n binary mixture: a \nthermodynamical approach to glass transition}, cond-mat/9904124.\n\n\\bi{LJ2} B. Coluzzi, G. Parisi and P. Verrocchio, {\\em The thermodynamical\nliquid-glass transition in a Lennard-Jones binary mixture}, \ncond-mat/9906124.\n\n\\bi{kepler} An introduction to recent work on Kepler's conjecture can be\nfound in:\nwww.math.lsa.umich.edu/~hales/countdown/.\n\n\\bi{angelani}\n L. Angelani, G. Parisi,\nG. Ruocco and G. Viliani, cond-mat/9904125.\n\n\\bi{theo}\nT.M. Nieuwenhuizen, Phys.Rev.Lett. {\\bf 79} (1997) 1317.\n\n\\bi{inherent}\nM. Goldstein, J. Chem. Phys. {\\bf 51}, 3728 (1969);\n F.H. Stillinger, Science {\\bf 267} (1995) 1935, and references \ntherein. Recent includes: S. Sastry, P.G. Debenedetti and\nF.H. Stillinger, Nature {\\bf 393}, 554 (1998),\nW. Kob, F. Sciortino and P. Tartaglia, cond-mat/9905090;\n F. Sciortino, W. Kob and P. Tartaglia, cond-mat/9906278;\nS. B\\\"uchner and A. Heuer, cond-mat/9906280.\n\n\n\\bi{remi} R. Monasson, {\\em Phys. Rev. Lett.} {\\bf 75}, 2847 (1995).\n\n\n\\bi{MEPAZEE} M. M\\'ezard, G. Parisi and A. Zee {\\em Spectra of Euclidean \nRandom\nMatrices}, \ncond-mat/9906135.\n\n\\bi{CAGIAPA} A. Cavagna, I. Giardina and G. Parisi, {\\em Analytic \ncomputation of\nthe Instantaneous \nNormal Modes spectrum in low density liquids} (cond-mat/9903155),\n Phys.Rev.Lett. to be \npublished.\n\n\\bi{taraz}\nY.Rosenfeld and P. Tarazona, Mol.Phys. {\\bf 95}, 141 (1998).\n\n\\bi{pedago}\nG.Parisi, cond-mat/9905318.\n\n\n\\bi{Han2}\nB. Bernu, J.-P. Hansen, Y. Hitawari and G. Pastore, {\\em Phys. Rev.} \n{\\bf A 36}, 4891 (1987). J.-L. Barrat, J.-N. Roux and J.-P. Hansen, \n{\\em Chem. Phys.} {\\bf 149}, 197 (1990). J.-P. Hansen and S. Yip, \n{\\em Trans. Theory and Stat. Phys.} {\\bf 24}, 1149 (1995).\n\n\n\\bi{SPEEDY} See the talk of Speedy at this conference and references therein. \n\n\\bi{pot} S. Franz ang G. Parisi, J. Physique I {\\bf 5} (1995) 1401; \n Phys.Rev.Lett. {\\bf 79} (1997) 2486.\n\n\n\\bi{card} \n M.Cardenas, S. Franz and G. Parisi, cond-mat/9712099.\n\n\\bi{INM} T. Keyes, J. Chem. Phys. A101 (1997) 2921.\n\n\\bi{SKT} F. Sciortino, W. Kob and P. Tartaglia, { \\em Inherent structure entropy\nof supercooled liquids}, cond-mat/9906081. See also Sciortino's contribution\nto this volume.\n\n\n\\end{thebibliography} \n\\end{document}\n\n\n"
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{
"name": "cond-mat0002128.extracted_bib",
"string": "\\begin{thebibliography}{999}\n\n\\bi{glass_revue}\nRecent reviews can be found in: C.A. Angell, Science, \n{\\bf 267}, 1924 (1995) and P.De Benedetti, `Metastable liquids', Princeton \nUniversity\nPress (1997). An introduction to the theory is: J.J\\\"ackle, Rep.Prog.\nPhys. {\\bf 49} (1986) 171.\n\\bi{kauzmann}\n A.W. Kauzman, Chem.Rev. {\\bf 43} (1948) 219.\n A nice recent discussion can be found \nin R. Richert and C.A. Angell, J.Chem.Phys. {\\bf 108} (1999) 9016.\n\n\\bi{AdGibbs}\nG. Adams and J.H. Gibbs J.Chem.Phys {\\bf 43} (1965) 139; J.H. Gibbs and E.A.\nDi \nMarzio, \nJ.Chem.Phys. {\\bf 28} (1958) 373.\n\n\\bi{rubber}\nIn this sense, the rubber is a structural glass which is much closer to\nspin glasses, because of the quenched random links between\nthe macromolecules. Theoretical studies of rubber\nare reviewed in P.M. Goldbart, H.E. Castillo and A. Zippelius\nAdv. Phys. {\\bf 45} (1996) 393.\n\n\\bi{MPV} For a review, see M. M\\'ezard, G. Parisi and M.A. Virasoro, {\\sl Spin glass theory \nand \nbeyond}, World Scientific (Singapore 1987)\n\n\\bi{REM}\nB. Derrida, {\\em Phys. Rev.} {\\bf B24}, 2613 (1981)\n\n\\bi{GrossMez}\nD.J. Gross and M. M\\'ezard, Nucl. Phys. {\\bf B240} (1984) 431.\n\n\n\\bi{KiThWo}\n T.R. Kirkpatrick and P.G. Wolynes, Phys. Rev. {\\bf\nA34}, 1045 (1986); T.R. Kirkpatrick and D. Thirumalai, Phys. Rev. Lett. {\\bf \n58},\n2091 (1987); T.R. Kirkpatrick and D. Thirumalai, Phys. Rev. {\\bf B36}, 5388 \n(1987); \nT.R. Kirkpatrick, D. Thirumalai and P.G. Wolynes, Phys. Rev. {\\bf\nA40}, 1045 (1989).\n\n\\bi{crisanti} A. Crisanti, H. Horner and H.J. Sommers,\nZ. Physik B {\\bf 92}, 257 (1993).\n\n\n\\bi{nodis1}\n J.-P. Bouchaud and M. M\\'ezard; J. Physique I (France) {\\bf \n4} (1994) 1109.\nE. Marinari, G. Parisi and F. Ritort; J. Phys. {\\bf A27} (1994) 7615; J. \nPhys. {\\bf A27} (1994) 7647.\n\n\\bi{nodis2}\nP.Chandra, L.B.Ioffe and D.Sherrington, Phys. Rev. lett. {\\bf 75} (1995) 713,\nand cond-mat/9809417.\nP.Chandra, M.V. Feigelman and L.B.Ioffe, Phys. Rev. lett. {\\bf 76} (1996) 4805.\n\n\\bi{nodis3}\n E. Marinari, G. Parisi and F. Ritort, cond-mat/9410089.\n S. Franz and J. Hertz, {\\it Phys. Rev. Lett.} {\\bf 74}, 2114 (1995).\n\n\\bi{cuku} L. F. Cugliandolo and J.Kurchan, Phys. Rev. Lett. {\\bf 71}, \n1 (1993).\n\n\\bibitem{fdr} S. Franz, M. M\\'ezard, G. Parisi and L. Peliti,\nPhys. Rev. Lett. {\\bf 81} 1758 (1998); {\\em The response of glassy systems\n to random perturbations: A bridge between equilibrium and \noff-equilibrium}, cond-mat/9903370, to appear in J.Stat.Phys.\n\n\\bi{gpglass}\n G. Parisi Phys.Rev.Lett. {\\bf 78}(1997)4581.\n\n\\bi{bk1}\n W. Kob and J.-L. Barrat, Phys.Rev.Lett. {\\bf 79} (1997) 3660.\n\n\\bi{bk2}\n J.-L. Barrat and W. Kob, cond-mat/9806027.\n\n\\bi{MePa1} M. M\\'ezard and G. Parisi, Phys. Rev. Lett. {\\bf 82}, 747 \n(1998).\n \n\\bi{Me} M. M\\'ezard, Physica A {\\bf 265}, 352 (1999).\n\n\\bi{MePa2} M. M\\'ezard and G. Parisi J. Chem. Phys. {\\bf 111}, 1076 (1999).\n\n\\bi{sferesoft} B. Coluzzi, M. M\\'ezard, G. Parisi and P. Verrocchio, {\\em\nThermodynamics of binary \nmixture glasses}, cond-mat/9903129.\n\n\\bi{LJ} B. Coluzzi, G. Parisi and P. Verrocchio, {\\em Lennard-Jones\n binary mixture: a \nthermodynamical approach to glass transition}, cond-mat/9904124.\n\n\\bi{LJ2} B. Coluzzi, G. Parisi and P. Verrocchio, {\\em The thermodynamical\nliquid-glass transition in a Lennard-Jones binary mixture}, \ncond-mat/9906124.\n\n\\bi{kepler} An introduction to recent work on Kepler's conjecture can be\nfound in:\nwww.math.lsa.umich.edu/~hales/countdown/.\n\n\\bi{angelani}\n L. Angelani, G. Parisi,\nG. Ruocco and G. Viliani, cond-mat/9904125.\n\n\\bi{theo}\nT.M. Nieuwenhuizen, Phys.Rev.Lett. {\\bf 79} (1997) 1317.\n\n\\bi{inherent}\nM. Goldstein, J. Chem. Phys. {\\bf 51}, 3728 (1969);\n F.H. Stillinger, Science {\\bf 267} (1995) 1935, and references \ntherein. Recent includes: S. Sastry, P.G. Debenedetti and\nF.H. Stillinger, Nature {\\bf 393}, 554 (1998),\nW. Kob, F. Sciortino and P. Tartaglia, cond-mat/9905090;\n F. Sciortino, W. Kob and P. Tartaglia, cond-mat/9906278;\nS. B\\\"uchner and A. Heuer, cond-mat/9906280.\n\n\n\\bi{remi} R. Monasson, {\\em Phys. Rev. Lett.} {\\bf 75}, 2847 (1995).\n\n\n\\bi{MEPAZEE} M. M\\'ezard, G. Parisi and A. Zee {\\em Spectra of Euclidean \nRandom\nMatrices}, \ncond-mat/9906135.\n\n\\bi{CAGIAPA} A. Cavagna, I. Giardina and G. Parisi, {\\em Analytic \ncomputation of\nthe Instantaneous \nNormal Modes spectrum in low density liquids} (cond-mat/9903155),\n Phys.Rev.Lett. to be \npublished.\n\n\\bi{taraz}\nY.Rosenfeld and P. Tarazona, Mol.Phys. {\\bf 95}, 141 (1998).\n\n\\bi{pedago}\nG.Parisi, cond-mat/9905318.\n\n\n\\bi{Han2}\nB. Bernu, J.-P. Hansen, Y. Hitawari and G. Pastore, {\\em Phys. Rev.} \n{\\bf A 36}, 4891 (1987). J.-L. Barrat, J.-N. Roux and J.-P. Hansen, \n{\\em Chem. Phys.} {\\bf 149}, 197 (1990). J.-P. Hansen and S. Yip, \n{\\em Trans. Theory and Stat. Phys.} {\\bf 24}, 1149 (1995).\n\n\n\\bi{SPEEDY} See the talk of Speedy at this conference and references therein. \n\n\\bi{pot} S. Franz ang G. Parisi, J. Physique I {\\bf 5} (1995) 1401; \n Phys.Rev.Lett. {\\bf 79} (1997) 2486.\n\n\n\\bi{card} \n M.Cardenas, S. Franz and G. Parisi, cond-mat/9712099.\n\n\\bi{INM} T. Keyes, J. Chem. Phys. A101 (1997) 2921.\n\n\\bi{SKT} F. Sciortino, W. Kob and P. Tartaglia, { \\em Inherent structure entropy\nof supercooled liquids}, cond-mat/9906081. See also Sciortino's contribution\nto this volume.\n\n\n\\end{thebibliography}"
}
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cond-mat0002129
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Cryptographical Properties of Ising Spin Systems
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[
{
"author": "Yoshiyuki~Kabashima$^{1}$"
},
{
"author": "Tatsuto Murayama$^{1}$ and David~Saad$^{2}$"
}
] |
The relation between Ising spin systems and public-key cryptography is investigated using methods of statistical physics. The insight gained from the analysis is used for devising a matrix-based cryptosystem whereby the ciphertext comprises products of the original message bits; these are selected by employing two predetermined randomly-constructed sparse matrices. The ciphertext is decrypted using methods of belief-propagation. The analyzed properties of the suggested cryptosystem show robustness against various attacks and competitive performance to modern cyptographical methods.
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[
{
"name": "PRL_final.tex",
"string": "%\\documentstyle[aps,preprint,prl,epsf]{revtex}\n\\documentstyle[aps,twocolumn,prl,epsf]{revtex}\n\\begin{document}\n\n\\draft\n\\input{epsf}\n\n\\newcommand{\\hz}{\\widehat{\\mbox{\\boldmath{$\\zeta$}}}}\n\\newcommand{\\btau}{\\mbox{\\boldmath{$\\tau$}}}\n\\newcommand{\\bxi}{\\mbox{\\boldmath{$\\xi$}}}\n\\newcommand{\\bS}{\\mbox{\\boldmath{$S$}}}\n\\newcommand{\\bz}{\\mbox{\\boldmath{$\\zeta$}}}\n\\newcommand{\\bJ}{\\mbox{\\boldmath{$J$}}}\n\\newcommand{\\JO}{\\mbox{\\boldmath{$J^{0}$}}}\n\\newcommand{\\cD}{{\\cal D}}\n\\newcommand{\\cJ}{{\\cal J}}\n\\newcommand{\\dashed}{\\mbox{-\\; -\\; -\\; -}}\n\\newcommand{\\dotted}{\\mbox{${\\mathinner{\\cdotp\\cdotp\\cdotp\\cdotp\\cdotp\\cdotp}}$}}\n\\newcommand{\\full}{\\mbox{------}}\n\n\n\\title{Cryptographical Properties of Ising Spin Systems}\n\\author{Yoshiyuki~Kabashima$^{1}$, Tatsuto Murayama$^{1}$ and\nDavid~Saad$^{2}$} \\address{$^{1}$ Department of Computational\nIntelligence and Systems Science, Tokyo Institute of Technology,\nYokohama 2268502, Japan. \\\\ $^{2}$The Neural Computing Research\nGroup, Aston University, Birmingham B4 7ET, UK.}\n\n\\maketitle\n\n\n\\begin{abstract}\nThe relation between Ising spin systems and public-key cryptography is\ninvestigated using methods of statistical physics. The insight gained\nfrom the analysis is used for devising a matrix-based cryptosystem\nwhereby the ciphertext comprises products of the original message\nbits; these are selected by employing two predetermined\nrandomly-constructed sparse matrices. The ciphertext is decrypted\nusing methods of belief-propagation. The analyzed properties of the\nsuggested cryptosystem show robustness against various attacks and\ncompetitive performance to modern cyptographical methods.\n\n\\end{abstract}\n\\pacs{89.90.+n, 02.50.-r, 05.50.+q, 75.10.Hk}\n\n\n\nPublic-key cryptography plays an important role in many aspects of\nmodern information transmission, for instance, in the areas of\nelectronic commerce and internet-based communication. It enables the\nservice provider to distribute a public key which may be used to\nencrypt messages in a manner that can only be decrypted by the service\nprovider. The on-going search for safer and more efficient\ncryptosystems produced many useful methods over the years such as RSA\n(by Rivest, Shamir and Adleman), elliptic curves, and the McEliece\ncryptosystem to name but a few.\n\nIn this Letter, we employ methods of statistical physics to study a\nspecific cryptosystem, somewhat similar to the one presented by\nMcEliece\\cite{McEliece}. These methods enable one to study the typical\nperformance of the suggested cryptosystem, to assess its robustness\nagainst attacks and to select optimal parameters.\n\n\nThe main motivation for the suggested cryptosystem comes from previous\nstudies of Gallager-type error-correcting\ncodes\\cite{Gallager,MacKay,Sourlas} and their physical\nproperties\\cite{us_gallager,us_sourlas}. The analysis exposes a\nsignificantly different behaviour for the two-matrix based codes (such\nas the MN code\\cite{MacKay}) and single-matrix codes\\cite{Sourlas},\nwhich may be exploited for constructing an efficient cryptosystem.\n\nIn the suggested cryptosystem, a plaintext represented by an\n$N$ dimensional Boolean vector $\\bxi \\in (0,1)^N$ is encrypted to the $M$\ndimensional Boolean ciphertext $\\bJ$ using a predetermined Boolean matrix $G$,\nof dimensionality $M\\times N$, and a corrupting $M$\ndimensional vector $\\bz$, whose elements are 1 with probability \n$p$ and 0 otherwise, in the following manner\n\\begin{equation}\n\\label{eq:ciphertext}\n\\bJ = G \\ \\bxi \\ + \\bz \\ , \n\\end{equation}\nwhere all operations are (mod 2). The matrix $G$ and the probability\n$p$ constitute the public key; the corrupting vector $\\bz$ is chosen at\nthe transmitting end. The matrix $G$, which is at the heart of the\nencryption/decryption process is constructed by choosing two\nrandomly-selected sparse matrices $A$ and $B$ of dimensionality\n$M\\!\\times \\!N$ and $M\\!\\times\\! M$ respectively, defining\n\\[ G \\! =\\! B^{-1}A \\ \\ \\mbox{(mod 2)} \\ . \\]\nThe matrices $A$ and $B$ are generally characterised by $K$ and $L$\nnon-zero unit elements per row and $C$ and $L$ per column\nrespectively; all other elements are set to zero. The finite, usually\nsmall, numbers $K$, $C$ and $L$ define a particular cryptosystem; both\nmatrices are known only to the authorised receiver.\nSuitable choices\nof probability $p$ will depend on the maximal achievable rate for the\nparticular cryptosystem as discussed below.\n\nThe authorised user may decrypt the received ciphertext $\\bJ$ by\ntaking the (mod 2) product $B \\bJ = A\\bxi\\! + \\! B\\bz$. Solving the\nequation\n\\begin{equation}\n\\label{eq:decoding}\nA\\bS + B\\btau =A\\bxi + B\\bz \\ \\ \\mbox{(mod 2)}, \\\n\\end{equation}\nis generally computationally hard. However, decryption can be carried\nout for particular choices of $K$ and $L$ via the iterative methods of\nBelief Propagation (BP)\\cite{MacKay}, where pseudo-posterior\nprobabilities for the decrypted message bits, $P(S_{i}\\!=\\!1 | \\bJ) \\\n1\\!\\le\\!i\\! \\le\\! N $ (and similarly for $\\btau$), are calculated by\nsolving iteratively a set of coupled equations for the conditional\nprobabilities of the ciphertext bits given the plaintext and vice\nversa. For details of the method used and the explicit equations see\n\\cite{MacKay}.\n\nThe unauthorised receiver, on the other hand, faces the task of\ndecrypting the ciphertext $\\bJ$ knowing only $G$ and $p$. The\nstraightforward attempt to try all possible $\\bz$ constructions is\nclearly doomed, provided that $p$ is not vanishingly small, giving\nrise to only a few corrupted bits; decomposing $G$ to the matrices $A$\nand $B$ is known to be a computationally hard problem\\cite{NP}, even if the\nvalues of $K,C$ and $L$ are known. Another approach to study the\nproblem is to exploit the similarity between the task at hand and the\nerror-correcting model suggested by Sourlas\\cite{Sourlas}, which we\nwill discuss below.\n\nThe treatment so far was completely general. We will now make\nuse of insight gained from our analysis of\nGallager-type\\cite{us_gallager} and Sourlas\\cite{us_sourlas}\nerror-correcting codes to suggest a specific cyptosystem construction\nand to assess its performance and capabilities. The method used in\nboth analyses \\cite{us_gallager,us_sourlas} is based on mapping the\nproblem onto an Ising spin system Hamiltonian, in the manner discovered\nby Sourlas\\cite{Sourlas}, which enables one to analyse\ntypical properties of such systems.\n\nTo facilitate the mapping we employ binary representations $(\\pm1)$ of\nthe dynamical variables $\\bS$ and $\\btau$, the vectors $\\bJ$, $\\bz$\nand $\\bxi$, and the matrices $A$, $B$ and $G$, rather than the Boolean\n$(0,1)$ ones.\n\nThe {\\em binary} ciphertext $\\bJ$ is generated by taking products of\nthe relevant binary plaintext message bits $J_{\\left\\langle i_{1},\ni_{2} \\ldots \\right\\rangle} \\! = \\! \\xi_{i_{1}} \\xi_{i_{2}} \\ldots\n\\zeta_{\\left\\langle i_{1}, i_{2} \\ldots \\right\\rangle}$, where the\nindices $i_{1},i_{2}\\ldots $ correspond to the non-zero elements of\n$B^{-1}A$, and $\\zeta_{\\left\\langle i_{1}, i_{2} \\ldots\n\\right\\rangle}$ is the corresponding element of the corrupting vector\n(the indices ${\\left\\langle i_{1}, i_{2} \\ldots\n\\right\\rangle}$ corresponds to the specific choice made for each\nciphertext bit). As we use statistical mechanics techniques, we\nconsider both plaintext ($N$) and ciphertext ($M$) dimensionalities to\nbe infinite, keeping the ratio between them $N/M$ finite. Using the\nthermodynamic limit is quite natural here as most transmitted\nciphertexts are long and finite size corrections are likely to be\nsmall.\n\nAn authorised user may use the matrix $B$ to obtain\nEq.(\\ref{eq:decoding}). To explore the system's capabilities one\nexamines the Gibbs distribution, based on the Hamiltonian\n\\begin{eqnarray}\n\\label{eq:Hamiltonian}\n{\\cal H} &=& \\!\\!\\!\\!\\! \\sum_{<i_1,..,i_K;j_1,..,j_L>}\n \\mbox{\\hspace*{-5mm}} \\cD_{<i_1,..,i_K;j_1,..,j_L>} \\ \\delta\n \\biggl[-1 \\ ; \\ \\cJ_{<i_1,..,i_K;j_1,..,j_L>} \\nonumber \\\\\n &\\cdot & S_{i_1}\\ldots S_{i_K} \\tau_{j_1}\\ldots\\tau_{j_L}\n \\biggr] - \\frac{F_s}{\\beta} \\sum_{i=1}^{N} S_i -\n \\frac{F_{\\tau}}{\\beta} \\sum_{j=1}^{M} \\tau_j \\ .\n\\end{eqnarray}\nThe tensor product $\\cD_{<i_1,..,i_K;j_1,..,j_L>}\n\\cJ_{<i_1,..,i_K;j_1,..,j_L>}$, where $\\cJ_{<i_1,..,j_L>} \\! = \\!\n\\xi_{i_{1}} \\xi_{i_{2}}.. \\xi_{i_{K}} \\zeta_{j_{1}} \\zeta_{j_{2}}\n.. \\zeta_{j_{L}}$, is the binary equivalent of $A\\bxi \\! + \\!\nB\\bz$, treating both signal ($\\bS$ and index $i$) and the corrupting\nnoise vector ($\\btau$ and index $j$) simultaneously. Elements of the\nsparse connectivity tensor $\\cD_{<i_1,..,j_L>}$ take the value 1 if\nthe corresponding indices of both signal and noise are chosen (i.e.,\nif all corresponding elements of the matrices $A$ and $B$ are 1) and 0\notherwise; it has $C$ unit elements per $i$-index and $L$ per\n$j$-index, representing the system's degree of connectivity. The\n$\\delta$ function provides $1$ if the selected sites' product\n$S_{i_1}..S_{i_K} \\tau_{j_1}..\\tau_{j_L}$ is in disagreement\nwith the corresponding element $\\cJ_{<i_1..j_L>}$, recording an\nerror, and $0$ otherwise. Notice that this term is not frustrated, and\ncan therefore vanish at sufficiently low temperatures ($T \\!=\\!\n1/\\beta \\!\\rightarrow\\! 0$), imposing the restriction of\nEq.(\\ref{eq:decoding}), while the last two terms, scaled with $\\beta$,\nsurvive. The additive fields $F_s$ and $F_{\\tau}$ are introduced to\nrepresent our prior knowledge on the signal and noise distributions,\nrespectively.\n\nThe random selection of elements in $\\cD$ introduces disorder to the\nsystem which is treated via methods of statistical physics. More\nspecifically, we calculate the partition function ${\\cal Z}\n({\\cD},\\mbox{\\boldmath $J$}) = \\mbox{Tr}_{\\{\\bS,\\btau\\}} \\exp [-\\beta\n{\\cal H}]$, which is then averaged over the disorder and the\nstatistical properties of the plaintext and noise, using the replica\nmethod\\cite{us_gallager,Wong_Sherrington}, to obtain the related free\nenergy ${\\cal F} = - \\langle \\ln {\\cal Z} \\rangle_{\\xi,\\zeta,{\\cal\nD}}$. The overlap between the plaintext and the dynamical vector\n$m\\!=\\!\\frac{1}{N}\\sum_{i=1}^N \\xi_i S_i$ will serve as a measure for\nthe decryption success.\n\nStudying this free energy for the case of $K\\!\\!=\\!\\! L\\!\\!=\\!\\! 2$ and in\nthe context of error-correcting codes\\cite{us_gallager}, indicates the\nexistence of paramagnetic and ferromagnetic solutions depicted in the\ninset of Fig.1. For corruption probabilities $p\\!>\\!p_{s}$ one obtains\neither a dominant paramagnetic solution or a mixture of ferromagnetic\n($m\\!=\\!\\pm 1$) and paramagnetic ($m\\!=\\! 0$) solutions as shown in\nthe inset; thin and thick lines correspond to higher and lower free\nenergies respectively, dashed lines represent unstable\nsolutions. Lines between the $m\\!=\\!\\pm 1$ and $m\\!=\\! 0$ axes\ncorrespond to sub-optimal ferromagnetic solutions.\nReliable decryption may only be obtained for $p\\!<\\!p_{s}$, which\ncorresponds to a spinodal point, where a unique ferromagnetic solution\nemerges at $m\\!=\\! 1$ (plus a mirror solution at $m\\!=\\! -1$).\n\nThe most striking result is the division of the complete space of\n$\\bS$ and $\\btau$ values to two basins of attraction for the\nferromagnetic solutions, for $p<p_{s}$, implying convergence from {\\em\nany} initialisation of the BP equations. Critical corruption rate\nvalues for $M/N=2$ were obtained from the analysis and via BP\nsolutions as shown in Fig.1, in comparison to the rate obtainable from\nShannon's channel capacity\\cite{Shannon} (solid line). The priors\nassumed for both the plaintext (unbiased in this case, $F_s=0$) and\nthe corrupting vector ($F_\\tau=(1/2) \\ln [(1-p)/p]$) correspond to\nNishimori's condition \\cite{Nishimori}, which is equivalent to having\nthe correct prior within the Bayesian framework\\cite{Sourlas_EPL}\n\nThe initial conditions for the BP-based decryption were chosen almost\nat random, with a very slight bias of ${\\cal O}(10^{-12})$ in the\ninitial magnetisation, corresponding to typical statistical\nfluctuation for a system size of $10^{24}$. Cryptosystems with other\n$K$ and $L$ values, e.g., $K,L \\ge 3$, seem to offer optimal\nperformance in terms of the corruption rate they accommodate\ntheoretically, but suffer from a decreasingly small basin of\nattraction as $K$ and $L$ increase. The co-existence of stable\nferromagnetic and paramagnetic solutions implies that the system will\nconverge to the undesired paramagnetic solution\\cite{us_gallager} from\nmost initial conditions which are typically of close-to-zero\nmagnetisation. It may still be possible to use successfully specific\nmatrices with higher $K$ and $L$ values (such as\nin\\cite{kanter_saad}); however, these cannot be justified\ntheoretically and there is no clear adventage in using them.\n\n\nTo conclude, for the authorised user, the $K\\!\\!=\\!\\! L\\!\\!=\\!\\! 2$\ncryptosystem offers a guaranteed convergence to the plaintext\nsolution, in the thermodynamic limit $N\\rightarrow \\infty$, as long as\nthe corruption process has a probability below $p_{s}$. The main\nconsequence of finite plaintexts would be a decrease in the allowed\ncorruption rate with little impact on the decoding success.\n\n\nThe task facing the unauthorised user, {i.e.}, \nfinding the plaintext from Eq. (\\ref{eq:ciphertext})\nwas investigated via similar methods\nby considering the Hamiltonian\n\\[\n{\\cal H}\\!=\\!-\\!\\!\\!\\! \\sum_{\\left\\langle i_{1},\n..i_{K'} \\right\\rangle} \\!\\!\\! {\\cal G}_{\\left\\langle i_{1},\n..i_{K'} \\right\\rangle} \\ J_{\\left\\langle i_{1},..i_{K'}\n\\right\\rangle} \\ S_{i_{1}}\\! .. \\! S_{i_{K'}} - \\frac{F_s}{\\beta} \n\\sum_{k=1}^{N} S_{k} \\ ,\n\\]\n%\nusing Nishimori's temperature $\\beta=(1/2) \\ln [(1-p)/p]$. The number\nof plaintext bits in each product is denoted $K'$, $\\mbox{\\boldmath\n$S$}$ is the $N$ dimensional binary vector of dynamical variables and\n${\\cal G}$ is a dense tensor with $C'$ unit elements per index\n(setting the rest of the elements to zero) and is the binary\nequivalent of the Boolean matrix $G$\\cite{us_sourlas}. The latter,\ntogether with the statistical properties of the corrupting vector\n$\\bz$ constitutes the public key used to determine the \nciphertext $\\mbox{\\boldmath $J$}$. The last term on the right is\nrequired in the case of sparse or biased messages and will require\nassigning a certain value to the additive field $F_s$.\n\nThe matrix $G$ generated in the case of $K\\!\\!=\\!\\! L\\!\\!=\\!\\! 2$ is dense\nand has a certain distribution of unit elements per row. The fraction\nof rows with a low (finite, not of ${\\cal O} (N)$) number of unit\nelements vanishes as $N$ increases, allowing one to approximate this\nscenario by the diluted Random Energy Model\\cite{REM} studied\nin\\cite{us_sourlas} where $K',C'\\rightarrow \\infty$ while\nkeeping the ratio $C'/K'$ finite.\n\nTo investigate the typical properties of this (frustrated) model, we\ncalculate again the partition function and the free energy by\naveraging over the randomness in choosing the plaintext, the\ncorrupting vector and the choice of the random matrix $G$ (being\ngenerated by a product of two sparse random matrices). To assess the\nlikelihood of obtaining spin-glass/ferromagnetic solutions, we\ncalculated the free energy landscape (per plaintext bit - $f$) as a\nfunction of overlap $m$. This can be carried out straightforwardly\nusing the analysis of \\cite{us_gallager}, and provides the energy\nlandscape shown in Fig.2. This has the structure of a golf-course with\na relatively flat area around the one-step replica symmetry breaking\n(frozen) spin-glass solution and a very deep but extremely narrow\narea, of ${\\cal O} (1/N)$, around the ferromagnetic solution. To\nvalidate the use of the random energy model we also added numerical\ndata ($+$, with error-bars), obtained by BP, which are consistent with\nthe theoretical results.\n\nThis free-energy landscape may be related directly to the marginal\nposterior $P(S_{i}\\!=\\!1 | \\bJ) \\ 1\\!\\le\\! i\\! \\le\\! N $ and is therefore\nindicative of the difficulties in obtaining ferromagnetic solutions\nwhen the starting point for the search is not infinitesimally close to\nthe original plaintext (which is clearly highly unlikely). It is\nplausible that any local search method, starting at some distance from\nthe ferromagnetic solution, will fail to produce the original\nplaintext. Similarly, any probabilistic method, such as simulated\nannealing, will require an exponentially long time for converging to\nthe $m\\!=\\!1$ solution. Numerical studies of similar energy landscapes show\nthat the time required increases exponentially with the system\nsize\\cite{Parisi}.\n\nMost attacks on this cryptosystems, by an unauthorised user, will face\nthe same difficulty: without explicit knowledge of the current\nplaintext and/or the decomposition of $G$ to the matrices $A$ and $B$\nit will require an exponentially long time to decipher a specific\nciphertext. Partial or complete knowledge of the ciphertext/plaintext\nas well as partial knowledge of the matrix $B$ (while ${\\cal O} (N)$\nof the elements remain unknown) will not be helpful for decomposing\n$G$ which will still require an exponentially long time to carry out.\n\nWe will consider here two attacks on specific plaintexts with partial\nknowledge of the corrupting vector $\\bz$ or of the matrix $B$. In the\nfirst case, knowing $p_a M$ of the $p M$ corrupting bits may allow one\nto subtract the approximated vector $\\hz$ from the ciphertext and take\nthe product of $G^{-1}$ and the resulting ciphertext. This attack is\nsimilar to the task facing an unauthorised user with a reduced\ncorruption rate of $(p\\!-\\! p_a)$. For any non-vanishing difference\nbetween $p_a$ and $p$, deciphering the transmitted message remains a\ndifficult task.\n\nA second attack is that whereby the matrix $B$ is known to some\ndegree; for instance, the location of a fraction of the unit elements,\nsay $1\\!-\\!\\rho$ is known. From Eq.(\\ref{eq:decoding}) one can\nidentify the absent information as having a higher effective\ncorruption level of $p \\!+\\! g(\\rho) $, where $g(\\cdot)$ is some\nnon-trivial function that depends on the actual scenario. To secure\nthe transmission one may work sufficiently close to the critical\ncorruption level $p_s$ such that the additional effective noise $\\rho$\nwill bring the system beyond the critical corruption rate and into the\nparamagnetic/spin-glass regime. However, the drawbacks of working {\\em\nvery close} to $p_s$ are twofold: Firstly, average decryption times\nusing BP methods ($\\tau$) will diverge proportionally to $1/(p_s \\!\n-\\!p)$ as demonstrated in the inset of Fig.2. Secondly, finite-size\neffects are expected to be larger close to $p_s$, which practically\nmeans that the system may not converge to the attractive optimal\nsolutions in some cases.\n \nWe will end this presentation with a short discussion on the\nadvantages and drawbacks of the suggested method in comparison with\nexisting techniques. Firstly, we would like to point out the\ndifferences between this method and the McEliece cryptosystem. The\nlatter is based on Goppa codes and is limited to relative low\ncorruption levels. These may allow for decrypting the ciphertext using\n(many) estimates of the corruption vector. Our suggestion allows for a\nsignificant corruption level, thus increasing the security of the\ncryptosystem. In addition, the suggested construction, $K\\!\\!=\\!\\!\nL\\!\\!=\\!\\! 2$, is not discussed in the information theory literature\n(e.g. in \\cite{MacKay}) which typically prefers higher $K,L$ value\nsystems. Secondly, in comparison to RSA where decryption takes ${\\cal\nO} (N^3)$ operations, our method only requires ${\\cal O} (N)$\noperations, multiplied by the number of BP iterations (which is\ntypically smaller than 100 for most system sizes examined except very\nclose $p_s$). Encryption costs are of ${\\cal O} (N^2)$ (as in RSA)\nwhile the inversion of the matrix $B$ is carried out only once and\nrequires $O(N^{3} )$ operations.\n\nThe two obvious drawbacks of our method are: 1) The transmission of\nthe public key, which is a dense matrix of dimensionality $M\\!\\times\\!\nN$. However, as public key transmission is carried out only once for\neach user we do not expect it to be of great significance. 2) The\nciphertext to plaintext bit ratio is greater than one to allow for\ncorruption, in contrast to RSA where it equals 1. Choosing the $N/M$\nratio is in the hands of the user and is directly related to the\nsecurity level required; we therefore do not expect it to be\nproblematic as the increased transmission time is compensated by a\nvery fast decryption.\n\nWe examine the typical performance of a new cryptosystem, based on\ninsight gained from our previous studies, by mapping it onto an Ising\nspin system; this complements the information theory approach which\nfocuses on rigorous worst-case bounds. We show that authorised\ndecryption is fast and simple while unauthorised decryption requires a\nprohibitively long time. Important aspects that are yet to be\ninvestigated include finite size effects and methods for alleviating\nthe drawbacks of the new method.\n\n\\vspace{-6mm}\n\n\\begin{thebibliography}{99}\n\n%\\vspace{-12mm}\n\n\\bibitem{McEliece} R.~McEliece, {\\em DSN Progress Report}, {\\bf 42-44}\n(JPL-Caltech, California), 114, 1978.\n%\n\\bibitem{Gallager} R.G.~Gallager, {\\em Low density parity check codes}\nResearch monograph series {\\bf 21} (MIT press), 1963 and\n{\\em IRE Trans.~Info.~Theory}, {\\bf IT-8} 21 (1962).\n%\n\\bibitem{MacKay} D.J.C.~MacKay, {\\em IEEE Trans. IT}, {\\bf 45}, 399\n(1999).\n%\n(1999).\n\\bibitem{Sourlas} N.~Sourlas, {\\em Nature}, {\\bf 339} 693 (1989).\n%\n\\bibitem{us_gallager} Y.~Kabashima, T.~Murayama and D.~Saad, {\\em Phys. Rev. Lett.} {\\bf 84}, 1355 (2000).\n%\n\\bibitem{us_sourlas} Y.~Kabashima and D.~Saad, {\\em Euro.Phys.Lett.},\n{\\bf 45} 97 (1999).\n%\n\\bibitem{NP} M.R.~Garey and D.S.~Johnson, {\\em Computers and\nIntractability} (Freeman), 251, 1979.\n%\n\\bibitem{Wong_Sherrington} K.Y.M.~Wong and D.~Sherrington, {\\em\nJ.Phys.A}, {\\bf 20}, L793 (1987).\n%\n\\bibitem{Shannon} C.E.~Shannon, {\\em Bell Sys.~Tech.~J.}, {\\bf 27} 379\n(1948); {\\bf 27} 623 (1948).\n%\n\\bibitem{REM} B.~Derrida, {\\em Phys.~Rev.~B}, {\\bf 24}, 2613 (1981).\n%\n\\bibitem{Nishimori} H.~Nishimori, {\\em Prog.Theo.Phys.}, {\\bf 66}, 1169\n(1981).\n%\n\\bibitem{Sourlas_EPL} N.~Sourlas, {\\em Euro.Phys.Lett.}, {\\bf 25}, 159\n(1994).\n%\n\\bibitem{kanter_saad} I.~Kanter and D.~Saad, {\\em Phys.~Rev.~Lett.}\n{\\bf 83}, 2660 (1999).\n%\n\\bibitem{Parisi} E.~Marinari, G.~Parisi and F.~Ritort, {\\em J.~Phys\nA}, {\\bf 27}, 7615 and 7647 (1994); E.~Marinari and F.~Ritort, {\\em\nJ.~Phys A}, {\\bf 27}, 7669 (1994).\n\n\\end{thebibliography}\n\n{\\footnotesize\n\\vspace*{0.1in} {\\bf \\hspace*{-1.5em} Acknowledgement} Support by\nJSPS-RFTF (YK), The Royal Society and EPSRC-GR/L52093 (DS) is\nacknowledged. We would like to thank Manfred Opper and\nHidetoshi Nishimori for reading the manuscript.}\n\n\\newpage\n\n% FIGURE 1\n% code rate for K=L=2\n\\begin{figure}[h]\n\\vspace*{-1.25cm}\n\\begin{center}\n\\begin{picture}(400,400)\n\\put(-40,200){\\epsfxsize=100mm \\epsfbox{Spinodal_K2.eps}}\n\\put(115,270){\\epsfxsize=40mm \\epsfbox{PT_K2L2.eps}}\n\\end{picture}\n\\end{center}\n\\vspace*{-7.3cm}\n\\caption{Critical transmission rate as a function of the corruption\nrate $p$ for an unbiased ciphertext. Numerical solutions (of the\nanalytically obtained equations - $\\Diamond$) and BP iterative\nsolutions (of system size $N\\!=\\! 10^4$, $+$), were averaged over 10\ndifferent initial conditions of almost zero magnetisation with error\nbars much smaller than the symbol size. Inset: The ferromagnetic\n({\\sf F}) (optimal/sub-optimal) and paramagnetic ({\\sf P}) solutions\nas functions of $p$; thick and thin lines denote stable solutions of\nlower and higher free energies respectively, dashed lines correspond\nto unstable solutions.}\n%\\end{center}\n\\end{figure}\n\n\n%\\newpage\n\n% FIGURE 2\n% REM \n\\begin{figure}[h]\n\\vspace*{-2.35cm}\n\\begin{center}\n\\begin{picture}(400,400)\n%\\put(-20,195){\\epsfxsize=95mm \\epsfbox{Free_Energy_new.ps}}\n\\put(-20,200){\\epsfxsize=95mm \\epsfbox{Free_Energy_new.ps}}\n%\\put(25,280){\\epsfxsize=40mm \\epsfbox{convtimes.eps}}\n%\\put(30,275){\\epsfxsize=55mm \\epsfbox{convtimes.eps}}\n\\end{picture}\n\\end{center}\n\\vspace*{-7.6cm}\n\\caption{The free energy landscape as a function of $m$ for the\ntransmission rate $N/M=1/2$ and flip rate $p=0.08$; theoretical values\nare represented by the solid line, numerical data, obtained by BP\n($N=200$) and averaged over 10 different initial conditions, are\nrepresented by symbols ($+$). The landscape is deep and narrow (of\nwidth ${\\cal O} (1/N)$) at $m=1$ and rather flat elsewhere. Inset -\nscattered plot of mean decryption times - $\\tau$. The optimal fitting\nof straight lines through the data provides another indication for the\ndivergence of decryption time for corruption rate close to\n$p_s=0.953\\pm5$ (in this example).}\n%\\end{center}\n\\end{figure}\n\n\\end{document}\n\n\n"
}
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[
{
"name": "cond-mat0002129.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n%\\vspace{-12mm}\n\n\\bibitem{McEliece} R.~McEliece, {\\em DSN Progress Report}, {\\bf 42-44}\n(JPL-Caltech, California), 114, 1978.\n%\n\\bibitem{Gallager} R.G.~Gallager, {\\em Low density parity check codes}\nResearch monograph series {\\bf 21} (MIT press), 1963 and\n{\\em IRE Trans.~Info.~Theory}, {\\bf IT-8} 21 (1962).\n%\n\\bibitem{MacKay} D.J.C.~MacKay, {\\em IEEE Trans. IT}, {\\bf 45}, 399\n(1999).\n%\n(1999).\n\\bibitem{Sourlas} N.~Sourlas, {\\em Nature}, {\\bf 339} 693 (1989).\n%\n\\bibitem{us_gallager} Y.~Kabashima, T.~Murayama and D.~Saad, {\\em Phys. Rev. Lett.} {\\bf 84}, 1355 (2000).\n%\n\\bibitem{us_sourlas} Y.~Kabashima and D.~Saad, {\\em Euro.Phys.Lett.},\n{\\bf 45} 97 (1999).\n%\n\\bibitem{NP} M.R.~Garey and D.S.~Johnson, {\\em Computers and\nIntractability} (Freeman), 251, 1979.\n%\n\\bibitem{Wong_Sherrington} K.Y.M.~Wong and D.~Sherrington, {\\em\nJ.Phys.A}, {\\bf 20}, L793 (1987).\n%\n\\bibitem{Shannon} C.E.~Shannon, {\\em Bell Sys.~Tech.~J.}, {\\bf 27} 379\n(1948); {\\bf 27} 623 (1948).\n%\n\\bibitem{REM} B.~Derrida, {\\em Phys.~Rev.~B}, {\\bf 24}, 2613 (1981).\n%\n\\bibitem{Nishimori} H.~Nishimori, {\\em Prog.Theo.Phys.}, {\\bf 66}, 1169\n(1981).\n%\n\\bibitem{Sourlas_EPL} N.~Sourlas, {\\em Euro.Phys.Lett.}, {\\bf 25}, 159\n(1994).\n%\n\\bibitem{kanter_saad} I.~Kanter and D.~Saad, {\\em Phys.~Rev.~Lett.}\n{\\bf 83}, 2660 (1999).\n%\n\\bibitem{Parisi} E.~Marinari, G.~Parisi and F.~Ritort, {\\em J.~Phys\nA}, {\\bf 27}, 7615 and 7647 (1994); E.~Marinari and F.~Ritort, {\\em\nJ.~Phys A}, {\\bf 27}, 7669 (1994).\n\n\\end{thebibliography}"
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cond-mat0002130
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The influence of critical behavior on the spin glass phase
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"author": "Hemant Bokil"
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We have argued in recent papers that the Monte Carlo results for the equilibrium properties of the Edwards-Anderson spin glass in three dimensions, which had been interpreted earlier as providing evidence for replica symmetry breaking, can be explained quite simply within the droplet model once finite size effects and proximity to the critical point are taken into account. In this paper we show that similar considerations are sufficient to explain the Monte Carlo data in four dimensions. In particular, we study the Parisi overlap and the link overlap for the four-dimensional Ising spin glass in the Migdal-Kadanoff approximation. Similar to what is seen in three dimensions, we find that temperatures well below those studied in the Monte Carlo simulations have to be reached before the droplet model predictions become apparent. We also show that the double-peak structure of the link overlap distribution function is related to the difference between domain-wall excitations that cross the entire system and droplet excitations that are confined to a smaller region. \pacs{PACS numbers: 75.50.Lk Spin glasses}
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"name": "4d_5.tex",
"string": "\\documentstyle[multicol,prl,aps,epsf]{revtex}\n\\begin{document}\n\n\\title{The influence of critical behavior on the spin glass \nphase}\n\n\\author{Hemant Bokil}\n\\address{Abdus Salam ICTP, Strada Costiera 11, 34100 Trieste, \nItaly}\n\\author{Barbara Drossel}\n\\address{School of Physics and Astronomy, Raymond and \nBeverley Sackler Faculty of Exact Sciences, Tel Aviv \nUniversity, Tel Aviv 69978, Israel}\n\\author{M. A. Moore}\n\\address{Department of Physics, University of Manchester, \nManchester M13 9PL, U.K.}\n\\date{\\today} \n\\maketitle\n\n\\begin{abstract}\n We have argued in recent papers that the Monte Carlo \nresults for the\n equilibrium properties of the Edwards-Anderson spin glass \nin three\n dimensions, which had been interpreted earlier as providing \nevidence\n for replica symmetry breaking, can be explained quite simply within \nthe\n droplet model once finite size effects and proximity to the \ncritical\n point are taken into account. In this paper we show that \nsimilar\n considerations are sufficient to explain the Monte Carlo \ndata in\n four dimensions. In particular, we study the Parisi overlap \nand the\n link overlap for the four-dimensional Ising spin glass in \nthe\n Migdal-Kadanoff approximation. Similar to what is seen in \nthree\n dimensions, we find that temperatures well below those \nstudied in\n the Monte Carlo simulations have to be reached before the \ndroplet model\n predictions become apparent. We also show that the double-peak\n structure of the link overlap distribution function is \nrelated to\n the difference between domain-wall excitations that cross \nthe entire\n system and droplet excitations that are confined to a \nsmaller\n region.\n\n \\pacs{PACS numbers:\n75.50.Lk Spin glasses}\n\\end{abstract}\n\\begin{multicols}{2}\n \n\\section{Introduction}\n\n % I rewrote this paragraph\n\n Despite over two decades of work, the controversy \nconcerning the\n nature of the ordered phase of short range Ising spin \nglasses\n continues. For a few years, Monte Carlo simulations \nappeared to be\n providing evidence for replica symmetry breaking (RSB) in \nthese\n systems\\cite{rby,marinari}. However, recent developments \nhave\n cast doubt on this interpretation of the Monte Carlo data.\n In a series of papers on the Ising spin glass within the\n Migdal-Kadanoff approximation (MKA), we showed that the \nequilibrium\n Monte Carlo data in three dimensions that had been \ninterpreted in\n the past as giving evidence for RSB can actually be \ninterpreted\n quite easily within the droplet picture, with apparent RSB \neffects\n being attributed to a crossover between critical behavior \nand the\n asymptotic droplet-like behavior for small system\n sizes\\cite{us1,us2,us3,us4}. We also showed that system \nsizes\n well beyond the reach of current simulations would probably \nbe \n required in order to unambiguously see droplet-like \nbehavior.\n The finding that the critical-point\n effects can still be felt at temperatures lower than those\n accessible by Monte Carlo simulations is supported by the \nMonte\n Carlo simulations of Berg and Janke\\cite{berg} who found \ncritical\n scaling working reasonably well down to $T=0.8T_c$ for \nsystem sizes\n upto $L=8$ in three dimensions. The zero temperature study \n of Pallasini and Young\\cite{py} also suggests that the \nground-\n state structure of three-dimensional Edwards-Anderson model \n is well described by droplet theory, though the existence \nof low \n energy excitations not included in\n the conventional droplet theory remains an open question. \nThus,\n while puzzles do remain, the weight of the evidence seems \nto be\n shifting towards a droplet-like description of the ordered \nphase in\n short range Ising spin glasses.\n \n However, it is expected that critical point effects are \nless\n dominant in four dimensions than in three dimensions. Our \naim in\n this paper is to quantify the extent of critical point \neffects in\n the low temperature phase of the four-dimensional \nEdwards-Anderson\n spin glass. We do this by providing results for the \nfour-dimensional\n Ising spin glass in the MKA and compare these with existing \nMonte\n Carlo work. In particular, we study the Parisi overlap \nfunction and\n the link overlap function for system sizes up to $L=16$ and\n temperatures as low as $T=0.16T_c$. We find that for \nsystem sizes\n and temperatures comparable to those of the Monte Carlo \nsimulations,\n the Parisi overlap distribution shows also in MKA the\n sample-to-sample fluctations and the stationary behavior at \nsmall\n overlap values, that are normally attributed to RSB. It is \nonly for\n larger system sizes (or for lower temperatures), that the \nasymptotic\n droplet-like behavior becomes apparent. For the link \noverlap, we\n find similar double-peaked curves as those found in \nMonte-Carlo\n simulations. This double peak structure is expected on \nquite general\n grounds independent of the nature of the low temperature \nphase.\n However, we show that two peaks in the link overlap in MKA \noccur\n because of a difference between domain-wall excitations \n(which cross\n the entire system) and droplet excitations (which do not \ncross the\n entire system). We argue that for small system sizes, the \neffect of\n domain walls increases with increasing dimension, making it\n necessary to go very far below $T_c$ to see the asymptotic \ndroplet\n behavior.\n \n This paper is organized as follows: in section \n\\ref{definitions}, we\n define the quantities discussed in this paper, and the \ndroplet-model\n predictions for their behavior. In section \\ref{MKA}, we \ndescribe\n the MKA, and our numerical methods of evaluating the \noverlap\n distribution. In section \\ref{Parisi}, we present our \nnumerical\n results for the Parisi overlap distribution, and compare to\n Monte-Carlo data. The following section studies the link \noverlap\n distribution. Finally, section \\ref{conclusions} contains \nthe\n concluding remarks, including some on the effects of \ncritical behavior on the\n dynamics in the spin glass phase. Again we suspect that \narguments which have\nbeen advanced against the droplet picture on the basis of \ndynamical studies\nhave failed to take into account the effects arising from \nproximity to the\ncritical point.\n\n\\section{Definitions and Scaling Laws}\n\\label{definitions}\n\nThe Edwards-Anderson spin glass in the absence of an external \nmagnetic field is\ndefined by\nthe Hamiltonian\n$$H=-\\sum_{\\langle i,j\\rangle} J_{ij} S_iS_j,$$ where the \nIsing spins can\ntake the values $\\pm 1$, and the nearest-neighbor couplings \n$J_{ij}$\nare independent from each other and Gaussian distributed with \na\nstandard deviation $J$. \n\nIt has proven useful to consider two identical copies \n(replicas) of\nthe system, and to measure overlaps between them. This gives\ninformation about the structure of the low-temperature phase, \nin\nparticular about the number of pure states. The quantities \nconsidered in this \npaper are the Parisi overlap function $P(q,L)$ and the\nlink overlap function $P(q_l,L)$. They are defined by\n\\begin{equation} \nP(q, L) = \\left[\\left<\\delta \\left(\\sum_{\\langle i j\\rangle} \n\\frac{{S_i^{(1)}S_i^{(2)} + S_j^{(1)}S_j^{(2)}}} \n{2N_L} - q\n\\right)\\right>\\right],\n\\label{p}\n\\end{equation}\nand \n\\begin{equation} \nP(q_l, L)= \\left[\\left<\\delta\\left(\\sum_{\\langle i j\\rangle} \n\\frac{S_i^{(1)}S_i^{(2)}S_j^{(1)}S_j^{(2)}}\n{N_L}-q_l\n\\right)\\right>\\right].\n\\label{pl}\n\\end{equation}\nHere, the superscripts $(1)$ and $(2)$ denote the two \nreplicas of the\nsystem, $N_L$ is the number of bonds, and $\\langle \n...\\rangle$ and\n$\\left[...\\right]$ denote the thermodynamic and disorder \naverage\nrespectively. We use $P(q, L)$ and $P(q_l,L)$ to denote the \noverlap functions\nfor a finite system of size $L$, reserving the more standard \nnotation\n$P(q)$ and $P(q_l)$ for the limit $\\lim_{L\\to \\infty} P(q, \nL)$\nand $\\lim_{L\\to \\infty} P(q_l, L)$.\n\nIn the mean-field RSB picture, $P(q)$ is nonzero in the spin \nglass\nphase in the entire interval $[-q_{EA},q_{EA}]$, while it is \ncomposed\nonly of two delta functions at $\\pm q_{EA}$ in the droplet \npicture.\nSimilarly, $P(q_l)$ is nonzero over a finite interval \n$[q_l^{min},\nq_l^{max}]$ in mean-field theory, while it is a \ndelta-function within\nthe droplet picture.\n\nMuch of the evidence for RSB for three- and four-dimensional \nsystems\ncomes from observing a stationary $P(q=0,L)$ for system sizes \nthat are\ngenerally smaller than 20 in 3D and smaller than 10 in 4D, \nand at\ntemperatures of the order of $0.7 T_c$. However, even within \nthe\ndroplet picture one expects to see a stationary $P(q=0, L)$ \nfor a\ncertain range of system sizes and temperatures. The reason \nis that at\n$T_c$ the overlap distribution $P(q,L)$ obeys the scaling law\n\\begin{equation}\nP(q,L)=L^{\\beta/\\nu} \\tilde P(q L^{\\beta/\\nu}),\n\\label{scaling}\n\\end{equation}\n$\\beta$ being the order parameter critical exponent, and \n$\\nu$ the\ncorrelation length exponent. Above the lower critical \ndimension (which\nis smaller than 3), $\\beta/\\nu$ is positive, leading to an \nincrease\n$P(q=0,L)$ as a function of $L$ (at $T=T_c$). On the other \nhand, for\n$T\\ll T_c$, the droplet model predicts a decay $$P(q=0,L) \n\\sim\n1/L^\\theta$$\non length scales larger than the (temperature--dependent)\ncorrelation length $\\xi$, $\\theta$ being the scaling exponent \nof the\ncoupling strength $J$. A few words are in order here on what \nwe mean\nby the correlation length.\nIn the spin glass phase, all correlation functions fall off\nas a power law at large distances. However, within the \ndroplet\nmodel, this is true only asymptotically, and the general\nform of the correlation function for\ntwo spins a distance $r$ apart, at a temperature $T\\le T_c$, \nis \n$\\sim r^{-\\theta} f(r/\\xi)$\nwhere $k_B$ is the Boltzmann constant and $f$ is a scaling \nfunction.\nThus, for $r\\le \\xi$ there are\ncorrections to the algebraic long-distance behavior and \nthe above expression defines\nthe temperature-dependent correlation length.\nNote that for $T\\to T_c$ this correlation length is \nexpected to diverge with the exponent $\\nu$.\n\nThus, for temperatures not too far below $T_c$, one can \nexpect an\nalmost stationary $P(q=0, L)$ for a certain range of system \nsizes. In\nthree dimensions both $\\beta/\\nu\\simeq0.3$\\cite{berg} and\n$\\theta\\simeq0.17$\\cite{bm} are rather small, this apparent\nstationarity may persist over a considerable range of system \nsizes\n$L$. However, in four dimensions, $\\beta/\\nu\\simeq 0.85$ \n\\cite{mar99}\nand $\\theta\\simeq 0.65$ \\cite{hartmann} and one would expect \nthe\ncrossover region to be smaller. In the present paper we shall\ninvestigate these crossover effects in four dimensions by \nstudying\n$P(q,L)$ for the Edwards-Anderson spin glass within the MKA. \nIt turns\nout that they are surprisingly persistent even at low \ntemperatures,\ndue to the presence of domain walls.\n\nMonte-Carlo simulations of the link overlap distribution show \na\nnontrivial shape with shoulders or even a double peak, which \nseems to\nbe incompatible with the droplet picture, where the \ndistribution\nshould tend towards a delta-function. For sufficiently low\ntemperatures and large length scales, the droplet picture \npredicts\nthat the width of the link overlap distribution scales as \n\\cite{us4}\n$$\\Delta q_l \\sim \\sqrt{kT} L^{d_s-d-\\theta/2}\\,,$$\nwhere $d_s$ is the\nfractal dimension of a domain wall. Below, we will show that \nthe\nnontrivial shape and the double peak reported from \nMonte-Carlo\nsimulations are also found in MKA in four dimensions, and we \nwill\npresent strong evidence that it is due to the different \nnature of\ndroplet and domain wall excitations. As the weight of domain \nwalls\nbecomes negligible in the thermodynamic limit, the droplet \npicture is\nregained on large scales.\n\n\n\n\\section{Migdal-Kadanoff approximation}\n\\label{MKA}\n\nThe Migdal-Kadanoff approximation (MKA) is a real-space \nrenormalization group the gives approximate recursion \nrelations for the various coupling constants. Evaluating a \nthermodynamic quantity in MKA\nin $d$ dimensions is equivalent to evaluating it on an\nhierarchical lattice that is constructed iteratively by \nreplacing each\nbond by $2^d$ bonds, as indicated in Fig.~\\ref{fig1}. The \ntotal number of bonds\nafter $I$ iterations is $2^{dI}$. $I=1$, the smallest \nnon-trivial system\nthat can be studied, corresponds to a system linear\ndimension $L=2$, $I=2$ corresponds to $L=4$, $I=3$ \ncorresponds to $L=8$\nand so on. Note that the number of bonds on hierarchical \nlattice after\n$I$ iterations is the same as the number of \n sites of a $d$-dimensional lattice of size $L=2^I$.\nThermodynamic quantities are then evaluated iteratively by \ntracing\nover the spins on the highest level of the hierarchy, until \nthe\nlowest level is reached and the trace over the remaining two \nspins is\ncalculated \\cite{southern77}. This procedure generates new\neffective couplings, which have to be included in the \nrecursion\nrelations.\n\\begin{figure}\n\\centerline{\n\\epsfysize=0.15\\columnwidth{\\epsfbox{4dfigures/fig1_4d.eps}}}\n\\narrowtext{\\caption{Construction of a hierarchical \nlattice.}\\label{fig1}}\n%\\caption{Construction of a hierarchical lattice.\\label{fig1} \n\\end{figure}\n\nIn\n\\cite{gardner84}, it was proved that in the limit of \ninfinitely many\ndimensions (and in an expansion away from infinite \ndimensions) the MKA\nreproduces the results of the droplet picture.\n\nAs was discussed in \\cite{us1}, the calculation of $P(q,L)$ \nis\nmade easier by first calculating its \nFourier transform $F(y,L)$, which is given by\n\\begin{equation}\nF(y,L)=\\left[\\left< \\exp[iy\\sum_{\\langle i j\\rangle}\n{(S_i^{(1)}S_i^{(2)}+S_j^{(1)}S_j^{(2)})\\over {2N_L}}] \n\\right>\\right] .\n\\end{equation}\nThe recursion relations for $F(y,L)$ involve two-\nand four-spin terms, and can easily be evaluated numerically \nbecause all\nterms are now in an exponential. Having calculated $F(y)$ one \ncan\nthen invert the Fourier transform to get $P(q,L)$. \n\nSimilarly, $P(q_l,L)$ is calculated by first evaluating \n\\begin{equation}\nF(y_l,L)=\\left[\\left< \\exp[iy_l\\sum_{\\langle i j\\rangle}\n{(S_i^{(1)}S_i^{(2)}S_j^{(1)}S_j^{(2)})\\over {N_L}}] \n\\right>\\right] .\n\\end{equation}\n\nBefore presenting our numerical results for the Parisi \noverlap and the\nlink overlap, let us discuss the flow of the coupling \nconstant $J$ in\nthe low-temperature phase, as obtained in MKA. In order to \nobtain this\nflow, we iterated the MKA recursion relation on a set of \n$10^6$ bonds.\nAt each iteration, each of the new set of $10^6$ bonds was \ngenerated\nby randomly choosing 16 bonds from the old set and taking the \ntrace\nover the inner spins (with a bond arrangement as in \nFig.~\\ref{fig1}).\nFigure \\ref{flow} shows $J/T$ as function of $L$ for \ndifferent initial\nvalues of the coupling strength. The critical point is at \n$T_c \\simeq\n2.1 J$. The first curve begins at $J/T=0.5$, which is close \nto the\ncritical point, and it reaches the low-temperature behavior \nonly at\nlengths around 1000. For an initial $J/T=0.7$, the \nasymptotic slope\nis already reached at $L$ around 40, and for $J/T=3.0$, which\ncorresponds to $T\\simeq 0.16T_c$ the entire curve shows the \nasymptotic\nslope. The asymptotic slope is identical to the \nabove-mentioned\nexponent $\\theta$ and has the value $\\theta \\simeq 0.75$. In \ncontrast to\n$d=3$ \\cite{us4}, we did not succeed in fitting the crossover \nregime\nby doing an expansion around the zero-temperature fixed \npoint. The\nreason is that dimension 4 is too far above the lower \ncritical\ndimension, such that the critical temperature is not small. \n\nNote that for each temperature the length scale beyond which \nthe flows\nof the coupling constants show the asymptotic behavior yields \none\nestimate for the correlation length mentioned above. We have\nconsidered the flow to be in the asymptotic regime when its \nslope was\nwithin 90\\% of its asymptotic value. However, this estimate \nis\nspecific to the flows of the coupling constant, and other\nquantities may show their asymptotic behavior later.\nIn fact, as we shall see below, the convergence of the\noverlap distributions is much slower than that of the \ncouplings, and\nwe will have to give reasons for this.\n\\begin{figure}\n \\centerline{\n \\epsfysize=0.7\\columnwidth{\\epsfbox{4dfigures/flow.eps}}}\n \\narrowtext{\\caption{Flow of the coupling strength $J$ in \nMKA. The\n curves correspond to $T/T_c=$0.96, 0.8, 0.68, 0.6, \n0.48, 0.33,\n 0.16 (from bottom to top). The correlation lengths, \nwhere the\n slope has reached $90\\%$ of the asymptotic slope, are \n960, 47,\n 24, 15, 8, 3, 1.}\\label{flow} }\n\\end{figure}\n\n\\section{The Parisi overlap}\n\\label{Parisi}\n\nWe now discuss our results for the Parisi overlap. First, \nlet us\nbriefly describe the critical behavior. Fig.~\\ref{fig2} shows \na\nscaling plot for $P(q,L)$ for $L=4,8,16$ at $T=T_c\\simeq \n2.1J$. We\nfind a good data collapse if we use the value \n$\\beta/\\nu=0.64$, thus\nconfirming the finite-size scaling ansatz Eq.~\\ref{scaling}.\n\\begin{figure}\n\\centerline{\n\\epsfysize=0.7\\columnwidth{\\epsfbox{4dfigures/pqcrit.eps}}}\n\\narrowtext{\\caption{Scaling collapse of $P(q,L)$ at $T=T_c$, \nwith \n$\\beta/\\nu\\simeq 0.64$. As $P(q,L)=P(-q,L)$, only the part \n$q\\ge 0$ is shown.\nFor each system size, we averaged over at least 5000 \nsamples.}\\label{fig2} \n}\n\\end{figure}\n\nWe next move on to the low-temperature phase. \nIn Fig.~\\ref{samples} we show $P(q,L)$ at $T=0.5T_c$\nand $L=8$ for three different samples. \n\\begin{figure}\n\\centerline{\n\\epsfysize=0.7\\columnwidth{\\epsfbox{4dfigures/pq_single_3_0.8.eps}}}\n\\narrowtext{\\caption{$P(q,L)$ for three different samples at \n$T=0.5T_c$\nand $L=8$.}\\label{samples} \n}\n\\end{figure}\nAs one can see there are substantial differences between the \nsamples.\nThis sensitivity to samples for system sizes around 10 is in\n\\cite{mar99} interpreted as evidence for RSB. In our case, \nwhere we\nknow that the droplet model is exact, it has to be considered \na finite\nsize effect. Note that we have not chosen the three samples \nin any\nparticular manner. By comparing to the curves obtained for \n$L=16$ (not\nshown), we can even see the trend to an increasing number of \npeaks,\njust as in \\cite{mar99}. Thus, one feature commonly \nassociated with\nRSB is certainly present in within the MKA for temperatures \nand system\nsizes comparable to those studied in simulations.\n\nLet us now focus on the behavior of $P(q=0,L)$ for\ndifferent system sizes and temperatures. But before \nexhibiting\nour own data, we discuss \nthe Monte Carlo data of Reger, Bhatt and Young\\cite{rby}\nwho were the first to study $P(q=0,L)$ for the \nEdwards-Anderson spin glass.\nThey studied system sizes $L=2,3,4,5,6$ at temperatures\ndown to $T=0.68T_c$. At $T=T_c$ they found the expected \ncritical\nscaling, $P(q=0)\\simeq L^{\\beta/\\nu}$ \nwith $\\beta/\\nu\\simeq 0.75$.\nThen, as the temperature was lowered, the curves\nfor $P(q=0)$ as a function of $L$ showed a downward curvature \nfor the\nlargest system sizes,\nwhich they interpreted as the beginning of the crossover \nbetween\ncritical behavior and the low temperature behavior. At \n$T=0.8T_c$,\n$P(q=0)$ seemed to be roughly constant or decreasing slowly. \nHowever,\nthe striking part of their data was that at $T=0.68T_c$ they\nfound that $P(q=0,L)$ initially decreased as a function of \nsystem\nsize for $L=2,3,4$ and then saturated for $L=4,5,6$. They\ninterpreted this as suggestive of RSB. They admitted however\nthat other explanations are possible. \n\nThe most recent Monte-Carlo simulation data for the 4d Ising \nspin\nglass are those in \\cite{mar99}. These authors focus on \n$T\\simeq\n0.6T_c$, and they find an essentially stationary $P(q=0,L)$ \nfor system\nsizes up to the largest simulated size $L=10$. They argue, \nthat\nstationarity over such a large range of $L$ values is most \nnaturally\ninterpreted as evidence for RSB. However, as can be seen from\nFig.~\\ref{fig2}, the correlation length is of the order of 16 \nfor\nthese temperatures and therefore comparable to the system \nsize.\n\nIn Fig.\\ref{fig4}, we show the MKA data for $P(q=0,L)$. We \nhave\ncalculated $P(q=0,L)$ for system sizes $L=4$,$8$,$16$ at \ntemperatures\n$T=T_c$, $0.68T_c$, $0.33T_c$, and $0.16T_c$. At $T=T_c$, \n$P(q=0,L)$\ngrows as $L^{\\beta/\\nu}$ with $\\beta/\\nu\\simeq 0.64$, in \nagreement\nwith Fig.\\ref{fig2}. At $T=0.68T_c$ (the lowest temperature \nstudied\nin \\cite{rby}, and not far from the lowest temperature \nstudied by\n\\cite{mar99}), we do not see a clear decrease even for \n$L=16$. The\ncurve for $P(q=0)$ looks more or less flat, though one could \nsay that\nthere is slight increase between $L=4$ and $L=8$ and a slight \ndecrease\nbetween $L=8$ and $L=16$. This flat behavior is similar to \nwhat was\nfound in \\cite{rby} and \\cite{mar99}. The deviation of the \n$L=2$ and\n$L=3$ data from the flat curve in \\cite{rby} can probably be \nascribed\nto artifacts at very small system sizes, which are also found\nelsewhere \\cite{bhatt88}. For lower temperatures, where the\ncorrelation length is smaller than the system size, there is \na clear\ndecrease of $P(q=0)$ although the decrease is not asympotic \neven at a\ntemperatures as low as $T_c/6$.\n\\begin{figure}\n\\centerline{\\epsfysize=0.7\\columnwidth{\\epsfbox{4dfigures/p0_fig.eps}}}\n\\narrowtext{\\caption{$P(q=0,L)$ at $T=T_c$, $0.68T_c$, \n$0.5T_c$, $0.33T_c$, $0.16 T_c$ for\nL=4,8,16. The error bars indicate the standard deviation of \nthe values. All data were obtained by averaging at least over \n5000 samples.}\\label{fig4} \n} \n\\end{figure}\nWe conclude that the observed stationarity of $P(q=0,L)$ in\nMonte-Carlo data is due to the effects of a finite system \nsize and\nfinite temperature. \nSimilarly, Monte-Carlo simulations at $T\\simeq 0.5\nT_c$ and at system sizes around 10, should be able to show \nthe\nnegative slope in $P(q=0,L)$. In the not too far future, it \nshould\nbecome possible to perform these simulations.\n\nThe fact that $P(q=0,L)$ does not show asymptotic behavior \neven at\n$T=T_c/6$ for the system sizes that we have studied, is \nsurprising,\nand is different from our findings in $d=3$ \\cite{us1}. That\n$P(q=0,L)$ converges slower towards the asymptotic behavior \nthan the\nflow of the coupling constant (see Fig.~\\ref{flow}), can be \nunderstood\nin the following way: A Parisi overlap value close to zero \ncan be\ngenerated by a domain wall excitation. For large system sizes \nand low\ntemperatures, such an excitation occurs with significant \nweight only\nin those samples where a domain wall excitation costs little \nenergy.\nThese are exactly the samples with a small renormalized \ncoupling\nconstant at system size $L$. As the width of the probability\ndistribution function of the couplings increases with \n$L^\\theta$, the\nprobability for obtaining a small renormalized coupling \ndecreases as\n$L^{-\\theta}$. This is the argument that predicts that \n$P(q=0,L) \\sim\nL^{-\\theta}$. However, for smaller system sizes and higher\ntemperatures, there are corrections to this argument. Thus, \neven\nsamples with a renormalized coupling that is not small can \ncontribute\nto $P(q=0,L)$ by means of large or multiple droplet \nexcitations, or of\nthermally activated domain walls. For this reason, $P(q=0,L)$ \ncan be\nexpected to converge towards asymptopia slower that the \ncoupling\nconstant itself. Furthermore, as we shall see in the next \nsection, the\nsuperposition of domain wall excitations and droplet \nexcitations leads\nto deviations from simple scaling, which may further slow \ndown the\nconvergence towards asymptotic scaling behavior.\n\n\\section{The Link Overlap}\n\nThe link overlap gives additional information about the spin \nglass phase that is\nnot readily seen in the Parisi overlap. The main qualitative\ndifferences between the Parisi overlap and the link overlap \nare (i)\nthat flipping all spins in one of the two replicas changes \nthe sign of\n$q$ but leaves $q_l$ invariant, and (ii) that flipping a \ndroplet of\nfinite size in one of the two replicas changes $q$ by an \namount\nproportional to the volume of the droplet, and $q_l$ by an \namount\nproportional to the surface of the droplet. Thus, the link \noverlap\ncontains information about the surface area of excitations. \n\nFirst, let us study $P(q_l,L)$ as function of temperature, \nfor a given\nsystem size $L=4$. Fig.~\\ref{pql2} shows our curves for \n$T=0.8T_c$,\n$0.67T_c$, $0.56T_c$, $0.48T_c$, and $0.33T_c$. They appear \nto result\nfrom the superposition of two different peaks, with their \ndistance\nincreasing with decreasing temperature, and the weight \nshifting from\nthe left peak to the right peak. \n\\begin{figure}\n\\centerline{\\epsfysize=0.7\\columnwidth{\\epsfbox{4dfigures/pql_2_all.eps}}}\n\\narrowtext{\\caption{$P(q_l,L)$ for $T=0.8T_c$, $0.67T_c$,\n $0.56T_c$, $0.48T_c$, $0.33T_c$ (from left to right), \nwith the\n system size $L=4$. }\\label{pql2} \n}\n\\end{figure}\nFig.~\\ref{pql_1.414} shows $P(q_l,L)$ for fixed $T=0.33T_c$ \nand for\ndifferent $L$. One can see that with increasing system size \nthe peaks\nmove closer together, and the weight of the left-hand peak \ndecreases.\n\\begin{figure}\n \n\\centerline{\\epsfysize=0.7\\columnwidth{\\epsfbox{4dfigures/pql_1.414.eps}}}\n \\narrowtext{\\caption{$P(q_l,L)$ for $L=2$, 4, 8 (from \nwidest to\n narrowest curve) and with $T=0.33T_c$. \n}\\label{pql_1.414} \n}\n\\end{figure}\n\nThese results are similar to what we found in MKA in three \ndimensions\n\\cite{us4}, however, in four dimensions the peaks are more \npronounced.\nMonte-Carlo simulations of the four-dimensional Ising spin \nglass also show two\npeaks for certain system sizes and temperatures \\cite{cir93}. \nThis feature is\nattributed by the authors to RSB. However, as it is also \npresent in MKA, there\nmust be a different explanation. The width of the curves \nshrinks with\nincreasing system size in \\cite{cir93}, just as it does in \nMKA and as is\nexpected from the droplet picture. If the RSB scenario were \ncorrect, the width\nwould go to a finite value in the limit $L\\to \\infty$.\n\nIn the following we present evidence that the left peak \ncorresponds to\nconfigurations where one of the two replicas has a domain \nwall\nexcitation, and the right peak to configurations where one of \nthe two\nreplicas has a droplet excitation. In MKA, domain wall \nexcitations\ninvolve flipping of one side of the system, including one of \nthe two\nboundary spins of the hierarchical lattice, while droplet \nexcitations\ninvolve flipping of a group of spins in the interior. If the \nsign of\nthe renormalized coupling is positive (negative), the two \nboundary\nspins are parallel (antiparallel) in the ground state. By \nplotting\nseparately the contributions from configurations with and \nwithout\nflipped boundary spins, we can separate domain wall \nexcitations from\ndroplet excitations. Fig.~\\ref{pql_bc} shows the three \ncontributions\nfrom configurations where none, one, or both replicas have a \ndomain\nwall. Clearly, the left peak is due to domain wall \nexcitations, and the right\npeak to droplet excitations.\n\\begin{figure}\n\\centerline{\\epsfysize=0.7\\columnwidth{\\epsfbox{4dfigures/pql_bc_2_1.414.eps}}}\n\\narrowtext{\\caption{Contribution of domain wall excitations \n(left curve) and\ndroplet excitations (right curve) to $P(q_l,L)$, for $L=4$ \nand $T=0.33T_c$. The\nthird, flat curve is due to configurations where both \nreplicas have a domain\nwall. }\\label{pql_bc} }\n\\end{figure}\nSimilar curves are obtained for other values of the \nparameters. We\nthus have shown that the qualitative differences between \ndroplet and\ndomain wall excitations are sufficient to explain the \nstructure of the\nlink overlap distribution, and no other low-lying excitations \nlike\nthose invoked by RSB are needed. \n\nThe weight with which domain-wall excitations occur is in agreement\nwith predictions from the droplet model.\nThe probability of having a domain wall in a system of size $L$ is according to the droplet picture of the order of \n$$\n(T/J)L^{-\\theta},$$\nwhich is $\\simeq 0.25$ at $T=0.33 T_c$ and\n$L=4$, and $\\simeq 0.15$ at $T=0.33T_c$ and $L=8$. From our\nsimulations, we find that the relative weights of domain walls for\nthese two situations are $\\simeq 0.12$ and $\\simeq 0.076$, which fits\nthe droplet picture very well if we include a factor 1/2 in the above\nexpression. Domain walls become negligible only when the product\n$(T/J)L^{-\\theta}$ becomes small. In higher dimensions, the critical\nvalue of $T/J$ becomes larger, and for a given relative distance from\nthe critical point, the weight of domain walls therefore also becomes\nlarger. This explains why the effect of domain walls is more visible\nin 4 dimensions than in 3 dimensions. However, with increasing system\nsize, domain walls should become negligible more rapidly in higher\ndimensions, due to the larger value of the exponent $\\theta$.\n\n\n\n\\section{Conclusions}\n\\label{conclusions}\n\nOur results for the Parisi overlap distribution in four \ndimensions\nshow that there are rather large finite size effects in four\ndimensions which give rise to phenomena normally attributed \nto RSB.\nThe system sizes needed to see the beginning of droplet like \nbehavior\nwithin the MKA are larger, and the temperatures are lower, \nthan those\nstudied by Monte Carlo simulations. However, at temperatures \nnot too\nfar below those studied in Monte Carlo simulations \n($T=0.5T_c$), the\nweight of the Parisi overlap distribution function $P(q=0,L)$ \nwithin\nthe MKA appears to decrease, albeit with an effective \nexponent\ndifferent from the asymptotic value. Thus, simulations at \nthese\ntemperatures for the Ising spin glass on a cubic lattice \nmight resolve\nthe controversy regarding the nature of the ordered state in \nshort\nrange spin glasses. However, the MKA is a low dimensional\napproximation and it is possible that the system sizes needed \nto see\nasymptotic behavior for a hypercubic lattice in four \ndimensions are\ndifferent from what is indicated by the MKA. So, any \ncomparison of the\nMKA with the Monte Carlo data should be taken with a pinch of \nsalt.\n\nRecently, a modified droplet picture was suggested by \nHoudayer and\nMartin \\cite{hou99}, and by Bouchaud \\cite{bou99}. Within this \npicture,\nexcitations on length scales much smaller than the system \nsize are\ndroplet-like, however, there exist large-scale excitations \nthat extend\nover the entire system and that have a small energy that does \nnot\ndiverge with increasing system size. As we have demonstrated \nwithin\nMKA, the double-peaked curves for the link overlap \ndistribution, can\nbe fully explained in terms of two types of excitations that\ncontribute to the low-temperature behavior, namely \ndomain-wall\nexcitations and droplet excitations. We therefore believe \nthat there\nis no need to invoke system-wide low-energy excitations that \nare\nmore relevant than domain walls.\n\nFinally, the whole field of dynamical studies of spin glasses \nis thought by\nmany \\cite{silvio.pc} to provide a strong reason for \nbelieving the RSB picture. \nA very recent study of spin glass dynamics on the \nhierarchical lattice \n\\cite{fredrico}, on which the MKA is exact, indicates that no \nageing occurs at \nlow temperatures in the response function whereas in \nMonte-Carlo simulations\non the Edwards-Anderson model and in spin glass experiments \nageing is seen in\nthe response function. We suggest that the ageing behavior \nfound in Monte-Carlo\nsimulations and experiment are in fact often dominated by \ncritical point\neffects,\nand not by droplet effects. Indeed we would expect that if \nthe simulations of\nRef. \\cite{fredrico} were performed at temperatures closer to \nthe critical\ntemperature then ageing effects would be seen in the response \nfunction, since\nnear the critical point of even a ferromagnet such ageing \neffects occur\n\\cite{Godreche}. The reason why experiments and simulations \non the\nEdwards-Anderson model see ageing in the response function \nis that they are\nprobing time scales that may be less than the critical time \nscale, which is\ngiven by $$\\tau = \\tau_0 (\\xi/a)^z,$$ with $a$ the lattice \nconstant and\n$\\tau_0$ the characteristic spin-flip time. The dynamical \ncritical exponent $z\n\\simeq 6$ in 3 dimensions \\cite{alan}. Only for droplet \nreversals which take\nplace on time scales larger than $\\tau$ (i.e for reversals of \ndroplets whose\nlinear dimensions exceed $\\xi$) will droplet results for the \ndynamics be\nappropriate. However, because of the large values of $\\xi$ \ndown to temperatures\nof at least $0.5T_c$ and the large value of $z$, $\\tau$ may \nbe very large in\nthe Monte-Carlo simulations and experiments. Thus if $\\xi/a$ \nis 100, then\n$\\tau/\\tau_0$ is $10^{12}$, which would make droplet like \ndynamics beyond the\nreach of a Monte-Carlo simulation. In practice, most data \nwill be in a crossover\nregime leading to an apparently temperature dependent \nexponent $z(T)$ (see for\nexample Ref \\cite{komori}).\n\n\\acknowledgements We thank A.~P.~Young for discussions and \nfor\nencouraging us to write this paper. Part of this work was \nperformed\nwhen HB and BD were at the Department of Physics, University \nof\nManchester, supported by EPSRC Grants GR/K79307 and \nGR/L38578. BD also\nacknowledges support from the Minerva foundation.\n\n\\begin{references} \n\\bibitem{rby} J.D. Reger, R.N. Bhatt, and A.P. Young, Phys. \nRev. Lett. \n{\\bf 64}, 1859 (1990).\n\\bibitem{marinari} E. Marinari, G. Parisi, F. \nRicci-Tersenghi, and J.J.\nRuiz-Lorenzo in ({\\it Spin Glasses and Random Fields}, ed: A. \nP. Young), \n(World Scientific, Singapore, 1997) and references therein.\n\\bibitem{us1} M. A. Moore, H. Bokil and B.Drossel, Phys. Rev. \nLett. {\\bf 81},\n4252 (1998).\n\\bibitem{us2} H. Bokil, A. J. Bray, B.Drossel, M. A. Moore, \nPhys. Rev. Lett. {\\bf 82}, 5174 (1999).\n\\bibitem{us3} H. Bokil, A. J. Bray, B.Drossel, M. A. Moore, \nPhys. Rev. Lett. {\\bf 82}, 5177 (1999).\n\\bibitem{us4} Barbara Drossel, Hemant Bokil, M. A. Moore, and \nA. J. Bray,\nEuropean Physical Journal B {\\bf 13}, 369-375 (2000).\n\\bibitem{berg} B.A. Berg and W. Janke, Phys. Rev. Lett. {\\bf \n80}, 4771 \n(1998). \n\\bibitem{py} M. Palassini and A. P. Young, \nPhys. Rev. Lett. {\\bf 83}, 5126 (1999), and unpublished.\n\\bibitem{bm} A. J. Bray and M. A. Moore, Phys. Rev. Lett. \n{\\bf 58}, 57 1987).\n\\bibitem{mar99} E. Marinari and F. Zuliani, cond-mat/9904303. \n\\bibitem{hartmann} A. Hartmann, Phys. Rev. E {\\bf 60}, 5035 \n(1999). \n\\bibitem{southern77} B.W. Southern and A.P. Young, J. Phys. C \n{\\bf 10}, 2179\n(1977). \n\\bibitem{gardner84} E. Gardner, J. Physique {\\bf 45}, 1755 \n(1984).\n\\bibitem{bhatt88} R.N. Bhatt and A.P. Young, Phys. Rev. B \n{\\bf 37}, 5606 \n(1988). \n\\bibitem{cir93} J.C. Ciria, G. Parisi, and F. Ritort, J. \nPhys. A: Math. Gen. {\\bf 26}, 6731 (1993). \n\\bibitem{hou99} J. Houdayer and O.C. Martin, cond-mat/990878.\n\\bibitem{bou99} J.P. Bouchaud, cond-mat/9910387.\n\\bibitem{silvio.pc} Silvio Franz, personal communication, \n(1999).\n\\bibitem{fredrico} F. Ricci-Tersenghi and F. Ritort, \ncond-mat/9910390.\n\\bibitem{Godreche} C. Godr\\`{e}che and J. M. Luck, cond-mat \n0001264.\n\\bibitem{alan} R.E. Blundell, K. Humayan and A.J. Bray, J. \nPhys. A: Math. Gen. \n {\\bf 25}, L733 (1992).\n\\bibitem{komori} T. Komori, H. Yoshino and H. Takayama, J. \nPhys. Soc. Jpn. {\\bf\n68}, 3387 (1999).\n\\end{references} \n\n\\end{multicols} \n\\end{document}\n\n"
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{
"name": "cond-mat0002130.extracted_bib",
"string": "\\bibitem{rby} J.D. Reger, R.N. Bhatt, and A.P. Young, Phys. \nRev. Lett. \n{\\bf 64}, 1859 (1990).\n\n\\bibitem{marinari} E. Marinari, G. Parisi, F. \nRicci-Tersenghi, and J.J.\nRuiz-Lorenzo in ({\\it Spin Glasses and Random Fields}, ed: A. \nP. Young), \n(World Scientific, Singapore, 1997) and references therein.\n\n\\bibitem{us1} M. A. Moore, H. Bokil and B.Drossel, Phys. Rev. \nLett. {\\bf 81},\n4252 (1998).\n\n\\bibitem{us2} H. Bokil, A. J. Bray, B.Drossel, M. A. Moore, \nPhys. Rev. Lett. {\\bf 82}, 5174 (1999).\n\n\\bibitem{us3} H. Bokil, A. J. Bray, B.Drossel, M. A. Moore, \nPhys. Rev. Lett. {\\bf 82}, 5177 (1999).\n\n\\bibitem{us4} Barbara Drossel, Hemant Bokil, M. A. Moore, and \nA. J. Bray,\nEuropean Physical Journal B {\\bf 13}, 369-375 (2000).\n\n\\bibitem{berg} B.A. Berg and W. Janke, Phys. Rev. Lett. {\\bf \n80}, 4771 \n(1998). \n\n\\bibitem{py} M. Palassini and A. P. Young, \nPhys. Rev. Lett. {\\bf 83}, 5126 (1999), and unpublished.\n\n\\bibitem{bm} A. J. Bray and M. A. Moore, Phys. Rev. Lett. \n{\\bf 58}, 57 1987).\n\n\\bibitem{mar99} E. Marinari and F. Zuliani, cond-mat/9904303. \n\n\\bibitem{hartmann} A. Hartmann, Phys. Rev. E {\\bf 60}, 5035 \n(1999). \n\n\\bibitem{southern77} B.W. Southern and A.P. Young, J. Phys. C \n{\\bf 10}, 2179\n(1977). \n\n\\bibitem{gardner84} E. Gardner, J. Physique {\\bf 45}, 1755 \n(1984).\n\n\\bibitem{bhatt88} R.N. Bhatt and A.P. Young, Phys. Rev. B \n{\\bf 37}, 5606 \n(1988). \n\n\\bibitem{cir93} J.C. Ciria, G. Parisi, and F. Ritort, J. \nPhys. A: Math. Gen. {\\bf 26}, 6731 (1993). \n\n\\bibitem{hou99} J. Houdayer and O.C. Martin, cond-mat/990878.\n\n\\bibitem{bou99} J.P. Bouchaud, cond-mat/9910387.\n\n\\bibitem{silvio.pc} Silvio Franz, personal communication, \n(1999).\n\n\\bibitem{fredrico} F. Ricci-Tersenghi and F. Ritort, \ncond-mat/9910390.\n\n\\bibitem{Godreche} C. Godr\\`{e}che and J. M. Luck, cond-mat \n0001264.\n\n\\bibitem{alan} R.E. Blundell, K. Humayan and A.J. Bray, J. \nPhys. A: Math. Gen. \n {\\bf 25}, L733 (1992).\n\n\\bibitem{komori} T. Komori, H. Yoshino and H. Takayama, J. \nPhys. Soc. Jpn. {\\bf\n68}, 3387 (1999).\n"
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cond-mat0002131
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On the evaluation of the specific heat and general off-diagonal n-point correlation functions within the loop algorithm
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"author": "J.V. Alvarez and Claudius Gros"
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We present an efficient way to compute diagonal and off-diagonal n-point correlation functions for quantum spin-systems within the loop algorithm. We show that the general rules for the evaluation of these correlation functions take an especially simple form within the framework of directed loops. These rules state that contributing loops have to close coherently. As an application we evaluate the specific heat for the case of spin chains and ladders.
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"name": "loop.tex",
"string": "\\documentstyle[preprint,aps,epsfig]{revtex}\n%\\documentstyle[prl,aps]{revtex}\n%\n% http://arXiv.org/abs/cond-mat/0002131\n% User: cond-mat/0002131, Password: sq7uu \n%\n%\n\\tighten\n \\newcommand \\be {\\begin{equation}}\n\\newcommand \\bea {\\begin{eqnarray} \\nonumber }\n\\newcommand \\ee {\\end{equation}}\n\\newcommand \\eea {\\end{eqnarray}}\n \\newcommand \\eps {\\epsilon}\n \\newcommand \\bi {\\bibitem}\n\\newcommand \\s {\\sigma}\n\\newcommand \\de {\\delta}\n\\newcommand \\De {\\Delta}\n\\newcommand \\g {\\gamma}\n\\newcommand \\G {\\Gamma}\n\\newcommand \\la {\\lambda}\n\\newcommand \\La {\\Lambda}\n \\newcommand \\al {\\alpha}\n \\newcommand \\NE {\\not=}\n \\newcommand \\N {{\\cal N}}\n\\newcommand \\R {{\\cal R}}\n\\newcommand \\LL{{\\cal L}}\n\\newcommand \\ba {\\overline}\n\\newcommand \\lan {\\langle}\n\\newcommand \\ran {\\rangle}\n\\newcommand \\r {\\rho}\n\\newcommand \\Tr {\\mbox{Tr}}\n\\newcommand{\\go}{\\rightarrow}\n\n%\\topmargin=-1.5cm\n%\\textheight=24.2cm\n%\\textwidth=16.2cm\n%\\oddsidemargin=0cm\n\\begin{document}\n\\draft\n\\preprint{Preprint no.}\n\\title{On the evaluation of the specific heat and\n general off-diagonal n-point correlation functions\n within the loop algorithm}\n\n\\author{J.V. Alvarez and Claudius Gros}\n\n\\address{Fachbereich Physik, Universit\\\"at des Saarlandes,\n Postfach 151150, 66041 Saarbr\\\"ucken, Germany}\n\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\n \nWe present an efficient way to compute diagonal and\noff-diagonal n-point correlation functions for \nquantum spin-systems \nwithin the loop algorithm. We show that the general rules\nfor the evaluation of these correlation functions\ntake an especially simple form within the framework of\ndirected loops. These rules state that contributing\nloops have to close coherently. As an application\nwe evaluate the specific heat for the case\nof spin chains and ladders.\n\\end{abstract} \n\n\\vfill\n\\pacs{64.70.Nr, 64.60.Cn}\n%{\\bf \\hfill cond-mat/9502045}\n\n\\vfill\n%\\newpage\n\n%\\baselineskip 6mm\n\\narrowtext\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Introduction}\n\nNumerical investigations of strongly correlated electron\nsystems \\cite{Dagotto} gained considerable importance\nin the last decade. The evaluation of non-diagonal\ncorrelation function and dynamical response function\nplays a major role in the context of correlated\nelectron systems \\cite{Dagotto,dyn_response}. On the\nother hand, there are only very few investigations of\nnon-diagonal and/or higher-order\ncorrelation function in the context of\nquantum spin-systems. Indeed, it has been realized only\nrecently, that non-diagonal correlation function might\nbe calculated efficiently within the loop-algorithm\n\\cite{WIESE}. The loop-algorithm \\cite{EVERTZ} has \nestablished itself as the method of choice for \nquantum-Monte Carlo (MC) simulations of non-frustrated\nquantum spin systems. \n\nThe key observation here is the fact, that local\nupdating dynamics in a MC simulation creates\nstrongly correlated configurations for \ngapless quantum spin systems at low temperatures. \nSince the samples are then not statistically\nindependent, the statistical error bars do decay\nonly very slowly with the number of samples.\nOne way to state this problem is to say, that\nthe autocorrelation time $\\tau_{auto}$ for the samples\nof spin-configurations created with the MC-walk\nincreases (in generally exponentially) at\nlow temperatures.\n\nMost efficient MC procedures implement consequently \nglobal update dynamics. Examples of these procedures are the \nclusters algorithms \\cite{SWENDSEN}.\nDesigned to circumvent the critical slowing down, these \nmethods have been intensively used to study classical statistical \nsystems near critical points, where the problem of large\n$\\tau_{auto}$ is very severe. \n \nThe loop algorithm \\cite{EVERTZ} can be considered \nas a generalization of classical clusters algorithms \nto quantum models. In fact, it gives a \nprescription on how global updates can be performed \nin quantum systems.\nAs we will see this prescription lays on the geometric interpretation \nof the transformation from a quantum system to a statistical \nmodel of oriented loops. The MC procedure can be implemented then\ndirectly on the loops. \nIt has the advantage that the updating dynamics \ndefined on the loops generates statistically nearly independent\nconfigurations. The autocorrelation time is therefore about\njust MC step and the corresponding operators can be measured \nat every MC step avoiding both 'waiting times' \nand substantial increments of the variance (statistical\nerror bars). \n\nIn addition, a loop has another remarkable property;\nstarting from an allowed spin configuration, \nconstructing a loop and then flipping all spins in one loop \n(flipping the orientation of the loop) \none obtains a new allowed configuration. \nThis observation allows to compute the expectation value\nof operators not only in one configuration per MC step \nbut in all configuration related to it \nby flipping any number of given loops.\nThis procedure is usually called improved estimator \\cite{WIESE}.\n\n \nThe purpose of this work is to extend the algorithm \nto the computation of higher order \n(and non-diagonal) correlations functions.\nAs we will see it involves dealing with two or more loop \ncontributions. In particular we will focus on the \nspecific heat $c_V$, which, in the past,\nhas been considered\na major challenge for Monte-Carlo simulations\n\\cite{Huscroft}. We will show, that the direct \nevaluation of the higher-order (non-diagonal)\ncorrelation functions contributing to $c_V$\nallows for improved estimators and such to gain\none order of magnitude in computational efficiency.\nThe method that we presented is valid in \nany dimension.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{The Loop Algorithm} \n \nA nice review of the loop algorithm can be found in\nRef.\\ \\cite{EVERTZ_rev}. Here we start with a short introduction\nin order to introduce the notation used further on for the\nevaluation of higher-order correlation functions.\n\nThe loop algorithm is most easily understood in the checkerboard\npicture for a discrete number of Trotter slices $N_T$; the generalization\nto continuous Trotter time \\cite{cont_time} \nis straightforward. \nThis picture, which is based on the Suzuki-Trotter\ndecomposition, describes in a graphical way how \nthe interacting spin system wave function evolves in \ndiscrete imaginary time.\n \nThe Suzuki-Trotter formula \\cite{Trotter}\nmaps a quantum spin system in dimension $d$\nonto a classical spin in dimension $d+1$. The partition function \nof the original quantum spin model is hereby written\nin terms of the trace of a product of \ntransfer matrices defined in the classical model.\n \nTo illustrate the method we consider an inhomogeneous \none-dimensional XXZ model $H=H_1+H_2$ on a bipartite chain\nof length $L$: \n\n%\n%\n\\[\nH_{1}=\\sum_{i=2m} H_{i},\\qquad \nH_{2}=\\sum_{j=2m+1} H_{j} \n\\]\n%\n%\n\\[ \nH_{i}=-\\frac{J_{i}^{XY}}{2}\n\\left(S_{i}^{+}S_{i+1}^{-}+S_{i}^{-}S_{i+1}^{+}\\right)\n+J_{i}^{Z}S_{i}^{Z}S_{i+1}^{Z}~,\n\\]\n%\n%\nwhere the sign of the term $\\sim J_{i}^{XY}$ has been choose\nto be negative by an appropriate rotation of the spins\non one of the two sublattices. This is always possible on a\nbipartite lattice and allows for positive transfer matrix elements \n(absence of the sign problem). The decomposition\n$H=H_1+H_2$ allows for the use of Totter-Suzuki formula\n\\cite{Trotter} for the representation of the partition\nfunction $Z= \\Tr \\left[\\exp(-\\frac{\\beta}{N_{T}}H)\\right]^{N_{T}}$,\n \n%\n% \n\\[\nZ=\\Tr \\prod_{n=1}^{N_{T}}\\sum_{\\alpha_n}\n \\lan \\phi_{\\alpha_n}^{(n)}|\n\\exp(-\\Delta\\tau H_1) \\exp(-\\Delta\\tau H_2)\n| \\phi_{\\alpha_{n+1}}^{(n+1)}\\ran + O({\\Delta\\tau^2})~, \n\\]\n%\n%\nwhere $\\Delta\\tau=\\beta/N_T$. Here we have introduced\nrepresentations of the unity operator\n$\\sum_{\\alpha_n}|\\phi_{\\alpha_n}^{(n)}\\ran\n\\lan \\phi_{\\alpha_n}^{(n)}|$ in between any of the\n$N_T$ imaginary time slices.\n\nSince $H_{1}$ and $H_{2}$ are sum of local operators\nthat commute with each other, we may write \nthe wave function as the product of the local basis\nin say z-component of spin,\n$|\\phi_{\\alpha_n}^{(n)}\\ran=\\otimes_i\n|\\sigma_i\\ran$, with\n$\\sigma_i=\\uparrow,\\downarrow$.\n\nIn the checkerboard lattice the interaction between two consecutive\npairs of spins\nis graphically denoted by shaded plaquettes \n(see Fig.\\ \\ref{plaquette}). \nThere are two spins interacting per plaquette so a 4x4 transfer matrix\n$T_i$ can be defined in each plaquette, which depends only\none the coupling constants.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\n\\epsfig{file=plaquette.ps, height=6cm,\nwidth = 4cm, angle=-90}\n\\medskip\n\n\\caption[]{Illustration of a plaquette with a spin flip process \n which corresponds to the matrix element \n$ \\lan \\uparrow \\downarrow | \\exp(-\\Delta\\tau H)\n|\\downarrow \\uparrow \\ran $ of the transfer matrix.}\n\\label{plaquette}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n \nFor the XXZ-model the transfer matrix $T_i$ is in the\nbasis $\\left( |\\uparrow,\\uparrow\\ran,\n|\\uparrow,\\downarrow\\ran,|\\downarrow,\\uparrow\\ran,\n|\\downarrow,\\downarrow\\ran\\right)$:\n%\n% \n\\[\nT_i= \\mbox{e}^{{\\Delta\\tau J_i^z\\over4}}\n\\left( \\begin{array}{cccc} \n\\exp(-\\frac{\\Delta\\tau J^{Z}_{i} }{2}) & 0 & 0 & 0 \\\\\n0 & \\cosh(\\frac{\\Delta\\tau J^{XY}_{i}}{2})\n&\\sinh(\\frac{\\Delta\\tau J^{XY}_{i}}{2}) & 0 \\\\\n 0 & \\sinh(\\frac{\\Delta\\tau J^{XY}_{i}}{2})& \n\\cosh(\\frac{\\Delta\\tau J^{XY}_{i}}{2}) & 0 \\\\\n 0 & 0 & 0 & \\exp(-\\frac{\\Delta\\tau J^{Z}_{i}}{2}) \n \\end{array} \\right)~. \n\\]\n%\n%\nThe partition function $Z$ is then, up to terms order \n$O(\\Delta\\tau^2)$,\nthe trace of a product of transfer matrices: \n\n%\n%\n\\[\nZ=\\Tr\\left[\\exp(-\\beta H)\\right]= \n\\Tr \\prod_{n=1}^{N_{T}}\n\\left( \\bigotimes_{i=2m} T_{i}\\right)\n\\left( \\bigotimes_{j=2m+1} T_{j }\\right)~. \n\\] \n%\n%\nAs a next step beyond this standard representation of\n$d$-dimensional quantum models\nin terms of classical statistical systems \\cite{Beard_96}\nwe expanded the transfer matrices \n$T_i=\\sum_{\\gamma} p_{i}^{(\\gamma)}M^{(\\gamma)}$\nin terms of certain matrices $M^{(\\gamma)}$ such that the\nweight $p^{(\\gamma)}_{i} \\ge 0$ are non-negative. This\nis, in general, not possible for all models.\nFor the XXZ with $J^{XY}_{i}\\ge J_{i}^{Z}$ we can choose:\n \n%\n%\n\\[ \nM^{(1)}=\\left( \\begin{array}{cccc} \n 1 & 0 & 0 & 0 \\\\ \n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1 \n \\end{array} \\right), \\quad \n M^{(2)}=\\left( \\begin{array}{cccc} \n 1 & 0 & 0 & 0 \\\\ \n 0 & 0 & 1 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 \n \\end{array} \\right), \\quad\n M^{(3)}=\\left( \\begin{array}{cccc} \n 0 & 0 & 0 & 0 \\\\ \n 0 & 1 & 1 & 0 \\\\\n 0 & 1 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 \n \\end{array} \\right)~,\n\\]\n%\n%\nwhere\n$p_{i}^{(1)}=\\frac{1}{2}(\\exp(-\\Delta\\tau J^z_i/2)\n +\\exp(-\\Delta\\tau J^{XY}_{i}/2)) \n\\exp(\\Delta\\tau J^{Z}_{i}/4)$,\n$ p_{i}^{(2)}=\\frac{1}{2}(\\exp(-\\Delta\\tau J^z_i/2)\n -\\exp(-\\Delta\\tau J^{XY}_{i}/2)) \n\\exp(\\Delta\\tau J^{Z}_{i}/4)$ and\n$p_{i}^{(3)}=\\frac{1}{2}(-\\exp(-\\Delta\\tau J^z_i/2)\n +\\exp(\\Delta\\tau J^{XY}_{i}/2)) \n\\exp(\\Delta\\tau J^{Z}_{i}/4)$. \nWe then obtain for the partition function\n \n%\n% \n\\be\nZ=\\Tr \\prod_{n=1}^{N_{T}} \\bigotimes_{i=2m}\\left(\n\\sum _{\\gamma} p_{i}^{(\\gamma)}M^{(\\gamma)} \\right)\n\\bigotimes_{j=2m+1}\\left(\n\\sum _{\\gamma} p_{j}^{(\\gamma)}M^{(\\gamma)} \\right)\n\\label{Z1}\n\\ee\n%\n%\n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\\epsfig{file=worldlines.ps, height=6cm,\nwidth = 6cm, angle=-90}\n\\medskip\n\n\\caption[]{Evolution of worldlines of up and down spins\nin imaginary time. Periodic boundary conditions are assumed in \nboth space and imaginary time. Note that we define worldlines\nfor both up- and down-spins, which differ by the direction\nin imaginary time.}\n\\label{wordlines}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n \n \nEq.\\ (\\ref{Z1}) can be interpreted in a geometrical way\n(see Fig.\\ \\ref{wordlines}).\nIn the checkerboard picture the \n$M^{(\\gamma)}$ matrices can be understood\nas different ways in which the worldlines \ncan be broken in every plaquette and are usually called \nbreakups. \nBy taking one breakup per every plaquette we \nforce the worldlines into closed paths which we call directed\nloops (see Fig.\\ \\ref{breakups}).\nA directed loop therefore follows the worldline of\nan up-spin when it evolves in positive Trotter-time\ndirection and the world-line of a down-spin when it\nevolves in negative Trotter-time direction.\n\nIn Fig.\\ \\ref{breakups} we show the graphic representation \nof the breakups $M^{(\\gamma)}$. \nThe lines now represent the directed loop segments.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=breakups.ps, height=15cm,\nwidth = 6cm, angle=-90}\n\\medskip\n\n\\caption[]{Illustration of loop breakups, the directed lines \nrepresent the loop segments. From left to right the\nvertical ($M^{(1)}$), diagonal ($M^{(2)}$) and the\nhorizontal breakup ($M^{(3)}$) are shown.}\n\\label{breakups} \n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nEq.\\ (\\ref{Z1}) states that the partition function can\nbe obtained as a sum over all breakups. As a sum over all\nbreakups is equivalent to a sum over all loop configurations\n$\\{l\\}$ we may rewrite Eq.\\ (\\ref{Z1}) as\n\n%\n%\n\\begin{equation}\nZ=\\sum_{\\{l\\}}\\rho(\\{l\\})\\\n\\Tr \\prod_{n=1}^{N_T}\n\\bigotimes_{i=2m}M^{(\\gamma_i)}\n\\bigotimes_{j=2m+1}M^{(\\gamma_j)}~,\n\\label{Z2}\n\\end{equation}\n%\n%\nwhere $\\rho(\\{l\\})=\\prod_i p_i^{(\\gamma_i)}\n \\prod_j p_j^{(\\gamma_j)}$.\nEq.\\ (\\ref{Z2}) leads to a very efficient \nMC-algorithm \\cite{EVERTZ}: (a) Choose loop-breakups\n$M^{(\\gamma_i)}$ with probabilities\n$p_i^{(\\gamma_i)}$. (b) Construct the loop\nconfiguration $\\{l\\}$ and flip all loops\nwith probability $1/2$. (c) Measure any\ndesired operator in all $2^{N_L(\\{l\\})}$\nspin configurations reachable with \nindependent loop flips (improved estimators),\nwhere $N_L(\\{l\\})$ is the number of loops\nin the loop configuration $\\{l\\}$.\n\n%The matrix elements of the\n%$M^{(\\gamma)}$ are just one or zero determining\n%which of the states on the plaquette is\n%allowed and which not. They have the property\n%\n%%\n%%\n%\\begin{equation}\n%\\langle\\sigma_3,\\sigma_4|M^{(1)}\n%|\\sigma_1,\\sigma_2\\rangle = \\delta_{\\sigma_1,\\sigma_3}\n% \\delta_{\\sigma_2,\\sigma_4},\\quad\n%\\langle\\sigma_3,\\sigma_4|M^{(2)}\n%|\\sigma_1,\\sigma_2\\rangle = \\delta_{\\sigma_1,\\sigma_4}\n% \\delta_{\\sigma_2,\\sigma_3}~,\n%\\label{Vicente}\n%\\end{equation}\n%%\n%%\n%%and \n%$\\langle\\sigma_3,\\sigma_4|M^{(3)}\n%|\\sigma_1,\\sigma_2\\rangle = \\delta_{\\sigma_1,-\\sigma_2}\n% \\delta_{\\sigma_3,-\\sigma_4}\n%$.\n%The trace over the product of transfer matrices\n%$\\Tr\\,\\prod_n\\bigotimes_iM^{(\\gamma_i)}\n% \\bigotimes_jM^{(\\gamma_j)}$\n%occurring in Eq (\\ref{Z2}) such determines which of\n%the $2^{2LN_T}$ states on the Trotter lattice\n%are allowed configuration for any given\n%loop configuration $\\{l\\}$. As every loop can\n%be flipped independently, there are \n%$2^{N_L(\\{l\\})}$ allowed spin configurations and \n%we may rewrite Eq.\\ (\\ref{Z2}) as\n%\n%%\n%%\n%\\be\n%Z= \\sum _{\\{l\\}}\\rho(\\{l\\})\\ 2^{N_L(\\{l\\})}~.\n%\\label{Z3}\n%\\ee\n%%\n%\n%Eq.\\ (\\ref{Z3}) represents the statistical model of \n%oriented loops which is equivalent to the original\n%quantum mechanical model \\cite{WIESE}. \n\nFor later use we rewrite Eq.\\ (\\ref{Z2}) in a\nform of traces over individual loops. \nNoting that vertical and diagonal loop segments\ndo not change the spin-direction\n(see Fig.\\ \\ref{breakups}), we\nmay associate the $2\\times2$ identity matrix\n$\\sigma^{0}=\\left(\\begin{array}{cc}1&0\\\\0&1\\end{array}\\right)$ \nwith vertical and diagonal loop segments. As horizontal\nloop segments do change the spin-direction, we\nassociate the Pauli-matrix\n$\\sigma^{x}=\\left(\\begin{array}{cc}0&1\\\\1&0\\end{array}\\right)$\nwith them. We then may rewrite\nEq.\\ (\\ref{Z2}) as\n\n%\n%\n\\be\nZ= \\sum _{\\{l\\}}\\rho(\\{l\\})\\,\\prod_{l\\in\\{l\\}}\n\\Tr_l \\prod_{\\mu} \\sigma^{\\gamma_\\mu}~,\n\\label{Z4}\n\\ee \n%\n%\nwhere $\\mu$ is an index running over loop $l$\nand $\\gamma_\\mu=0,x$. $\\Tr_l$ denotes the trace\nover loop $l$.\nSince $\\Tr_l \\prod_{\\mu} \\sigma^{\\gamma_\\mu}=2$,\nEq.\\ (\\ref{Z4}) is equivalent to a statistical mechanical\nmodel of oriented loops,\n$Z= \\sum _{\\{l\\}}\\rho(\\{l\\})\\ 2^{N_L(\\{l\\})}$.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n \n\\section{Correlation functions, improved estimators}\n \nThe expectation value of an operator ${\\cal O}$ is\n\n%\n%\n\\be \n \\lan {\\cal O} \\ran = \\Tr ({\\cal O} \\exp(-\\beta H) )=\n\\sum_{\\alpha,\\beta} \\lan \\phi_{\\alpha}|{\\cal O}|\\phi_{\\beta}\\ran\n \\lan \\phi_{\\beta}|\\exp(-\\beta H)| \\phi_{\\alpha}\\ran \n\\label{Nondiagonal} \n\\ee \n%\n% \n\nIf ${\\cal O}$ is diagonal in the basis $ \\{|\\phi_{\\alpha} \\ran \\}$\nthen this procedure is straightforward.\nThe updating procedure generates a sequence of configurations \n$c_{i_{MC}}$ ($i_{MC}=1\\dots N_{MC}$),\naccording with the distribution function of the system. \nIn these configurations ${\\cal O}$ takes a well defined \nvalue ${\\cal O}(c_{i_{MC}})$, therefore: \n\n%\n%\n\\be\n\\lan {\\cal O}\\ran =\\frac{1}{N_{MC}}\\sum_{i_{MC}}\n{\\cal O}(c_{i_{MC}})~. \n\\label{MC_expect}\n\\ee\n%\n%\n\n\n\n \nThe loop algorithm allows to measure an operator not only in \n$c_{i_{MC}}$\nbut in all configurations related by loop flippings. \nWe illustrate the use of these improved estimators by computing \n${\\cal O}=S^{z}_{\\bf x} S^{z}_{\\bf y}$ \n(here indices $\\bf x$ and $\\bf y$ label both\nspace and Trotter time (see Fig.\\ \\ref{diagonal}).\nWhen $\\bf x$ and $\\bf y$ belong to different loops the orientation \ncan be changed independently and the total contribution cancels.\nBy the contrary when $\\bf x$ and $\\bf y$ are on the same loop \nthe orientations of the loop in both sites are linked and \nthese terms contribute for the two possible orientations of the loop.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=diagonal.ps, height=6cm,\nwidth = 4cm, angle=-90}\n\\medskip\n\n\\caption[]{Summing over all possible loop orientations, the\ntwo-loop contributions cancel each other \nfor $S^{z}_{\\bf x} S^{z}_{\\bf y}$. \nOnly when the two spin-operators act \non the same loop we get a non-vanishing contribution.}\n\\label{diagonal}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe will consider now the problem of non diagonal operators. \nThe expectation value of a non diagonal operator \n${\\cal O'}$ in the loop picture is, see Eq.\\ (\\ref{Z2}): \n\n%\n%\n\\be \n \\lan {\\cal O'} \\ran= \\sum _{\\{l\\}}\\rho(\\{l\\})\\,\n\\Tr{\\cal T}\\left( {\\cal O'} \n\\prod_{n=1}^{N_T}\n\\bigotimes_{i}M^{(\\gamma_i)}\n\\bigotimes_{j}M^{(\\gamma_j)}\n\\right)~,\n\\label{Oprime} \n\\ee \n%\n%\n\nwhere ${\\cal T}()$ means proper imaginary time ordering.\nLet us take as an example the two-point correlator \n${\\cal O'}=S^{+}_{\\bf x} S^{-}_{\\bf y}$, \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=insert.ps, height=12cm,\nwidth = 9cm, angle=-90}\n\\medskip\n\n\\caption[]{The action of $S^{-}_{\\bf x}S^{+}_{\\bf y}$ \ncan be represented as the insertion\nof a new kind of plaquette, here depicted in black, which acts \non two loop segments. This operator flips the spins and\nchanges therefore the orientation of the two loops\nfor the remaining segments. Left-picture: The arrows denote\nthe spin-direction. Right-picture: The arrows denote the\ndirection of the loops.}\n\\label{insert}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n \nGraphically the evaluation of an operator can be \ninterpreted on the checkerboard framework\nas the insertion of a new kind of plaquette. \nIn Fig.\\ \\ref{insert} we show the action of that operator in the \ncheckerboard picture. We note that an off-diagonal operator\nin general reverses the direction of one or more loops.\nThe loop configurations generated by the MC\nupdating-procedure does, on the other hand, only generate\nloops with well defined loop orientations.\nNevertheless there is a close connection between \nthese two types of configurations which is\neasy to understand in graphical terms.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=1loop.ps, height=10cm,\nwidth = 6cm, angle=-90}\n\\medskip\n\n\\caption[]{\nIf $S^{+}_{\\bf x}$ and $S^{-}_{\\bf y}$ operate on different loops\nnone of them can be closed consistently in terms of loop \norientation, represented here by arrows.\nWhen both operators act on the same loop \nthat configuration \ncontributes to $\\lan S^{+}_{\\bf x} S^{-}_{\\bf y} \\ran$.}\n\\label{1loop}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\nIn Fig.\\ \\ref{insert} it is shown how the\nflipping of one spin 'propagates' through \nthe loop, changing the orientation of the loop from that \npoint. Thinking in terms of oriented loops it is \nobvious that with only one of these \nflipping processes ($S_x^{+}$ or $S_y^{-}$) per loop,\nit is not possible to\nclose the loop consistently. To reestablish the original loop orientation\nit is necessary to have an \neven number of properly ordered $S^{-}$ or $S^{+}$ \noperators on the same loop \nto close it consistently in terms of loop orientation variables.\nA loop which is not properly closed does not contribute \nto $ \\lan {\\cal O'} \\ran $. Then we can establish \nthat for a two-point correlation function we only \nobtain a contribution when $x$ and $y$ belong to the \nsame loop (see Fig.\\ \\ref{1loop}).\nEq.\\ (\\ref{Nondiagonal}) could suggest that measurements \nof non diagonal operators consume more \ncomputing time than diagonal operators, but\nusing this graphical picture we note that \nboth computations can be implemented \nin an equivalent way.\n \nThese ideas can be justified in formal terms \nusing Eq.\\ (\\ref{Z4}) and Eq.\\ (\\ref{Oprime}).\nThe $S^{+}_{\\bf x}$ and $S^{-}_{\\bf y}$ operators \nare placed in between of two $\\sigma^{\\gamma}$ matrices\nbelonging to neighboring plaquettes and\ntraces can be taken again independently in each loop.\nWe define the $2\\times2$ matrices \n$\\sigma^{+}=\\left(\\begin{array}{cc}0&1\\\\0&0\\end{array}\\right)$\nand\n$\\sigma^{-}=\\left(\\begin{array}{cc}0&0\\\\1&0\\end{array}\\right)$.\nFor positive loop-direction (with respect to the Trotter\ndirection) $S^+$ is equivalent to $\\sigma^+$, for\na directed loop segment with negative loop-direction\n$S^+$ is equivalent to $\\sigma^-$. For $S^-$ it is\njust the other way round. The loop direction of relevance\nhere is the one before the insertion of either a\n$S^+$ or a $S^-$ operator.\n\nWe start considering contributions to\n$\\langle S^{+}_{\\bf x} S^{-}_{\\bf y}\\rangle$ where the\nloop-direction at site $\\bf x$ is up and down \nat site $\\bf y$ (see Fig.\\ \\ref{1loop}).\nThe expectation value of the non-diagonal operator\n$S^{+}_{\\bf x} S^{-}_{\\bf y}$ then becomes\n(compare Eq.\\ (\\ref{Z4}))\n\n%\n%\n\\be\n\\langle S^{+}_{\\bf x} S^{-}_{\\bf y}\\rangle\n\\rightarrow{1\\over Z} \\sum _{\\{l\\}}\\rho(\\{l\\})\\,\n{\\cal T}\\left(\\sigma_{\\bf x}^{+} \\sigma_{\\bf y}^{+}\n\\prod_{l\\in\\{l\\}}\n\\Tr_l \\prod_{\\mu} \\sigma^{\\gamma_\\mu}\\right)~.\n\\label{O4}\n\\ee \n%\n%\n\nHere ${\\cal T}$ means proper time and space ordering.\nWhen $\\sigma^{+}_{\\bf x}$ and $\\sigma^{+}_{\\bf y}$ \nare placed in different loops, the traces taken \nin these two loops cancel. If they are in the same loop \nthe trace taken in that loop equals 1 (and not 2),\nindependently of the spin-configuration.\nWe will prove this last point now.\nWe start by writing the partial trace of the loop containing\n$\\sigma^{+}_{\\bf x}$ and $\\sigma^{-}_{\\bf y}$ as\n\n%\n%\n\\[\nT^{(++)} =\\Tr_l\\, \\sigma_{\\bf x}^{+} \\left(\\sigma^{x}\\right)^{z_1} \n \\sigma_{\\bf y}^{+} \\left(\\sigma^{x}\\right)^{z_2}~,\n\\]\n%\n%\nwhere we neglected the $\\sigma^{0}$ matrices, as they are\njust the identity matrices. We note that\n$z_1+z_2$ is even since \n$\\left(\\sigma^{x}\\right)^2=\\sigma^{0}$\nand because we are considering a loop which did\ncontribute to the partition function $Z$ before \nthe $S_{\\bf x}^+S_{\\bf y}^-$ operators were inserted.\nThe $\\sigma^x$ matrix corresponds to a horizontal\nloop segment and such to a change in loop direction.\n$z_1$ needs therefore to be odd \n(and therefore also $z_2$), since one needs an odd number\nof directional inversions to arrive to a negative\nloop direction at site $\\bf y$, starting from a positive\ndirection at site $\\bf x$. We may therefore rewrite\n$T^{(++)}$ \n(using again $\\left(\\sigma^{x}\\right)^2=\\sigma^{0}$) as\n\n%\n%\n\\[\nT^{(++)} =\\Tr_l\\, \\sigma_{\\bf x}^{+} \\sigma^x \n\\sigma_{\\bf y}^{+} \\sigma^x \\equiv1~,\n\\]\n%\n%\nas one can easily evaluate. Similarly one can consider the\ncase when the initial loop directions are both positive\nat sites $\\bf x$ and $\\bf y$. \nThe expectation value of the non-diagonal operator\n$S^{+}_{\\bf x} S^{-}_{\\bf y}$ becomes then in this case\n\n%\n%\n\\be\n\\langle S^{+}_{\\bf x} S^{-}_{\\bf y}\\rangle\n\\rightarrow{1\\over Z} \\sum _{\\{l\\}}\\rho(\\{l\\})\\,\n{\\cal T}\\left(\\sigma_{\\bf x}^{+} \\sigma_{\\bf y}^{-}\n\\prod_{l\\in\\{l\\}}\n\\Tr_l \\prod_{\\mu} \\sigma^{\\gamma_\\mu}\\right)~.\n\\label{O5}\n\\ee \n%\n%\nThe corresponding one-loop contributions then have the\nform\n%\n%\n\\[\nT^{(+-)} =\\Tr_l\\, \\sigma_{\\bf x}^{+} \\left(\\sigma^{x}\\right)^{z_1} \n \\sigma_{\\bf y}^{-} \\left(\\sigma^{x}\\right)^{z_2}=\n\\Tr_l\\, \\sigma_{\\bf x}^{+} \\sigma_{\\bf y}^{-}\\equiv 1~,\n\\]\n%\n%\nsince both $z_1$ and $z_2$ have to be even in this case.\nSimilarly one can consider the two remaining cases\nof loop directions down/up and down/down at the sites\n$\\bf x$ and $\\bf y$. It is worthwhile noting, that one easily\nproves along these lines the expected result\n$\\langle S^{+}_{\\bf x} S^{+}_{\\bf y}\\rangle=0$.\n\n\n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\n\\section{General case n-point correlation functions}\n \nIn the last section we have shown how \nthe loop orientation is the fundamental variable\nto deal with the computation of \ncorrelation functions using improved estimators. \nIn fact the problem of n-point correlation functions\ncan also be reduced to the study of\nhow the loop orientation is changed by the action \nof some operators. \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\\epsfig{file=2loopind.ps, height=6cm,\nwidth = 6cm, angle=-90}\n\\medskip\n\n\\caption[]{Schematic example of two disconnected one-loop \ncontributions to $\\langle{\\cal O}''\\rangle$.\nThe dotted line in between two operators illustrates the\ncase of two operators at the same Trotter time.}\n\\label{2loopind}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n \nWe illustrate the case of two-loop terms for the\nfour-point correlation function \n${\\cal O}''=S_{\\bf x}^{+} S_{\\bf y}^{-} \nS_{{\\bf x}'}^{+} S_{{\\bf y}'}^{-}$.\nHere we consider the case relevant for the specific \nheat were $({\\bf x},{\\bf y})$ and \n$({\\bf x}',{\\bf y}')$ are pairs of\nreal-space nearest neighbor (n.n.) sites at the same\nTrotter time. \nThis operator can generate several different kinds\nof contributions. The first one is the case of\ntwo disconnected one-loop contributions \n(see Fig.\\ \\ref{2loopind}).\nThis is the case if $S_{\\bf x}^{+}$ and $S_{\\bf y}^{-}$\nact in one loop and \n$S_{{\\bf x}'}^{+}$ and $S_{{\\bf y}'}^{-}$ in a second loop.\nA second contribution arises if \n$S_{\\bf x}^{+}$ and $S_{{\\bf y}'}^{-}$\nact in one loop and \n$S_{\\bf y}^{-}$ and $S_{{\\bf x}'}^{+}$ in a second loop\n(see Fig.\\ \\ref{2loopdep}). We call this contribution\na connected two-loop contribution.\nA third contribution arises when all four sites act on\nthe same loop. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\\epsfig{file=2loop.ps, height=6cm,\nwidth = 6cm, angle=-90}\n\\medskip\n\n\\caption[]{Schematic example of connected two-loop contribution.\nTwo n.n.\\ operators at the same Trotter time \nare connected with a dotted line.}\n\\label{2loopdep}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe evaluation of a single off-diagonal four-point\noperator ${\\cal O}''$ does not pose a problem\nwithin the loop algorithm. For the case of interest,\nthe specific heat a few additional points need to be\nkept in mind. The specific heat $c_V$ is given by\n\n%\n%\n\\be\nc_{V}= \\frac{\\beta^{2}}{L\\,N_T^{2}}\\left[\n \\sum_{{\\bf x},{\\bf x}'}\n\\lan\\, ( {\\bf S}_{\\bf x}\\cdot{\\bf S}_{\\bf y})\n ( {\\bf S}_{{\\bf x}'}\\cdot{\\bf S}_{{\\bf y}'})\n\\,\\ran\n-\\left(\\sum_{\\bf x}\n\\lan {\\bf S}_{\\bf x}\\cdot{\\bf S}_{\\bf y}\\ran\\right)^{2} \n\\right]~,\n\\label{c_V}\n\\ee\n%\n%\nwhere, again, $({\\bf x},{\\bf y})$ and\n $({\\bf x}',{\\bf y}')$ are pairs of (real-space) \nn.n.\\ sites on the Trotter lattice.\nThe first term of Eq.\\ (\\ref{c_V}) \nis a local energy-energy correlation function. When,\n$\\bf x$ and $\\bf y$ belong to a loop and \n${\\bf x}'$ and ${\\bf y}'$ to another, we generate \ntwo-loop \ndisconnected terms (as the one illustrated in\nFig.\\ \\ref{2loopind})\nthat can be computed from the \nexpectation value of the internal energy, the \nsecond term of specific heat.\nThe energy in a given MC-configuration, $E_{i_{MC}}$,\ncan be written \nas a sum of the energy in the $N_{L}(i_{MC})$ loops in this\nMC-configuration: \n \n%\n%\n\\[ \nE_{i_{MC}}=\\sum_{l=1}^{N_L(i_{MC})}E_{i_{MC}}^{l}~.\n\\]\n%\n%\n\nWith this definition we obtain\n \n%\n%\n\\[ \nc_V^{(ind)}={1\\over N_{MC}}\\sum_{i_{MC}}\n\\sum_{l \\neq k}E_{i_{MC}}^{l}E_{i_{MC}}^{k}\n={1\\over N_{MC}}\\sum_{i_{MC}} \\left[\n\\left(\\sum_{l} E_{i_{MC}}^{l}\\right)^{2}-\n\\sum_{l} \\left(E_{i_{MC}}^{l}\\right)^{2} \n\\right]~,\n\\]\n%\n%\nwhere $c_V=c_V^{(conn)}+c_V^{(ind)}$. \nFor the evaluation of the connected term\n$c_V^{(conn)}$ one has to evaluate the off-site terms,\n$c_V^{(off)}$,\nwhere the pairs $({\\bf x},{\\bf y})$ and \n$({\\bf x}',{\\bf y}')$ are disjunct,\nseparately from the on-site terms,\n$c_V^{(on)}$, where they are not disjunct:\n$c_V^{(conn)}=c_V^{(off)}+c_V^{(on)}$.\nBy spin-algebra the on-site terms reduce to\ngeneral two-point correlation functions.\nThe (connected) off-site contributions fall in\nthree categories, depending on the number $S^z$\noperators involved (four, two or zero). The\ncontributions with four $S^z$ operators\nhave one and two loop contributions. A connected term\nwith two $S^z$ operators has no two-loop contribution.\nEvery correlation with two $S^z$ operators has the form\n$S^{z}_{\\bf x}S^{z}_{\\bf y}S^{+}_{{\\bf x}'}S^{-}_{{\\bf y}'}$.\nIf the indices ${\\bf x}$ and ${\\bf y}$\nare not in the same loop the two $S^{z}$ operators act \nin different loops and their traces \ncancel for the reason explained in section III.\nThe same reasoning is valid for ${\\bf x}'$ and ${\\bf y}'$ \nwith the operators $S^{+}$ and $S^{-}$.\nFinally, terms with no $S^{z}$ operators can have \ntwo loop contributions (see Fig.\\ \\ref{2loopdep})\nand also one-loop contributions when the \n$S^{+}$ and $S^{-}$ are properly ordered \nalong the loop to close the loop coherently \nin terms of loop orientation. On the left of Fig.\\ \\ref{no1loop}\nwe see that an arbitrary insertion of the operators\n$S^{+}$ and $S^{-}$ can produce a conflict \non the orientation of the loop. Technically, \nthe value of the trace taken along the loop will depend \non the structure of the correlator. This structure determines the order\nof the insertion of the $\\sigma^{+}$ and $\\sigma^{-}$ matrices.\nFor example the trace along the loop on the left of \nFig.\\ \\ref{no1loop} is: \n%\n%\n\\[ \n\\Tr(\\sigma^{-}_{\\bf x} \\sigma^{x} \\sigma^{-}_{{\\bf x}'}\n\\sigma^{-}_{\\bf y}\\sigma^{x}\\sigma^{+}_{{\\bf y}'})=0~.\n\\]\n%\n%\nFor the loop on the right of Fig.\\ \\ref{no1loop} it is:\n%\n%\n\\[ \n\\Tr(\\sigma^{-}_{\\bf x}\\sigma^{x}\\sigma^{-}_{\\bf y}\n\\sigma^{+}_{{\\bf x}'}\\sigma^{x}\\sigma^{+}_{{\\bf y}'})=1~.\n\\]\n%\n% \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\\epsfig{file=no1loop.ps, height=10cm,\nwidth = 6cm, angle=-90}\n\\medskip\n\n\\caption[]{On the left we show an example of \nnon-contributing configuration to the\nspecific heat. The loop orientation is ill defined and \ntherefore this configuration does not contribute. On the right \nwe see a contributing one-loop configuration. }\n\\label{no1loop}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nIt is possible to evaluate certain off-diagonal operators $\\cal O$ \nby an alternative method. The condition is, that the\noperator can be expressed by a sum of local operators which\ndo involve the same pairs of sites $\\lan l,l'\\ran$ as the\nHamilton-operator $H=\\sum_{\\lan l,l'\\ran} H_{l,l'}$:\n${\\cal O}=\\sum_{\\lan l,l'\\ran} {\\cal O}_{l,l'}$.\nIt is then possible to compute $\\lan \\cal O\\ran$ by a reweighting\nmethod. The idea is to extend the plaquette of the \ncheckerboard representation by new internal degrees of\nfreedom, $\\sum_{\\beta} |\\phi_{\\beta}\\ran \\lan \\phi_{\\beta}|$\n(see Fig.\\ \\ref{reweight}). The reweighted matrix element\nof $\\lan {\\cal O}_{x,x'} \\ran$ is then\n%\n%\n\\be\n{\\cal O}_{\\alpha_n,\\alpha_{n+1}}^{(n)}(x,x') = \n\\sum_{\\beta}\\,{\n\\lan \\phi_{\\alpha_n}^{(n)}|{\\cal O}_{x,x'}|\\phi_{\\beta}\\ran\n\\, \\lan \\phi_{\\beta}|\\exp(-\\Delta\\tau H_{x,x'})\n |\\phi_{\\alpha_{n+1}}^{(n+1)} \\ran\\over\n \\lan \\phi_{\\alpha_n}^{(n)}|\\exp(-\\Delta\\tau H_{x,x'})\n |\\phi_{\\alpha_{n+1}}^{(n+1)}\\ran\n\t } ~,\n\\ee \n%\n% \nwhere $x$ and $x'$ denote combined space-time indices.\nFor a given spin-configuration \n$c_{i_{MC}}=\\{\\phi_{\\alpha_n}^{(n)}|(n=1,\\dots,N_T)\\}$ \nthe\noff-diagonal expectation value of ${\\cal O}(c_{i_{MC}})$ is\n${\\cal O}(c_{i_{MC}})=1/N_T\\sum_{\\lan x,x'\\ran,(n)}\n{\\cal O}_{\\alpha_n,\\alpha_n+1}^{(n)}(x,x')\n$ and \n$\\lan {\\cal O}\\ran=1/N_{MC}\\sum_{i_{MC}} {\\cal O}(c_{i_{MC}})$\n(see Eq.\\ (\\ref{MC_expect})).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\\epsfig{file=reweighting.ps, height=6cm,\nwidth = 6cm, angle=-90}\n\\medskip\n\n\\caption[]{New structure of the plaquette in the reweighting method.\nThe grey plaquette is the conventional plaquette where the \nevolution in imaginary time takes place, the black plaquette represents \nthe operator $\\cal{O}$ on the basis of $\\sigma_{z}$. The product \nof the two matrices generates a new composite plaquette where \nthe new weight is defined.}\n\\label{reweight}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe reweighting method may also be applied to\nspecific heat, which is the sum of products of local operators. \n\nFrom the point of view of the complexity of the algorithm,\nmeasuring four-point correlation functions\nrequires more computing time than \ntwo-point correlation functions. \nFor the latter is only necessary to know whether or not \ntwo sites are in the same loop. This information \ncan be obtained at the same time the loop is constructed \nand consequently the computing time remains proportional \nto $LN_T$. For n-point correlation functions the situation\nis more complex. In this case, there are contributions\ninvolving two or more loops and at the same time \nnon-diagonal operators give different contributions \ndepending on how they are ordered on the loop.\nIn practice this depends on the shape of the loops \nin each configuration. \nA rigorous study of the performance of the method\nmust include an analysis of the behavior of the \nstatistical errors as a function of the\ntemperature, size, number and type \nof operator involved in the correlation functions\nand the details of the Hamiltonian. \nThis detailed analysis of technical aspects \nof n-point correlations will be presented elsewhere.\n \n\\section{Results}\n\nAs an application of the rules explained in this paper\nwe have computed the specific\nheat for a Heisenberg chain and for a ladder with\n$J_{\\perp}=0.5J$ (which corresponds to the ratio\nfor the ladder-compound Sr$_{14}$Cu$_{24}$O$_{41}$\n\\cite{Dagotto_Rice}).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\\epsfig{file=L_8.ps, height=12cm,\nwidth = 12cm, angle=-90}\n\\medskip\n\n\\caption[]{Specific heat for the Heisenberg model in the 8-site chain\nwith $J=2$, as a function of temperature in units of $J$.\nThe two sets of data correspond to the QMC simulations with \nimproved estimators (open squares) and with the reweighting \nmethod (filled triangles), for the same number of QMC steps. \nThe solid line is the exact diagonalization data.\n}\n\\label{L_8}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn Fig.\\ \\ref{L_8} we compare exact diagonalization results\nwith the results using the method described above and the reweighting method \nfor the same number of\nMC steps. The error bars in these two methods \nare also compared. For the lowest temperature \nthe error bar with improved estimators are \n6 times smaller. Taking into account that \nerror bars decay as $\\frac{1}{\\sqrt{N_{MC}}}$\nwe expect that without using improved estimators \n36 times more MC steps are necessary to get \nequal size error bars. The statistical errors \nare amplified by the factor $\\beta^2$. This factor\nand the substraction of similarly large numbers\nlead to large error bars at low temperatures. \n\nIn the Fig.\\ \\ref{L_100} we present results\nfor the specific heat of a 100-site Heisenberg\nchain. To reproduce the linear regime at low \ntemperatures it is necessary to perform a careful\nextrapolation to $\\Delta \\tau \\go 0 $ taking \nhalf a million of MC steps for each $\\Delta \\tau$\nvalues and 10 different $N_T$ values \nranging from 20 to 200. \n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\\epsfig{file=L_100.ps, height=12cm,\nwidth = 12cm, angle=-90}\n\\medskip\n\n\\caption[]{ Specific heat for a 100 sites Heisenberg with J=2.0\nchain using improved estimators, as a function of temperature,\nin units of $J$. The error-bars are smaller than the symbol\nsizes. The solid line is the\nexact Bethe-Ansatz result for the infinite-chain \n\\protect\\cite{Kluemper}.}\n\\label{L_100}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\nIn Fig.\\ \\ref{L_2x201} we present results for the\n two-leg ladder of $2\\times201$ sites with twisted\nboundary conditions (i.e.\\ for $J_{\\perp}=0$ this\nsystem corresponds to a L=402-site Heisenberg chain).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[thb]\n\\begin{center}\n\\epsfig{file=L_2x201.ps, height=12cm,\nwidth = 12cm, angle=-90}\n\\medskip\n\n\\caption[]{Specific heat for the $2\\times201$ ladder with \ntwisted boundary conditions as a function of temperature\nin units of $J$.\nThe values of the couplings are $J=1.0$ \nand $J_{\\perp}=0.5J$.}\n\\label{L_2x201}\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\n\n\\section{Conclusions}\n\nWe have presented detailed rules on how to evaluate general,\noff-diagonal n-point Greens functions within the\nloop algorithm. These rules have a very simple interpretation\nin the picture of oriented loops. They state that the\nloop-orientation has to close coherently whenever a certain\nnumber of non-diagonal operators are inserted. We have shown\nhow to apply these rules to the case of the specific heat and\npresented results for the 1D-Heisenberg model and a\nladder system.\n\n\\section{Acknowledgments}\n\n\nWe would like to acknowledge discussions with\nMatthias Troyer, Naoki Kawashima and Andreas\nKl\\\"umper and the support of the German Science\nFoundation. We acknowledge the hospitality of the\nITP in Santa Barbara. This research was supported\nby the National Science Foundation under Grant\nNo. PHY94-07194.\n\n\n\\vfill\\eject\n\\newpage\n\\begin{thebibliography}{99}\n\n\\bibitem{Dagotto} See for instance E. Dagotto, \n Rev. Mod. Phys. {\\bf 66}, 763 (1994) and references therein. \n\n\\bibitem{dyn_response} J.E. Hirsch and R.M. Feye,\n Phys. Rev. Lett. {\\bf 56}, 2521 (1986).\n\n\\bi{WIESE} R. Brower, S. Chandrasekaran, U.-J. Wiese, \n Physica A {\\bf 261}, 520 (1998).\n\n\\bi{EVERTZ} H.G. Evertz, G. Lana and M. Marcu,\n Phys. Rev. Lett. {\\bf 70}, 875 (1993).\n\n\\bi{EVERTZ_rev} H.G. Evertz in ``Numerical Methods for Lattice \nQuantum Many-Body Problems.'' \ned. D.J. Scalapino, Addison Wesley Longman, Frontiers in Physics.\n\n\n\n\\bi{SWENDSEN} R.H. Swendsen, J.S. Wang, \n Phys. Rev. Lett. {\\bf 58},86 (1987);\n U. Wolff, \n Phys. Rev. Lett. {\\bf 62}, 361 (1989).\n\n\\bibitem{Huscroft} C. Huscroft, R. Gass, M. Jarrell, \n Cond-mat 9906155;\n\t\t R. Fey and R. Scalettar,\n\t\t Phys. Rev. B {\\bf 36}, 3833 (1987).\n\n\\bibitem{cont_time} B.B. Beard and U.-J. Wiese,\n Phys. Rev. Lett. {\\bf 77}, 5130 (1996).\n\n\\bibitem{Trotter} H.F. Trotter, \n Proc. Am. Math. Soc. {\\bf 10}, 545 (1959);\n M. Suzuki, Prog. Theor. Phys. {\\bf 56}, 1454 (1976).\n\n\\bi{Beard_96} B.B. Beard and U.-J. Wiese,\n Phys. Rev. Lett. {\\bf 77}, 5130 (1996).\n\n\\bi{Kluemper} A. K\\\"umper and D.C. Johnston,\n connd-mat/0002140;\n A. Kl\\\"umper, personal communication.\n\n\\bibitem{Dagotto_Rice} E. Dagotto and T.M. Rice,\n Science {\\bf 271}, 618 (1996);\n E. Dagotto, cond-mat/9908250. \n\n%\\bi{KAWASHIMA_I} N. Kawashima, J.E. Gubernatis, \n% Phys. Rev. Lett. {\\bf 73} (1994) 1295 .\n%\t\t ------ NOT CITED !!!\n\n%\\bi{KAWASHIMA_III} N. Kawashima, J.E. Gubernatis, \n% J.Stat. Phys. {\\bf 80} (1995) 169.\n%\t\t ------ NOT CITED !!!\n\n\n\\end{thebibliography}\n\n\\end{document}\n \n"
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"name": "cond-mat0002131.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Dagotto} See for instance E. Dagotto, \n Rev. Mod. Phys. {\\bf 66}, 763 (1994) and references therein. \n\n\\bibitem{dyn_response} J.E. Hirsch and R.M. Feye,\n Phys. Rev. Lett. {\\bf 56}, 2521 (1986).\n\n\\bi{WIESE} R. Brower, S. Chandrasekaran, U.-J. Wiese, \n Physica A {\\bf 261}, 520 (1998).\n\n\\bi{EVERTZ} H.G. Evertz, G. Lana and M. Marcu,\n Phys. Rev. Lett. {\\bf 70}, 875 (1993).\n\n\\bi{EVERTZ_rev} H.G. Evertz in ``Numerical Methods for Lattice \nQuantum Many-Body Problems.'' \ned. D.J. Scalapino, Addison Wesley Longman, Frontiers in Physics.\n\n\n\n\\bi{SWENDSEN} R.H. Swendsen, J.S. Wang, \n Phys. Rev. Lett. {\\bf 58},86 (1987);\n U. Wolff, \n Phys. Rev. Lett. {\\bf 62}, 361 (1989).\n\n\\bibitem{Huscroft} C. Huscroft, R. Gass, M. Jarrell, \n Cond-mat 9906155;\n\t\t R. Fey and R. Scalettar,\n\t\t Phys. Rev. B {\\bf 36}, 3833 (1987).\n\n\\bibitem{cont_time} B.B. Beard and U.-J. Wiese,\n Phys. Rev. Lett. {\\bf 77}, 5130 (1996).\n\n\\bibitem{Trotter} H.F. Trotter, \n Proc. Am. Math. Soc. {\\bf 10}, 545 (1959);\n M. Suzuki, Prog. Theor. Phys. {\\bf 56}, 1454 (1976).\n\n\\bi{Beard_96} B.B. Beard and U.-J. Wiese,\n Phys. Rev. Lett. {\\bf 77}, 5130 (1996).\n\n\\bi{Kluemper} A. K\\\"umper and D.C. Johnston,\n connd-mat/0002140;\n A. Kl\\\"umper, personal communication.\n\n\\bibitem{Dagotto_Rice} E. Dagotto and T.M. Rice,\n Science {\\bf 271}, 618 (1996);\n E. Dagotto, cond-mat/9908250. \n\n%\\bi{KAWASHIMA_I} N. Kawashima, J.E. Gubernatis, \n% Phys. Rev. Lett. {\\bf 73} (1994) 1295 .\n%\t\t ------ NOT CITED !!!\n\n%\\bi{KAWASHIMA_III} N. Kawashima, J.E. Gubernatis, \n% J.Stat. Phys. {\\bf 80} (1995) 169.\n%\t\t ------ NOT CITED !!!\n\n\n\\end{thebibliography}"
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cond-mat0002132
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Long-time-tail Effects on Lyapunov Exponents of a Random, Two-dimensional Field-driven Lorentz Gas
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[
{
"author": "D. Panja$^{1}$"
},
{
"author": "J. R. Dorfman$^{1}$"
},
{
"author": "and Henk van Beijeren$^{2}$"
}
] |
\noindent We study the Lyapunov exponents for a moving, charged particle in a two-dimensional Lorentz gas with randomly placed, non-overlapping hard disk scatterers placed in a thermostatted electric field, $\vec E $. The low density values of the Lyapunov exponents have been calculated with the use of an extended Lorentz-Boltzmann equation. In this paper we develop a method to extend these results to higher density, using the BBGKY hierarchy equations and extending them to include the additional variables needed for calculation of Lyapunov exponents. We then consider the effects of correlated collision sequences, due to the so-called ring events, on the Lyapunov exponents. For small values of the applied electric field, the ring terms lead to non-analytic, field dependent, contributions to both the positive and negative Lyapunov exponents which are of the form ${\tilde{\varepsilon}}^{2}\,\ln\tilde{\varepsilon}$, where $\tilde{\varepsilon}$ is a dimensionless parameter proportional to the strength of the applied field. We show that these non-analytic terms can be understood as resulting from the change in the collision frequency from its equilibrium value, due to the presence of the thermostatted field, and that the collision frequency also contains such non-analytic terms. \vspace{5mm} \noindent{KEYWORDS} : Lyapunov exponents; Lorentz gas; Extended Lorentz-Boltzmann equation; BBGKY hierarchy equations; Long time tail effect.
|
[
{
"name": "final.tex",
"string": "\\documentclass[12pt]{article} \\linespread{2}\n\\usepackage{amstex,amsfonts} \\hsize=2in \\tolerance=10000\n\\marginparwidth 0pt \\oddsidemargin 0pt \\evensidemargin 0pt\n\\marginparsep 0pt \\textwidth 6.5in \\textheight 8.9in \\topmargin 0pt\n\\voffset -0.5in \\usepackage{graphicx}\n\\baselineskip = 1.3\\normalbaselineskip\n\n\\begin{document}\n\\title{Long-time-tail Effects on Lyapunov Exponents of a Random,\nTwo-dimensional Field-driven Lorentz Gas} \\author{D. Panja$^{1}$,\nJ. R. Dorfman$^{1}$, and Henk van Beijeren$^{2}$ \\\\} \\maketitle \n\\begin{center}\n{\\em $^{1}$Institute for Physical Science and Technology and\nDepartment of Physics,\\\\ University of Maryland, College Park, MD -\n20742, USA \\\\ $^{2}$Institute for Theoretical Physics, University of\nUtrecht \\\\ Postbus 80006, Utrecht 3508 TA, The Netherlands}\n\\end{center}\n\\maketitle\n\n\\begin{abstract}\n\\noindent We study the Lyapunov exponents for a moving, charged\nparticle in a two-dimensional Lorentz gas with randomly placed,\nnon-overlapping hard disk scatterers placed in a thermostatted\nelectric field, $\\vec E $. The low density values of the Lyapunov\nexponents have been calculated with the use of an extended\nLorentz-Boltzmann equation. In this paper we develop a method to\nextend these results to higher density, using the BBGKY hierarchy\nequations and extending them to include the additional variables\nneeded for calculation of Lyapunov exponents. We then consider the\neffects of correlated collision sequences, due to the so-called ring\nevents, on the Lyapunov exponents. For small values of the applied\nelectric field, the ring terms lead to non-analytic, field \ndependent, contributions to both the positive and negative Lyapunov \nexponents which are of the form \n${\\tilde{\\varepsilon}}^{2}\\,\\ln\\tilde{\\varepsilon}$, where \n$\\tilde{\\varepsilon}$ is a dimensionless parameter proportional to \nthe strength of the applied field. We show that these non-analytic \nterms can be understood as resulting from the change in the collision \nfrequency from its equilibrium value, due to the presence of the \nthermostatted field, and that the collision frequency also contains \nsuch non-analytic terms. \n \n\\vspace{5mm} \n\\noindent{\\bf KEYWORDS} : Lyapunov exponents; Lorentz gas; Extended \nLorentz-Boltzmann equation; BBGKY hierarchy equations; Long time tail \neffect. \n\\end{abstract} \n \n\\vspace{1.5cm} \n\\section{Introduction} \n \n\\noindent \nThe Lorentz gas has proved to be a useful model for studying the \nrelations between dynamical systems theory and non-equilibrium \nproperties of many body systems. This model consists of a set of \nscatterers that are fixed in space together with moving particles that \ncollide with the scatterers. Here we consider the version of the \nmodel in two dimensions where the scatterers are fixed hard disks, \nplaced at random in the plane without overlapping. Each of the moving \nparticles is a point particle with a mass and a charge, and is \nsubjected to an external, uniform electric field as well as a Gaussian \nthermostat which is designed to keep the kinetic energy of the moving \nparticle at a constant value. The particles make elastic, specular \ncollisions with the scatterers, but do not interact with each \nother. The interest in the Lorentz gas model stems from the fact that \nits chaotic properties can be analyzed in some detail, at least if the \nscatterers form a sufficiently dilute, quenched gas, so that the \naverage distance between scatterers is large compared to their \nradii. The interest in a thermostatted electric field arises from the \nfact that at small fields a transport coefficient, the electrical \nconductivity of the particles, is proportional to the sum of the \nLyapunov exponents describing the chaotic motion of the moving \nparticle \\cite{EM_ap_book}. The Lyapunov exponents are to be \ncalculated for the case where the charged particle is described by a \nnon-equilibrium steady state phase-space distribution function which \nis reached from some typical initial distribution function after a \nsufficiently long period of time. In this state, the distribution \nfunction for an ensemble of moving particles (all interacting with the \nscatterers and the field, but not with each other) is independent of \ntime and its average over the distribution of scatterers is spatially \nhomogeneous. It is known from computer simulations \n\\cite{EHFML_pra_83,Evans_jcp_83} and theoretical discussions \n\\cite{CELS_cmp_93,Dettmann_preprint} that in the stationary state the \ntrajectories of the moving particles in phase space lie on a fractal \nattractor of lower dimension than the dimension of the constant energy \nsurface, which is three dimensional for the constant energy Lorentz \ngas in two dimensions. There can be at most two non-zero Lyapunov \nexponents for this model since the Lyapunov exponent in the direction \nof the phase-space trajectory is zero. Also, the relation between \nthe Lyapunov exponents and the electrical conductivity requires that \nthe sum of the non-zero exponents should be negative due to the \npositivity of electrical conductivity \\cite{CELS_cmp_93}. \n \nThe case of the dilute, random Lorentz gas has already been studied in \ndetail. Van Beijeren and coworkers \\cite{vBD_prl_95,vBLD_pre_98} have \ncalculated the Lyapunov spectrum for an equilibrium Lorentz gas in two \nand three dimensions using various kinetic theory methods including \nBoltzmann equation techniques. These methods were also applied to the \ndilute, random Lorentz gas in a thermostatted electric field with \nresults for the Lyapunov exponents that are in excellent agreement \nwith computer simulations \\cite{vBDCPD_prl_96,LvBD_prl_97}. Moreover, \nthe results for the field-dependent case were in accord with the \nrelation between the electrical conductivity and the Lyapunov \nexponents for the moving particle. \n \nThe purpose of this paper is to extend the results obtained for the\nLyapunov exponents for dilute Lorentz gases to higher densities. Our\ncentral themes will be: (a) to describe a general method, based upon\nthe BBGKY hierarchy equations, for accomplishing this task, and (b) to\nexamine the effects on the Lyapunov exponents of long range in time\ncorrelations between the moving particle and the scatterers produced\nby correlated collision sequences where the particle collides with a\ngiven scatterer more than once and the time interval between such\nre-collisions is on the order of several mean free times, with an\narbitrary number of intermediate collisions with other\nscatterers. These correlated collision sequences are of particular\ninterest in kinetic and transport theory because they are responsible\nfor the ``long-time-tail'' effects in the Green-Kubo time correlation\nfunctions, which lead to various divergences in the transport\ncoefficients for two and three dimensional gases, where all of the\nparticles move \\cite{DvB_berne_book}. In the case of a Lorentz gas in\n$d$ dimensions, the Green-Kubo velocity correlation functions decay\nwith time, $t$, as $t^{-(d/2 +1)}$ \\cite{EW_pl_71} and the diffusion\ncoefficient is finite in both two and three dimensions. Here we\ndescribe the effects of these type of correlations on the Lyapunov\nexponents for the two-dimensional Lorentz gas, in equilibrium, where\nwe find no effect, and in a thermostatted electric field, where we\nfind a small, logarithmic dependence of the Lyapunov exponents upon\nthe applied field. This logarithmic effect is an indicator for\nsimilar effects to be expected when one calculates Lyapunov exponents\nassociated with more general transport in two-dimensional gases,\nwhereas in three dimensional systems one would expect corresponding\nnon-analytic terms proportional to $\\tilde{\\epsilon}^{\\,5/2}$. In the\ncase of the Lorentz gas, at least, the logarithmic terms can easily be\nassociated with the logarithmic terms that appear in the field\ndependent collision frequency, and a very simple argument can be used\nto establish this relation between logarithmic terms in the Lyapunov\nexponents and in the collision frequency. \n \nIn Section 2 of this article we describe the general theory of\nLyapunov exponents of a two-dimensional thermostatted electric\nfield-driven Lorentz gas and quote the results within the scope of\nthe Boltzmann equation. In Section 3, we generalize the theory to\nincorporate the effect of correlated collision sequences. In Section\n4, we outline the calculation of the effects of the correlated\ncollision sequences on the non-zero Lyapunov exponents using the BBGKY\nequations discussed in Section 3 and obtain the non-analytic\nfield-dependent term in the Lyapunov exponents, originating from the\ncorrelated collision sequences, along with other analytic\nfield-dependent terms. In Section 5, we present some simple arguments\nexplaining the field-dependence of the collision frequency and show\nthat this is the sole origin of the non-analytic, field-dependent\nterms in the Lyapunov exponents. Notice that the arguments given in\nSection 5 are independent of and much simpler to follow than the\nformalism developed in Sections 3 and 4. We conclude in Section 6\nwith a discussion of the results obtained here, and with a\nconsideration of open questions. Methods for determination of the\nfield-dependence of the collision frequency are outlined in the\nAppendix. \n \n\\section{Lyapunov exponents of field driven Lorentz gases in two \ndimensions} \n \n\\subsection{General theory} \n \n\\noindent The random Lorentz gas consists of point particles of mass \n$m$ and charge $q$ moving in a random array of fixed scatterers. In \ntwo dimensions, each scatterer is a hard disk of radius $a$. The disks \ndo not overlap with each other and are distributed with number density \n$n$, such that at low density $na^{2}<1$. The point particles are acted \nupon by a uniform, constant electric field $\\vec{E}$ in the $\\hat{x}$ \ndirection, but there is no interaction between any two point particles. \nThere is also a Gaussian thermostat in the system to keep the speed of \neach particle constant at $v$ by means of a dynamical friction during \nflights between collisions with the scatterers. The collisions between \na point particle and the scatterers are instantaneous, specular and \nelastic. During a flight, the equations of motion of a point particle \nare \n\\begin{eqnarray} \n\\dot{\\vec{r}}\\,=\\,{\\vec{v}}\\,=\\,\\frac{\\vec{p}}{m}, {\\hspace{1cm}} \n\\dot{\\vec{p}}\\,=\\,m\\dot{\\vec{v}}\\,=\\,q\\vec{E}\\,-\\, \\alpha \\vec{p} \n\\label{e1} \n\\end{eqnarray} \n\\hspace{4.9cm}{\\includegraphics[width=2.6in]{fig1}} \n\\begin{center} \nFig. 1 : Collision between a point particle and a scatterer.\\\\ \n\\end{center} \nand at a collision with a scatterer, the post-collisional position and \nvelocity, $\\vec{r}_{+}$ and $\\vec{v}_{+}$, are related to the \npre-collisional position and velocity, $\\vec{r}_{-}$ and \n$\\vec{v}_{-}$, by \n\\begin{eqnarray} \n\\vec{r}_{+}\\,=\\,\\vec{r}_{-}\\,, {\\hspace{1cm}} \n\\vec{v}_{+}\\,=\\,\\vec{v}_{-}\\,-\\,2\\,(\\vec{v}_{-}\\cdot\\hat{\\sigma})\\,\\hat{\\sigma}\\,, \n\\label{e2} \n\\end{eqnarray} \nwhere $\\hat{\\sigma}$ is the unit vector from the center of the \nscatterer to the point of collision (see Fig. 1). The fact \nthat each particle has a constant speed $v$ determines the value of \n$\\alpha$ : \n\\begin{eqnarray} \n\\alpha\\,=\\,\\frac{q\\vec{E}\\cdot\\vec{p}}{p^{2}} \n{\\hspace{0.5cm}}\\Rightarrow{\\hspace{0.5cm}}\\dot{\\vec{p}}\\,=\\,q\\vec{E}\\,-\\,\\frac{q\\vec{E}\\cdot\\vec{p}}{p^{2}}\\,\\vec{p}\\,. \n\\label{e3} \n\\end{eqnarray} \nEquivalently, in polar coordinates, the velocity direction with \nrespect to the field, defined through\n$\\hat{v}\\cdot\\hat{x}\\,=\\,\\cos\\theta$, changes between collisions as \n\\begin{eqnarray} \n\\dot{\\theta}\\,=\\,-\\,\\varepsilon\\sin\\theta\\,, \n\\label{e4} \n\\end{eqnarray} \nwhere $\\varepsilon\\,=\\,\\displaystyle{\\frac{q |\\vec{E}|}{mv}}$ and we \ndefine the dimensionless electric field parameter \n$\\tilde{\\varepsilon}\\,=\\,\\displaystyle{\\frac{\\varepsilon l}{v}}$, \nwhere $l =(2na)^{-\\,1}$ is the mean free path length for the particle \nin the dilute Lorentz gas. To denote the electric field, we will normally \nuse $\\varepsilon$, though from time to time we will use \n$\\tilde{\\varepsilon}$, too. \n \nTreating this two-dimensional Lorentz gas as a dynamical system, we \ndefine the Lyapunov exponents in the usual way: a point particle in \nits phase space $(\\vec{r}, \\vec{v}) =\\vec{{\\bf X}}$ starts at time \n$t_{0}$ at a phase space location $\\vec{{\\bf X}}(t_{0})$. Under time \nevolution, $\\vec{{\\bf X}}(t)$ follows a trajectory in this phase space \nwhich we call the ``reference trajectory''. We consider an \ninfinitesimally displaced trajectory which starts at the same time \n$t_{0}$, but at $\\vec{\\bf \nX^{\\begin{Sp}\\prime\\end{Sp}}}(t_{0})\\,=\\,\\vec{{\\bf \nX}}(t_{0})\\,+\\,{\\bf\\delta\\vec{X}}(t_{0})$. Under time evolution, \n$\\vec{{\\bf X^{\\begin{Sp}\\prime\\end{Sp}}}}(t)$ follows another \ntrajectory, always staying infinitesimally close to the reference \ntrajectory. This trajectory we call the ``adjacent \ntrajectory''. Typically the two trajectories will separate in time due \nto the convex nature of the collisions. Thus, we can define the \npositive Lyapunov exponent as \n\\begin{eqnarray} \n\\lambda_{+}\\,=\\,\\lim_{\\begin{Sb} T \\rightarrow \\infty \n\\end{Sb}}\\,\\lim_{\\begin{Sb} \n|\\delta{\\bf\\vec{X}}(t_{0})|\\rightarrow 0 \n\\end{Sb}} \n\\,\\frac{1}{T} \\ln \n\\frac{|\\delta{\\bf\\vec{X}}(t_{0}\\,+\\,T)|}{|\\delta{\\bf\\vec{X}}(t_{0})|} \n\\,. \n\\label{e5} \n\\end{eqnarray} \nfor a typical trajectory of the system. \n \nWe assume that, for small fields, this Lorentz gas system is \nhyperbolic. Since the two-dimensional Lorentz gas can have at most two \nnonzero Lyapunov exponents, we denote the negative Lyapunov exponent \nby $\\lambda_{-}$. Without any loss of generality, we can choose to \nmeasure the separation of the reference and adjacent trajectories \nequivalently in $\\vec{r}$-space, thereby reducing the definition of \nthe positive Lyapunov exponent to \n\\begin{eqnarray} \n\\lambda_{+}\\,=\\,\\lim_{\\begin{Sb} T \\rightarrow \\infty \n\\end{Sb}}\\,\\lim_{\\begin{Sb} \n|\\delta\\vec{r}(t_{0})|\\rightarrow 0 \n\\end{Sb}} \n\\,\\frac{1}{T} \\ln \n\\frac{|\\delta{\\vec{r}}(t_{0}\\,+\\,T)|}{|\\delta{\\vec{r}}(t_{0})|}\\,. \n\\label{e6} \n\\end{eqnarray} \nIn order to calculate the right hand side of Eq. (\\ref{e6}), we \nintroduce another dynamical quantity, the radius of curvature $\\rho$, \ncharacterizing the spatial separation of the two trajectories (see \nFig. 2) : \n \n\\begin{center} \n\\hspace{2cm}{\\includegraphics[width=3.67in]{fig2}}\\\\ Fig. 2 : \n$\\rho(t)\\,=\\,\\displaystyle{\\frac{\\delta S(t)}{\\delta\\theta(t)}}\\,=\\,|AP|\\,$. \n\\end{center} \nIn Fig. 2, a particle on the reference trajectory would be at point A \nat time $t$. At the same time, a particle on the adjacent trajectory \nwould be at B. A local perpendicular on the reference trajectory at A \nintersects the adjacent trajectory at C. The backward extensions of \ninstantaneous velocity directions on the reference and adjacent \ntrajectories at A and C, respectively, intersect each other at point \nP. We denote the length of the line segment AC by $\\delta S(t)$ and \n$\\angle APC$ by $\\delta\\theta(t)$. The radius of curvature associated \nwith the particle on the reference trajectory at time $t$ is then \ngiven by \n\\begin{eqnarray} \n\\rho(t)\\,=\\,\\frac{\\delta S(t)}{\\delta \\theta (t)}\\,=\\,|AP|\\,. \n\\label{e7} \n\\end{eqnarray} \nHaving defined $\\rho(t)$, one can make a simple geometric argument to \nshow that \n\\begin{equation} \n{\\delta\\dot{S}}(t)\\,=\\, v\\,{\\delta \\theta (t)}\\,=\\,\\frac{v\\,\\delta S(t)}{\\rho(t)}, \n\\label{e7a} \n\\end{equation} \n so as to obtain a version of Sinai's formula \\cite{Sinai_rms_70}, \n\\begin{eqnarray} \n\\lambda\\,=\\,\\lim_{T\\rightarrow\\infty}\\,\\frac{v}{T}\\int_{\\begin{Sb}t_{0}\\end{Sb}}^{\\begin{Sp}t_{0}\\,+\\,T\\end{Sp}}\\,\\frac{dt}{\\rho(t)}. \n\\label{e8} \n\\end{eqnarray} \n \nDuring a flight, the equation of motion for $\\rho$ is given by \n\\cite{vBDCPD_prl_96} \n\\begin{eqnarray} \n\\dot{\\rho}\\,=\\,v\\,+\\,\\rho\\varepsilon\\cos\\theta\\,+\\,\\frac{\\rho^{2}\\varepsilon^{2}\\sin^{2}\\theta}{v}\\,. \n\\label{e9} \n\\end{eqnarray} \nAt a collision with a scatterer, the post-collisional velocity angle \n$\\theta_{+}$ and radius of curvature $\\rho_{+}$ are related to the \npre-collisional velocity angle $\\theta_{-}$ and radius of curvature \n$\\rho_{-}$ by \\cite{Kovacs_priv_disc,Kovacs_preprint} : \n\\begin{eqnarray} \n\\theta_{+}\\,=\\,\\theta_{-}-\\pi+2\\phi\\,,\\hspace{1cm} \n\\frac{1}{\\rho_{+}}\\,=\\,\\frac{1}{\\rho_{-}}\\,+\\,\\frac{2}{a\\cos\\phi}\\,+\\,\\frac{\\varepsilon}{v}\\,\\tan\\phi\\,(\\sin\\theta_{-}\\,+\\,\\sin\\theta_{+})\\,, \n\\label{e10} \n\\end{eqnarray} \nwhere $\\phi$ is the collision angle, i.e, \n$\\cos\\phi\\,=\\,|\\hat{v}_{-}\\cdot\\hat{\\sigma}|\\,=\\,|\\hat{v}_{+}\\cdot\\hat{\\sigma}|$ \n(see Fig. 1). \n \nNow we assume that, for sufficiently weak electric field, the \nfield-driven Lorentz gas in two dimensions is ergodic, and that we can \nreplace the long time average in Eq. (\\ref{e6}) by a non-equilibrium \nsteady state (NESS) average, including an average over all allowed \nconfigurations of scatterers, to obtain \n\\begin{eqnarray} \n\\lambda = {\\bigg <}\\frac{v}{\\rho}{\\bigg >}_{\\mbox{\\scriptsize NESS}}\\,. \n\\label{e11} \n\\end{eqnarray} \nThe electric field is considered weak if the work done by the electric \nfield on the point particle over a flight of one mean free path is \nmuch smaller than the particle's kinetic energy, i.e, \n$\\displaystyle{\\frac{q|\\vec{E}|\\,l}{mv^{2}}}\\,=\\,\\displaystyle{\\frac{\\varepsilon l}{v}}\\,=\\,\\tilde{\\varepsilon}\\,<<\\,1$. \n \nWe note for future reference, that the sum of the two nonzero Lyapunov \nexponents is related to the average of the friction coefficient \n$\\alpha$, by \\cite{EM_ap_book,Hoover_elsevier_book,Panja_preprint} \n\\begin{equation} \n\\lambda_{+}\\,+\\,\\lambda_{-}\\,=\\,-\\,\\big<\\alpha\\big>_{\\mbox{\\tiny NESS}}\\,=\\,-\\,\\bigg<\\frac{q\\vec{E}\\cdot\\vec{v}}{mv^{2}}\\bigg>_{\\mbox{\\scriptsize NESS}}\\,=\\,-\\,\\frac{\\vec{J}\\cdot\\vec{E}}{mv^{2}}\\,=\\,-\\,\\frac{\\sigma E^{2}}{mv^{2}}. \n\\label{e12} \n\\end{equation} \nHere the electric current $\\vec{J}\\,=\\,\\langle \nq\\vec{v}\\rangle_{\\mbox{\\tiny NESS}}$ is, for small fields, assumed to \nsatisfy Ohm's law, $\\vec{J}=\\sigma\\vec{E}$, and $\\sigma$ is the \nelectrical conductivity. \n \n \n\\subsection{Results obtained using the Lorentz-Boltzmann equation} \n \nTo the lowest order in density, one can assume that the collisions \nsuffered by the point particle are uncorrelated, and use an extended \nLorentz-Boltzmann equation (ELBE) for the distribution function of the \nmoving particle, $f_{1}(\\vec{r},\\vec{v},\\rho,t)$ in $(\\vec{r}, \n\\vec{v}, \\rho)$-space \\cite{vBDCPD_prl_96} needed for the evaluation \nof the averages appearing in Eqs. (\\ref{e11}) and (\\ref{e12}). To \ncalculate the positive Lyapunov exponent, one needs to consider the \nforward-time ELBE while to calculate the negative Lyapunov exponent \none needs the time reversed ELBE. To the leading order in density, the \nLyapunov exponents are then given by \\cite{vBDCPD_prl_96} : \n\\begin{eqnarray} \n\\lambda^{\\mbox{\\tiny \n(B)}}_{+}\\,=\\,\\lambda_{0}\\,-\\,\\frac{11}{48}\\frac{l}{v}\\varepsilon^{2}\\,+\\,O(\\varepsilon^{4})\\hspace{1cm}{\\mbox{and}}\\hspace{1cm}\\lambda^{\\mbox{\\tiny(B)}}_{-}\\,=\\,-\\,\\lambda_{0}\\,-\\,\\frac{7}{48}\\frac{l}{v}\\varepsilon^{2}\\,+\\,O(\\varepsilon^{4})\\,. \n\\label{e13} \n\\end{eqnarray} \nThe superscript, B, indicates that these are results obtained from the \nLorentz-Boltzmann equation. Here $\\lambda_{0}$ is the positive \nLyapunov exponent for a field-free Lorentz gas (see for example \n\\cite{D_cup_book}) given by \n\\begin{equation} \n\\lambda_0 = 2nav\\,[\\,1 - {\\cal{C}} -\\ln(2na^{2})\\,], \n\\label{e131} \n\\end{equation} \nwhere ${\\cal{C}}$ is Euler's constant, ${\\cal C}=0.5772...$. From \nEqs. (\\ref{e12}) and (\\ref{e13}), using Einstein's relation between \ndiffusion constant and conductivity, one gets the correct diffusion \ncoefficient within the Boltzmann regime, $D^{\\mbox{\\tiny (B)}}\\,=\\,\\displaystyle{\\frac{3}{8}\\,lv}$. \n \nTo derive the results in Eq. (\\ref{e13}), one uses Eq. (\\ref{e10}) \nwith $\\varepsilon=0$. The $\\varepsilon$-dependent term in Eq. \n(\\ref{e10}) can be explicitly shown to be of higher order in the \ndensity than the terms present in Eq. (\\ref{e13}) \n\\cite{PD_unpublished}. In the following sections we will investigate \nthe effect of sequences of correlated collisions between the point \nparticle and the scatterers on the Lyapunov exponents. However, \nthe $\\varepsilon$-dependent term in Eq. (\\ref{e10}) will again be \nneglected since we will present the effect of these correlated \ncollision sequences in leading order in the density of scatterers \nonly. Thus, instead of Eq. (\\ref{e10}), we will use \n\\begin{eqnarray} \n\\frac{1}{\\rho_{+}}\\,=\\,\\frac{1}{\\rho_{-}}\\,+\\,\\frac{2}{a\\cos\\phi}\\,. \n\\label{e14} \n\\end{eqnarray} \n \n \n \n\\section{The extension of the ELBE to higher density} \n \n\\subsection{Binary collision operators in $(\\vec{r}, \\vec{v}, \n\\rho)$-space and\\\\ the BBGKY hierarchy equations} \n \nThe Boltzmann theory for the Lyapunov exponents assumes that the \nscatterers form a dilute, but quenched system and that the collisions \nof the point particles with the scatterers are uncorrelated. To \nincorporate the effects of correlated collisions on the Lyapunov \nexponents, we will use a method based on the BBGKY hierarchy equations, \nfamiliar from the kinetic theory of moderately dense gases \n\\cite{DvB_berne_book}. Since the moving particles do not interact with \neach other, it is sufficient to consider the distribution functions \nfor just one of them, together with a number of scatterers. One starts \nfrom a fundamental equation for an $(N+1)$-body distribution function, \n$f_{N+1}\\,=\\,f_{N+1}(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{1}, \n\\vec{R}_{2},.,.,\\vec{R}_{N}; t)$, which is the probability density \nfunction in the entire extended phase space $\\Gamma$ spanned by the \nvariables $\\vec{r}, \\vec{v}, \\rho, \\vec{R}_{1}, \n\\vec{R}_{2},.,.,\\vec{R}_{N}$, describing our system of $N$ scatterers \nand one moving particle. We require that $f_{N+1}$ satisfies the \nnormalization condition \n\\begin{eqnarray} \n\\int \nd\\vec{r}\\,d\\vec{v}\\,d\\rho\\,d\\vec{R}_{1}\\,d\\vec{R}_{2}\\,.\\,.d\\vec{R}_{N}\\,\\,f_{N+1}(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{1}, \\vec{R}_{2},.,.,\\vec{R}_{N}; t)\\,=\\,1\\,. \n\\label{e15} \n\\end{eqnarray} \nThis $(N+1)$-body distribution function satisfies a Liouville-like \nequation determined by the collisions of the moving particles with the \nscatterers and by the motion of the particles in the thermostatted \nelectric field, between collisions. Since the time evolution of \n$\\vec{r}$ and $\\vec{v}$ in this field is not Hamiltonian, we must use \nthe Liouville equation in the form of a conservation law, rather than \nthe usual form for Hamiltonian systems, to obtain \n\\begin{eqnarray} \n\\frac{\\partial f_{N\\,+\\,1}}{\\partial \nt}\\,+\\,\\vec{\\nabla}_{\\vec{r}}\\cdot(\\dot{\\vec{r}}\\,f_{N\\,+\\,1})\\,+\\,\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,f_{N\\,+\\,1})\\,+\\,\\frac{\\partial}{\\partial\\rho}(\\dot{\\rho}f_{N\\,+\\,1})\\,=\\,\\sum_{i\\,=\\,1}^{N}\\tilde{T}_{-,\\,i}\\,f_{N\\,+\\,1}\\,. \n\\label{e22} \n\\end{eqnarray} \nHere the operators $\\tilde{T}_{-,\\,i}$ are binary collision operators \nwhich describe the effects on the distribution function due to an \ninstantaneous, elastic collision between the moving particle and the \nscatterer labeled by the index $i$. The explicit form of the binary \ncollision operators may be easily obtained by a slight modification of \nthe methods used by Ernst {\\it et al.} \\cite{DE_jsp_89}, in order to \ninclude the radius of curvature as an additional variable. One finds \nthat the action of this operator on any function $f(\\vec{r}, \\vec{v}, \n\\rho; \\vec{R}_{1}, \\vec{R}_{2},.,.,\\vec{R}_{j}; t)$ is \n\\begin{eqnarray} \n\\tilde{T}_{-,\\,i}\\,\\,f\\,=\\,a\\int_{\\vec{v}\\cdot\\hat{\\sigma}_{i}\\,>\\,0}d\\hat{\\sigma}_{i}\\,|\\vec{v}\\cdot\\hat{\\sigma}_{i}|\\,\\bigg\\{\\int_{0}^{\\infty}d\\rho'\\,\\delta\\bigg(\\rho\\,-\\,\\frac{\\rho'\\,a\\,\\cos\\phi_{i}}{a\\,\\cos\\phi_{i}\\,+\\,2\\rho'}\\bigg)\\,\\delta(a\\hat{\\sigma}_{i}\\,-\\,(\\vec{r}\\,-\\,\\vec{R}_{i}))\\times\\nonumber\\\\&& \n{\\hspace{-8cm}}\\times\\,b_{\\sigma_{i},\\,\\rho'}\\,-\\,\\delta(a\\hat{\\sigma}_{i}\\,+\\,(\\vec{r}\\,-\\,\\vec{R}_{i}))\\bigg\\}\\,f\\,, \n\\label{e17} \n\\end{eqnarray} \nwhere $\\hat{\\sigma}_{i}$ is the unit vector from the center of the \nscatterer fixed at $\\vec{R}_{i}$ to the point of collision. The action \nof the operator $b_{\\sigma_{i},\\,\\rho'}$ on the function $f(\\vec{r}, \n\\vec{v}, \\rho; \\vec{R}_{1}, \\vec{R}_{2},.,.,\\vec{R}_{j}; t)$ is \ndefined by \n\\begin{eqnarray} \nb_{\\sigma_{i},\\,\\rho'}\\,\\,f(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{1}, \\vec{R}_{2},.,.,\\vec{R}_{j}; t)\\,=\\,f(\\vec{r},\\,\\vec{v}-2\\,(\\vec{v}\\cdot\\hat{\\sigma}_{i})\\,\\hat{\\sigma}_{i},\\,\\rho';\\vec{R}_{1}, \\vec{R}_{2},.,.,\\vec{R}_{j}; t)\\,; \n\\label{e18} \n\\end{eqnarray} \nthat is, $b_{\\sigma_{i}\\rho'}$ is a substitution operator that \nreplaces $\\rho$ by $\\rho'$ and the velocity $\\vec{v}$ by its \nrestituting value, i.e, the value it should have before collision \nso as to lead to the value $\\vec{v}$ after collision. It is often \nuseful to express the binary collision operators as as sum of two \nterms such that \n\\begin{eqnarray} \n\\tilde{T}_{-,\\,i}\\,=\\,\\tilde{T}^{(+)}_{-,\\,i}\\,-\\,\\tilde{T}^{(-)}_{-,\\,i}\\,, \n\\label{e21} \n\\end{eqnarray} \nwhere \n\\begin{eqnarray} \n\\tilde{T}^{(+)}_{-,\\,i}\\,=\\,a\\,\\int_{\\vec{v}\\cdot\\hat{\\sigma}_{i}\\,>\\,0}d\\hat{\\sigma}_{i}\\,|\\vec{v}\\cdot\\hat{\\sigma}_{i}|\\,\\int_{0}^{\\infty}d\\rho'\\,\\delta\\bigg(\\rho\\,-\\,\\frac{\\rho'\\,a\\,\\cos\\phi_{i}}{a\\,\\cos\\phi_{i}\\,+\\,2\\rho'}\\bigg)\\,\\delta(a\\hat{\\sigma}_{i}\\,-\\,(\\vec{r}\\,-\\,\\vec{R}_{i}))\\,b_{\\sigma_{i},\\,\\rho'}\\, \n\\label{e19} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\tilde{T}^{(-)}_{-,\\,i}\\,=\\,a\\,\\int_{\\vec{v}\\cdot\\hat{\\sigma}_{i}\\,>\\,0}d\\hat{\\sigma}_{i}\\,|\\vec{v}\\cdot\\hat{\\sigma}_{i}|\\,\\delta(a\\hat{\\sigma}_{i}\\,+\\,(\\vec{r}\\,-\\,\\vec{R}_{i}))\\,. \n\\label{e20} \n\\end{eqnarray} \nOne sees that $\\tilde{T}^{(+)}_{-,\\,i}\\,f$ and \n$\\tilde{T}^{(-)}_{-,\\,i}\\,f$ respectively describe the rate of \n``gain'' and the rate of ``loss'' of $f$ due to a collision of the \npoint particle with the scatterer fixed at $\\vec{R}_{i}$. \n \nThe BBGKY hierarchy equations are then obtained from Eq. (\\ref{e22}) \nby integrating over scatterer coordinates, as a set of equations for \nthe reduced distributions $f_{j}$ for the moving particle and $(j-1)$ \nscatterers, defined by \n\\begin{eqnarray} \nf_{j}(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{1},\\vec{R}_{2},.,.,\\vec{R}_{j-1};t)\\,\\nonumber\\\\&&{\\hspace{-2.5cm}}=\\,\\frac{N!}{(N-j+1)!}\\int d\\vec{R}_{j}..d\\vec{R}_{N}\\,f_{N\\,+\\,1}(\\vec{r}, \\vec{v}, \\rho;\\vec{R}_{1}, \\vec{R}_{2},.,.,\\vec{R}_{N}; t)\\,. \n\\label{e16} \n\\end{eqnarray} \nOne then easily obtains the BBGKY hierarchy equations \n($1\\,\\leq\\,j\\,\\leq\\,N$) \n\\begin{eqnarray} \n\\frac{\\partial f_{j}}{\\partial \nt}\\,+\\,\\vec{\\nabla}_{\\vec{r}}\\cdot(\\dot{\\vec{r}}\\,f_{j})\\,+\\,\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,f_{j})\\,+\\,\\frac{\\partial}{\\partial\\rho}(\\dot{\\rho}f_{j})\\,-\\,\\sum^{j\\,-\\,1}_{k\\,=\\,1}\\,\\tilde{T}_{-,\\,k}\\,f_{j}\\,=\\,\\int d\\vec{R}_{j}\\,\\tilde{T}_{-,\\,j}\\,f_{j\\,+\\,1}\\,. \n\\label{e23} \n\\end{eqnarray} \n \n\\subsection{Cluster expansions and truncation of the hierarchy \nequations} \n \nThe usual procedure for truncating the hierarchy equations in order to \nobtain the Boltzmann equation and its extension to higher densities is \nto make cluster expansions of the distribution functions, $f_2, \nf_3\\,.\\,.\\,.$ in terms of a set of correlation functions, $g_2, \ng_3\\,.\\,.\\,.$ as follows: \n\\begin{eqnarray} \nf_{2}(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{1}; t)\\,=\\,nf_{1}(\\vec{r}, \n\\vec{v}, \\rho; t)\\,+\\,g_{2}(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{1}; t), \n\\label{e24} \n\\end{eqnarray} \n\\begin{eqnarray} \nf_{3}(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{1}, \\vec{R}_{2}; \nt)\\,=\\,n^{2}f_{1}(\\vec{r}, \\vec{v}, \\rho; t)\\,+\\,ng_{2}(\\vec{r}, \n\\vec{v}, \\rho; \\vec{R}_{1}; t)\\,+\\,ng_{2}(\\vec{r}, \\vec{v}, \\rho; \n\\vec{R}_{2}; t)\\,\\nonumber\\\\&&{\\hspace{-3.7cm}}+\\,g_{3}(\\vec{r}, \n\\vec{v}, \\rho; \\vec{R}_{1}, \\vec{R}_{2}; t)\\,, \n\\label{e25} \n\\end{eqnarray} \nand so on. Hereafter, to save writing, we denote $g_{2}(\\vec{r}, \n\\vec{v}, \\rho; \\vec{R}_{1}; t)$ as $g_{2,\\,\\vec{R}_{1}}$, \n$g_{2}(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{2}; t)$ as \n$g_{2,\\,\\vec{R}_{2}}$, $f_{3}(\\vec{r}, \\vec{v}, \\rho; \\vec{R}_{1}, \n\\vec{R}_{2}; t)$ as $f_{3}$ and $g_{3}(\\vec{r}, \\vec{v}, \\rho; \n\\vec{R}_{1}, \\vec{R}_{2}; t)$ as $g_{3}$. The first terms in each of \nthese expansions represent the totally uncorrelated situation, where \nthere are independent probabilities of finding the moving particle and \nthe scatterers at the designated coordinates. The next terms involving \nthe pair correlation functions $g_{2,\\,\\vec{R}_{i}}$ in Eqs. \n(\\ref{e24}) and (\\ref{e25}) take into account possible dynamical and \nexcluded volume correlations between the point particle and the \nscatterer at $\\vec{R}_{i}$. If one replaces $f_2$ by $nf_1$ in the \nfirst BBGKY hierarchy equation, Eq. (\\ref{e23}) with $j=1$, reduces to \nthe ELBE. To find the corrections to the ELBE for higher \ndensities, one must keep the $g_2$ term in Eq. (\\ref{e24}) and use the \nsecond hierarchy equation to determine $g_2$. However, in order to \nsolve the second equation, we have to say something about $g_3$. A \ncareful examination of the second and higher equations shows that \n$g_3$ contains, of course, the effects of three-body correlations, \ni.e, correlated collisions involving the point particle, a scatterer \nfixed at $\\vec{R}_{1}$ and another scatterer fixed at $\\vec{R}_{2}$, \nas well as excluded volume corrections due to the non-overlapping \nproperty of the scatterers. Here we will be primarily interested in \nthe effects of the so called ``ring'' collisions on the Lyapunov \nexponents. These collision sequences are composed of one collision of \nthe moving particle with a given scatterer, followed by an arbitrary \nnumber of collisions with a succession of different scatterers, and \ncompleted by a final re-collision of the moving particle with the \nfirst scatterer in the sequence, as illustrated in Fig 3. \n \n \n\\begin{center} \n\\unitlength=0.1mm \n\\begin{picture}(1650,400)(0,0) \n\\thicklines \\put(80,300){\\circle{200}} \\put(134,100){\\circle{200}} \n\\put(21.72,109.44){\\line(1,2){60}} \n\\put(21.72,109.44){\\vector(1,2){45}} \n\\put(83.39,226.77){\\line(1,-2){29}} \n\\put(83.39,226.77){\\vector(1,-2){20}} \n\\put(113.36,165.83){\\line(0,1){72}} \n\\put(113.36,165.83){\\vector(0,1){60}} \n\\put(113.36,237.83){\\line(2,-1){110}} \n\\put(113.36,237.83){\\vector(2,-1){80}} \\put(71,287){\\Large{1}} \n\\put(125,87){\\Large{2}} \\put(240,200){\\Large {$+$}} \n\\put(390,300){\\circle{200}} \\put(390,230){\\line(-1,-1){80}} \n\\put(355,195){\\vector(1,1){20}} \\put(390,230){\\line(1,-1){49}} \n\\put(390,230){\\vector(1,-1){35}} \\put(439,110){\\circle{140}} \n\\put(439,180){\\line(1,1){60}} \\put(439,180){\\vector(1,1){40}} \n\\put(571,240){\\circle{200}} \\put(501,240){\\line(-1,1){43}} \n\\put(500,240){\\vector(-1,1){35}} \\put(459,283){\\line(2,1){140}} \n\\put(459,283){\\vector(2,1){60}} \\put(381,287){\\Large{1}} \n\\put(430,97){\\Large{2}} \\put(562,227){\\Large{3}} \\put(670,200){\\Large \n{$+$}} \\put(825,238){\\circle{200}} \\put(816,225){\\Large 1} \n\\put(825,166){\\line(-2,-1){120}} \\put(735,121){\\vector(2,1){70}} \n\\put(825,166){\\line(2,-1){50}} \\put(825,166){\\vector(2,-1){40}} \n\\put(877,70){\\circle{200}} \\put(867,57){\\Large 2} \n\\put(877,142){\\line(2,1){81}} \\put(877,142){\\vector(2,1){55}} \n\\put(1017,143){\\circle{200}} \\put(1007,130){\\Large 3} \n\\put(958,181){\\line(0,1){110}} \\put(958,181){\\vector(0,1){70}} \n\\put(1010,337){\\circle{200}} \\put(1000,325){\\Large 4} \n\\put(960,287){\\line(-1,0){83}} \\put(960,287){\\vector(-1,0){60}} \n\\put(876.5,286.5){\\line(0,1){120}} \\put(876.5,286.5){\\vector(0,1){75}} \n\\put(1090,200){\\Large {+ .... = }} \\put(1480,250){\\Large 1} \n\\end{picture} \n\\end{center} \n\\vspace{-4cm} \\hspace{13.2cm} \n{\\includegraphics[width=1.267in]{diagram1}} \n \n\\begin{center} \nFig. 3 : Sequential collisions with scatterers at $\\vec{R}_{2},\\vec{R}_{3},.,.$ adding up to the ring diagram. \n\\end{center} \n \nThe ring diagrams, taken individually, are the most divergent terms \nthat appear in the expansion of dynamical properties of the Lorentz \ngas as a {\\it power} series in the density of scatterers. They lead to \nthe logarithmic terms in the density expansion of the diffusion \ncoefficient of the moving particle \\cite{vLW_physica_67}, and to the \nalgebraic long time tails in the velocity time correlation function \nof the moving particle \\cite{EW_pl_71}. While many other dynamical \nevents and excluded volume effects contribute to the Lyapunov \nexponents, and must be included for a full treatment, we concentrate \nhere on the effects of these most divergent collision sequences, since \nin other contexts, they are responsible for the most dramatic higher \ndensity corrections to the Boltzmann equation results. \n \nThus we drop $g_{3}$ in Eq. (\\ref{e25}) and obtain a somewhat \nsimplified cluster expansion of $f_{3}$, given by \n\\begin{eqnarray} \nf_{3}\\,=\\,n^{2}f_{1}\\,+\\,ng_{2,\\,\\vec{R}_{1}}\\,+\\,ng_{2,\\,\\vec{R}_{2}} \n\\label{e26} \n\\end{eqnarray} \nUsing Eqs. (\\ref{e24}) and (\\ref{e26}) and the first two of the BBGKY \nhierarchy equations, we obtain a closed set of two equations involving \ntwo unknowns, $f_{1}$ and $g_{2,\\,\\vec{R}_{1}}$ given by \n \n\\begin{eqnarray} \n\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,f_{1})\\,+\\,\\frac{\\partial}{\\partial\\rho}(\\dot{\\rho}\\,f_{1})\\,=\\,\\int d\\vec{R}_{1}\\,\\tilde{T}_{-,\\,1}\\,[\\,nf_{1}\\,+\\,g_{2,\\,\\vec{R}_{1}}]\\, \n\\label{e33} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\vec{\\nabla}_{\\vec{r}}\\cdot(\\dot{\\vec{r}}\\,g_{2,\\,\\vec{R}_{1}})\\,+\\,\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,g_{2,\\,\\vec{R}_{1}})\\,+\\,\\frac{\\partial}{\\partial\\rho}(\\dot{\\rho}\\,g_{2,\\,\\vec{R}_{1}})\\,-\\,n\\int d\\vec{R}_{2}\\,\\tilde{T}_{-,\\,2}\\,g_{2,\\,\\vec{R}_{1}}\\,=\\,n\\,\\tilde{T} \n_{-,\\,1}\\,f_{1}\\,. \n\\label{e34} \n\\end{eqnarray} \n \nIn the derivation of Eq. (\\ref{e34}) from the second hierarchy \nequation not only have we dropped $g_3$ as discussed above, we also \ndropped a term of the form \n$\\tilde{T}_{-,\\,1}\\,g_{2,\\,\\vec{R_1}}$. This term provides ``repeated \nring'' corrections to the ring contributions to \n$g_{2,\\,\\vec{R_1}}$. These are of the same order as terms neglected \nby dropping $g_{3}$ \\cite{ED_physica_72,PD_unpublished}, and should be \nneglected for consistency. We also dropped the time derivatives in the \nequations, so we are now looking for the distribution and correlation \nfunctions appropriate for the NESS. \n \nIn Section 4, we will solve Eqs. (\\ref{e33}) and (\\ref{e34}) in order \nto calculate the ring contributions to the positive Lyapunov exponent. \nBefore doing so, it is useful to write down the usual form of the ring \nequations in $(\\vec{r}, \\vec{v})$-space, which can be obtained by \nintegrating Eqs. (\\ref{e33}) and (\\ref{e34}) over all values of the \nradius of curvature, $0\\leq \\rho < \\infty$. We define the usual \nsingle-particle distribution function by, \n$F_{1}=\\int_{\\rho>0}d\\rho\\,f_{1}$ and the pair-correlation function \n$G_{2,\\,\\vec{R}_{1}}=\\int_{\\rho>0}d\\rho\\,g_{2,\\,\\vec{R}_{1}}$. By \nimposing the boundary conditions that both $f_{1}$ and \n$g_{2,\\,\\vec{R}_{1}}$ go to zero as $\\rho\\rightarrow 0$ and as \n$\\rho\\rightarrow\\infty$, we obtain \n\\begin{eqnarray} \n\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,F_{1})\\,=\\,\\int d\\vec{R}_{1}\\,\\overline{T}_{-,\\,1}\\,[\\,nF_{1}\\,+\\,G_{2,\\,\\vec{R}_{1}}]\\, \n\\label{e35} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\vec{\\nabla}_{\\vec{r}}\\cdot(\\dot{\\vec{r}}\\,G_{2,\\,\\vec{R}_{1}})\\,+\\,\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,G_{2,\\,\\vec{R}_{1}})\\,-\\,n\\int d\\vec{R}_{2}\\,\\overline{T}_{-,\\,2}\\,G_{2,\\,\\vec{R}_{1}}\\,=\\,n\\,\\overline{T}_{-,\\,1}\\,F_{1}\\,. \n\\label{e36} \n\\end{eqnarray} \nThe actions of $\\overline{T}_{-,\\,1}$ or $\\overline{T}_{-,\\,2}$ on \n$F_{1}$ and $G_{2,\\,\\vec{R}_{1}}$ can be obtained by appropriately \nintegrating $\\tilde{T}_{-,\\,1}\\,f_{1}$, \n$\\tilde{T}_{-,\\,1}\\,g_{2,\\,\\vec{R}_{1}}$ or \n$\\tilde{T}_{-,\\,2}\\,g_{2,\\,\\vec{R}_{1}}$ over $\\rho$ from $0$ to \n$\\infty$ using the definitions in Eqs. (\\ref{e17}) and \n(\\ref{e18}). $\\overline{T}_{-,\\,1}$ and $\\overline{T}_{-,\\,2}$ are \nthe analogs in ($\\vec{r},\\,\\vec{v}$) space of $\\tilde{T}_{-,\\,1}$ and \n$\\tilde{T}_{-,\\,2}$ (see Eqs. (\\ref{e17}) and (\\ref{e18})), i.e., \n\\begin{eqnarray} \n\\overline{T}_{-,i}\\,=\\,a\\int_{\\vec{v}\\cdot\\hat{\\sigma}_{i}\\,>\\,0}d\\hat{\\sigma}_{i}\\,|\\vec{v}\\cdot\\hat{\\sigma}_{i}|\\,\\bigg\\{\\,\\delta\\,(a\\hat{\\sigma}_{i}\\,-\\,(\\vec{r}\\,-\\,\\vec{R}_{i}))\\,b_{\\sigma_{i}}\\,-\\,\\delta(a\\hat{\\sigma}_{i}\\,+\\,(\\vec{r}\\,-\\,\\vec{R}_{i}))\\bigg\\}\\,. \n\\label{e37} \n\\end{eqnarray} \nIn future applications however, we will drop the $a\\hat{\\sigma}_{i}$ \nterms from the arguments of both \n$\\delta\\,(a\\hat{\\sigma}_{i}\\,\\pm\\,(\\vec{r}\\,-\\,\\vec{R}_{i}))$ in \n$\\tilde{T}_{-,\\,i}$ and $\\overline{T}_{-,\\,i}$ operators since they \nlead to corrections similar to excluded volume terms, neglected already. \n \n\\section{Effects of long range time correlation on $\\lambda_{+}$ and $\\lambda_{-}$} \n \nWe now concentrate on the solution of the BBGKY equations for the \ndistribution functions that determine the Lyapunov exponents. The \nsolutions of Eqs. (\\ref{e33}) and (\\ref{e34}) are to be obtained as \nexpansions in two small variables, $na^2$ and \n$\\tilde{\\varepsilon}$. The density expansion will give the corrections \nto the previously obtained Boltzmann regime results from the ELBE, and \nthe $\\tilde{\\varepsilon}$ expansion will provide the field dependence \nof these corrections. We therefore write the density expansions of \n$f_{1}$ and $g_{2}$ (hereafter we drop the subscript $\\vec{R}_{1}$ \nfrom $g_{2,\\,\\vec{R}_{1}}$) as \n\\begin{eqnarray} \nf_{1}\\,=\\,f_{1}^{\\mbox{\\tiny(B)}}\\,+\\,f_{1}^{\\mbox{\\tiny(R)}}\\,+\\,.\\,.\\,. \n\\hspace{1cm}{\\mbox{and}}\\hspace{1cm}g_{2}\\,=\\,g_{2}^{\\mbox{\\tiny{(R)}}}\\,+\\,.\\,.\\,.\\,, \n\\label{e38} \n\\end{eqnarray} \nwhere the superscript B indicates the lowest density result for \n$f_{1}$ as given by the ELBE, and the superscript R denotes the ring \ncontribution. At the order in density of interest here, the quantities \nindicated explicitly in the above equations satisfy \n\\begin{eqnarray} \n\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,f^{\\mbox{\\tiny{(B)}}}_{1})\\,+\\,\\frac{\\partial}{\\partial\\rho}(\\dot{\\rho}\\,f^{\\mbox{\\tiny(B)}}_{1})\\,=\\,n\\int d\\vec{R}_{1}\\,\\tilde{T}_{-,\\,1}\\,f^{\\mbox{\\tiny (B)}}_{1}, \n\\label{e39} \n\\end{eqnarray} \n\\begin{eqnarray} \n\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,f^{\\mbox{\\tiny(R)}}_{1})\\,+\\,\\frac{\\partial}{\\partial\\rho}(\\dot{\\rho}\\,f^{\\mbox{\\tiny{(R)}}}_{1})\\,=\\,\\int d\\vec{R}_{1}\\,\\tilde{T}_{-,\\,1}\\,[\\,nf^{\\mbox{\\tiny{(R)}}}_{1}\\,+\\,g^{\\mbox{\\tiny(R)}}_{2}] \n\\label{e40} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\vec{\\nabla}_{\\vec{r}}\\cdot(\\dot{\\vec{r}}\\,g^{\\mbox{\\tiny{(R)}}}_{2})\\,+\\,\\vec{\\nabla}_{\\vec{v}}\\cdot(\\dot{\\vec{v}}\\,g^{\\mbox{\\tiny{(R)}}}_{2})\\,+\\,\\frac{\\partial}{\\partial\\rho}(\\dot{\\rho}\\,g^{\\mbox{\\tiny(R)}}_{2})\\,-\\,n\\int d\\vec{R}_{2}\\,\\tilde{T}_{-,\\,2}\\,g^{\\mbox{\\tiny(R)}}_{2}\\,=\\,n\\,\\tilde{T}_{-,\\,1}\\,f^{\\mbox{\\tiny(B)}}_{1}\\,. \n\\label{e41} \n\\end{eqnarray} \nOur aim here is to solve Eqs. (\\ref{e40}) and (\\ref{e41}) using the \nresults of the ELBE for $f^{\\mbox{\\tiny (B)}}_1$. We suppose further \nthat each of these functions possesses an expansion in powers of \n$\\varepsilon$ as \n\\begin{eqnarray} \nf^{\\mbox{\\tiny (B,\\,R)}}_{1}&=&f^{\\mbox{\\tiny(B,\\,R)}}_{1,\\,0}\\,+\\,\\varepsilon\\,f^{\\mbox{\\tiny(B,\\,R)}}_{1,\\,1}\\,+\\,\\varepsilon^{2}\\,f^{\\mbox{\\tiny(B,\\,R)}}_{1,\\,2}\\,+\\,.\\,.\\,. \n\\label{e42} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \ng^{\\mbox{\\tiny (R)}}_{2}&=&g^{\\mbox{\\tiny(R)}}_{2,\\,0}\\,+\\,\\varepsilon\\,g^{\\mbox{\\tiny(R)}}_{2,\\,1}\\,+\\,\\varepsilon^2\\,g^{\\mbox{\\tiny(R)}}_{2,\\,2}\\,+\\,.\\,.\\,.\\, \n\\label{e43} \n\\end{eqnarray} \nThe functions $f^{\\mbox{\\tiny (B)}}_{1,\\,i}$ have been previously \nobtained as the $\\varepsilon$ solutions of the ELBE. Since we will be \ndealing with $g_{2}$ only in the context of of the ring term, we drop \nthe superscript R from now on. \n \nAs mentioned above, we will neglect the term $a\\hat{\\sigma}$ within \nthe arguments of the $\\delta$-functions appearing in each of the \nbinary collision operators $\\tilde{T}_{-}$ and $\\overline{T}_{-}$, \nso as to take the moving particle to be located at the same point as \nthe center of the appropriate scatterer at collision. The terms \nneglected by this approximation lead to higher density corrections to \nthe terms we will obtain below. Secondly, an inspection of the radius \nof curvature delta function in the expression for the ``gain'' part of \nthe binary collision operator, Eq. (\\ref{e19}), shows that this term \nis only non-vanishing when $\\rho \\leq \\displaystyle{\\frac{a}{2}}$, and \nthat the dominant contribution to the $\\rho'$ integration comes from \nthe region $\\rho' \\sim l$. Naturally, \n$\\displaystyle{\\frac{\\rho'\\,a\\,\\cos\\phi_{i}}{a\\,\\cos\\phi_{i}\\,+\\,2\\rho'}\\sim\\frac{a\\,\\cos\\phi_{i}}{2}(1+O(n))}$ \nin the argument of the delta function. In the Boltzmann level \napproximation this $O(n)$ term may therefore be neglected and it can \nbe shown not to contribute to the leading field-dependent ring term \neffects on the Lyapunov exponents. Therefore, we will neglect it in \nwhat follows. Under this approximation, the gain part of the binary \ncollision operator \\cite{vBDCPD_prl_96} acts on an arbitrary function \n$h(\\vec{r},\\vec{v},\\rho)$ as \n\\begin{eqnarray} \n\\tilde{T}^{(+)}_{-,\\,i}\\,h(\\vec{r},\\vec{v},\\rho)\\,\\approx\\,\\delta(\\vec{r}\\,-\\,\\vec{R}_i)\\,\\Theta\\big(\\frac{a}{2}\\,-\\,\\rho\\big)\\,I(\\rho)\\,[\\,H(\\vec{r},\\vec{v}_{+})\\,+\\,H(\\vec{r},\\vec{v}_{-})\\,]\\,\\equiv\\,\\delta(\\vec{r}-\\vec{R}_{i})\\,\\Gamma(\\rho, H)\\,. \n\\label{e1new} \n\\end{eqnarray} \nHere \n\\begin{eqnarray} \n\\vec{v}_{\\pm}\\,=\\,\\vec{v}\\,-\\,2\\,(\\vec{v}\\cdot\\hat{\\sigma}_{i,\\,\\pm})\\,\\hat{\\sigma}_{i,\\,\\pm}\\,, \n\\label{e2new} \n\\end{eqnarray} \nand $\\hat{\\sigma}_{i,\\,\\pm}$ is defined by the condition that the \nscattering angle \n$\\phi=\\pm\\cos^{-1}\\bigg(\\displaystyle{\\frac{2\\rho}{a}}\\bigg)$. Also \n\\begin{eqnarray} \nH(\\vec{r},\\vec{v})\\,=\\,\\int_{0}^{\\infty} d\\rho'\\,h(\\vec{r}, \\vec{v}, \n\\rho') \n\\label{e3new} \n\\end{eqnarray} \nand \n\\begin{equation} \nI(\\rho)\\,=\\,\\frac{4v\\rho}{a\\,\\sqrt{1 -\\big(\\frac{2\\rho}{a}\\big)^{2}}}. \n\\label{e4new} \n\\end{equation} \nFinally, we express the velocity vector $\\vec{v}$ in terms of the \nangle $\\theta$ that it makes with the direction of the electric field \nso as to obtain the following set of equations for the terms in the \n$\\varepsilon$-expansion of $g_{2}$ \n\\begin{eqnarray} \n\\bigg[\\,\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,+\\,v\\frac{\\partial}{\\partial\\rho}\\,+\\,2nav\\,\\bigg]\\,g_{2,\\,0}\\,=\\,-\\,2nav\\,\\delta(\\vec{r}\\,-\\,\\vec{R}_{1})\\,f^{\\mbox{\\tiny(B)}}_{1,\\,0}\\,+\\,n\\,\\delta(\\vec{r}-\\vec{R}_{1})\\,\\Gamma(\\rho,\\,F^{\\mbox{\\tiny (B)}}_{1,0})\\,, \n\\label{e44} \n\\end{eqnarray} \n\\begin{eqnarray} \n\\bigg[\\,\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,+\\,v\\frac{\\partial}{\\partial\\rho}\\,+\\,2nav\\,\\bigg]\\,g_{2,\\,1}\\,=\\,-\\,\\frac{\\partial}{\\partial\\rho}\\bigg(\\rho\\cos\\theta\\,g_{2,\\,0}\\bigg)\\,+\\,\\frac{\\partial}{\\partial\\theta}\\bigg(\\sin\\theta\\,g_{2,\\,0}\\bigg)\\,\\nonumber \\\\ \n&&\\hspace{-7.5cm}-\\,2nav\\,\\delta(\\vec{r}\\,-\\,\\vec{R}_{1})\\,f^{\\mbox{\\tiny(B)}}_{1,\\,1}\\,+\\,n\\,\\delta(\\vec{r}-\\vec{R}_{1})\\,\\Gamma(\\rho,F^{\\mbox{\\tiny(B)}}_{1,1})\\,, \n\\label{e45} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\bigg[\\,\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,+\\,v\\frac{\\partial}{\\partial\\rho}\\,+\\,2nav\\,\\bigg]\\,g_{2,\\,2}\\,=\\,-\\,\\frac{\\partial}{\\partial\\rho}\\bigg(\\rho\\cos\\theta\\,g_{2,\\,1}\\,+\\,\\frac{\\rho^2\\sin^{2}\\theta}{v}\\,g_{2,\\,0}\\bigg)\\,+\\,\\frac{\\partial}{\\partial\\theta}\\bigg(\\sin\\theta g_{2,\\,1}\\bigg)\\,\\nonumber \n\\\\&&\\hspace{-9.5cm}-\\,2nav\\,\\delta(\\vec{r}\\,-\\,\\vec{R}_{1})\\,f^{\\mbox{\\tiny(B)}}_{1,\\,2}\\,+\\,n\\,\\delta(\\vec{r}-\\vec{R}_{1})\\,\\Gamma(\\rho,F^{\\mbox{\\tiny(B)}}_{1,2})\\,. \n\\label{e46} \n\\end{eqnarray} \nWe notice that the equations thus obtained are linear inhomogeneous \ndifferential equations of the form $L\\,g_{2,\\,j}=b_{j}$ ($j=0,1,2$), \nwhere \n$L\\,=\\,\\big[\\,\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,+\\,v\\displaystyle{\\frac{\\partial}{\\partial\\rho}}\\,+\\,2nav\\,\\big]$ \nis a linear differential operator and $b_{j}$ for $j=0,1,2$ are the \ninhomogeneous terms on the r.h.s. of Eqs. (\\ref{e44}), (\\ref{e45}) \nand(\\ref{e46}) respectively. \n \nWe will need to solve Eqs. (\\ref{e44}-\\ref{e46}), in conjunction with \nthe equations for $G_{2,\\,0}$, $G_{2,\\,1}$ and $G_{2,\\,2}$ obtained by \ndirectly integrating Eqs. (\\ref{e44}-\\ref{e46}) over $\\rho$ from 0 to \n$\\infty$. The equations for the corresponding $G_{2,\\,i}$ are then \n\\begin{eqnarray} \n\\bigg[\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,-\\,n\\int \n\\vec{dR_{2}}\\,\\overline{T}_{-,\\,2}\\bigg]\\,G_{2,\\,0}&=&n\\,\\overline{T}_{-,\\,1}\\,F^{\\mbox{\\tiny (B)}}_{1,\\,0}\\,, \n\\label{e47} \n\\end{eqnarray} \n\\begin{eqnarray} \n\\bigg[\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,-\\,n\\int\\vec{dR_{2}}\\,\\overline{T}_{-,\\,2}\\bigg]\\,G_{2,\\,1}&=&\\frac{\\partial}{\\partial\\theta}\\,\\bigg(\\sin\\theta\\,G_{2,\\,0}\\bigg)\\,+\\,n\\,\\overline{T}_{-,\\,1}\\,F^{\\mbox{\\tiny(B)}}_{1,\\,1} \n\\label{e48} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\bigg[\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,-\\,n\\int\\vec{dR_{2}}\\,\\overline{T}_{-,\\,2}\\bigg]\\,G_{2,\\,2}&=&\\frac{\\partial}{\\partial\\theta}\\,\\bigg(\\sin\\theta\\,G_{2,\\,1}\\bigg)\\,+\\,n\\,\\overline{T}_{-,\\,1}\\,F^{\\mbox{\\tiny (B)}}_{1,\\,2}\\,. \n\\label{e49} \n\\end{eqnarray} \nThe equations for $G_{2,\\,0}$, $G_{2,\\,1}$ and $G_{2,\\,2}$ are also \nlinear differential equations of the form \n$L^{\\begin{Sp}\\prime\\end{Sp}}\\,G_{2,\\,j}=B_{j}$, where \n$L^{\\begin{Sp}\\prime\\end{Sp}}\\,=\\,\\big[\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,-\\,n\\int\\vec{dR_{2}}\\,\\overline{T}_{-,\\,2}\\big]$ and $B_{j}$ for $j\\,=\\,0, 1, \n2$ are the inhomogeneous terms on the r.h.s. of Eqs. (\\ref{e47}), \n(\\ref{e48}) and (\\ref{e49}) respectively. \n \nTo solve these equations, we take Fourier transforms of $g_{2}$ and of \n$G_2$ in the variables $\\vec{r}$ and in the velocity angle, \n$\\theta$. That is, we define \n$\\tilde{g}_{2}(\\vec{k})\\,=\\,\\displaystyle{\\frac{1}{\\sqrt{V}}}\\,\\int_{V}\\vec{dr}\\,g_{2}\\,e^{-\\,i\\vec{k}\\,\\cdot\\,(\\vec{r}\\,-\\,\\vec{R}_{1})}$ \nand calculate $\\tilde{g}_{2,\\,0}(\\vec{k})$, \n$\\tilde{g}_{2,\\,1}(\\vec{k})$ and $\\tilde{g}_{2,\\,2}(\\vec{k})$ in the \n$\\vec{k}$-basis, using periodic boundary conditions. Similarly, we \ndefine the $m$-th angular mode of $\\tilde{g}_{2}(\\vec{k})$ as \n$\\tilde{g}^{(m)}_{2}(\\vec{k})\\,=\\,\\displaystyle{\\frac{1}{\\sqrt{2\\pi}}}\\int d\\theta\\,e^{-\\,im\\theta}\\,\\tilde{g}_{2}(\\vec{k})$. We also \ndefine $\\tilde{G}^{(m)}_{2}(\\vec{k})$ in an analogous way. Thus, \ncorresponding to Eqs. (\\ref{e44}-\\ref{e46}) and Eqs. \n(\\ref{e47}-\\ref{e49}), we have two sets of three equations to be \nsolved, one involving $\\tilde{g}_{2,\\,0}(\\vec{k})$, \n$\\tilde{g}_{2,\\,1}(\\vec{k})$ and $\\tilde{g}_{2,\\,2}(\\vec{k})$, and the \nother involving $\\tilde{G}_{2,\\,0}(\\vec{k})$, \n$\\tilde{G}_{2,\\,1}(\\vec{k})$ and $\\tilde{G}_{2,\\,2}(\\vec{k})$, in \n$(\\vec{k}, m)$ basis. In this basis, the operators \n$L_{\\vec{k}}\\,=\\,\\big[\\,i\\vec{k}\\cdot\\vec{v}\\,+\\,v\\displaystyle{\\frac{\\partial}{\\partial \\rho}}\\,+\\,2nav\\big]$ and \n$L^{\\begin{Sp}\\prime\\end{Sp}}_{\\vec{k}}\\,=\\,\\big[\\,i\\vec{k}\\cdot\\vec{v}\\,-\\,n\\int\\vec{dR_{2}}\\,\\overline{T}_{-,\\,2}\\big]$ \nare both infinite dimensional matrices in $m$-space and both of them \nhave non-zero off-diagonal elements due to the term \n$i\\vec{k}\\cdot\\vec{v}$ generated from the operator \n$\\vec{v}\\cdot\\vec{\\nabla}_{\\vec{r}}\\,$. However, it is easily seen \nthat these off-diagonal elements are proportional to \n$\\delta_{m,\\,m+1}$ and $\\delta_{m,\\,m-1}$ and they are easily treated. \n \nA further simplification can be made by noticing that the schematic \nforms of the solutions are \n$\\tilde{g}_{2,\\,j}(\\vec{k})=[L_{\\vec{k}}]^{-1}\\,b_{j}(\\vec{k})$ and \n$\\tilde{G}_{2,\\,j}(\\vec{k})=[L^{\\begin{Sp}\\prime\\end{Sp}}_{\\vec{k}}]^{-1}\\,B_{j}(\\vec{k})$ \nand hence the dominant parts of $\\tilde{g}_{2,\\,j}(\\vec{k})$ and \n$\\tilde{G}_{2,\\,j}(\\vec{k})$ will come, loosely speaking, from the \neigenfunctions of $L_{\\vec{k}}$ and \n$L^{\\begin{Sp}\\prime\\end{Sp}}_{\\vec{k}}$ having the smallest \neigenvalues. The lowest eigenvalues of \n$L^{\\begin{Sp}\\prime\\end{Sp}}_{\\vec{k}}$ are $\\propto k^{2}$ due to \nthe contributions from the hydrodynamic modes \\cite{EW_pl_71}. Thus, \nto capture the dominant part of the solutions we should solve the \nequations in the range $k = |\\vec{k}| << l^{-1}$, the inverse mean \nfree path and use perturbation expansions in the small parameter $kl$. \nWe will not, in our analysis, follow the mode expansion technique, as \nit is simpler to calculate $G_{2}$ directly. However, one can use mode \nexpansions and one finds that the results of both the methods agree. \n \n\\subsection{Solution for $G_{2}$} \n \n\\noindent To solve for $G_{2}$ first we need to know the solutions of \nthe Lorentz-Boltzmann equation for $F^{\\mbox{\\tiny (B)}}_{1,\\,0}$, \n$F^{\\mbox{\\tiny (B)}}_{1,\\,1}$ and $F^{\\mbox{\\tiny (B)}}_{1,\\,2}$. \nThese are given by \\cite{vBDCPD_prl_96} \n\\begin{eqnarray} \nF^{\\mbox{\\tiny \n(B)}}_{1,\\,0}\\,=\\,\\frac{1}{2\\pi},\\hspace{0.8cm}F^{\\mbox{\\tiny(B)}}_{1,\\,1}\\,=\\,\\frac{3}{16\\pi nav}\\,\\cos\\theta\\hspace{0.6cm}{\\mbox{and}}\\hspace{0.6cm}F^{\\mbox{\\tiny(B)}}_{1,\\,2}\\,=\\,\\frac{45}{512\\pi (nav)^{2}}\\,\\cos 2\\theta\\,. \n\\label{e50} \n\\end{eqnarray} \n \nWe note that in $m$-space, defined above, the $m$-th diagonal element \nof the infinite matrix $L^{'}_{\\vec{k}}$ is \n$\\displaystyle{\\frac{4m^{2}}{(4m^{2}\\,-\\,1)}}\\displaystyle{\\frac{v}{l}}$ \nwhile the off-diagonal elements are $\\,ikv\\, \\delta_{m,m\\pm1}$. Thus \nan expansion in $\\tilde{k}=kl$ can be easily obtained by considering \nsuccessively larger parts of the matrix $L^{'}_{\\vec{k}}$ in the \nindex $m$, starting with $3\\times 3$, $5\\times 5$ matrices and so on, \nchosen in such a way that the element of $L^{'}_{\\vec{k}}$ \ncorresponding to $m=0$ appears as the center element of these \nmatrices. As we want to make our results correct up to \n$O(\\tilde{k}^{0})$, we need to increase the size of these matrices \ntill the expressions of $\\tilde{G}_{2,\\,j}(\\vec{k})$ obtained from \n$\\tilde{G}_{2,\\,j}(\\vec{k})=[L^{\\begin{Sp}\\prime\\end{Sp}}_{\\vec{k}}]^{-1}\\,B_{j}(\\vec{k})$ \n(for $j=0, 1, 2$) converges up to $O(\\tilde{k}^{0})$. Also, as we want \nto obtain the expression of $\\lambda_{+}$ and $\\lambda_{-}$ in the \nleading field-dependent order, which is $ \\varepsilon^{2}$; we need \nthe solutions of all the $m$-modes of $\\tilde{G}_{2,\\,0}(\\vec{k})$, \n$\\tilde{G}_{2,\\,1}(\\vec{k})$ and $\\tilde{G}_{2,\\,2}(\\vec{k})$ that are \nnecessary to obtain all the terms of $f^{\\mbox{\\tiny (R)}}_{1}$ that \nare $\\propto\\varepsilon^{2}$ and contribute to this leading \nfield-dependent order of $\\lambda_{+}$ and $\\lambda_{-}$. In more \nexplicit form, this means that we definitely need the solutions of \n$\\tilde{G}^{(m\\,=\\,0)}_{2,\\,0}(\\vec{k})$, \n$\\tilde{G}^{(m\\,=\\,0)}_{2,\\,1}(\\vec{k})$, \n$\\tilde{G}^{(m\\,=\\,\\pm1)}_{2,\\,1}(\\vec{k})$, \n$\\tilde{G}^{(m\\,=\\,\\pm2)}_{2,\\,1}(\\vec{k})$ and \n$\\tilde{G}^{(m\\,=\\,0)}_{2,\\,2}(\\vec{k})$ up to \n$O(\\tilde{k}^{0})$. However, once we present these solutions, from \nthe structure and properties of them, it will turn out that we will \nalso need the expressions of $\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})$ \nin the leading order of $\\tilde{k}\\,$ for $j\\,=\\,2, 3,\\,.\\,.$, to \nconsistently obtain all the terms, that are $\\propto\\varepsilon^{2}$. \n \nAt the $\\varepsilon^{0}$ or equilibrium order, we find \n\\begin{eqnarray} \n\\tilde{G}^{(m)}_{2,\\,0}(\\vec{k})\\,=\\,0 \\hspace{1cm}\\forall m\\,. \n\\label{e51} \n\\end{eqnarray} \nProceeding to order $\\varepsilon$, we find that \n$\\tilde{G}^{(m)}_{2,\\,1}(\\vec{k})$'s obey \n\\begin{eqnarray} \n\\frac{iv}{2}\\,\\bigg[(k_{x}\\,+\\,ik_{y})\\,\\tilde{G}^{(m+1)}_{2,\\,1}(\\vec{k})\\,+\\,(k_{x}\\,-\\,ik_{y})\\,\\tilde{G}^{(m-1)}_{2,\\,1}(\\vec{k})\\bigg]\\,+\\,\\frac{8navm^{2}}{4m^{2}\\,-\\,1}\\,\\tilde{G}^{(m)}_{2,\\,1}(\\vec{k})\\nonumber\\\\ \n&&\\hspace{-4cm}\\,=\\,-\\,\\frac{1}{2\\,\\sqrt{2\\pi V}}\\,\\big(\\delta_{m,\\,1}\\,+\\,\\delta_{m,\\,-1}\\big)\\,, \n\\label{e52} \n\\end{eqnarray} \nwhere it turns out that we need a $5\\times 5$ matrix block \ncorresponding to $m=-2,\\,-1,\\,0,\\,1,$ and $2$ to get the solutions of \n$\\tilde{G}^{(m)}_{2,\\,1}(\\vec{k})$ up to $O(k^{0})$ that are relevant \nfor us, yielding \n\\begin{eqnarray} \n\\tilde{G}^{(m=\\,0)}_{2,\\,1}(\\vec{k})\\,=\\,\\frac{ik_{x}}{vk^{2}\\,\\sqrt{2\\pi V}}\\,, \\nonumber \n\\end{eqnarray} \n\\begin{eqnarray} \n\\tilde{G}^{(m=1)}_{2,\\,1}(\\vec{k})\\,=\\,-\\,\\frac{3ik_{y}(k_{x}\\,-\\,ik_{y})}{16navk^{2}\\,\\sqrt{2\\pi V}}\\,,\\hspace{1.5cm}\\tilde{G}^{(m=\\,-1)}_{2,\\,1}(\\vec{k})\\,=\\,\\frac{3ik_{y}(k_{x} + ik_{y})}{16navk^{2}\\,\\sqrt{2\\pi V}} \\nonumber \n\\end{eqnarray} \n\\begin{eqnarray} \n\\tilde{G}^{(m=\\,2)}_{2,\\,1}(\\vec{k})\\,=\\,-\\,\\frac{45k_{y}(k_{x}\\,-\\,ik_{y})^{2}}{1024(na)^{2}vk^{2}\\,\\sqrt{2\\pi V}}\\hspace{0.7cm}{\\mbox{and}}\\hspace{0.7cm}\\tilde{G}^{(m=\\,-2)}_{2,\\,1}(\\vec{k})\\,=\\,\\frac{45k_{y}(k_{x}\\,+\\,ik_{y})^{2}}{1024(na)^{2}vk^{2}\\,\\sqrt{2\\pi V}}\\,. \n\\label{e53} \n\\end{eqnarray} \nNotice that we have also calculated \n$\\tilde{G}^{(m=\\,-\\,2)}_{2,\\,1}(\\vec{k})$ and \n$\\tilde{G}^{(m=\\,2)}_{2,\\,1}(\\vec{k})$, even though they are $O(k)$, \nbecause they affect the $O(k^{0})$ solution for \n$\\tilde{G}^{(m=\\,0)}_{2,\\,2}(\\vec{k})$. \n \nFor order $\\varepsilon^2$, the relevant \n$\\tilde{G}^{(m)}_{2,\\,2}(\\vec{k})$'s are then calculated using Eq. \n(\\ref{e53}) and considering a $5\\times 5$ matrix block of \n$L^{\\begin{Sp}\\prime\\end{Sp}}_{\\vec{k}}$. There we need only the \nsolution for $\\tilde{G}^{(m=\\,0)}_{2,\\,2}(\\vec{k})$ : \n\\begin{eqnarray} \n\\tilde{G}^{(m=\\,0)}_{2,\\,2}(\\vec{k})\\,=\\,\\frac{k^{2}_{x}}{v^{2}k^{4}\\,\\sqrt{2\\pi V}}\\,+\\,\\frac{45\\,(2k_{x}^{2}\\,-\\,5k_{y}^{2})}{1024(nav)^{2}k^{2}\\,\\sqrt{2\\pi V}}\\,. \n\\label{e54} \n\\end{eqnarray} \nExamining the properties of the solutions, Eqs. (\\ref{e53}-\\ref{e54}) \nand observing from Eqs. (\\ref{e47}-\\ref{e49}) the way the solution of \n$G_{2,\\,j}$ affects the solution of $G_{2,\\,(j\\,+\\,1)}$, one sees that \nthe leading power of $k$ in the expression of \n$\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})$ for ($j=1, 2, 3....$) is \n$k^{-\\,2j}$. However, in the expression of $G_{2}$, \n$\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})$ appears with a factor of \n$\\varepsilon^{2j}$. When $G_{2}$ is finally calculated, after a \nsummation of the appropriate $\\vec{k}$-values\\footnote{To see how the \n$\\vec{k}$-integration is performed, see the last paragraph of Section \n4.2}, the contribution of the sum of all the effects coming from the \n$O(k^{-\\,2j})$ terms of the $\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})$'s \nis seen to be in the same order of density of scatterers as the \n$O(k^{0})$ term on the r.h.s. of Eq. (\\ref{e54}). In fact, it also \nturns out that the $O(k^{-\\,2j})$ terms of the \n$\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})$'s are the only ones among the \n$\\tilde{G}^{(m)}_{2,\\,j}(\\vec{k})$'s that contribute to \n$f^{\\mbox{\\tiny (R)}}_{1}$ in the order of $\\varepsilon^{2}$. This \nimplies that along with the solutions, Eqs. (\\ref{e51}), (\\ref{e53}) \nand (\\ref{e54}), we also need to include the $O(k^{-\\,2j})$ term of \n$\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})$ to be consistent. If one just \nconsiders this $O(k^{-\\,2j})$ term in \n$\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})$, then it is easy to see that \nthey satisfy a recurrence relation for $j\\geq 1$ : \n\\begin{eqnarray} \n\\tilde{G}^{(m=\\,0)}_{2,\\,2\\,(j\\,+\\,1)}(\\vec{k})&=&-\\,\\frac{k^{2}_{x}}{v^{2}k^{4}}\\,\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})\\,, \n\\label{e55} \n\\end{eqnarray} \ni.e, \n\\begin{eqnarray} \n\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})&=&\\frac{(-1)^{\\,j\\,-\\,1}}{\\sqrt{2\\pi V}}\\,\\bigg(\\frac{k^{2}_{x}}{v^{2}k^{4}}\\bigg)^{j}\\,. \n\\label{e56} \n\\end{eqnarray} \nThe solutions, Eqs. (\\ref{e51}), (\\ref{e53}), (\\ref{e54}) and \n(\\ref{e56}), are then used to determine the integration constants that \narise when we solve the differential Eqs. (\\ref{e44}-\\ref{e46}). It \nis important to note that the first term on the right hand side of \nEq. (\\ref{e54}) is inversely proportional to $k^2$. This is the origin \nof the logarithmic terms we find below. \n \n\\subsection{Solution for $g_{2}$} \n \nHere we apply the same procedure to solve for the \n$\\tilde{g}_{2}(\\vec{k})$'s from the equations \n$L_{\\vec{k}}\\,\\tilde{g}_{2}(\\vec{k})\\,=\\,b_{j}(\\vec{k})$ for \n$j\\,=\\,0,\\,1,\\,2$. This time, the elements of $L_{\\vec{k}}$ are \ndifferential operators in the variable $\\rho$ and the corresponding \nconstants of integrations are determined using the solutions of \n$\\tilde{G}^{(m)}_{2}(\\vec{k})$'s while maintaining that \n$\\tilde{g}^{(m)}_{2}(\\vec{k})$'s go to zero as $\\rho\\rightarrow 0$ and \nas $\\rho\\rightarrow\\infty$. We also note that for our purpose, \nsolutions of the $\\tilde{g}_{2}(\\vec{k})$'s are only needed for \n$\\rho>\\displaystyle{\\frac{a}{2}}$ as the solution of the \n$\\tilde{g}_{2}(\\vec{k})$'s for $\\rho<\\displaystyle{\\frac{a}{2}}$ gives \nrise to higher order density corrections than under consideration \nhere. These solutions can also be obtained by the mode expansion \ntechnique discussed above in the paragraph preceding Section 4.1. \nHowever, as it is fairly straightforward to solve the differential \nEqs. (\\ref{e44}-\\ref{e46}) for $\\rho>\\displaystyle{\\frac{a}{2}}$, we \ndirectly write down the necessary solutions up to $O(k^{0})$. \n \nWe obtain, for $\\rho>\\displaystyle{\\frac{a}{2}}$, \n\\begin{eqnarray} \n\\tilde{g}^{(m)}_{2,\\,0}(\\vec{k})\\,=\\,\\frac{2na}{\\sqrt{2\\pi V}}\\,(1\\,-\\,2na\\rho)\\,e^{-\\,2na\\rho}\\,, \\nonumber \n\\end{eqnarray} \n\\begin{eqnarray} \n\\tilde{g}^{(m=\\,-1)}_{2,\\,1}(\\vec{k})\\,=\\,\\frac{1}{v\\,\\sqrt{2\\pi V}}\\,\\bigg[-\\,\\frac{k_{x}(k_{x}\\,+\\,ik_{y})}{8k^{2}}\\,+\\,\\frac{k_{x}(k_{x}\\,+\\,ik_{y})}{2k^{2}}\\,2na\\rho\\,+\\,\\frac{2na\\rho}{8}\\,\\nonumber\\\\ \n&& {\\hspace{-4cm}}+\\,\\frac{(2na\\rho)^{2}}{2}\\,-\\,\\frac{(2na\\rho)^{3}}{4}\\,\\bigg]\\,e^{-\\,2na\\rho}\\,, \\nonumber \n\\end{eqnarray} \n\\begin{eqnarray} \n\\tilde{g}^{(m=\\,0)}_{2,\\,1}(\\vec{k})\\,=\\,\\frac{ik_{x}}{vk^{2}\\,\\sqrt{2\\pi V}}\\,2na\\,e^{-\\,2na\\rho}\\,, \\nonumber \n\\end{eqnarray} \n\\begin{eqnarray} \n\\tilde{g}^{(m=\\,1)}_{2,\\,1}(\\vec{k})\\,=\\,\\frac{1}{v\\,\\sqrt{2\\pi V}}\\,\\bigg[-\\,\\frac{k_{x}(k_{x}\\,-\\,ik_{y})}{8k^{2}}\\,+\\,\\frac{k_{x}(k_{x}\\,-\\,ik_{y})}{2k^{2}}\\,2na\\rho\\,+\\,\\frac{2na\\rho}{8}\\,\\nonumber\\\\ \n&& {\\hspace{-4cm}}+\\,\\frac{(2na\\rho)^{2}}{2}\\,-\\,\\frac{(2na\\rho)^{3}}{4}\\,\\bigg]\\,e^{-\\,2na\\rho}\\,, \\nonumber \n\\end{eqnarray} \n\\begin{eqnarray} \n\\tilde{g}^{(m=\\,0)}_{2,\\,2}(\\vec{k})\\,=\\,\\frac{k^{2}_{x}}{v^{2}k^{4}\\,\\sqrt{2\\pi V}}\\,2na\\,e^{-\\,2na\\rho}\\nonumber\\\\ \n&& \n{\\hspace{-4.5cm}}+\\,\\frac{1}{2v^{2}\\,\\sqrt{2\\pi V}}\\bigg[\\frac{45}{128na}\\,-\\,\\frac{315k_{y}^{2}}{256nak^{2}}\\,+\\,\\frac{k_{x}^{2}}{4k^{2}}\\,\\rho\\,-\\,\\frac{11}{8}\\,2na\\rho^{2}\\,\\nonumber\\\\ \n&&{\\hspace{-1.5cm}}-\\,\\frac{13}{8}\\,\\frac{k_{x}^{2}}{k^{2}}\\,2na\\rho^{2}\\,+\\,\\frac{19}{24}\\,(2na)^{2}\\rho^{3}\\,+\\,\\frac{k_{x}^{2}}{2k^{2}}\\,(2na)^{2}\\rho^{3}\\,\\nonumber\\\\ \n&& \n+\\,\\frac{13}{24}\\,(2na)^{3}\\rho^{4}\\,-\\,\\frac{1}{8}\\,(2na)^{4}\\rho^{5}\\bigg]\\,e^{-2na\\rho} \n\\label{e61} \n\\end{eqnarray} \nand for the $O(k^{-\\,2j})$ terms in \n$\\tilde{G}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})$ we have \n\\begin{eqnarray} \n\\tilde{g}^{(m=\\,0)}_{2,\\,2j}(\\vec{k})&=&\\frac{(-1)^{\\,j\\,-\\,1}}{\\sqrt{2\\pi \n V}}\\,\\bigg(\\frac{k^{2}_{x}}{v^{2}k^{4}}\\bigg)^{j}\\,2na\\,e^{-\\,2na\\rho}\\,. \n\\label{e62} \n\\end{eqnarray} \nWe point out that all of the\nterms in each of the square brackets, in each of the above three\nequations, are of the same order in the density. This can be seen\neasily by noting that $\\rho$ is typically of order $(2na)^{-1}$, so\nthat $(2na\\rho)$ is typically independent of the density.\nThe solutions, Eqs. (\\ref{e51}), (\\ref{e53}), (\\ref{e54}), \n(\\ref{e56}), (\\ref{e61}) and (\\ref{e62}) now can be assembled to \ncalculate $G_{2}$ and $g_{2}$ in $(\\vec{r}, \\vec{v})$ and $(\\vec{r}, \n\\vec{v}, \\rho)$ space respectively and feed the results into the \nr.h.s. of Eqs. (\\ref{e35}) and (\\ref{e40}) to obtain $f^{\\mbox{\\tiny(R)}}_{1}$. This involves a summation of different $m$ and \n$\\vec{k}$-values. In the infinite volume limit the $\\vec{k}$-sum can \nbe converted to an integration over $\\vec{k}$. The sum over $m$ is \nstraightforward, but we have to remember that the integration over \n$\\vec{k}$ has to be carried out in a range $k\\leq \nk_{0}\\sim l^{-1}$. Secondly, since we have expanded the distribution\nfunctions in powers of $\\varepsilon$ and then subsequently in powers\nof $k$, the lower limit of $k$ for the $\\vec{k}$-integration cannot be\ntaken to be zero. To determine this lower limit of $k$ for the\n$k$-integration, we observe that the expansion in $\\varepsilon$ cannot\nbe carried out for those values of $k$ where\n$k<\\displaystyle{\\frac{\\varepsilon}{2v}}$, so that the value\n$\\displaystyle{\\frac{\\varepsilon}{2v}}$ forms a natural lower cut-off\nfor the Fourier transform. Our solutions of $G_{2}$ and $g_{2}$\ntherefore do not hold for $k<\\displaystyle{\\frac{\\varepsilon}{2v}}$\nand to do a satisfactory perturbation theory in the range\n$k<\\displaystyle{\\frac{\\varepsilon}{2v}}$, one needs to consider both\nthe $\\varepsilon$ and $\\vec{k}$-dependent terms together. After doing\nso, one finds that such a perturbation theory does not affect our\nresults at the present density order \\cite{PD_unpublished}. \n \nBefore performing the integration over $\\vec{k}$, we notice that in\ntwo dimensions, the numerator of the $\\vec{k}$-integral is\nproportional to $k\\,dk$. This means that any part of the solutions of\n$\\tilde{g}_{2}(\\vec{k})$ or $\\tilde{G}_{2}(\\vec{k})$ having a leading\npower of $k$ of order 2 or higher in the denominator gives rise to a\nsingularity at $k\\rightarrow 0$ for the $\\vec{k}$-integral. First, the\nhighest leading power of $k$ in the denominators of Eqs. (\\ref{e51}),\n(\\ref{e53}), (\\ref{e54}) and (\\ref{e61}) is $k^{2}$, occurring in\n$\\tilde{G}^{(m\\,=\\,0)}_{2,\\,2}(\\vec{k})$ and\n$\\tilde{g}^{(m\\,=\\,0)}_{2,\\,2}(\\vec{k})$ respectively. These terms\nproportional to $k^{-\\,2}$ give rise to a logarithmic electric field\ndependence once the $\\vec{k}$-integration is performed for\n$\\displaystyle{\\frac{\\varepsilon}{2v}}\\leq k\\leq k_{0}$. The rest of\nthe terms in these solutions supply only analytic field dependences\nthat can be expressed as power series in $\\varepsilon$. Secondly, even\nthough the solutions given in Eqs. (\\ref{e56}) and (\\ref{e62}) have\nhigher powers of $k$ than $k^{2}$ in the denominators, they also come\nwith subsequently higher powers of $\\varepsilon$ in their\nnumerators. Thus, when the $\\vec{k}$-integration is performed, they\ncontribute terms proportional to $\\varepsilon^{2}$ or higher, to\n$g_{2}$ or $G_{2}$. Consequently, in addition to analytic field\ndependent terms, in our present approximation we have only one\nnon-analytic field dependent term appearing in $g_{2}$ or $G_{2}$\nand that is proportional to\n$\\tilde{\\varepsilon}^{2}\\,\\ln\\,\\tilde{\\varepsilon}$. No doubt there\nexist further non-analytic terms in higher orders in\n$\\tilde{\\varepsilon}$, but their calculation would require a careful\nconsideration of various terms we have neglected here, such as the\nrepeated ring contributions. \n \n\\subsection{Solution for $f^{\\mbox{\\tiny (R)}}_{1}$ and the \ncalculation of $\\lambda^{\\mbox{\\tiny (R)}}_{+}$} \n \nOnce the solutions, Eqs. (\\ref{e51}), (\\ref{e53}), (\\ref{e54}), \n(\\ref{e56}), (\\ref{e61}) and (\\ref{e62}) are inserted in \nEqs. (\\ref{e35}) and (\\ref{e41}) and the $\\vec{k}$-integration is \nperformed in the range \n$\\displaystyle{\\frac{\\varepsilon}{2v}}<k<k_{0}$, we get, by the \nmethod described in (\\ref{e1new}-\\ref{e4new}), the following equations \nto be solved to obtain \n$f^{\\mbox{\\tiny (R)}}_{1}$ and $F^{\\mbox{\\tiny(R)}}_{1}$ respectively : \n\\begin{eqnarray} \n-\\,\\varepsilon\\,\\frac{\\partial}{\\partial\\theta}\\,(\\sin\\theta\\,f^{\\mbox{\\tiny(R)}}_{1})\\,+\\,\\frac{\\partial}{\\partial\\rho}\\,\\bigg\\{\\bigg(v\\,+\\,\\rho\\varepsilon\\cos\\theta\\,+\\,\\frac{\\rho^{2}\\varepsilon^{2}\\sin^{2}\\theta}{v}\\bigg)\\,f^{\\mbox{\\tiny (R)}}_{1}\\bigg\\}\\,+\\,2navf^{\\mbox{\\tiny(R)}}_{1}\\nonumber\\\\ \n&&{\\hspace{-14cm}}=\\,\\Theta\\big(\\frac{a}{2}\\,-\\,\\rho\\big)\\,\\frac{4v\\rho}{a\\sqrt{1\\,-\\,\\big(\\frac{2\\rho}{a}\\big)^{2}}}\\times\\nonumber \n\\\\&&{\\hspace{-13.7cm}}\\times\\!\\int^{\\frac{\\pi}{2}}_{-\\,\\frac{\\pi}{2}}\\!d\\phi\\,\\cos\\phi\\,b_{\\sigma}\\bigg[\\,\\frac{\\varepsilon^{2}}{8\\pi^{2}v^{2}}\\,\\bigg\\{\\ln\\frac{2vk_{0}}{\\varepsilon}\\bigg\\}\\!-\\!\\frac{k^{2}_{0}}{8\\pi^{2}}\\bigg\\{\\!\\frac{3}{16nav}\\,\\varepsilon\\cos\\theta\\,+\\,\\frac{135}{2048(nav)^{2}}\\varepsilon^{2}\\bigg\\}\\!+\\!\\frac{A\\,\\varepsilon^{2}}{16\\pi^{3}v^{2}}\\,\\bigg]\\,\\nonumber\\\\ \n&&{\\hspace{-14cm}}-\\,2av\\,\\bigg[\\,\\frac{\\varepsilon^{2}}{8\\pi^{2}v^{2}}\\,\\bigg\\{\\ln\\,\\frac{2vk_{0}}{\\varepsilon}\\bigg\\}\\,2na\\,e^{-2na\\rho}\\,\\nonumber\\\\ \n&&{\\hspace{-12.5cm}}+\\,\\frac{k^{2}_{0}}{8\\pi^{2}}\\,e^{-2na\\rho}\\,\\bigg\\{2na\\,(1\\,-\\,2na\\rho)\\,-\\,\\frac{\\varepsilon}{v}\\,\\bigg[\\frac{(2na\\rho)^{3}}{2}\\,-\\,(2na\\rho)^{2}\\,-\\,\\frac{3}{4}\\,2na\\rho\\,+\\,\\frac{1}{8}\\bigg]\\cos\\theta\\nonumber\\\\ \n&&{\\hspace{-12.5cm}}-\\frac{\\varepsilon^{2}}{4nav^{2}}\\,\\bigg[\\frac{(2na\\rho)^{5}}{8}\\,-\\,\\frac{13}{24}\\,(2na\\rho)^{4}\\,-\\,\\frac{25}{24}\\,(2na\\rho)^{3}\\,+\\,\\frac{35}{16}\\,(2na\\rho)^{2}\\,-\\,\\frac{2na\\rho}{8}\\,+\\,\\frac{135}{256}\\bigg]\\bigg\\}\\nonumber \\\\&& \n{\\hspace{-5.3cm}}+\\,\\frac{1}{16\\pi^{3}v^{2}}\\,A\\,\\varepsilon^{2}\\,2na\\,e^{-\\,2na\\rho}\\,\\bigg]\\,+\\,.\\,.\\,.\\,. \n\\label{e63} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n-\\,\\varepsilon\\,\\frac{\\partial}{\\partial\\theta}\\,(\\sin\\theta\\,F^{\\mbox{\\tiny(R)}}_{1})\\,-\\,nav\\int^{\\frac{\\pi}{2}}_{-\\,\\frac{\\pi}{2}}d\\phi\\,\\cos\\phi\\,(b_{\\sigma}\\,-\\,1)\\,F^{\\mbox{\\tiny (R)}}_{1}\\,\\nonumber\\\\ \n&&{\\hspace{-8cm}}=\\,\\int^{\\frac{\\pi}{2}}_{-\\,\\frac{\\pi}{2}}d\\phi\\,\\cos\\phi\\,(b_{\\sigma}\\,-\\,1)\\bigg[\\,\\frac{a\\varepsilon^{2}}{8\\pi^{2}v}\\,\\ln\\bigg(\\frac{2vk_{0}}{\\varepsilon}\\bigg)\\,\\nonumber\\\\ \n&&{\\hspace{-2.5cm}}-\\,\\frac{avk^{2}_{0}}{8\\pi^{2}}\\bigg\\{\\frac{3}{16nav}\\,\\varepsilon\\cos\\theta\\,+\\,\\frac{135}{2048(nav)^{2}}\\varepsilon^{2}\\bigg\\}\\,\\nonumber\\\\ \n&&{\\hspace{1.5cm}}+\\,\\frac{a}{16\\pi^{3}v}\\,A\\,\\varepsilon^{2}\\,+\\,. \n\\,.\\,.\\,.\\,\\bigg]\\nonumber\\\\ \n&&{\\hspace{-8cm}}=\\,\\frac{ak^{2}_{0}}{16\\pi^{2}na}\\,\\varepsilon\\cos\\theta\\,+\\,.\\,.\\,.\\,.\\,, \n\\label{e64} \n\\end{eqnarray} \nwhere $b_{\\sigma}$ has been defined in Eq. (\\ref{e37}). The \n$A$-dependent terms in Eqs. (\\ref{e63}) and (\\ref{e64}) originate from \nEqs. (\\ref{e62}) and (\\ref{e56}) respectively after the \n$\\vec{k}$-integration is carried out. Here $A$ is the integral\\footnote{We\nthank the referee for pointing out an error in a previous calculation of\nthis integral.} \n\\begin{eqnarray} \nA\\,=\\,\\int_{0}^{2\\pi}d\\phi\\,\\cos^{2}\\phi\\,\\ln\\,\\bigg[\\,1\\,+\\,\\frac{1}{4}\\,\\cos^{2}\\phi\\,\\bigg]\\,=\\,0.53536\\,.\\,.\\,.\\,. \n\\label{e65} \n\\end{eqnarray} \n \nThe dominant effect of the ring term on the single particle \ndistribution function, i.e, $f^{\\mbox{\\tiny (R)}}_{1}$, can now be \ndetermined from Eqs. (\\ref{e63}) and (\\ref{e64}). It is also of some \ninterest to give a crude estimate of the terms that we have \nneglected. One knows from other studies in the kinetic theory of gases \n\\cite{DvB_berne_book,CC_cup_book} that excluded volume corrections to \nBoltzmann equation results are the numerically most important \ncorrections, until the density of the system becomes high enough that \nthe mean free path of a particle is less than the size of the particle \nitself. These excluded volume corrections are provided by the Enskog \ntheory, and this theory can be applied to the Lorentz gas, as well \n\\cite{vLW_physica_67}. In our case, the Enskog corrections can be \nincluded by replacing the density parameter $n$ by $n\\,(1\\,-\\,\\pi \nna^{2})^{-\\,1}\\approx n\\,(1\\,+\\,\\pi na^{2})$ in the Boltzmann \nequation. The Enskog correction affects both $\\lambda_{0}$ and the \n$\\varepsilon$-dependent terms in the expressions for $\\lambda_{\\pm}$ \nin Eq. (\\ref{e13}). Along with the Enskog correction there are other \ncorrection terms that affect both $\\lambda_{0}$ and the \nfield-dependent terms $\\lambda_{\\pm}$ \n\\cite{PD_unpublished,Kruis_thesis} at the same density order as the \nEnskog correction. Also, the terms that have been dropped to obtain \nEq. (\\ref{e14}) from Eq. (\\ref{e10}), contribute to $\\lambda_{\\pm}$ \nat the same density order as the Enskog correction. However, since the \nprincipal objective of this paper is to investigate the non-analytic \ncontribution of the ring term to the Lyapunov exponents, we will \nignore the Enskog and related corrections from our consideration. \nThus, using Eqs. (\\ref{e63}) and (\\ref{e64}), one can express the full \nsolutions of $f_{1}$ and $F_{1}$ as sums of a solution in the \nBoltzmann regime, a correction due to the ring term and a correction \ndue to the Enskog term, plus all of the other terms we have \nneglected, as \n\\begin{eqnarray} \nf_{1}\\,=\\,f^{\\mbox{\\tiny (B)}}_{1}\\,+\\,f^{\\mbox{\\tiny(R)}}_{1}\\,+\\,.\\,.\\,.\\,. \n\\label{e66} \n\\end{eqnarray} \n\\begin{eqnarray} \nF_{1}\\,=\\,F^{\\mbox{\\tiny (B)}}_{1}\\,+\\,F^{\\mbox{\\tiny(R)}}_{1}\\,+\\,.\\,.\\,.\\,. \n\\label{e67} \n\\end{eqnarray} \nConsequently, for the positive Lyapunov exponent $\\lambda_{+}$ we \nhave, \n\\begin{eqnarray} \n\\lambda_{+}\\,=\\,\\lambda^{\\mbox{\\tiny(B)}}_{+}\\,+\\,\\lambda^{\\mbox{\\tiny (R)}}_{+}\\,+\\,.\\,.\\,.\\,. \n\\label{e68} \n\\end{eqnarray} \n \nThe solution of $F^{\\mbox{\\tiny (R)}}_{1}$ is quite straightforward, \n\\begin{eqnarray} \nF^{\\mbox{\\tiny \n(R)}}_{1}\\,=\\,\\frac{3ak^{2}_{0}}{128\\pi^{2}(na)^{2}v}\\,\\varepsilon\\cos\\theta\\,+\\,.\\,.\\,.\\,. \n\\label{e69} \n\\end{eqnarray} \nHowever, to solve for $f^{\\mbox{\\tiny (R)}}_{1}$ we find that in \naddition to the analytic field-dependent terms which can be expressed \nas a power series in $\\varepsilon$, there is a non-analytic \nfield-dependent term in $f^{\\mbox{\\tiny (R)}}_{1}$ proportional to \n$\\tilde{\\varepsilon}^{2}\\,\\ln\\,\\tilde{\\varepsilon}$. Thus, with \n\\begin{eqnarray} \nf^{\\mbox{\\tiny (R)}}_{1}\\,=\\,f^{\\mbox{\\tiny (R)}}_{1,\\,\\mbox{\\tiny analytic}}\\,+\\,f^{\\mbox{\\tiny (R)}}_{1,\\,\\mbox{\\tiny non-analytic}}\\,, \n\\label{e70} \n\\end{eqnarray} \nwe have \n\\begin{eqnarray} \nf^{\\mbox{\\tiny (R)}}_{1,\\,\\mbox{\\tiny analytic}}=-\\frac{ak^{2}_{0}}{4\\pi^{2}}\\bigg[\\,2na\\rho\\,-\\,\\frac{(2na\\rho)^{2}}{2}\\, -\\,\\frac{\\varepsilon\\cos\\theta}{2nav}\\,\\bigg\\{\\frac{(2na\\rho)^{4}}{4}\\,-\\, (2na\\rho)^{3} + \\frac{(2na\\rho)^{2}}{8} + \\frac{2na\\rho}{8}\\bigg\\}\\,\\nonumber\\\\ \n&&{\\hspace{-12.7cm}} +\\,\\frac{\\varepsilon^{2}}{4(nav)^{2}}\\,\\bigg\\{-\\,\\frac{(2na\\rho)^{6}}{32}\\,+\\, \\frac{11}{48}\\,(2na\\rho)^{5}\\,-\\,\\frac{(2na\\rho)^{4}}{96}\\,-\\,\\frac{79}{96}\\,(2na\\rho)^{3}\\,\\nonumber\\\\ \n&&{\\hspace{-9.8cm}}+\\,\\frac{3}{32}\\,(2na\\rho)^{2}\\,-\\,\\frac{135}{512}\\,(2na\\rho)\\,+\\,\\frac{135}{512}\\bigg\\}\\,\\bigg]\\,e^{-2na\\rho}\\nonumber\\\\ \n&&{\\hspace{-11.8cm}}+\\,\\frac{a\\varepsilon^{2}}{(2\\pi)^{3}v^{2}}\\,A\\,(1\\,-\\,2na\\rho)\\,e^{-2na\\rho}\\,+\\,.\\,.\\,.\\,.\\hspace{1.95cm} \n\\mbox{for $\\rho>\\displaystyle{\\frac{a}{2}}$}\\nonumber\\\\ \n&&\\hspace{-14.7cm}=\\bigg[\\frac{aA}{(2\\pi)^{3}v^{2}}\\,-\\,\\frac{135 ak^{2}_{0}}{512\\, (4\\pi nav)^{2}}\\bigg]\\bigg\\{1-\\sqrt{1\\,-\\,\\big(\\frac{2\\rho}{a}\\big)^{2}}\\bigg\\}\\, \\varepsilon^{2}\\,+\\,.\\,.\\,.\\,.\\hspace{1cm}\\mbox{for $\\rho<\\displaystyle{\\frac{a}{2}} \n$} \n\\label{e71} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \nf^{\\mbox{\\tiny (R)}}_{1,\\,\\mbox{\\tiny non-analytic}}\\,=\\,\\frac{a\\varepsilon^{2}}{4\\pi^{2}v^{2}}\\,\\bigg\\{\\ln\\,\\frac{2vk_{0}}{\\varepsilon}\\bigg\\}\\,(1\\,-\\,2na\\rho)\\,e^{-2na\\rho}\\hspace{1.8cm}\\mbox{for $\\rho>\\displaystyle{\\frac{a}{2}}$}\n\\nonumber\\\\\n&&\\hspace{-11.15cm}=\\,\\frac{a\\varepsilon^{2}}{4\\pi^{2}v^{2}}\\,\n\\bigg\\{\\ln\\,\\frac{2vk_{0}}{\\varepsilon}\\bigg\\}\\bigg[\\,1-\\sqrt{1\\,-\\,\n\\big(\\frac{2\\rho}{a}\\big)^{2}}\\,\\bigg]\\hspace{1.4cm} \\mbox{for\n$\\rho<\\displaystyle{\\frac{a}{2}}$}\\,. \n\\label{e72} \n\\end{eqnarray} \nNotice that the ring contribution to the distribution function in\nEqs. (\\ref{e71}) and (\\ref{e72}) satisfies the boundary conditions\nthat $f^{\\mbox{\\tiny (R)}}_{1}\\rightarrow 0$ as $\\rho\\rightarrow0$ and\n$\\rho\\rightarrow\\infty$. Equations (\\ref{e71}) and (\\ref{e72}) also\nsatisfy continuity at $\\rho=\\displaystyle{\\frac{a}{2}}$ at the leading\ndensity order. The distribution functions, Eqs. (\\ref{e69}),\n(\\ref{e71}) and (\\ref{e72}), are all the ones that we need to\ncalculate $\\lambda^{\\mbox{\\tiny (R)}}_{+}$. Consequently, \n\\begin{eqnarray} \n\\lambda^{\\mbox{\\tiny\n(R)}}_{+}\\,=\\,\\lambda^{\\mbox{\\tiny(R)}}_{+,\\,\\mbox{\\tiny\nanalytic}}\\,+\\,\\lambda^{\\mbox{\\tiny(R)}}_{+,\\,\\mbox{\\tiny\nnon-analytic}} \n\\label{e74} \n\\end{eqnarray} \nand using the the definition of Lyapunov exponents in Eq. (\\ref{e11}),\nwe have \n\\begin{eqnarray} \n\\lambda^{\\mbox{\\tiny (R)}}_{+,\\,\\mbox{\\tiny\nanalytic}}\\,=\\,\\int_{0}^{2\\pi}d\\theta\\,\\int_{\\frac{a}{2}}^{\\infty}d\\rho\\,\\frac{f^{\\mbox{\\tiny(R)}}_{1,\\,\\mbox{\\tiny\nanalytic}}}{\\rho}\\,\\nonumber\\\\ &&{\\hspace{-5.3cm}}=\\,-\\,\\frac{a\nk^{2}_{0} v}{4\\pi}\\,-\\,\\frac{ak^{2}_{0} l^{2} \\varepsilon^{2}}{2\\pi\nv}\\,\\bigg\\{\\,\\frac{13}{96}\\,-\\,\\frac{135}{512}\\,\\big(\\ln\n2na^{2}\\,+\\,{\\cal C}\\,\\big)\\,\\,\\bigg\\}\\,\\nonumber\\\\\n&&{\\hspace{-1cm}}-\\,0.53536\\,\\frac{a\\varepsilon^{2}}{(2\\pi)^{2}v}\\,\\big(\\ln\\,2na^{2}\\,+\\,{\\cal\nC}\\,\\big)\\,+\\,.\\,.\\,.\\,. \n\\label{e75} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\lambda^{\\mbox{\\tiny (R)}}_{+,\\,\\mbox{\\tiny\nnon-analytic}}\\,=\\,\\int_{0}^{2\\pi}d\\theta\\,\\int_{\\frac{a}{2}}^{\\infty}d\\rho\\,\\frac{f^{\\mbox{\\tiny(R)}}_{1,\\,\\mbox{\\,\\tiny\nnon-analytic}}}{\\rho}\\,\\nonumber\\\\\n&&{\\hspace{-6.2cm}}=\\,-\\,\\frac{a\\varepsilon^{2}}{2\\pi\nv}\\,\\bigg\\{\\ln\\frac{2 k_{0} v}{\\varepsilon}\\bigg\\}\\,\\big(\\ln\n2na^{2}\\,+\\,{\\cal C}\\,\\big)\\,, \n\\label{e76} \n\\end{eqnarray} \nwhere $l$ is the mean free path and ${\\cal C}$ is Euler's constant,\n${\\cal C}\\,=\\,0.5772\\,.\\,.\\,.\\,$. \n \n \n\\subsection{Calculation of $\\lambda^{\\mbox{\\tiny (R)}}_{-}$} \n \nTo calculate the corresponding effect of the ring term on\n$\\lambda_{-}$, we make use of the relation Eq. (\\ref{e12}). It is easy\nto calculate the effect of the ring term on\n$\\big<\\alpha\\big>_{\\mbox{\\tiny NESS}}$ using $F^{\\mbox{\\tiny\n(R)}}_{1}$ already determined in the previous section. Thus, using \n\\begin{eqnarray} \n\\lambda_{+}\\,+\\,\\lambda_{-}\\,=\\,-\\,\\big<\\alpha\\big>_{\\mbox{\\tiny\nNESS}}\\,, \n\\label{e88} \n\\end{eqnarray} \nand a complete analogy to Eqs. (\\ref{e66}-\\ref{e68}), we can calculate\nthree terms of $\\big<\\alpha\\big>_{\\mbox{\\tiny NESS}}$: \n\\begin{eqnarray} \n\\big<\\alpha\\big>_{\\mbox{\\tiny NESS}}\\,=\\,\\big<\\alpha\\big>^{\\mbox{\\tiny\n(B)}}_{\\mbox{\\tiny NESS}}\\,+\\,\\big<\\alpha\\big>^{\\mbox{\\tiny\n(R)}}_{\\mbox{\\tiny NESS}}\\,+\\,.\\,.\\,.\\,.\\,, \n\\label{e89} \n\\end{eqnarray} \nwith \n\\begin{eqnarray} \n\\big<\\alpha\\big>^{\\mbox{\\tiny (R)}}_{\\mbox{\\tiny\nNESS}}\\,=\\,\\frac{3ak^{2}_{0} l^{2} \\varepsilon^{2}}{32\\pi\nv}\\,+\\,\\,.\\,.\\,.\\,. \n\\label{e90} \n\\end{eqnarray} \n \nFollowing Eqs. (\\ref{e66}-\\ref{e68}), we now express $\\lambda_{-}$ as\n$\\lambda_{-}\\,=\\,\\lambda^{\\mbox{\\tiny(B)}}_{-}\\,+\\,\\lambda^{\\mbox{\\tiny\n(R)}}_{-}\\,+\\,.\\,.\\,.\\,.\\,$, satisfying $\\lambda^{\\mbox{\\tiny\n(R)}}_{+}\\,+\\,\\lambda^{\\mbox{\\tiny(R)}}_{-}\\,=\\,-\\,\\big<\\alpha\\big>^{\\mbox{\\tiny\n(R)}}_{\\mbox{\\tiny NESS}}$. This leads us to \n\\begin{eqnarray} \n\\lambda^{\\mbox{\\tiny (R)}}_{-,\\,\\mbox{\\tiny\nanalytic}}\\,=\\,\\frac{ak^{2}_{0} v}{4\\pi}\\,-\\,\\frac{a k^{2}_{0} l^{2}\n\\varepsilon^{2}}{2\\pi\nv}\\,\\bigg\\{\\,\\frac{5}{96}\\,+\\,\\frac{135}{512}\\,(\\,\\ln2na^{2}\\,+\\,{\\cal\nC}\\,\\big)\\,\\bigg\\}\\,\\nonumber\\\\&&{\\hspace{-6cm}}+\\,\\,0.53536\\,\\frac{a\\varepsilon^{2}}{(2\\pi)^{2}v}\\,(\\,\\ln\\,2na^{2}\\,+\\,{\\cal\nC}\\,\\big)\\,+\\,.\\,.\\,.\\,.\\,, \n\\label{e92} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\lambda^{\\mbox{\\tiny (R)}}_{-,\\,\\mbox{\\tiny\nnon-analytic}}\\,=\\,\\frac{a\\varepsilon^{2}}{2\\pi v}\\bigg\\{\\ln\n\\frac{2k_{0} v}{\\varepsilon}\\bigg\\}\\,(\\,\\ln 2na^{2}\\,+\\,{\\cal\nC}\\,\\big). \n\\label{e93} \n\\end{eqnarray} \nwhere $l$ is the mean free path and ${\\cal C}$ is Euler's constant,\n${\\cal C}\\,=\\,0.5772\\,.\\,.\\,.\\,$. \n \n\\section{The field-dependent collision frequency and its \\\\effects on \nthe Lyapunov exponents} \n \nAs stated before, our second main purpose was the derivation of the\nleading non-analyticity in the field dependence of the Lyapunov\nexponents. In analogy with the transport coefficients, we expected\nthese non-analyticities to result from the long time behavior of the\nring terms, which we found confirmed in the preceding section. Some\nfurther thought reveals we can estimate the non-analytic field\ndependence in a simple way. \n \nIn the presence of a thermostatted field there are two types of\ncontributions to the positive Lyapunov exponent of the two-dimensional\nLorentz gas:\\\\ 1) contributions from the bending of the trajectories\nby the fields and\\\\ 2) contributions from the divergence of trajectory\npairs at collisions. \n \nThe first type of contributions are of order $\\tilde{\\varepsilon}^2$\nin the Boltzmann approximation. We expect that the coefficient of this\nterm will pick up higher density corrections and there will be\nadditional terms of higher orders in $\\tilde{\\varepsilon}$. But we\nhave not found any indications for corrections of lower order than\n$\\tilde{\\varepsilon}^2$ resulting from the field-bending\ncontributions. \n \nThe collisional contributions can be generally expressed as an\naverage of the form\n$\\nu\\,\\langle\\,\\ln\\displaystyle{\\frac{|\\delta\\vec{v'}|}{|\\delta\\vec{v}|}}\\,\\rangle_{{\\mbox{\\scriptsize\nc}}}$, with $\\delta \\vec{v'}$ and $\\delta \\vec{v}$ the velocity\ndifferences between the adjacent trajectories just after and just\nbefore a collision, respectively, $\\nu$ the average collision\nfrequency, and the angular brackets,\n$\\langle\\,\\rangle_{{\\mbox{\\scriptsize c}}}$, indicating an average\nover collisions. At low densities even correlated collisions happen\nat large distances, i.e. in the order of a mean free path length apart\nfrom each other. Therefore their distribution of collision angles and \nhence their contribution to the average\n$\\langle\\,\\rangle_{{\\mbox{\\scriptsize c}}}$, to the leading order in\ndensity remains the same as for uncorrelated collisions. We should\nthen expect that at low densities the main effect of the correlated\ncollisions on the Lyapunov exponents should be due to a change of the\ncollision frequency $\\nu$ as a result of correlated collisions taking\nplace in the presence of the field. If the latter changes from\n$\\nu_0$ to $\\nu_0+\\delta\\nu$, then Eq. (\\ref{e131}) predicts a change\nof the positive Lyapunov exponent of magnitude \n\\begin{eqnarray} \n\\delta\\lambda_{+}\\,=\\,-\\,\\delta\\nu\\,\\bigg\\{\\ln\n\\frac{a\\nu_{0}}{v}\\,+\\cal{C}\\bigg\\}. \n\\label{deltalambda} \n\\end{eqnarray} \nTo obtain this result we have used the fact that the equilibrium, low\ndensity Lyapunov exponent, Eq. \\ref{e131}) can be written in the form \n\\begin{eqnarray} \n\\lambda_0\\,=\\,\\nu_{0}\\,\\bigg\\{1\\,-\\,{\\cal\nC}\\,-\\,\\ln\\frac{a\\nu_{0}}{v}\\bigg\\}, \n\\label{e1311} \n\\end{eqnarray} \nwhere $\\nu_{0}=2nav$. In order to understand why and how the\nthermostatted field changes the collision frequency we first recall\nthat in equilibrium the collision frequency can be obtained simply\nby using the uniformity of the equilibrium distribution for the point\nparticle in available phase space, with the result that\n$\\nu=\\displaystyle{\\frac{2nav}{1-\\pi n a^2}}$. One just has to\nconsider the probability that the light particle during an\ninfinitesimal time $d\\,t$ will hit one of the scatterers. On the other\nhand, at a time $t$ after a given initial time, the probability for a\ncollision may be considered to be a sum of three contributions: the\ncollision frequency obtained by assuming that all collisions are\nuncorrelated and independent of each other, {\\em plus} the\nprobability for a recollision with a scatterer with which it has\ncollided before, {\\em minus} the reduction of the collision\nprobability due to any collected knowledge of where no scatterers\nare present. In equilibrium the last two contributions have to\ncancel, as we demonstrate in the Appendix. In the presence of a field,\nhowever, this cancellation does not occur. This can easily be\nunderstood in a qualitative way following the argument that the\ncancellation in equilibrium occurs because the probability for return\nto the boundary of a scatterer is exactly the same as that for return\nto the boundary of a region where a scatterer could be, but in fact is\nnot present (a virtual scatterer). In the presence of a field, the\naverage velocity of the point particle before collision with a real\nscatterer will be in the direction of the field, and after the\ncollision the average velocity will be anti-parallel to the\nfield. The field will then tend to turn the particle around and have\nit move back in the direction of the scatterer. This effect enhances\nthe probability of a recollision in comparison to that for an\nisotropic distribution around the scatterer. In a ``virtual\ncollision\", in which the velocity does not change, the particle, on\naverage, ends up downstream (i.e. in the direction of the applied\nfield) from the virtual scatterer and its recollision probability is\ndecreased compared to that for an isotropic distribution. \n \nIn the Appendix, a quantitative calculation is given based on the\nfollowing two assumptions:\\\\ 1) After the real or virtual collision\nthe spatial distribution of the point particle becomes centered around\na point at a distance of a diffusion length from the scatterer\nand\\\\ 2) for long times this distribution can be found by solving the\ndiffusion equation. The resulting expression for $\\delta\\nu$ is \n\\begin{eqnarray} \n\\delta\\nu\\,=\\,\\frac{a\\varepsilon^2}{2\\pi\nv}\\ln\\frac{\\nu_0}{\\varepsilon}\\,. \n\\label{delnu} \n\\end{eqnarray} \n \nA more formal, but equivalent, way to obtain this result is by\nextending the method described by Latz, van Beijeren and Dorfman\n\\cite{LvBD_preprint} for the low density distribution of time of free\nflights of the moving particle to include the contribution from ring\nevents, so as to apply to a system in a thermostatted electric\nfield. The main idea is to solve a kinetic equation for\n$f(\\vec{r},\\vec{v},t,\\tau)$, the distribution of particles at a phase\npoint $(\\vec{r},\\vec{v})$ at time $t$ such that their last collision\ntook place at a time $\\tau$ earlier, i.e., at time $t-\\tau$. It is\nthen easy to argue that the distribution of free flight times is\nsimply the derivative of this (``last collision'') distribution with\nrespect to $t-\\tau$. We can then obtain a NESS average of the time \nof free flight and thereby calculate the field dependent collision frequency\n$\\nu(\\varepsilon)=\\nu_{0}+\\delta\\nu$. Since we want to show that the\norigin of the non-analytic field dependence of both $\\lambda_{+}$ and\n$\\lambda_{-}$ is rooted in the non-analytic field dependence of\ncollision frequency $\\delta\\nu$, let us keep only the non-analytic\nfield-dependent term as the leading term of the expansion of\n$\\delta\\nu$ in the density of scatterers and in the electric field\nstrength and write \n\\begin{eqnarray} \n\\delta\\nu\\,=\\,\\beta\\,{\\varepsilon^{2}}\\ln\\bigg\\{\\frac{2k_{0}v}{\\varepsilon}\\bigg\\}\\,+\\,\\cdots\\,, \n\\label{nu1} \n\\end{eqnarray} \nwhere the quantity $\\beta$ has to be determined from the NESS average\nof $\\tau$, using the effect of the ring term on the NESS distribution\nfunction $f(\\vec{r},\\vec{v},\\tau)$ with $k_{0}$ of the order of\n$\\displaystyle{\\frac{1}{\\nu_{0}v}}\\,$. To obtain this distribution\nfunction, we follow exactly the same procedure as outlined in Sections\n4 and 5, but this time, with the variable $\\tau$ instead of\n$\\rho$. Notice that, this time, even though the equations for\ncorresponding $f_{1}$ and $g_{2}$'s are different, due to the\ndifference in the dynamical equations for $\\dot{\\rho}$ and\n$\\dot{\\tau}$ during free flights and at collisions, the equations\ninvolving $F_{1}$ and $G_{2}$'s remain the same. The source of the\nnon-analytic field-dependent term will surface again exactly from the\n$O(k^{-2})$ term in Eq. (\\ref{e54}). As far as this non-analytic\nfield-dependent term is concerned, at the lowest order of density, the\nvariables $\\rho$ and $\\tau$ are identical up to a multiplicative\nfactor $v$. Both grow linearly with time in between collisions and are\nset back to (for $\\rho$, almost) zero at each collision with a\nscatterer. One then recovers the corresponding non-analytic part of\nthe NESS distribution function \\cite{Panja_thesis}, analogous to\nEq. (\\ref{e72}), \n\\begin{eqnarray} \nf^{\\mbox{\\tiny (R)}}_{1,\\,\\mbox{\\tiny \nnon-analytic}}(\\vec{v},\\tau)\\,=\\,\\frac{a\\varepsilon^{2}}{4\\pi^{2}v}\\,\\bigg\\{\\ln\\,\\frac{2vk_{0}}{\\varepsilon}\\bigg\\}\\,(1\\,-\\,2nav\\tau)\\,e^{-2nav\\tau}\\hspace{1.8cm}\\mbox{for\\, $\\tau>0$}\\,, \n\\label{nu2} \n\\end{eqnarray} \nfrom which $\\beta$ can be obtained to be \n\\begin{eqnarray} \n\\beta\\,=\\,\\frac{a}{2\\pi v}\\,, \n\\label{nu3} \n\\end{eqnarray} \nafter which, one easily recovers the result of Eq. (\\ref{delnu}). \n \n\\section{Discussion} \n \nWhile much of this paper is quite technical, there are two main points\nthat we would like to emphasize: (1) We have developed a method which\nallows an extension of the calculation of the Lyapunov exponents for a\ntwo-dimensional Lorentz gas to higher densities than is possible by\nmeans of the ELBE. (2) The logarithmic terms obtained here, while\nsmall, are indicators of similar logarithmic terms which are certain\nto appear when these calculations are extended to general\ntwo-dimensional gases, where all of the particles move. \n \nThe first point allows one to contemplate a general kinetic theory for\nthe calculation of of sums, at least, of all positive, or of all\nnegative Lyapunov exponents. Such an approach was also indicated by\nDorfman, Latz, and van Beijeren \\cite{DLvB_chaos_98}, for the\nKS-entropy of a dilute gas in equilibrium, but the theory there has\nnot yet been developed beyond the Boltzmann equation. The relevance\nof the second point can be seen if one realizes that the linear\nNavier-Stokes transport coefficients of a two-dimensional gas diverge\nbecause of long time tail effects, of the type discussed here\n\\cite{DvB_berne_book}. In the general gas case therefore the\nlogarithmic terms in the positive and negative Lyapunov exponents will\nnot cancel as they do here, because the transport coefficients\nthemselves should diverge as $\\ln\\tilde{\\varepsilon}$ as $\\varepsilon$\napproaches zero. Thus the logarithmic terms obtained here should be\nseen as precursors of the more important logarithmic terms that will\nappear in the theory of two-dimensional gases. \n \nIt is worth noting that the\n$\\tilde{\\varepsilon}^{2}\\ln{\\tilde{\\varepsilon}}$ term results from a\nlong range correlation in time between the moving particle and the\nscatterers that is present in both the pair correlation functions,\n$G_2$, and $g_2$, either of which is proportional to the square of\nthe electric field strength and the inverse square of the wave number,\nat small wave numbers and fields. This dependence is not present in\nthe Lorentz gas in equilibrium, of course, but similar collision\nfrequency arguments to those given here suggest that non-analytic\nterms may be present in the ring contributions to the positive\nLyapunov exponent for trajectories on the fractal repeller for an open\nLorentz gas. In this case the inverse system size, $L^{-1}$, plays the\nrole of $\\displaystyle{\\frac{\\varepsilon}{2v}}$, the lower limit of\n$k$ for the integration over $\\vec{k}$ and one would expect to find\nterms of order $L^{-2}\\ln L$ in the ring term for this case. This\npoint is currently under investigation. \n \nFinally we mention that neither the non-analytic terms found here, nor\nthe excluded volume corrections included in the Enskog terms are able\nto account for the field dependence of the Lyapunov exponents as\nobserved in the computer simulations by Dellago and Posch\n\\cite{vBDCPD_prl_96}. This is not unexpected since we have not been\nsystematic in computing the density dependence of the coefficient of\n$\\varepsilon^{2}$, nor have we considered higher order terms in\n$\\varepsilon$ beyond order $\\varepsilon^{2}\\ln\\varepsilon$. All of the\nneglected terms are likely to be numerically more important than the\nones we have kept. There is also no indication in the simulation data\nfor the Lyapunov exponents of a clear presence of the interesting\nlogarithmic term in the applied field. Such logarithmic terms are\ntypically difficult to detect in simulation data, without a careful\nhunt for them\\cite{vLW_physica_67}. However, it may be easier to\ncheck, by means of computer simulation, the existence of the\n$\\tilde{\\varepsilon}^{2}\\ln{\\tilde{\\varepsilon}}$ term in the\ncollision frequency than in the Lyapunov exponents. In any case, we\nwould like to emphasize that computer simulation studies of\nthermostatted systems provide very useful ways to check a number of\nphenomena predicted by the kinetic theory of moderately dense gases. \n \nACKNOWLEDGEMENTS: This paper is dedicated to George Stell on the\noccasion of his 60-th birthday. We would like to thank Kosei Ide,\nZolt\\'{a}n Kov\\'{a}cs, Herman Kruis, Arnulf Latz, Luis Nasser, and\nRamses van Zon for many valuable conversations and suggestions. J.\\\nR.\\ D.\\ wishes to thank the National Science Foundation for support\nunder Grant PHY-96-00428. H.\\ v.\\ B.\\ acknowledges support by FOM,\nSMC and by the NWO Priority Program Non-Linear Systems, which are\nfinancially supported by the \"Nederlandse Organisatie voor\nWetenschappelijk Onderzoek (NWO)\". \n \n \n\\appendix \n \n\\section*{Appendix\\\\Derivation of the field-dependent collision \nfrequency} \n \n\\setcounter{equation}{0} \n \n\\renewcommand{\\theequation}{A\\arabic{equation}} \n \nTo derive the field dependence of the collision frequency we first\napproximate the probability of a recollision at time $t$ as \n\\begin{eqnarray} \nP^{\\mbox{\\scriptsize\nrec}}(t)\\,=\\,\\frac{\\nu}{2}\\,\\int_0^{2\\pi}d\\,\\theta\\int_0^{\\infty}d\\tau\\,\\int_{\\vec{v}\\cdot\\hat{\\sigma}>0}d\\hat{\\sigma}\\,|\\vec{v}\\cdot\n\\hat{\\sigma}|\\,R(\\tau,\\theta,\\sigma)\\,b_{\\hat{\\sigma}}\\,F^{\\mbox{\\tiny(B)}}(\\theta)\\,. \n\\label{Prec} \n\\end{eqnarray} \nHere $F^{\\mbox{\\tiny (B)}}(\\theta)$ describes the Boltzmann\ndistribution for the velocity in the NESS. The function\n$R(\\tau,\\theta,\\sigma)$ describes the probability density for return\nto the circumference of a given scatterer in a time $\\tau$ just after\ncolliding with this scatterer with scattering vector $\\hat{\\sigma}$\nand post-collisional velocity described by $\\theta$ (see Fig. 4). We\nhave ignored a possible dependence of the collision frequency $\\nu$ on\n$\\hat{v}$, which would only play a role at higher orders in the\ndensity. \n \n\\vspace{5mm} \n\\hspace{0.8in}{\\includegraphics[width=4.267in]{fig5}}\n\\vspace{-3mm} \n\\begin{center} \nFig. 4 : A recollision taking place after a real (solid line) or\ncorresponding virtual (dashed line) collision, followed by a\npost-collisional excursion maintaining on average the direction of\nvelocity over a persistence length $l_p$. \n\\end{center} \n\nSimilarly the reduction of the collision frequency at time $t$ due to\nvirtual recollisions can be estimated as \n\\begin{equation} \nP^{\\mbox{\\scriptsize nc}}(t)\\,=\\,-\\,\\frac{\\nu}{2}\\int_0^{2\\pi}\nd\\theta\\int_0^{\\infty}d\\tau\\int_{\\vec{v}\\cdot\\hat{\\sigma}>0}d\\hat{\\sigma}\\,|\\vec{v}\\cdot\n\\hat{\\sigma}|\\,R(\\tau,\\theta,\\sigma)\\,F^{\\mbox{\\tiny (B)}}(\\theta)\\,. \n\\label{Prec1} \n\\end{equation} \n \nIn equilibrium $F^{\\mbox{\\tiny (B)}}(\\theta)$ is independent of\n$\\theta$, so one sees immediately that both terms cancel, as they\nshould. In the presence of a thermostatted field we need the explicit\nform of $F^{\\mbox{\\tiny (B)}}_{1}(\\theta)$ up to the first field-dependent\norder, given in Eq. (\\ref{e50}) as \n\\begin{equation} \nF^{\\mbox{\\tiny(B)}}_{1}(\\theta)\\,=\\,\\frac{1}{2\\pi}\\left[\\,1\\,+\\,\\frac{3\\varepsilon}{8nav}\\,\\cos\\theta\\,\\right]. \n\\label{A2} \n\\end{equation} \nThe function $R(\\tau,\\theta,\\sigma)$ for large enough $\\tau$ may be\napproximated by the product of $2av$ (velocity times cross section)\nand the probability density for finding the point particle at the\nposition of the scatterer. For weak fields the latter may be\napproximated by the solution of a diffusion equation with a drift\nvelocity $u\\hat{x}$ in the $+x$-direction and an initial density\nlocalized at the position $l_p \\hat{\\theta}$ with respect to the\ncenter of the scatterer. Here $l_p$ is the persistence length, that\nis, the average distance traveled by a point particle in an\nequilibrium system in the direction of its initial velocity and\n$\\hat{\\theta}$ is the unit vector in the direction of the velocity\nright after the initial collision at $t-\\tau$. The persistence length\nmay be expressed as $\nl_{p}=\\displaystyle{\\int_0^{\\infty}dt\\,\\langle\\hat{v}\\cdot\n\\vec{v}(t)\\rangle}$. Multiplying this by the constant speed $v$ one\nfinds with the aid of the Green-Kubo expression for the diffusion that\n$l_p=\\displaystyle{\\frac{2D}{v}}$ in two dimensions. This assumption\nfor the long time distribution may be understood by imagining that the\nfirst few free flights after the initial collision of the particle\nmove it over a distance in the order of a mean free path in the\ndirection of its initial postcollisional velocity before it starts to\ndiffuse by virtue of further collisions with scatterers. Thus for\nlarge $\\tau$ the distribution of the light particle will be centered\naround the point $l_p\\hat{\\theta}$ with respect to the center of the\nscatterer, and the final point, on the surface of the scatterer, may\nbe approximated to be at the center of the scatterer as well, because\nof low density. These arguments lead to the explicit form for the\nrecollision probability given by \n\\begin{equation} \nR(\\tau,\\theta,\\sigma)\\,=\\,2av\\,\\frac{e^{-\\frac{[\\,l_p\\hat{\\theta}\\,+\\,u\\tau\\hat{x}\\,]^{2}}{4D\\tau}}}{4\\pi\nD\\tau}\\,. \n\\label{A3} \n\\end{equation} \nFinally we need the explicit form $u=\\displaystyle{\\frac{3\\varepsilon\nv}{8\\nu_0}}$ for the drift velocity to leading order in the density,\nand the identity that \n\\begin{equation} \n\\frac{1}{2}\\,\\int_{\\vec{v}\\cdot\\hat{\\sigma}>0}d\\hat{\\sigma}\\,|\\vec{v}\\cdot\\hat{\\sigma}|\\,b_{\\hat{\\sigma}}\\,\\cos\\theta\\,=\\,-\\,\\frac{v}{3}\\,\\cos\\theta\\,. \n\\end{equation} \nThen, after expanding \n\\begin{equation} \ne^{-\\,\\frac{2l_p\nu\\tau\\hat{\\theta}\\cdot\\hat{x}}{4D\\tau}}\\,=\\,1\\,-\\,\\frac{l_p\nu}{2D}\\cos\\theta\\,+\\,.\\,.\\,.\\,, \n\\end{equation} \nwe can now do all the calculations needed to obtain the leading\nnon-analytic term in the field expansion of the collision frequency.\nWe find that \n\\begin{eqnarray} \n\\delta\\nu\\,=\\,av\\nu\\int_{0}^{2\\pi}d\\theta\\int_{0}^{\\infty}d\\tau\\,\\frac{e^{-\\frac{[\\,l_{p}^{2}\\,+\\,(u\\tau)^{2}]}{4D\\tau}}}{4\\pi\nD\\tau}\\,\\bigg[\\,1\\,-\\,\\frac{l_{p}u}{2D}\\,\\cos\\theta\\,\\bigg]\\int_{\\vec{v}\\cdot{\\hat{\\sigma}}>0}d\\hat{\\sigma}\\,|\\vec{v}\\cdot\\hat{\\sigma}|\\,\n(b_{\\hat{\\sigma}}-1)\\,\\frac{3\\varepsilon\\cos\\theta}{16\\pi nav}\\,. \n\\label{A4} \n\\end{eqnarray} \nAfter performing the integrations, we recover\nEq. (\\ref{delnu}). 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Ergodic Properties of Dispersing Billiards},\nRuss.\\ Math.\\ Surv.\\ {\\bf 25}, 137 (1970). \n \n\\bibitem{Kovacs_priv_disc} Z.\\ Kov\\'{a}cs, private discussion. \n \n\\bibitem{Kovacs_preprint} Z.\\ Kov\\'{a}cs, {\\it Orbit Stability in\nBilliards in Magnetic Field}, Phys.\\ Rep.\\ {\\bf 290}, 49 (1997). \n \n\\bibitem{Hoover_elsevier_book} W.\\ G.\\ Hoover, {\\it Computational\nStatistical Mechanics}, Elsevier Science Publishers, Amsterdam, 1991. \n \n\\bibitem{Panja_preprint} D.\\ Panja, {\\it An Elementary Proof of\nLyapunov Exponent Pairing for Hard-Sphere Systems at Constant Kinetic\nEnergy} (preprint). \n \n\\bibitem{D_cup_book} J.\\ R.\\ Dorfman, {\\it An Introduction to Chaos in\nNon-Equilibrium Statistical Mechanics}, Cambridge Univ.\\ Press, 1999. \n \n\\bibitem{PD_unpublished} D.\\ Panja and J.\\ R.\\ Dorfman (unpublished). \n \n\\bibitem{DE_jsp_89} J.\\ R.\\ Dorfman and M.\\ H.\\ Ernst, {\\it\nHard-Sphere Binary Collision Operators}, J.\\ Stat.\\ Phys.\\ {\\bf 57},\n581 (1989). See also M.\\ H.\\ Ernst, J.\\ R.\\ Dorfman, W.\\ R.\\ Hoegy and\nJ.\\ M.\\ J.\\ van Leeuwen, {\\it Hard-Sphere Dynamics and\nBinary-Collision Operators}, Physica {\\bf 45}, 127 (1969). \n \n\\bibitem{vLW_physica_67} J.\\ M.\\ J.\\ van Leeuwen and A.\\ Weijland, \n{\\it Non-analytic Density Behaviour of the Diffusion Coefficient of a\nLorentz Gas}, Physica {\\bf 36}, 457 (1967). See also C.\\ Bruin, {\\it A\nComputer Experiment on Diffusion in the Lorentz Gas}, Physica\n(Utrecht) {\\bf 72}, 261 (1974). \n \n\\bibitem{ED_physica_72} M.\\ H.\\ Ernst and J.\\ R.\\ Dorfman, {\\it\nNonanalytic Dispersion Relations in Classical Fluids. I. The Hard\nSphere Gas}, Physica {\\bf 61}, 157 (1972). \n \n\\bibitem{CC_cup_book} S.\\ Chapman and T.\\ G.\\ Cowling, {\\it The \nMathematical Theory of Non-uniform Gases}, Cambridge University \nPress, 1970. \n \n\\bibitem{Kruis_thesis} H.\\ Kruis, Masters thesis, University of \nUtrecht, The Netherlands (1997). \n \n\\bibitem{DLvB_chaos_98} J.\\ R.\\ Dorfman, A.\\ Latz and H.\\ van\nBeijeren, {\\it Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy Methods\nfor Sums of Lyapunov Exponents for Dilute Gases}, Chaos {\\bf 8}, 444\n(1998). \n \n\\bibitem{PD_prl_97} H.\\ A.\\ Posch and Ch.\\ Dellago, {\\it Lyapunov\nSpectrum and the Conjugate Pairing Rule for a Thermostatted Random\nLorentz Gas: Numerical Simulations}, Phys.\\ Rev.\\ Lett.\\ {\\bf 78}, \n211 (1997). \n \n\\bibitem{LvBD_preprint} A.\\ Latz, H.\\ van Beijeren, and J.\\ R.\\ Dorfman, \n{\\it Chaotic Properties of Dilute Two- and Three-Dimensional Random \nLorentz Gases II: Open Systems} (preprint). \n \n\\bibitem{Panja_thesis} D.\\ Panja, Ph.D.\\ thesis, University\\ of Maryland, \nCollege Park, USA (2000). \n \n \n\\end{thebibliography} \n \n \n\\end{document}\n"
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{
"name": "cond-mat0002132.extracted_bib",
"string": "\\begin{thebibliography}{99} \n\n \n\\bibitem{EM_ap_book} D.\\ J.\\ Evans and G.\\ P.\\ Morriss, {\\it\nStatistical Mechanics of Non-equilibrium Liquids}, Academic Press,\nLondon, 1990. \n \n\\bibitem{EHFML_pra_83} D.\\ J.\\ Evans, W.\\ G.\\ Hoover, B.\\ H.\\ Failor,\nB.\\ Moran and A.\\ J.\\ C.\\ Ladd, {\\it Nonequilibrium Molecular\nDynamics via Gauss's Principle of Least Constraint}, Phys.\\ Rev.\\ A\n{\\bf 28}, 1016 (1983). \n \n\\bibitem{Evans_jcp_83} D.\\ J.\\ Evans, {\\it Computer ``Experiment'' for\nNonlinear Thermodynamics of Couette Flow}, J.\\ Chem.\\ Phys.\\ {\\bf 78}, 3297 \n(1983). \n \n\\bibitem{CELS_cmp_93} N.\\ I.\\ Chernov, G.\\ L.\\ Eyink, J.\\ L.\\ Lebowitz and \nYa.\\ G.\\ Sinai, {\\it Steady-State Electrical Conduction in Periodic\nLorentz Gas}, Comm.\\ Math.\\ Phys.\\ {\\bf 154}, 569 (1993). \n \n\\bibitem{Dettmann_preprint} C.\\ Dettmann, {\\it The Lorentz Gas: A \nParadigm for Non-Equilibrium Stationary States} (preprint). \n \n\\bibitem{vBD_prl_95} H.\\ van Beijeren and J.\\ R.\\ Dorfman, {\\it\nLyapunov Exponents and Kolmogorov-Sinai Entropy for the Lorentz Gas at\nLow Densities}, Phys.\\ Rev.\\ Lett.\\ {\\bf 74}, 4412 (1995) \n \n\\bibitem{vBLD_pre_98} H.\\ van Beijeren, A.\\ Latz and J.\\ R.\\ Dorfman,\n{\\it Chaotic Properties of Dilute Two- and Three-Dimensional Random\nLorentz Gases: Equilibrium Systems}, Phys.\\ Rev.\\ E {\\bf 57}, 4077 (1998). \n \n\\bibitem{vBDCPD_prl_96} H.\\ van Beijeren, J.\\ R.\\ Dorfman, \nE.\\ G.\\ D.\\ Cohen, H.\\ A.\\ Posch and Ch.\\ Dellago, {\\it Lyapunov\nExponents from Kinetic Theory for a Dilute, Field-Driven Lorentz Gas},\nPhys.\\ Rev.\\ Lett.\\ {\\bf 77}, 1974 (1996). \n \n\\bibitem{LvBD_prl_97} A.\\ Latz, H.\\ van Beijeren and J.\\ R.\\ Dorfman,\n{\\it Lyapunov Spectrum and the Conjugate Pairing Rule for a\nThermostatted Random Lorentz Gas: Kinetic Theory}, Phys.\\ Rev.\\ Lett.\\\n{\\bf 78}, 207 (1997). \n \n\\bibitem{DvB_berne_book} J.\\ R.\\ Dorfman and H.\\ van Beijeren, {\\it \nStatistical Mechanics, Part B.\\ Time-dependent Processes}, Bruce H.\\ \nBerne Ed., Plenum Press, New York, 1977. \n \n\\bibitem{EW_pl_71} M.\\ H.\\ Ernst and A.\\ Weijland, {\\it Long Time\nBehaviour of the Velocity Auto-Correlation Function in a Lorentz Gas},\nPhys.\\ Lett., {\\bf 34A}, 39 (1971). \n \n\\bibitem{Sinai_rms_70} Ya.\\ G.\\ Sinai,{\\it Dynamical Systems with\nElastic Reflections. Ergodic Properties of Dispersing Billiards},\nRuss.\\ Math.\\ Surv.\\ {\\bf 25}, 137 (1970). \n \n\\bibitem{Kovacs_priv_disc} Z.\\ Kov\\'{a}cs, private discussion. \n \n\\bibitem{Kovacs_preprint} Z.\\ Kov\\'{a}cs, {\\it Orbit Stability in\nBilliards in Magnetic Field}, Phys.\\ Rep.\\ {\\bf 290}, 49 (1997). \n \n\\bibitem{Hoover_elsevier_book} W.\\ G.\\ Hoover, {\\it Computational\nStatistical Mechanics}, Elsevier Science Publishers, Amsterdam, 1991. \n \n\\bibitem{Panja_preprint} D.\\ Panja, {\\it An Elementary Proof of\nLyapunov Exponent Pairing for Hard-Sphere Systems at Constant Kinetic\nEnergy} (preprint). \n \n\\bibitem{D_cup_book} J.\\ R.\\ Dorfman, {\\it An Introduction to Chaos in\nNon-Equilibrium Statistical Mechanics}, Cambridge Univ.\\ Press, 1999. \n \n\\bibitem{PD_unpublished} D.\\ Panja and J.\\ R.\\ Dorfman (unpublished). \n \n\\bibitem{DE_jsp_89} J.\\ R.\\ Dorfman and M.\\ H.\\ Ernst, {\\it\nHard-Sphere Binary Collision Operators}, J.\\ Stat.\\ Phys.\\ {\\bf 57},\n581 (1989). See also M.\\ H.\\ Ernst, J.\\ R.\\ Dorfman, W.\\ R.\\ Hoegy and\nJ.\\ M.\\ J.\\ van Leeuwen, {\\it Hard-Sphere Dynamics and\nBinary-Collision Operators}, Physica {\\bf 45}, 127 (1969). \n \n\\bibitem{vLW_physica_67} J.\\ M.\\ J.\\ van Leeuwen and A.\\ Weijland, \n{\\it Non-analytic Density Behaviour of the Diffusion Coefficient of a\nLorentz Gas}, Physica {\\bf 36}, 457 (1967). See also C.\\ Bruin, {\\it A\nComputer Experiment on Diffusion in the Lorentz Gas}, Physica\n(Utrecht) {\\bf 72}, 261 (1974). \n \n\\bibitem{ED_physica_72} M.\\ H.\\ Ernst and J.\\ R.\\ Dorfman, {\\it\nNonanalytic Dispersion Relations in Classical Fluids. I. The Hard\nSphere Gas}, Physica {\\bf 61}, 157 (1972). \n \n\\bibitem{CC_cup_book} S.\\ Chapman and T.\\ G.\\ Cowling, {\\it The \nMathematical Theory of Non-uniform Gases}, Cambridge University \nPress, 1970. \n \n\\bibitem{Kruis_thesis} H.\\ Kruis, Masters thesis, University of \nUtrecht, The Netherlands (1997). \n \n\\bibitem{DLvB_chaos_98} J.\\ R.\\ Dorfman, A.\\ Latz and H.\\ van\nBeijeren, {\\it Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy Methods\nfor Sums of Lyapunov Exponents for Dilute Gases}, Chaos {\\bf 8}, 444\n(1998). \n \n\\bibitem{PD_prl_97} H.\\ A.\\ Posch and Ch.\\ Dellago, {\\it Lyapunov\nSpectrum and the Conjugate Pairing Rule for a Thermostatted Random\nLorentz Gas: Numerical Simulations}, Phys.\\ Rev.\\ Lett.\\ {\\bf 78}, \n211 (1997). \n \n\\bibitem{LvBD_preprint} A.\\ Latz, H.\\ van Beijeren, and J.\\ R.\\ Dorfman, \n{\\it Chaotic Properties of Dilute Two- and Three-Dimensional Random \nLorentz Gases II: Open Systems} (preprint). \n \n\\bibitem{Panja_thesis} D.\\ Panja, Ph.D.\\ thesis, University\\ of Maryland, \nCollege Park, USA (2000). \n \n \n\\end{thebibliography}"
}
] |
cond-mat0002133
|
Wetting-induced effective interaction potential between spherical particles
|
[
{
"author": "C. Bauer"
},
{
"author": "T. Bieker"
},
{
"author": "and S. Dietrich"
}
] |
Using a density functional based interface displacement model we determine the effective interaction potential between two spherical particles which are immersed in a homogeneous fluid such as the vapor phase of a one-component substance or the A-rich liquid phase of a binary liquid mixture composed of A and B particles. If this solvent is thermodynamically close to a first-order fluid-fluid phase transition, the spheres are covered with wetting films of the incipient bulk phase, i.e., the liquid phase or the B-rich liquid, respectively. Below a critical distance between the spheres their wetting films snap to a bridgelike configuration. We determine phase diagrams for this morphological transition and analyze its repercussions on the effective interaction potential. Our results are accessible to force microscopy and may be relevant to flocculation in colloidal suspensions.
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[
{
"name": "paper.tex",
"string": "%\\documentstyle[pre,twocolumn,eqsecnum,aps,amssymb,epsfig]{revtex}\n\\documentstyle[preprint,pre,eqsecnum,aps,amssymb,epsfig]{revtex}\n\n\\begin{document}\n\\tightenlines\n\\newcommand{\\mS}{{\\mathcal S}}\n\\newcommand{\\mV}{{\\mathcal V}}\n\\newcommand{\\mL}{{\\mathcal L}}\n\\newcommand{\\mM}{{\\mathcal M}}\n\\newcommand{\\mK}{{\\mathcal K}}\n\\newcommand{\\rv}{{\\mathbf r}}\n\n\\draft\n\\title{Wetting-induced effective interaction potential between spherical\n particles} \n\\author{C. Bauer, T. Bieker, and S. Dietrich}\n\\address{Fachbereich Physik, Bergische Universit\\\"at Wuppertal,\\\\\nD-42097 Wuppertal, Germany}\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\nUsing a density functional based interface displacement model we\ndetermine the effective interaction potential between two spherical\nparticles which are immersed in a homogeneous fluid such as the\nvapor phase of a one-component substance or the A-rich liquid phase of a\nbinary liquid mixture composed of A and B particles. If this solvent\nis thermodynamically close to a first-order fluid-fluid phase\ntransition, the spheres are covered with wetting films of the incipient\nbulk phase, i.e., the liquid phase or the B-rich liquid,\nrespectively. Below a critical distance between the spheres their\nwetting films snap to a bridgelike configuration. We determine phase\ndiagrams for this morphological transition and analyze its\nrepercussions on the effective\ninteraction potential. Our results are accessible to force microscopy\nand may be relevant to flocculation in colloidal suspensions.\n\\end{abstract}\n\n\\pacs{68.45.Gd,68.10.-m,82.70.Dd}\n\n\\section{Introduction}\n\\label{s:intro}\n\nIn view of understanding a particular phenomenon in condensed matter,\ntheory is supposed to identify the corresponding relevant degrees of\nfreedom and to provide the effective interaction between them by,\napproximately, integrating out the remaining ones so that one is left\nwith a manageable model. It is a major challenge to determine the\neffective interactions because that requires to calculate the\npartition function of the whole system under the constraint of a fixed\nconfiguration of the relevant degrees of freedom. The benefit for\ncarrying out this constrained calculation, which in general is more\ndifficult than the original full problem, is twofold. First, there is\na gain in transparency by describing the system in terms of relevant\ndegrees of freedom. Secondly, it is typically less risky to apply\napproximations for the partial trace because they only concern the\nless relevant degrees of freedom.\n\nThe determination of the phase behavior and of the structural\nproperties of multi-component fluids represents a case study for this\ngeneral approach. If the composing particles of the mixture are of\ncomparable size and shape their degrees of freedom have to be treated\non equal footing. The well developed machinery of liquid state\ntheory~\\cite{hansenmcdonald} \noffers various techniques to cope with this problem. However, these\ntechniques fail to yield reliable results if, e.g., one component is\nmuch larger than the others; in this case numerical simulations become\ninefficient and integral theories lose their accuracy. Colloidal\nsuspensions are a paradigmatic case for such highly asymmetric\nsolutions. For their description these difficulties can be overcome by\nresorting to the general scheme laid out at the beginning with the \npositions of the\ncolloidal particles as the relevant degrees of freedom. Accordingly\nthe degrees of freedom of the small solvent particles are to be\nintegrated out for a fixed configuration of the colloidal particles\nwhich we assume to be smooth, monodisperse spheres. At sufficiently\nlow concentrations of the suspended particles this leads to an\neffective pair potential between them. In many cases the effective\npotential resembles the bare one, i.e., the one in the absence of the solvent,\nbut with modified, effective interaction parameters which depend on\nthe thermodynamic variables of the system such as pressure and\ntemperature.\n\nThe effective pair potential acquires additional new features if the\nsolvent is enriched with particles of medium size such as, e.g.,\npolymers. If the colloidal particles come close to each other the\ndepletion zones around them, generated by the finite size of the\nmedium particles, overlap leading to an entropically driven attraction\nof the colloidal particles~\\cite{asakuraoosawa,maoetal}. Correlation\neffects can modify the form and the range of these depletion forces\nconsiderably~\\cite{goetzelmannetal,rothetal}. These effective\npotentials have indeed turned out to be successful in describing the\nphase behavior of colloidal suspensions~\\cite{dijkstraetal}.\n\nQualitatively new aspects arise if the solvent particles exhibit a\nstrong cooperative behavior of their own such as a phase transition\nwhich proliferates to the effective potential between the large \nparticles. If the solvent undergoes a continuous phase transition,\nthermal Casimir forces between the large particles are induced due to\nthe geometrical constraint they pose for the critical\nfluctuations~\\cite{eisenriegleretal,hankeetal}. Such forces are\nlong-ranged and have a strong influence on the phase behavior of the\ncolloidal particles~\\cite{loewen,netz}. If the solvent is\nthermodynamically close to a first-order phase transition, wetting\nphenomena~\\cite{sdreview} can occur at the surfaces of the dissolved\nparticles (see Ref.~\\cite{bieker} and references therein). If the bulk\nphase of the solvent is the vapor phase of a one-component fluid, the\nsurfaces of the large spheres can be covered by a liquidlike wetting\nfilm. This situation corresponds to aerosol particles floating in a\nvapor. If the bulk phase of the solvent is the A-rich liquid phase of a\nbinary liquid mixture composed of (small) A and B molecules, the\ndissolved colloidal particles can be coated by the B-rich liquid\nphase of the mixture. If the wet spheres approach each other, at a\ncritical distance the two wetting films snap to a bridgelike\nstructure. This morphological phase transition is expected to yield a\nnonanalytic form of the effective interaction potential between the\nlarge spheres. This nonanalyticity demonstrates that cooperative\nphenomena among those degrees of freedom which are integrated out can\nleave clearly visible fingerprints on the effective interaction\nbetween the remaining relevant degrees of freedom. The study of this\nkind of profileration is not only of theoretical interest in its own\nright but seems to play an important (albeit not\nexclusive~\\cite{lawetal}) role for the experimentally observed\nflocculation of colloidal particles dissolved in a binary liquid\nmixture close to its demixing transition into an A-rich and a B-rich\nliquid\nphase~\\cite{beysensetc,makeretal,kiralyetal,jayalakshmikaler,gruellwoermann}.\nThis observation has triggered numerous theoretical efforts devoted to\nvarious possible explanations of it. Since they are reviewed in Sec.~I\nof Ref.~\\cite{bieker} and more recently in Ref.~\\cite{gilipsentejero}\nthe interested reader is referred to there and \nwe refrain from repeating this discussion here.\n\nIn our present analysis of this problem we apply density functional\ntheory~\\cite{evans} which offers two advantages. First, this technique\nis particularly well suited to calculate, as required here, free\nenergies under constraints. Secondly, it allows one to keep track of\nthe basic molecular interaction potentials of the system. We\nfocus our interest on thermodynamic states of the solvent which are\nsufficiently far away from its critical point so that the emerging\nliquid-vapor interfaces of the wetting films exhibit only a small\nwidth. Therefore we can apply the so-called sharp-kink approximation\nwhich considers only steplike variations of the solvent density\ndistribution and thus leaves the interface position as the main\nstatistical variable. This approximation has turned out to be\nsurprisingly accurate for the description of wetting\nphenomena~\\cite{dietnap}. Our analysis extends and goes beyond\nprevious efforts~\\cite{dobbsetal,dobbsyeomans} which are based on a\nsimilar interface displacement model grounded on a phenomenological\nansatz. Whereas Refs.~\\cite{dobbsetal} and \\cite{dobbsyeomans} are\naimed at mapping out the phase diagram in terms of interaction\nparameters for the bridging transition\nmentioned above, we focus on the effective interaction potentials\nbetween the wet spheres, which are not presented in\nRefs.~\\cite{dobbsetal} and \\cite{dobbsyeomans}, and on their\nmicroscopic origin. Inter alia, this allows us to compare the\neffective interaction potential between the colloidal particles with\nthe bare one, i.e., in the absence of the solvent, and thus to comment\non the quantitative relevance of the solvent-mediated interaction.\nMoreover, we present the phase diagram of the system in terms of the\nthermodynamic variables temperature and chemical potential which is\nalso not contained in Refs.~\\cite{dobbsetal} and \\cite{dobbsyeomans}.\n\nIn Sec.~\\ref{s:theory} we describe the implementation of a simple\nversion of density functional theory for the present problem. For\nreasons of simplicity we confine our analysis to liquid-vapor\ncoexistence of a one-component solvent; the generalization to a binary\nsolvent is straightforward. In Sec.~\\ref{s:morphology} we present some\nexamples for the numerically calculated wetting film morphologies and\ndiscuss a phase diagram for the aforementioned morphological transition, and\nin Sec.~\\ref{s:eip} we analyze the effective wetting-induced interaction potential\nbetween the spheres as a function of the distance between the spheres and\nthe undersaturation. The experimental relevance of our model\ncalculations is discussed in Sec.~\\ref{s:discussion} and\nSec.~\\ref{s:summary} summarizes our main results. The Appendix\ncontains some technical details.\n\n\\section{Density functional theory}\n\\label{s:theory}\n\n\\subsection{Model}\n\nWe consider two identical, homogeneous, and smooth spherical particles of radius\n$R$ whose centers of mass are separated by a distance $D$ (see\nFig.~\\ref{f:system}). They are immersed in a fluid of particles of number\ndensity $\\rho(\\rv)$ which interact via a Lennard-Jones potential \n\\begin{equation}\\label{e:ljff}\n\\phi(r) =\n4\\epsilon\\left(\\left(\\frac{\\sigma}{r}\\right)^{12} -\n \\left(\\frac{\\sigma}{r}\\right)^6\\right).\n\\end{equation}\nThe system is symmetric with respect to a rotation around the axis\nwhich connects the centers of mass of the spheres\n(Fig.~\\ref{f:system}) and with respect to a reflection at a plane in\nthe middle between the spheres that is perpendicular to the symmetry\naxis. Since we work in a grand canonical ensemble and \nthe fluid particles are subject to the external potential exerted by\nthe spheres, the equilibrium number density profile of the fluid\nparticles exhibits these symmetries, too. Therefore we describe the\nsystem in cylindrical coordinates $(r_{\\perp},\\phi,z)$, with the $z$ \naxis being the symmetry axis of the system. The \ntwo centers of mass of the spheres are located at $(r=0,z=\\pm D/2)$\nsuch that the spheres occupy the volumes \n$\\mS_{\\pm} = \\{\\rv(r_{\\perp},\\phi,z)\n=(x,y,z)=(r_{\\perp}\\cos\\phi,r_{\\perp}\\sin\\phi,z)\\in {\\mathbb\n R}^3|\\pm D/2-R\\leq z\\leq\\pm D/2+R,\n\\sqrt{r_{\\perp}^2+(z\\mp D/2)^2}\\leq R\\}$.\nThe external potential exerted by both spheres on each individual fluid\nparticle is \n\\begin{equation}\\label{e:vtot}\nv_{tot}(r_{\\perp},z;R) = v\\left(\\sqrt{r_{\\perp}^2+(z-D/2)^2};R\\right) +\nv\\left(\\sqrt{r_{\\perp}^2+(z+D/2)^2};R\\right) \n\\end{equation}\nwhere (see Eq.~(A.4) in Ref.~\\cite{bieker})\n\\begin{eqnarray}\\label{e:vsphere}\nv(r;R) & = &\n\\frac{9}{8}u_9\\left(\\frac{1}{r(r+R)^8}-\\frac{1}{r(r-R)^8}\\right)\n-u_9\\left(\\frac{1}{(r+R)^9}-\\frac{1}{(r-R)^9}\\right) \\nonumber\\\\\n& & -\\frac{3}{2}u_3\\left(\\frac{1}{r(r+R)^2}-\\frac{1}{r(r-R)^2}\\right)\n+u_3\\left(\\frac{1}{(r+R)^3}-\\frac{1}{(r-R)^3}\\right)\n\\end{eqnarray}\nis the interaction potential between a single sphere of radius $R$ and a fluid\nparticle at a distance $r>R$ from the center of mass of the\nsphere. In a continuum description, $v(r;R)$ follows from\nan integration of the Lennard-Jones potential\n\\begin{equation}\\label{e:ljsphere}\n\\phi_{sf}(r) = 4\\epsilon_{sf}\\left(\\left(\\frac{\\sigma_{sf}}{r}\\right)^{12} -\n \\left(\\frac{\\sigma_{sf}}{r}\\right)^6\\right)\n\\end{equation}\nbetween a molecule of the spherical \\emph{s}ubstrate and a \\emph{f}luid particle.\nThe subscript $sf$ denotes the parameters\nof the dispersion interaction between a particle in the fluid and a\nparticle in the spheres. One has $u_3 =\n\\frac{2\\pi}{3}\\epsilon_{sf}\\rho_s\\sigma_{sf}^6$ and $u_9 =\n\\frac{4\\pi}{45}\\epsilon_{sf}\\rho_s\\sigma_{sf}^{12}$ where $\\rho_s$ is\nthe number density of the particles forming the spheres. (Many colloidal\nparticles exhibit an even more complicated substrate potential because\nthey are coated by a material different from their core so that they\nare no longer radially homogeneous as assumed for Eq.~(\\ref{e:vsphere}).)\n\nWithin our density functional approach the equilibrium particle\nnumber density distribution of \nthe inhomogeneous fluid surrounding the spheres in a grand canonical\nensemble minimizes the functional~\\cite{evans}\n\\begin{eqnarray}\\label{e:ldafunc}\n\\Omega([\\rho(\\rv)];T,\\mu) & = & \\int\\limits_{\\mV_f} d^3r\n\\Big(f_{HS}(\\rho(\\rv),T) + \\big(v_{tot}(\\rv)-\\mu\\big)\\rho(\\rv)\\Big) \\nonumber\\\\\n& & + \\frac{1}{2}\\int\\limits_{\\mV_f}\\int\\limits_{\\mV_f} d^3r\\,d^3r'\\,\nw(|\\rv-\\rv'|)\\rho(\\rv)\\rho(\\rv'). \n\\end{eqnarray}\n$\\mV_f = \\mV\\setminus(\\mS_+\\cup\\mS_-)$ is the\nvolume accessible for the fluid particles, $\\mV$ is the total volume of\nthe system; $\\mV \\to {\\mathbb R}^3$ in the thermodynamic\nlimit. Equation~(\\ref{e:ldafunc}) does not include the bare\ninteraction potential $\\Phi(D;R)$ (see, c.f., Sec.~\\ref{s:discussion})\nbetween the solid spheres, separated by vacuum, generated by the\ndispersion forces between the molecules forming the two spheres.\n$f_{HS}(\\rho,T)$ is the free energy density\nof a hard-sphere fluid of number density $\\rho$ at temperature $T$. In\nEq.~(\\ref{e:ldafunc}), the hard-sphere reference fluid is treated in\nlocal density approximation. We apply the\nWeeks-Chandler-Andersen procedure \\cite{wca} to split up \n$\\phi(r)$ into an attractive part $\\phi_{att}(r)$ and a repulsive\npart $\\phi_{rep}(r)$. The latter gives rise to an effective,\ntemperature dependent hard-sphere diameter\n\\begin{equation}\nd(T) = \\int\\limits_0^{2^{1/6}\\sigma}dr\\,\n\\left(1-\\exp\\left(-\\frac{\\phi_{rep}(r)}{k_BT}\\right)\\right) \n\\end{equation}\nwhich is inserted into the Carnahan-Starling expression~\\cite{cs}\n\\begin{equation}\\label{e:cs}\nf_{HS}(\\rho,T) =\nk_BT\\rho\\left(\\ln(\\rho\\lambda^3)-1+\\frac{4\\eta-3\\eta^2}{(1-\\eta)^2}\\right)\n\\end{equation}\nfor the free energy density $f_{HS}$ of the \\emph{h}ard-\\emph{s}phere fluid, where\n$\\eta=\\frac{\\pi}{6}\\rho(d(T))^3$ is the dimensionless packing \nfraction and $\\lambda$ is the thermal de Broglie wavelength.\nWe approximate the attractive part of the interaction $\\phi_{att}(r)$ by\n\\begin{equation}\\label{e:attractivepot}\nw(r) = \\frac{4w_0\\sigma^3}{\\pi^2}(r^2+\\sigma^2)^{-3}\n\\end{equation}\nwith\n\\begin{equation}\\label{e:w0}\nw_0 = \\int_{{\\mathbb R}^3}d^3r\\,w(r) = \\int_{{\\mathbb\nR}^3}d^3r\\,\\phi_{att}(r) = -\\frac{32}{9}\\sqrt{2}\\pi\\epsilon\\sigma^3\n\\end{equation}\nin order to simplify subsequent analytical calculations. The double\nintegral in Eq.~(\\ref{e:ldafunc}) takes into account this attractive\ninteraction within mean-field approximation.\n\nIn the bulk the particle density $\\rho_{\\gamma}$ (where\n$\\gamma=l,g$ denotes the liquid and vapor phase, respectively) is\nspatially constant, leading to (see Eq.~(\\ref{e:ldafunc}))\n\\begin{equation}\\label{e:bfedensity}\n\\Omega_b(\\rho_{\\gamma},T,\\mu) = f_{HS}(\\rho_{\\gamma},T) +\n\\frac{1}{2}w_0\\rho_{\\gamma}^2 - \\mu\\rho_{\\gamma}\n\\end{equation}\nfor the grand canonical free energy density of the \\emph{b}ulk\nfluid. Minimization of \n$\\Omega_b$ with respect to $\\rho_{\\gamma}$ yields the equilibrium densities.\nThe line $\\mu=\\mu_0(T)$ of bulk liquid-vapor coexistence and the two\nbulk densities $\\rho_l$ and $\\rho_g$ at coexistence follow from\n\\begin{equation}\\label{e:coexcond}\n\\left.\\frac{\\partial\\Omega_b}{\\partial\\rho}\\right|_{\\rho=\\rho_g} = \n\\left.\\frac{\\partial\\Omega_b}{\\partial\\rho}\\right|_{\\rho=\\rho_l} = 0\n\\quad \\mbox{and} \\quad \\Omega_b(\\rho_g) = \\Omega_b(\\rho_l).\n\\end{equation}\nFor $\\mu\\neq\\mu_0$, i.e., off coexistence, only the liquid or the\nvapor phase is stable. In this case the density of the metastable\nphase corresponds to the second local minimum of $\\Omega_b$.\n\n\\subsection{General expressions for the contributions to the effective\n interaction potential}\n\nHenceforth we consider the case that the substrate is sufficiently\nattractive so that the liquid phase\nis preferentially adsorbed. Therefore, if in the bulk the vapor phase is stable\n($\\mu\\leq\\mu_0$), the fluid density is significantly increased in\nthe vicinity of both spheres. In the spirit of the so-called\nsharp-kink approximation (see Sec.~\\ref{s:intro} and\nRef.~\\cite{dietnap}) we assume that a thin film of constant density \n$\\rho_l$ but with locally varying thickness is adsorbed at the\nsurfaces of the spheres, separating the \nspheres from the bulk vapor phase of density $\\rho_g$. This wetting film\nencapsulating both spheres is characterized by a function $h(z)$:\n\\begin{equation}\\label{e:hz}\n\\rho(\\rv) = \\rho(r_{\\perp},\\phi,z) = \n\\Theta(r_{\\perp}-(R+d_s))\\bigg(\\Theta(h(z)-r_{\\perp})\\rho_l +\n\\Theta(r_{\\perp}-h(z))\\rho_g\\bigg) \n\\end{equation}\nwhere $\\Theta$ denotes the Heaviside step function. The length $d_s$\ntakes into account the excluded volume at the surfaces of the spheres\nwhich the centers of the fluid particles cannot penetrate due to repulsive\nforces. The profile $h(z)$ as given by\nEq.~(\\ref{e:hz}) can describe both a \nconfiguration in which the wetting films surrounding each sphere are\nconnected by a liquid bridge as well as the configuration in which\nboth single spheres are surrounded by disjunct wetting layers. In the\nlatter configuration there is a region around $z=0$ with\n$h(z)=0$.\n\nInserting $\\rho(r_{\\perp},\\phi,z)$ from Eq.~(\\ref{e:hz}) into the\nfunctional $\\Omega$ in Eq.~(\\ref{e:ldafunc}) leads to a decomposition of\n$\\Omega = \\mbox{Vol}(\\mV_f)\\Omega_b(\\rho_g)+\\Omega_S$ into a bulk\nand subdominant contributions. The bulk \ncontribution is $\\mbox{Vol}(\\mV_f)\\Omega_b(\\rho_g)$ (with\n$\\Omega_b$ given by Eq.~(\\ref{e:bfedensity})) and\ncorresponds to the vapor phase which is stable in the bulk. The subdominant\ncontribution is \n\\begin{equation}\\label{e:decomp}\n\\Omega_S[h] = \\Omega_{sl} + \\Omega_{ex}[h] + \\Omega_{ei}[h] +\n\\Omega_{lg}[h]\n\\end{equation}\nwhere only $\\Omega_{sl}$ is independent of $h(z)$ and all the other three\ncontributions are functionals of $h(z)$. Since we have not found an\nindication for spontaneous symmetry breaking, in the following we\ndiscuss only symmetric configurations with $h(z) = h(-z)$.\n\\begin{equation}\\label{e:excess}\n\\Omega_{ex}[h(z)] = \\mbox{Vol}(\\mL)\\Big(\\Omega_b(\\rho_l)-\\Omega_b(\\rho_g)\\Big) \n\\end{equation}\nwith \n\\begin{equation}\\label{e:volliq}\n\\mbox{Vol}(\\mL) = 2\\pi\\int\\limits_0^{L_z}dz\\,h^2(z) -\n\\frac{8\\pi}{3}R^3\n\\end{equation}\nis an \\emph{ex}cess contribution which takes into account that the volume\n$\\mL = \\mK\\setminus(\\mS_-\\cup \\mS_+)$ is filled with the metastable\nliquid instead of the vapor phase; $\\mK = \\{ \\rv(r_{\\perp},\\phi,z) \\in {\\mathbb\n R}^3 | r_{\\perp}\\leq h(z)\\}$ \nis the volume enclosed by the liquid-vapor interface. (The excluded\nvolume due to $d_s$ enters into $\\Omega_{sl}$ (see, c.f.,\nEq.~(\\ref{e:omegasl})).) This free energy\ncontribution $\\Omega_{ex}$ vanishes at two-phase coexistence\n$\\mu=\\mu_0(T)$ (compare Eq.~(\\ref{e:coexcond})).\n$2L_z$ is the extension of the total volume of the system $\\mV$ in $z$ direction;\n$L_z\\to\\infty$ in the thermodynamic limit and $h(z>z_{max}) = 0$ with\n$z_{max} \\ll L_z$.\n\\begin{equation}\\label{e:omegaei}\n\\Omega_{ei}[h(z)] = 2\\Delta\\rho\\int\\limits_{\\mV_-\\setminus \\mK_-}\nd^3r\\,\\Big(\\rho_l(t(\\rv,\\mS_-)+t(\\rv,\\mS_+)) -\nv_{tot}(\\rv)\\Big) \n\\end{equation}\ncan be interpreted as the integrated \\emph{e}ffective \\emph{i}nteraction between the\nspheres and the liquid-vapor interface described by $h(z)$;\n$\\Delta\\rho = \\rho_l-\\rho_g$. $\\mV_-$ is that part of the volume\n$\\mV$ with $z<0$ (we note again that $\\mV_-\\to{\\mathbb R}_-^3$ in the\nthermodynamic limit which is always considered here), \nanalogously $\\mK_-$ is the part of the set $\\mK$ with $z<0$. In\nEq.~(\\ref{e:omegaei}) we have introduced the interaction potential\n\\begin{equation}\nt(\\rv;\\mM) = \\int\\limits_{\\mM} d^3r'\\,w(|\\rv-\\rv'|)\n\\end{equation}\nbetween a fluid particle at $\\rv$ and a region $\\mM$ (with\n$\\rv\\not\\in\\mM$) homogeneously filled with the same fluid\nparticles (analogous to the function $t(z)$ introduced in\nRefs.~\\cite{sdreview} and \\cite{dietnap} in the case of a planar\nsubstrate). $v_{tot}$ is the total interaction potential between a \nfluid particle and both spheres (see Eq.~(\\ref{e:vtot})).\nFinally,\n\\begin{equation}\\label{e:omegalg}\n\\Omega_{lg}[h(z)] = -(\\Delta\\rho)^2\\int\\limits_{\\mV_-\\setminus\n \\mK_-} d^3r\\,\\Big(t(\\rv;\\mK_-)+t(\\rv;\\mK_+)\\Big)\n\\end{equation}\nis the free energy contribution from the free \\emph{l}iquid-\\emph{g}as interface. It\nis a \\emph{nonlocal} functional of $h(z)$ in contrast to $\\Omega_{ex}$\nand $\\Omega_{ei}$ whose dependence on $h(z)$ enters only via the\nintegration volume $\\mK_-$. The \\emph{local} approximation thereof,\nwhich is provided by the gradient expansion of\nEq.~(\\ref{e:omegalg}), is\n\\begin{equation}\\label{e:local}\n\\Omega_{lg}^{loc} = 4\\pi\\sigma_{lg}^{(p)}\\int\\limits_0^{L_z} dz\\, h(z)\\,\n\\sqrt{1+\\left(\\frac{dh}{dz}\\right)^2}.\n\\end{equation}\nIn Eq.~(\\ref{e:local})\n\\begin{equation}\\label{e:sigmalgplanar}\n\\sigma_{lg}^{(p)} =\n-\\frac{1}{2}(\\Delta\\rho)^2\\int\\limits_0^{\\infty}dz\\,\n\\int\\limits_z^{\\infty}dz'\\int\\limits_{{\\mathbb R}^2}d^2r_{\\parallel}\nw\\left(\\sqrt{r_{\\parallel}^2+z'^2}\\right)\n\\end{equation}\nis the interfacial tension of a planar, free liquid-vapor interface in\nsharp-kink approximation. We note that, strictly speaking, the\nsurface tension of a curved liquid-vapor interface depends on the local\nradius of curvature (see Fig.~2 in Ref.~\\cite{bieker} and the\nreferences therein concerning the Tolman length). This curvature\ndependence is omitted in the local model presented here. However, for\nspheres of radius $R\\geq20\\sigma$ as considered henceforth the curvature \ncorrection is less than $1\\%$. Similar arguments hold for the\ndeviation of the actual liquidlike density in the wetting film from\nthe bulk value $\\rho_l$.\n\nFor our choice of interaction potentials (Eq.~(\\ref{e:ljff}) and \n(\\ref{e:ljsphere})) a tedious calculation leads to explicit\nexpressions for the contributions $\\Omega_{ei}$ and\n$\\Omega_{lg}$ which are given in the Appendix. The remaining contribution \n\\begin{equation}\\label{e:omegasl}\n\\Omega_{sl} = -\\rho_l\\int\\limits_{\\mV_-\\setminus \\mS_-}\nd^3r\\,\\Big(\\rho_l(t(\\rv,\\mS_-)+t(\\rv,\\mS_+)) -\n2v_{tot}(\\rv)\\Big) - \\Omega_b(\\rho_l)\\frac{8\\pi}{3}((R+d_s)^3-R^3),\n\\end{equation}\nwhich is independent of $h(z)$, is the \\emph{s}phere-\\emph{l}iquid interfacial free\nenergy corresponding to the interface between the spheres and the\nliquid phase. The last term in Eq.~(\\ref{e:omegasl}) takes into\naccount the excluded volumes at the surfaces of the spheres. In the\nlimit of large separations $D$ one has\n\\begin{equation}\\label{e:omegasl_asymp}\n\\Omega_{sl}(D\\to\\infty) - 2\\Omega_{sl}^{(1)} \\sim D^{-6}\n\\end{equation}\nwith the sphere-liquid interfacial free energy $\\Omega_{sl}^{(1)}$ of\na single sphere immersed in the liquid phase. The\nleading power law $\\sim D^{-6}$ in Eq.~(\\ref{e:omegasl_asymp}) can be\ninferred from the following consideration: \nif present, the second sphere displaces a spherical volume from the\nhomogeneous liquid phase so that the free energy of the interaction of\nthe first sphere with the bulk liquid is reduced by the interaction\nfree energy of that sphere with the displaced spherical liquid volume. This\nlatter interaction decays as $D^{-6}$ for large separations $D$, at\nwhich the dispersion interaction between two spherical objects resembles the\ndispersion interaction between two pointlike particles. (Here, as\nbefore, we have not yet taken into account the bare interaction\npotential $\\Phi(D;R)$ between the two solid spheres; but see, c.f.,\nSec.~\\ref{s:discussion}.) \n\nUp to the bulk contribution the grand canonical potential of the\nsystem is the minimum of $\\Omega_S[h(z)]$ with respect to the profile $h(z)$:\n\\begin{equation}\\label{e:gcpismin}\n\\Omega_S = \\Omega_S(D;R) = \\min_{\\{h(z)\\}}(\\Omega[h(z)]).\n\\end{equation}\nThus the equilibrium interface morphology $h(z)$ minimizes\n$\\Omega_S[h(z)]$ which includes the contributions $\\Omega_{ex}[h(z)]$,\n$\\Omega_{ei}[h(z)]$, $\\Omega_{lg}[h(z)]$, and $\\Omega_{sl}$.\nThe functional used in Refs.~\\cite{dobbsetal} and\n\\cite{dobbsyeomans} (Eq.~(1) in both references) is, albeit formulated\nin another coordinate system and using a more phenomenological ansatz\nfor the basic interaction potentials, essentially identical with the sum\n$(\\Omega_{lg}^{loc}+\\Omega_{ex}+\\Omega_{ei})[h(z)]$. However, this model \ndescription does neither incorporate the bare dispersion\ninteraction of the two spheres (c.f., Sec.~\\ref{s:discussion}) nor the\nfree energy contribution \n$\\Omega_{sl}$ which describes the sphere-liquid interfacial\nfree energy. We emphasize that the consideration of the contribution\n$\\Omega_{sl}$ -- which does not depend on $h(z)$ -- is not essential\nfor the determination of the equilibrium wetting film morphology and\nhence it is not relevant for the thermodynamic phase diagram of\nthin-thick and bridging \ntransitions (Fig.~2 in Ref.~\\cite{dobbsyeomans}) for a \\emph{fixed}\nseparation $D$ between the spheres. But the term $\\Omega_{sl}$ is crucial to\nthe \\emph{shape} of the effective,\nwetting-induced interaction potential between the spheres, i.e., its\ndependence on $D$ (see Eq.~(\\ref{e:omegasl_asymp})).\n\n\\section{Morphology of the wetting layers}\n\\label{s:morphology}\n\n\\subsection{Interface profiles}\n\nThe actual wetting layer morphology $h(z)$ follows from numerical\nminimization of the functional $\\Omega_S[h(z)]$ (Eq.~(\\ref{e:decomp}))\nfor a given temperature $T$ and undersaturation $\\Delta\\mu=\\mu_0(T)-\\mu$,\nwith the contributions $\\Omega_{ex}$ (Eq.~(\\ref{e:excess})),\n$\\Omega_{ei}$ (Eq.~(\\ref{e:omegaei})), $\\Omega_{sl}$\n(Eq.~(\\ref{e:omegasl})), and $\\Omega_{lg}$ (Eq.~(\\ref{e:omegalg}) within\nthe nonlocal and Eq.~(\\ref{e:local}) for the local theory). Within\na range of parameters $(T,\\Delta\\mu)$ the numerical minimization\nyields two different solutions for $h(z)$, one with a liquid bridge and one\nwithout, depending on the initial function $h(z)$ used in the\niteration scheme for the minimization. For small separations\n$a\\ll2R$ only the solution which exhibits\na liquid bridge is stable whereas for large separations\n$a\\gg2R$ only the solution without bridge minimizes $\\Omega_S$. For\nlarge distances $D$ the minimization consistently\nyields twice the result known for a single individual sphere enclosed by a wetting\nfilm (compare Ref.~\\cite{bieker}). This observation amounts to a\nuseful check of the numerical procedure.\n\nAs a first example, in Fig.~\\ref{f:example1} we present the numerical\nresults for a wetting layer enclosing two spheres of radius\n$R=20\\sigma$. For our particular choice of interaction parameters, at\ncoexistence $\\Delta\\mu=0$ the wetting film on each of the single spheres alone\nexhibits a first-order \\emph{t}hin-\\emph{t}hick transition (which is the remnant\nof the first-order wetting transition on the corresponding planar\nsubstrate, see Fig.~8(a) in Ref.~\\cite{bieker}) at $T_{tt}^* =\nk_BT/\\epsilon \\approx 1.271$ \n(which corresponds to $T_{tt}/T_c \\approx 0.9$ where $T_c$ is the\ncritical temperature of gas-liquid coexistence in the bulk). The planar substrate,\ni.e., a single sphere in the limit $R\\to\\infty$, exhibits a genuine\nfirst-order wetting transition (with the film thickness jumping to\na macroscopic value) at $T_w^* \\approx 1.053$ ($T_w/T_c \\approx 0.75$,\n$T_{tt}/T_w\\approx1.21$). Figure~\\ref{f:example1}(a) depicts a typical\nsolution with a bridge, here for a separation $a = D-2R = 10\\sigma$ ($D = 50\\sigma$)\nand the thermodynamic parameters $T^*=1.3>T_{tt}^*$ and\n$\\Delta\\mu=0$, i.e., at liquid-vapor coexistence.\nThe solution without a bridge for the same\nchoice of parameters is shown in Fig.~\\ref{f:example1}(b). The latter\nsolution has a higher free energy than the former one. Therefore the\nsolution with bridge is thermodynamically stable whereas the solution\nwithout bridge is metastable. For \nthe solution without a bridge the distortion of the liquidlike layer\naround one sphere due to the presence of the other sphere is not visible. Finally,\nFig.~\\ref{f:example1}(c) displays the wetting film morphology for the\nstable state with bridge at the temperature $T^*=1.2$, i.e., below the\nthin-thick transition temperature $T_{tt}^*$. (We note that the\nthin-thick transition temperature $T_{tt}$ for each sphere is slightly shifted\nby the presence of the second sphere. However, as already pointed out\nin Ref.~\\cite{dobbsyeomans}, this effect is negligibly small.) In\nany case, the difference between the nonlocal and the\nlocal theory is very small. This latter result is in accordance\nwith the findings for the comparison between the nonlocal and the local\ndescription of the three-phase\ncontact line on a homogeneous substrate and of the wetting layer\nmorphology on a chemically structured substrate (compare\nRef.~\\cite{bauer1}). For this reason, henceforth we only consider\nthe local theory.\n\nFigure~\\ref{f:example2} shows another pertinent example. Here we study\nthe wetting layer morphology for two larger spheres of radius $R=50\\sigma$ as\na function of the undersaturation $\\Delta\\mu$ along the isotherm\n$T^*=1.2$. The interaction potential parameters are the same as for\nthe previous first example and the separation of the surfaces $a$ is\n$20\\sigma$ ($D=120\\sigma$). At coexistence each single\nsphere exhibits a first-order thin-thick transition \nat $T_{tt}^* \\approx 1.191$ (i.e., $T_{tt}/T_c \\approx 0.84$ and\n$T_{tt}/T_w \\approx 1.13$). In analogy to the prewetting line on a\nhomogeneous substrate there is a line of thin-thick transitions\n$(T,\\Delta\\mu_{tt}(T))$ which intersects the liquid-vapor coexistence\nline at $(T=T_{tt},\\Delta\\mu=0)$ (compare with, c.f.,\nFig.~\\ref{f:btandttt} and Fig.~8(a) in \nRef.~\\cite{bieker}). At the temperature $T^*=1.2>T_{tt}^*$ considered\nhere the thin-thick transition occurs at\n$\\Delta\\mu_{tt}^* = \\Delta\\mu_{tt}/\\epsilon \\approx 0.0103$. Upon reducing the\nundersaturation along the isotherm, starting at, e.g., $\\Delta\\mu^*\n= 0.05$, first the \nconfiguration with thin films and without bridge is stable\n(Fig.~\\ref{f:example2}(a)). For \n$\\Delta\\mu\\leq\\Delta\\mu_{bt}$ (\\emph{b}ridging \\emph{t}ransition) with\n$\\Delta\\mu_{bt}^*\\approx 0.0235 > \\Delta\\mu_{tt}^*(T)$ the solution\nwith bridge becomes \nstable, but the layers enclosing the spheres still remain thin\n(Fig.~\\ref{f:example2}(b)). Upon\nfurther reduction of $\\Delta\\mu$, at $\\Delta\\mu_{tt}(T)$ \nthe second transition from a solution with bridge and thin films to\na solution with bridge and thick films (Fig.~\\ref{f:example2}(c))\ntakes place. (As before, concerning the value of $T_{tt}^*$ at\ncoexistence, also the value $\\Delta\\mu_{tt}^*(T)$ is practically\nunchanged by the presence of the second sphere -- even for the bridge\nconfiguration.) We note that for this \nchoice of parameters and in the case of a solution with bridge and \\emph{thin}\nfilms (Fig.~\\ref{f:example2}(b)) the profile $h(z)$ exhibits\n\\emph{six} turning points instead of only two as for the case of a solution with\nbridge and \\emph{thick} films (Fig.~\\ref{f:example2}(c)). This rich curvature\nbehavior is caused by the details of the \neffective interaction potential between the spherical substrate surfaces and the\nliquid-vapor interface (see Sec.~2.3 in\nRef.~\\cite{bieker}), similar to the curvature behavior of the \nliquid-vapor interface when it meets a homogeneous, planar substrate\nforming a three-phase contact line (compare Ref.~\\cite{bauer1}). These features\nmay also occur for a bridge configuration with thin films at\ncoexistence and $T<T_{tt}$. \n\n\\subsection{Phase diagram}\n\nThe example presented in the previous paragraph shows that besides\nthe gas-liquid coexistence curve $\\Delta\\mu=0$ the\n$T$-$\\Delta\\mu$ phase diagram of the system contains two distinct\nlines of first-order phase \ntransitions: a line of thin-thick transitions $(T,\\Delta\\mu_{tt}(T))$\non the single spheres (which is the remnant of the line of prewetting\ntransitions on the corresponding flat substrate and which is, as\nstated above, practically unshifted by the presence of the second\nsphere) and a second, \\emph{independent} line of bridging transitions\n$(T,\\Delta\\mu_{bt}(T))$. If one crosses the latter along an isotherm $T=T_0$\napproaching coexistence $(T_0,\\Delta\\mu\\to0)$, at $\\Delta\\mu=\\Delta\\mu_{bt}(T_0)$\na transition from the configuration without bridge\n$(\\Delta\\mu>\\Delta\\mu_{bt}(T_0))$ to a configuration with bridge\n$(\\Delta\\mu<\\Delta\\mu_{bt}(T_0))$ occurs. The derivative\n$\\partial\\Omega_S/\\partial\\Delta\\mu$ is discontinuous at\n$\\Delta\\mu_{bt}$, indicating that the bridging transition is first\norder. Figure~\\ref{f:btandttt} shows the \n$T$-$\\Delta\\mu$ phase diagram for the two spheres with $R=20\\sigma$\nfor $D=50\\sigma$ ($a=10\\sigma$). The line of thin-thick transitions intersects the\nliquid-vapor coexistence line at $T_{tt}^*\\approx 1.271$ with a\nfinite, negative slope (compare Fig.~8(a) in\nRef.~\\cite{bieker}). It extends into the vapor phase region\n($\\Delta\\mu>0$) of the phase diagram and ends at a critical point. The\nline of bridging transitions intersects the coexistence line also with\na finite, negative slope. On the other end, within our sharp-kink\ninterface model, it happens to be cut off at that metastability line\nin the phase diagram at which the second minimum of \nthe bulk free energy at high fluid density (Eq.~(\\ref{e:bfedensity})) ceases to\nexist so that for larger undersaturations the liquid phase is not even\nmetastable. Within a more sophisticated approach, e.g., by seeking the\nfull minimal density distributions of Eq.~(\\ref{e:ldafunc}), the line\nof bridging transitions is expected to end in a critical point,\ntoo. (Concerning the effect of fluctuations on these mean field\npredictions see the following paragraph.) The line of bridging\ntransitions is entirely located in the region \nwhere the liquidlike films on the spheres are thin. Moreover, the\neffect of the presence of the liquid bridge on the line of thin-thick\ntransitions is negligibly small. In Fig.~\\ref{f:btandttt} the relative\nlocation of the bridging transition line and the thin-thick transition\nline corresponds to our specific choice of the interaction potential\nparameters as well as the chosen size of and distance between the\nspheres. Changing these parameters will lead to shifts of these lines\nand, possibly, to different topologies of the phase diagram. Here\nwe refrain from exhaustingly presenting all possibilities which can\noccur according to Refs.~\\cite{dobbsetal} and \\cite{dobbsyeomans}.\n\nSince the liquid volume enclosed by the interface $h(z)$ is\nquasi-zerodimensional, fluctuation effects destroy the sharp\nfirst-order phase transition (see\nRefs.~\\cite{privmanfisher} and \\cite{gelfandlipowsky}). In Sec.~4 of\nRef.~\\cite{bieker} it has been extensively discussed how finite size\neffects smear out the thin-thick transition such that the thickness\nincreases sharply but continuously within a range $\\delta\\mu$ around\n$\\Delta\\mu_{tt}(T)$; these results apply analogously to the present case.\nUsing similar approximations we obtain a range $\\delta\\mu$\nbetween $\\delta\\mu^*\\approx0.004$ for $T^*=1.16$ and\n$\\delta\\mu^*\\approx0.02$ for $T^*=1.26$\nover which the \\emph{bridging} transitions shown in\nFig.~\\ref{f:btandttt} are smeared out around $\\Delta\\mu_{bt}(T)$. Thus\nclose to $\\Delta\\mu=0$ the quasi-first-order thin-thick transitions\nare clearly visible. However, for larger values of $\\Delta\\mu$ they\nbecome progressively smeared out such that their critical points\npredicted by mean field theory are erased by fluctuations.\n\n\\section{Effective film-induced interaction potential}\n\\label{s:eip}\n\n\\subsection{Shape of the effective potential, metastability, and\n asymptotic behavior}\n\nIn the following we change our point of view: we vary the\ndistance $D$ between the centers of mass of the spheres instead of\nthe thermodynamic parameters $T$ and $\\Delta\\mu$.\nFigure~\\ref{f:eipforexample1} shows the grand canonical potential\n$\\Omega_S$ corresponding to the wetting layer morphologies for the\ncase $R=20\\sigma$ and $T^*=1.2$ (Fig.~\\ref{f:example1}(c)) as a\nfunction of the separation $a = D-2R$ for several values of\n$\\Delta\\mu$. $\\Omega_S$ is the \nminimum of $\\Omega_S[h(z)]$ (Eq.~(\\ref{e:decomp})) for the given set\nof parameters $T$, $\\Delta\\mu$, and $D = 2R+a$. For each value of\n$\\Delta\\mu$ there are two branches of the free energy, one\ncorresponding to the solution without bridge, which for the case\n$R=20\\sigma$ considered here exists only for\n$a\\gtrsim 0.15R$, and the other corresponding to the solution with \nbridge which exists up to $a\\approx 0.65R$ and $a\\approx\n0.6R$ for $\\Delta\\mu^*=0$ and $\\Delta\\mu^*=0.01$, respectively.\nAt a certain value $D=D_{bt}$ or, equivalently, $a=a_{bt}$, which are\nfunctions of $\\Delta\\mu$, a \nfirst-order phase transition occurs with discontinuous derivative\n$\\partial\\Omega_S/\\partial D$ between the solutions with and without\nbridge. The main effect of increasing the undersaturation\n$\\Delta\\mu$ is that the free-energy curves are rigidly shifted upwards.\nThis shift is approximately proportional to $\\Delta\\mu$ and larger in the\ncase of the solution with bridge, resulting in the dependence of $D_{bt}$ on\n$\\Delta\\mu$. The values of $\\Omega_S$ shown in\nFig.~\\ref{f:eipforexample1} are obtained within the local theory. The\nnonlocal theory yields the same functional dependence\n$\\Omega_S(D)$ but with a slight and rigid shift of the free-energy\ncurves, relative to the results of the local theory,\nof the order of $0.1\\%$ and of the same sign and size for both\nthe solutions with and without bridge. Finite-size effects again\ndestroy the sharp first-order bridging transition; we obtain a range\n$\\delta D\\sim0.1\\sigma$ (corresponging to $\\delta D\\sim0.005R$) over\nwhich the bridging transitions shown in Fig.~\\ref{f:eipforexample1}\nare smeared out. \n\nThe thermodynamic states which are located on the metastable branches\nof the free energy curves survive during an average lifetime\n$\\tau\\approx\\tau_0\\exp(\\Delta\\Omega_S/k_BT)$ where \n$\\Delta\\Omega_S$ is the height of the energy barrier that separates\nthe metastable from the stable branch and $\\tau_0$ is a characteristic\nmicroscopic time scale for the dynamics associated with the transition\nfrom a metastable to a stable wetting layer configuration. The energy\nbarrier is highest \nin the vicinity of the bridging transition and vanishes near the ends\nof the metastable branches. An estimation of the energy barrier\nheight yields, e.g., $\\Delta\\Omega_S\\approx75\\epsilon$ for\n$\\Delta\\mu=0$ and $D=50\\sigma$ ($a=0.5R$), and with\n$k_BT\\sim\\epsilon$ it follows that $\\exp(\\Delta\\Omega_S/k_BT)\\sim\n10^{32}$, i.e., the metastable unbridged state for $a=0.5R$ near the\nbridging transition remains stable practically forever. However, at, e.g.,\n$a=0.2R$ one has $\\exp(\\Delta\\Omega_S/k_BT)\\sim10^{11}$ so that\nwith $\\tau_0\\sim 1$ps$\\dots 1$ns one may observe a decay of the\nmetastable states near the ends of the metastable branches within\nseconds or minutes. Thus the change of the morphology of the wetting\nfilms is expected to exhibit pronounced hysteresis effects as function\nof $D$.\n\nObviously, in the limit of large separation $D\\to\\infty$ (in which only the\nconfiguration without a bridge is stable) the grand canonical\npotential $\\Omega_S(D)$ approaches the \nlimiting value $2\\Omega_S^{(1)}$ corresponding to the free energy of\ntwo individual spheres, each surrounded by a wetting layer.\nIt is convenient to separate this constant contribution\n$2\\Omega_S^{(1)}$ from the grand canonical potential $\\Omega_S$ of the\nsystem and thus to define an \\emph{e}xcess free energy\n$\\Omega_E(D) = \\Omega_S(D)-2\\Omega_S^{(1)}$ which contains all\ncontributions from the wetting-layer induced interaction between the two spheres.\nIn the limit $D\\to\\infty$, i.e., in the absence of a liquid bridge,\nthis excess free energy $\\Omega_E(D)$ decays as\n$D^{-6}$ (see Eq.~(\\ref{e:omegasl_asymp}) and, c.f.,\nSec.~\\ref{s:discussion}). We note that for the example \nshown in Fig.~\\ref{f:eipforexample1} the coefficient of this leading\norder is \\emph{positive}, i.e., the effective potential in the\nabsence of a liquid bridge is \\emph{repulsive}. This is owed to the\nchoice $T<T_{tt}$ for this example: the spheres disfavor the\nadsorption of thick liquid \nfilms and the presence of the second sphere with its surrounding liquidlike\nlayer leads to an additional cost in free energy which diminishes for\nincreasing $D$. For the choice $T>T_{tt}$, i.e., if the spheres favor\nthe adsorption of liquid (e.g., for $T^*=1.3$ as in\nFigs.~\\ref{f:example1}(a) and (b)) the coefficient of $D^{-6}$ is\n\\emph{negative} and the effective interaction is \\emph{attractive}.\nHowever, in the presence of a liquid bridge, i.e., for sufficiently\nsmall values of $D$, the effective potential\nshows the same qualitative behavior as in Fig.~\\ref{f:eipforexample1} also for the\ncase of thick wetting layers ($T^*=1.3>T_{tt}^*$) as well as for the larger\nspheres ($R=50\\sigma$) with thin or thick films.\n\n\\subsection{Effective interaction potential for large spheres}\n\\label{ss:eipforlargespheres}\n\nIn this subsection we consider the limiting case that the sphere\nradius $R$ is much larger than the diameter $\\sigma$ of the solvent\nparticles and that the separations $a$ between the surfaces of the\nspheres are proportional to $R$: $R\\gg\\sigma$, $\\sigma\\ll a \\approx\nR$. For such large separations as compared to $\\sigma$ the\ncontributions $\\Omega_{sl}$ (Eq.~(\\ref{e:omegasl})), $\\Omega_{ei}$\n(Eq.~(\\ref{e:omegaei})), and $\\Phi$ (c.f., Eq.~(\\ref{e:vdwbetwsph}))\nbecome vanishingly small relative to the contributions $\\Omega_{lg}$\n(Eq.~(\\ref{e:omegalg})) and $\\Omega_{ex}$ (Eq.~(\\ref{e:excess})). For\nthe case described above $\\Omega_{lg}$ and $\\Omega_{ex}$ scale\nproportional to the surface area of the spheres, i.e., $\\sim R^2$,\nwhereas for $a/\\sigma\\to\\infty$, $R/\\sigma\\to\\infty$, $a/R$ finite,\n$\\Phi(D;R)$ remains finite $\\sim\\epsilon_{ss}\\sigma_{ss}^6\\rho_s^2$\nwith a proportionality constant of the order $1$. Analogously, in the\nsame limit $\\Omega_{ei}-2\\Omega_{ei}^{(1)}$ (Eq.~(\\ref{e:omegaei}))\nand $\\Omega_{sl}-2\\Omega_{sl}^{(1)}$ (Eq.~(\\ref{e:omegasl})) are\ndetermined by finite terms \\mbox{$\\sim\\Delta\\rho\\rho_l\\epsilon\\sigma^6$} and\n\\mbox{$\\sim\\Delta\\rho\\rho_s\\epsilon_{sf}\\sigma_{sf}^6$} and of terms\n\\mbox{$\\sim\\rho_l^2\\epsilon\\sigma^6$} and\n\\mbox{$\\sim\\rho_l\\rho_s\\epsilon_{sf}\\sigma_{sf}^6$}, respectively, each with\na proportionality constant of the order $1$. Therefore measured in\nunits of $8\\pi R^2$ the unbridged branch of $\\Omega_E$ in\nFig.~\\ref{f:eipforexample1}(b) vanishes in the limit\n$R\\to\\infty$. Moreover, on this scale the excluded \nvolume at small $a$ disappears from the figure, too, because $d_s/R\\to0$. \n\nFig.~\\ref{f:eip_largespheres} shows the excess effective interaction\npotential $\\Omega_E$ in the limit of large spheres for the case\n$\\Delta\\mu=0$, i.e., at two-phase coexistence in the solvent. In this\nlimit and for $\\Delta\\mu=0$, $\\Omega_{lg}$ is the only \nrelevant contribution to $\\Omega_S$ because\n$\\Omega_{ex}(\\Delta\\mu=0)=0$. Accordingly, in this case the bridging\ntransition is determined by the equality of the surface areas of the\nliquid-vapor interfaces for the unbridged and bridged\nconfiguration. From this condition and from dimensional analysis it\nfollows that for large spheres $D_{bt}(\\Delta\\mu=0)$ is determined by\nthe equation\n\\begin{equation}\\label{e:surfaceequality}\n8\\pi(R+l_0)^2\\sigma_{lg}^{(p)} = 8\\pi(R+l_0)^2\\sigma_{lg}^{(p)}\nf\\left(\\frac{D_{bt}}{R+l_0}\\right) \n\\end{equation}\nwhere $f$ is, for\ndimensional reasons, a universal function of $D/(R+l_0)$\nalone which describes the surface area of the bridged\nconfiguration; $l_0$ is the equilibrium wetting layer thickness on a\nsingle sphere. Since the line of bridging transitions lies below the\nline of thin-thick transitions, $l_0$ remains microscopicly small at\nthe bridging transition (Fig.~\\ref{f:btandttt}). Therefore one has \n\\begin{equation}\\label{e:Dbt}\nD_{bt}(\\Delta\\mu=0) = \\lambda(R+l_0)\n\\end{equation}\nwith a universal number \n\\begin{equation}\\label{e:lambda}\n\\lambda\\approx2.32\n\\end{equation}\ndetermined by $f(\\lambda)=1$ (compare\nFig.~\\ref{f:eip_largespheres}). If one applies this reasoning to \nFig.~\\ref{f:eipforexample1} one finds $\\lambda\\approx2.39$. Therefore\neven for $R=20\\sigma$ this macroscopic approximation leads to a\nsurprisingly small error of only $3\\%$ for\n$D_{bt}(\\Delta\\mu=0)$. Accordingly, in Fig.~\\ref{f:eipforexample1} the\nfull curves corresponding to $\\Delta\\mu=0$ closely resemble the ones\nin Fig.~\\ref{f:eip_largespheres} describing the case of large\nspheres. The only differences appear for small separations $a$ where\nthe bridged branch linearly extends down to its minimum value\n$\\Omega_E/(8\\pi(R/\\sigma)^2) \\approx -0.0227\\epsilon$ at $a/R=0$\n(Fig.~\\ref{f:eip_largespheres}). Only in this range of separations the\neffect of the contributions $\\Omega_{ei}$ and $\\Omega_{sl}$ becomes\nsignificant, leading to the deeper minimum visible in\nFig.~\\ref{f:eipforexample1}. Thus for $\\Delta\\mu=0$ and large $R$\nthe dependence of the effective interaction potential on $R$ for the\nbridged configuration is captured by the indicated rescaling of the\naxes in Fig.~\\ref{f:eipforexample1}(b). However, our numerical\nanalysis shows that the smallness of the deviations between the\nmacroscopic description valid for $R\\gg\\sigma$ and the actual results\nfor $R=20\\sigma$ is somewhat fortuitous. Whereas the dependence of\n$D_{bt}(\\Delta\\mu=0)$ on $R$ is indeed weak, the shape of the\npotential (for $\\Delta\\mu=0$) reduces to that shown in\nFig.~\\ref{f:eip_largespheres} only for $R$ larger than several hundred\n$\\sigma$ and, surprisingly, for $R$ up to $20\\dots30\\sigma$ with the\ndeviations being maximal for $R\\approx100\\sigma$.\n\nOff coexistence $\\Delta\\Omega_b = \\Omega_b(\\rho_l)-\\Omega_b(\\rho_g)\n\\approx \\Delta\\mu\\Delta\\rho$ is positive so that\nEq.~(\\ref{e:surfaceequality}) has to be augmented correspondingly:\n\\begin{equation}\n8\\pi(R+l_0)^2 +\n\\frac{8\\pi\\Delta\\Omega_b}{3\\sigma_{lg}^{(p)}}\\left((R+l_0)^3-R^3\\right)\n= {\\mathcal A}+\\frac{\\Delta\\Omega_b}{\\sigma_{lg}^{(p)}}\\mbox{Vol}(\\mL)\n\\end{equation}\nwhere ${\\mathcal A}$ and $\\mbox{Vol}(\\mL)$ (Eq.~(\\ref{e:volliq})) are\nthe area of the liquid-vapor interface and the volume of the liquid,\nrespectively, for the bridged configuration. They are obtained by\ninserting into Eqs.~(\\ref{e:local}) and (\\ref{e:volliq}) that profile\n$h(z)$ which solves the differential equation determining the minimum\nof $\\Omega_{lg}[h]+\\Omega_{ex}[h]$ together with the appropriate\nboundary conditions. By splitting off a factor $(R+l_0)^2$ from\n${\\mathcal A}$ and $(R+l_0)^3$ from $\\mbox{Vol}(\\mL)$ dimensional\nanalysis shows that up to terms $\\sim l_0/R$ the critical distance for\nthe bridging transition is given by a universal scaling function\n$\\Lambda$:\n\\begin{equation}\\label{e:Lambda}\nD_{bt}(\\Delta\\mu) =\n\\Lambda\\left(\\frac{\\Delta\\rho\\Delta\\mu R}{\\sigma_{lg}^{(p)}}\\right) R\n\\end{equation}\nwith $\\Lambda(0) = \\lambda$. Thus off coexistence the critical\nbridging transition depends, apart from an explicit factor $R$, on $R$\nand $\\Delta\\mu$ via the scaling variable\n$\\Delta\\rho\\Delta\\mu R/\\sigma_{lg}^{(p)}$. This property is shared by\nthe whole bridged branch of the effective interaction potential. Thus\nincreasing $R$ for fixed undersaturation $\\Delta\\mu$ has the same\neffect as increasing $\\Delta\\mu$ for fixed $R$. From\nFig.~\\ref{f:eipforexample1}(b), in which the unbridged branch will\ndisappear in the limit $R\\gg\\sigma$, one infers that the range and the depth \nof $\\Omega_E$ decrease for increasing $R$ at fixed undersaturation\n$\\Delta\\mu$. The behavior of $D_{bt}$ and of the bridged branch of the\neffective interaction potential off coexistence and for $R\\to\\infty$\nis determined by the behavior of the scaling function $\\Lambda(x)$ in the limit\n$x\\to\\infty$. Our numerical data indicate that $\\Lambda(x\\to\\infty)<2$\nso that due to the geometric constraint $D\\geq2R$ there is no bridging\ntransition and the bridged branch \nof the effective potential vanishes for any value of $\\Delta\\mu$ \nin the limit $R\\to\\infty$. The cost in free energy due to the excess\ncontribution $\\Omega_{ex}$ suppresses the formation of a liquidlike\nbridge in the case of macroscopicly large spheres. In turn, this\nmeans that for any finite \nvalue of $\\Delta\\mu$ there is a large but finite critical radius $R_c$ for\nwhich the critical separation $a_{bt}$ for the bridging transition\nattains the value $a_{bt}=0$, such that for $R>R_c$ there is no\nbridging transition. The determination of $R_c$ requires to analyze\nthe full dependence of \n$\\Lambda$ on the scaling variable $x$. This, however, implies such a large\nnumerical effort that it is beyond the scope of the present paper.\n\n\\section{Discussion}\n\\label{s:discussion}\n\n\\subsection{Total interaction potential}\n\nThe bare dispersion interaction between the two spheres is not included\nin Eq.~(\\ref{e:ldafunc}). According to Hamaker~\\cite{hamaker} this\ncontribution is given by\n\\begin{equation}\\label{e:vdwbetwsph}\n\\Phi(D;R) = -\\frac{A_{ss}}{12}\\left(\\frac{4R^2}{(D-2R)(D+2R)} +\n\\frac{4R^2}{D^2} + 2\\ln\\left(\\frac{(D-2R)(D+2R)}{D^2}\\right)\\right)\n\\end{equation}\nas the dispersion interaction between two\nidentical spheres of radius $R$ at center-of-mass distance $D$. In\nthe limit $a/R\\ll1$, where $a = \nD-2R$ (see Fig.~\\ref{f:system}(a)) is the smallest separation between the\nsurfaces of the spheres, Eq.~(\\ref{e:vdwbetwsph}) reduces to\n\\begin{equation}\\label{e:barevdw_smalla}\n\\Phi(D=2R+a;R\\gg a) \\approx -\\frac{A_{ss}}{12}\\,\\frac{R}{a}\n\\end{equation}\nwhich corresponds to the Derjaguin approximation whereas $\\Phi(D\\gg\nR;R)=-16A_{ss}R^6/(9D^6)$. Thus except for the $D$-independent bulk\ncontribution $\\mbox{Vol}(\\mV_f)\\Omega_b(\\rho_g)$\nthe total grand canonical potential of the system is\n\\begin{equation}\\label{e:omegatot}\n\\Omega_{tot}(D;R) = \\Omega_S(D;R) + \\Phi(D;R)\n\\end{equation}\nwhere $\\Omega_S(D;R)$ is given by the minimum value\n$\\min_{\\{h(z)\\}}(\\Omega_S[h(z)])$ for given $D$ and $R$\n(Eqs.~(\\ref{e:decomp}) and (\\ref{e:gcpismin})); in analogy to\n$\\Omega_E$ we define the excess total free energy $\\Omega_{E,tot} =\n\\Omega_{tot}-2\\Omega_S^{(1)}$. $A_{ss}$ is the Hamaker \nconstant appertaining to the bare dispersion interaction between the\nparticles in the spheres. In the case of pairwise additivity of the molecular \ninteractions and in the absence of retardation effects one has $A_{ss} =\n4\\pi^2\\epsilon_{ss}\\sigma_{ss}^6\\rho_s^2$ if the interaction potential\nbetween two individual molecules in the spheres is given by a\nLennard-Jones potential \n(Eq.~(\\ref{e:ljsphere})) with the parameters $\\epsilon_{ss}$ and\n$\\sigma_{ss}$. Typically $A_{ss}$ is of the order of $10^{-19}$J or, equivalently,\n$10\\dots100\\epsilon$. If the vacuum between the spheres is replaced by\na medium of condensed matter the interaction between the\nspheres is screened~\\cite{israelachvili}. In our present model this medium\nis the bulk vapor phase modified by the presence of the liquidlike films\nadsorbed on the spheres and the screening effect is\ndescribed microscopicly by the functional $\\Omega[\\rho(\\rv)]$.\n\nIn Refs.~\\cite{vold} and \\cite{vincentetal} this\nadditional screening effect, due to spherical shells of adsorbed,\nhomogeneous layers surrounding\nspherical particles, on the dispersion interaction between the\nlatter immersed in another homogeneous medium has been calculated\nmacroscopicly. Beyond molecular scales these results should closely\ncorrespond to the \nconfiguration without liquid bridge discussed herein because the\ndeviation of the spherical shape of one wetting layer due to the\npresence of the second sphere is very small. Indeed, the\ninteraction energy calculated in Refs.~\\cite{vold} and \n\\cite{vincentetal} is practically the same as the sum of the $D$-dependent\ncontributions in $\\Omega_{ei}$ (Eq.~(\\ref{e:omegaei})) and\n$\\Omega_{sl}$ (Eq.~(\\ref{e:omegasl})) for the configurations without\nbridge -- for these configurations $\\Omega_{lg}$ and $\\Omega_{ex}$ do\nnot contribute to the dependence of $\\Omega_S$ on $D$ -- and the direct \ndispersion interaction $\\Phi(D;R)$. In Ref.~\\cite{vold} the total dispersion \ninteraction is shown to be always attractive if the Hamaker constants\n$A_{ij}$ corresponding to the interaction between any two media $i$ and\n$j$ are chosen such that $A_{ij} = \\sqrt{A_{ii}\\,A_{jj}}$. Although the\neffective interaction induced by the wetting layers shown in\nFigs.~\\ref{f:eipforexample1}(b) and \\ref{f:eipforexample1plusbare} for the\nconfiguration without bridge is repulsive, we note that the sum\n$\\Omega_{tot}$ of this interaction and of the bare dispersion \npotential $\\Phi(D;R)$ is also \\emph{attractive} if we choose the Hamaker\nconstant in Eq.~(\\ref{e:vdwbetwsph}) accordingly, i.e., $A_{ss} =\nA_{sf}^2/A$ (Fig.~\\ref{f:eipforexample1plusbare}). Therefore \nour results are consistent with those obtained in Ref.~\\cite{vold}. Since only\neffective interactions between finite volumes enter into the total\nexcess interaction potential $\\Omega_{E,tot}$ and these\neffective interactions decay as $D^{-6}$ in the limit of large\nseparations $D$, the same holds for $\\Omega_{E,tot}$.\n\nFigure~\\ref{f:eipforexample1} shows that as soon as the wetting films\nsnap to a liquidlike bridge, whether it is stable or metastable, there\nis an attractive wetting-layer-induced force\n$-\\partial\\Omega_E/\\partial D$ that pulls the \nspheres together. From Fig.~\\ref{f:eipforexample1} one can infer that\nthis attractive force is of the order of \n$40\\epsilon/\\sigma$ in the range between $a\\approx4\\sigma$ (i.e.,\n$0.2R$ for $R=20\\sigma$ discussed in this figure) and $a\\approx\n10\\sigma$ ($0.5R$) where the effective potential varies almost\nlinearly. At the small separation $a_{min}\\approx2.5\\sigma$ \nthe effective potential $\\Omega_E$ induced by\nthe bridgelike wetting layer is minimal and the wetting-induced force\nis zero. Finally, at smaller separations the interaction is repulsive\nleading to a stabilization of the spheres at $D=D_{min}=2R+a_{min}$. Within the\nrange $a\\ll R$ the bare, direct\ndispersion interaction between the spheres\n(Eqs.~(\\ref{e:vdwbetwsph}) and (\\ref{e:barevdw_smalla})) gives rise to a\nforce $F_{bare}(a) \\approx -A_{ss}R/12a^2$. The estimate\n$A_{ss}\\approx4\\pi^2\\epsilon_{sf}^2\\sigma_{sf}^{12}\\rho_{s}^2/\\epsilon\\sigma^6 \n\\sim 400\\epsilon$ \nfor the case of pairwise additive interactions without retardation\nfollows from the ansatz $A_{sf} = \\sqrt{A\\,A_{ss}}$, so that \nthe bare dispersion force in our example with $R=20\\sigma$ is\n$F_{bare}(a)\\approx -670\\epsilon\\sigma/a^2$. Therefore in the range where the\nbridge-induced force is almost constant ($4\\sigma\\lesssim a\\lesssim10\\sigma$)\nthe direct, bare dispersion\nforce decays from approximately $-40\\epsilon/\\sigma$ (which is of the same order\nof magnitude as the bridge-induced force) to approximately $-6\\epsilon/\\sigma$,\nwhereas for smaller separations it becomes the dominant force. \n\n\\subsection{Relevance for force microscopy}\n\\label{ss:forcemicroscopy}\n\nOur model calculations can be tested experimentally by force\nmicroscopy. This can be done by suitably fixing one sphere in the\nfluid and by attaching the second one to the tip of a force\nmicroscope. Alternatively, both spheres can be positioned by optical\ntweezers and the force law can be inferred by monitoring optically\ntheir dynamics after switching off the tweezers. At separations\nbetween the spheres which are comparable with the diameter $\\sigma$ of\nthe solvent particles the actual effective interaction potential will\nexhibit an additional oscillatory contribution\ndue to packing effects which decays exponentially on the scale\n$\\sigma$~\\cite{kinoshita}. In order to obtain these \noscillations one would have to resort to density functional theories\nwhich are more sophisticated than the one in\nEq.~(\\ref{e:ldafunc}). This, in turn, would make it much more\ndifficult to obtain the bridgelike configuration, to map out the\ncomplete phase diagram, and to obtain results for large\nspheres. According to Subsec.~\\ref{ss:eipforlargespheres}, for\n$R\\gg\\sigma$ and at two-phase coexistence $\\Delta\\mu=0$ the bridging\ntransition occurs at distances $a$ which are proportional to $R$. In\nthis case, due to $R\\gg\\sigma$, the effective interaction potential\nwill be practically unaffected by this oscillatory contribution for\nthe vast portion $\\sigma\\ll a\\ll a_{bt}$ of the range of the effective\ninteraction potential.\n\n\\subsection{Relevance for charge stabilized colloidal suspensions}\n\\label{ss:cscs}\n\nWhereas the kinds of experiments considered in the previous subsection\nare focused on two individual\nspherical particles, we discussed in the Introduction that the\neffective interaction potential enters into the collective behavior of\ncolloidal suspensions such that the bridging transition may trigger\nflocculation. If colloidal suspensions would be governed by dispersion\nforces alone, most of them would flocculate even in the absence of the\nwetting-induced forces discussed here because the dispersion forces\ngenerate the so-called primary minimum in the effective interaction\npotential close to contact. Since this minimum is much deeper than\n$k_BT$ the colloidal particles would simply stick together\npermanently. This effect, which is undesired for many applications,\ncan be avoided by endowing the particles with electrical charges which\nadds a screened Coulomb repulsion between the charged particles. As\na result, such charge stabilized colloidal suspensions are characterized\nby effective interaction potentials in which a substantial energy\nbarrier separates the aforementioned primary minimum from a second,\nmuch more shallow minimum at larger distances. Since this potential\nbarrier is typically large compared with $k_BT$ the phase behavior of\nthe colloidal particles is practically independent of the primary\nminimum formed by the dispersion forces and determined by the\nshape of the potential \\emph{outside} the barrier. As demonstrated by\nFigs.~\\ref{f:eipforexample1} and \\ref{f:eipforexample1plusbare} the\nrange of the wetting-induced forces is about $0.55R$, in good agreement\nwith $D_{bt}\\approx2.32(R+l_0)$ (see Eqs.~(\\ref{e:Dbt}) and (\\ref{e:lambda})).\nOn the other hand the position (and height) of the aforementioned\nenergy barrier depends sensitively on the size of the total charge on\nthe spheres, the amount of salt in the solvent, and the dispersion\nforces and can be varied over a wide range. With a high salt\nconcentration the barrier position can be as small as a few tens of\nnm. Thus under such circumstances the wetting-induced interaction\npotentials would be relevant even for colloidal particles whose radii\nare only a few tens of nm.\n\n\\subsection{Relevance for stericly stabilized colloidal suspensions}\n\\label{ss:sscs}\n\nThere is another class of colloidal suspensions for which the\nwetting-induced forces can be of practical importance. By coating the\ncolloidal particles with polymers and by matching the refractive\nindices of the colloidal particles and the bulk fluid (in our\ncase study the vapor phase or, more realisticly in the present\ncontext, the A-rich liquid phase of a binary liquid mixture acting as\nthe solvent) the colloidal particles behave effectively like hard spheres (see,\ne.g., Refs.~\\cite{puseyvanmegen} and\n\\cite{poonpuseylekkerkerker}). Through this index matching the sum \nof the bare interaction potential $\\Phi(D;R)$ and the effective\ninteraction potential $\\Omega_{sg}$, which would arise if the spheres\nwere immersed in the homogeneous and unperturbed bulk solvent,\nvanishes. Within our model $\\Omega_{sg}$ is given by the expression in\nEq.~(\\ref{e:omegasl}) with $\\rho_l$ replaced by $\\rho_g$, which is the\ndensity of the bulk phase. Since the index matching works for the bulk\nphase, it does not work for the wetting phase. As a consequence the\nwetting-induced forces appear against a background effective potential\nof hard spheres. Therefore for this class of colloidal suspensions the\nwetting phenomena discussed here are expected to have a pronounced\neffect on their phase behavior. Within our model, for\n\\emph{i}ndex-\\emph{m}atched suspensions the \\emph{total} effective\ninteraction potential is given by\n\\begin{equation}\n\\Omega_{tot,im}(D;R) = \\Omega_{tot}(D;R)-(\\Phi(D;R)+\\Omega_{sg}(D;R))\n= \\Omega_S(D;R)-\\Omega_{sg}(D;R)\n\\end{equation}\nand in analogy to $\\Omega_E$ and $\\Omega_{E,tot}$ we define\n\\begin{equation}\n\\Omega_{E,im}(D;R) = \\Omega_{tot,im}(D;R)-2\\Omega_{im}^{(1)}(R)\n\\end{equation}\nwith $\\Omega_{E,im}(D\\to\\infty;R) = 0$ for the unbridged\nsolutions. Figure~\\ref{f:indexmatch} displays $\\Omega_{tot,im}$ and\n$\\Omega_{E,im}$ as function of $a=D-2R$ for the same system as in\nFigs.~\\ref{f:eipforexample1} and \\ref{f:eipforexample1plusbare}.\n$\\Omega_{sg}$ is about $30\\%$ smaller than $\\Omega_S$ for the unbridged\nsolution and also approaches its asymptote $2\\Omega_{sg}^{(1)}$ from\nabove. As before (see the discussion of\nFig.~\\ref{f:eipforexample1plusbare} above), the resulting total\neffective interaction between spheres in an \nindex-matched bulk fluid for the state with liquid bridge is still\nattractive and of the same order of magnitude as the bare dispersion\ninteraction between the spheres, i.e., in the absence of the solvent.\n\n\\section{Summary}\n\\label{s:summary}\n\nWe have obtained the following main results:\n\\begin{enumerate}\n\\item Based on microscopic interaction potentials and within a simple\n version of density functional theory\n (Eqs.~(\\ref{e:ldafunc})--(\\ref{e:w0})) we have calculated the grand\n canonical potential of a system of two spheres immersed in a bulk\n fluid phase (Fig.~\\ref{f:system}). The microscopic interactions are\n chosen such that the spheres\n prefer the adsorption of a second fluid phase which is\n thermodynamically close to the bulk fluid phase. Accordingly, a\n single sphere immersed in the fluid is covered by a homogeneous\n wetting layer of this second phase of thickness $l_0$. These thin wetting\n layers covering the spheres lead to an effective wetting-induced\n interaction potential $\\Omega_S(D)$ between the spheres. We have\n systematically determined the dependence of $\\Omega_S$ on the\n distance $D$ between the spheres in terms of the morphology\n $h(z)$ of the wetting film enclosing the spheres\n (Eqs.~(\\ref{e:decomp})--(\\ref{e:omegasl})). We find that the shape\n of the effective interaction potential $\\Omega_S(D)$ depends, inter\n alia, on the effective\n interaction of two spheres immersed in the homogeneous \\emph{wetting}\n phase (Eq.~(\\ref{e:omegasl})). This contribution, which is\n independent of $h(z)$, is not incorporated in previous\n phenomenological models for this system~\\cite{dobbsetal,dobbsyeomans}.\n\\item The equilibrium interfacial profiles of the wetting\n layers are determined numerically by minimizing the free\n energy functional $\\Omega_S[h(z)]$ in\n Eqs.~(\\ref{e:decomp})--(\\ref{e:omegasl}). We have calculated the\n rich structure of these \n equilibrium profiles (Fig.~\\ref{f:example2}) for spheres of radii $R=20\\sigma$\n (Fig.~\\ref{f:example1}) and $R=50\\sigma$ (Fig.~\\ref{f:example2})\n where $\\sigma$ denotes the diameter of the solvent\n particles. As function of distance $D$, temperature $T$, and\n undersaturation $\\Delta\\mu$ the system undergoes a first-order\n ``bridging transition'' between the two configurations shown in\n Fig.~\\ref{f:system}. For a fixed distance $D$ we have\n mapped out the phase diagram of bridging transitions in the $T$-$\\Delta\\mu$ plane\n (Fig.~\\ref{f:btandttt}). It turns out that the bridging transition\n differs from and to a large extent is independent of the\n thin-thick transition of the wetting layer on \n each single sphere which is a remnant of the prewetting transition\n on the corresponding flat substrate. Thus one has to distinguish\n between the prewetting line for a first-order wetting transition on\n a planar substrate, the thin-thick transition line for wetting on a\n single sphere, and the bridging transition line for two spheres\n (Fig.~\\ref{f:btandttt}). At two-phase coexistence $\\Delta\\mu=0$ and\n for $R\\gg\\sigma$ the bridging transition is determined by the equality of the\n surface areas of the interfaces in the bridged and the unbridged configuration,\n leading to a universal ratio $D_{bt}(\\Delta\\mu=0)/(R+l_0)\\approx2.32$ for the\n critical distance $D_{bt}(\\Delta\\mu=0)$ of the bridging transition\n at coexistence (Fig.~\\ref{f:eip_largespheres} and\n Subsec.~\\ref{ss:eipforlargespheres}). Off \n coexistence $D_{bt}(\\Delta\\mu,R)$ is described by a universal\n scaling function (Eq.~(\\ref{e:Lambda})).\n\\item At large distances and depending on the temperature relative to\n the thin-thick transition temperature on a single sphere the\n wetting-induced effective interaction potential can be either\n attractive or repulsive; in both cases it decays $\\sim D^{-6}$ for\n large $D$. The bridging transition leads to a strong break in slope of the\n effective interaction potential at $D=D_{bt}$. This is the\n fingerprint of a cooperative phenomenon among the fluid particles\n whose degrees of freedom have been integrated out (see\n Sec.~\\ref{s:intro}). Metastable branches\n of the effective potential give rise to pronounced hysteresis\n effects (Fig.~\\ref{f:eipforexample1}).\n\\item In the case that a bridge of the wetting phase connects the\n spheres (i.e., $D<D_{bt}$) there is an attractive wetting-induced interaction\n (Fig.~\\ref{f:eipforexample1}) that pulls the\n spheres together. Within a wide range of separations $a=D-2R$ of the\n spherical surfaces this force is of the same order of magnitude as\n the bare dispersion interaction potential $\\Phi$\n (Eq.~(\\ref{e:vdwbetwsph})) between the spheres. This bare\n interaction of two spheres (corresponding to the case that they are\n separated by vacuum) has to be added to\n the effective potential $\\Omega_S$ to yield the total interaction\n potential $\\Omega_{tot}$ between the spheres which is attractive at\n large distances\n (Eqs.~(\\ref{e:vdwbetwsph}), (\\ref{e:omegatot}), and\n Fig.~\\ref{f:eipforexample1plusbare}). \n\\item The wetting-induced force between spherical particles is\n experimentally accessible \\emph{directly} through suitable force\n microscopy techniques (Subsec.~\\ref{ss:forcemicroscopy}). Moreover,\n in Subsec.~\\ref{ss:sscs} we argue \n that this force influences the phase behavior of \\emph{stericly}\n stabilized, index-matched colloidal suspensions. The total effective\n interaction potential for such a case is shown in\n Fig.~\\ref{f:indexmatch}; it is repulsive at large\n distances. The phase behavior of \\emph{charge} stabilized \n colloidal suspensions (Subsec.~\\ref{ss:cscs}) is only affected by\n the wetting-induced interaction potential if the screening length of the Coulomb\n repulsion in the solvent is smaller than\n $a_{bt}=D_{bt}-2R\\approx0.32R$. Depending on the size of the\n charges, the salt concentration of the solvent, and the underlying\n dispersion forces this criterion may be fulfilled even for colloidal\n particles whose radii are only a few tens of nm.\n\\end{enumerate}\n\n\\acknowledgements\n\nWe gratefully acknowledge financial support by the German Science\nFoundation within the special research initiative\n\\emph{Wetting and Structure Formation at Interfaces}.\n\n\\appendix\n\n\\section*{Contributions to the free energy}\n\nOur choice of interaction potentials $\\phi(r)$ (Eq.~(\\ref{e:ljff}))\nand $\\phi_s(r)$ (Eq.~(\\ref{e:ljsphere})) leads to the following\nexpressions for the contributions to the free energy\n$\\Omega_S$ (with the thermodynamic limit already carried out):\n\n\\begin{equation}\n\\Omega_{ei}[h(z)] = 2\\Delta\\rho\\left(\\rho_l \\int\\limits_0^{\\infty}\n dz\\, (g_+(z)+g_-(z)) - \\int\\limits_0^{\\infty} dz\\,\n (f_+(z)+f_-(z))\\right)\n\\end{equation}\nwith\n\\begin{eqnarray}\ng_{\\pm}(z) & = & 2w_0\\sigma^2\\frac{R_1}{\\sigma} -\n w_0\\sigma^2\\left(\\frac{h^2(z)}{\\sigma^2} +\n \\left(\\frac{z}{\\sigma}\\pm\\frac{D}{2\\sigma}\\right)^2 -\n \\frac{R_1^2}{\\sigma^2}+1\\right) \\\\ \n& &\n \\times\\left(\\arctan\\left(\\sqrt{\\frac{h^2(z)}{\\sigma^2} +\n \\left(\\frac{z}{\\sigma}\\pm\\frac{D}{2\\sigma}\\right)^2}\n + \\frac{R_1}{\\sigma}\\right)\\right. - \\nonumber \\\\\n& & \n \\left.\\arctan\\left(\\sqrt{\\frac{h^2(z)}{\\sigma^2}+\\left(\\frac{z}{\\sigma}\n \\pm\\frac{D}{2\\sigma}\\right)^2} \n - \\frac{R_1}{\\sigma}\\right)\\right) \\nonumber\n\\end{eqnarray}\nwhere $R_1 = R+d_s$ and\n\\begin{eqnarray}\nf_{\\pm}(z) & = & \\frac{\\pi u_9}{4}\\left(\\frac{1}{7}\\left\n ( \\frac{1}{(k_{\\pm}+R)^7} - \\frac{1}{(k_{\\pm}-R)^7}\\right) + R\n \\left(\\frac{1}{(k_{\\pm}+R)^8} +\n \\frac{1}{(k_{\\pm}-R)^8}\\right)\\right) \\nonumber\\\\\n& & - \\pi u_3 \\left( \\frac{1}{k_{\\pm}+R} - \\frac{1}{k_{\\pm}-R} + R\n \\left(\\frac{1}{(k_{\\pm}+R)^2} +\n \\frac{1}{(k_{\\pm}-R)^2}\\right)\\right)\n\\end{eqnarray}\nwith $k_{\\pm} = \\sqrt{h^2(z)+(z\\pm D/2)^2}$. The contribution\n$\\Omega_{lg}$ is given by\n\\begin{eqnarray}\\label{e:app_omegalg}\n\\Omega_{lg}[h(z)] & = & -w_0\\sigma^3(\\Delta\\rho)^2 \\int\\limits_0^{\\infty} dz\n\\int\\limits_0^{\\infty} dz' \\,\\left(\n\\frac{1}{y_-^4}\\left(\\frac{qy_-^6+y_-^4(2q^2+p^2)+3p^2qy_-^2+p^4}\n {(p^2+2y_-^2q+y_-^4)^{3/2}} - p\\right)\\right. \\nonumber\\\\\n& & + \\left.\\frac{1}{y_+^4}\\left(\\frac{qy_+^6+y_+^4(2q^2+p^2)+3p^2qy_+^2+p^4}\n {(p^2+2y_+^2q+y_+^4)^{3/2}} - p\\right)\\right)\n\\end{eqnarray}\nwhere the abbreviations $y_{\\pm}$, $p$, and $q$ are defined by\n\\begin{equation}\ny_{\\pm}^2 = \\sigma^2+(z\\pm z')^2,\n\\end{equation}\n\\begin{equation}\np = h^2(z)-h^2(z'),\n\\end{equation}\nand\n\\begin{equation}\nq = h^2(z)+h^2(z').\n\\end{equation}\nThe double integral in Eq.~(\\ref{e:app_omegalg}) demonstrates the\nnonlocal functional dependence of $\\Omega_{lg}$ on $h(z)$.\n\n\\begin{references}\n\n\\bibitem{hansenmcdonald} See, e.g., J.~P.~Hansen and I.~R.~McDonald,\n \\emph{Theory of Simple Liquids}, 2nd. 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Phys. \\textbf{94}, 485 (1998);\n and references therein.\n\n\\bibitem{puseyvanmegen} P.~N.~Pusey and W.~van~Megen, Nature\n \\textbf{320}, 340 (1986).\n\n\\bibitem{poonpuseylekkerkerker} W.~Poon, P.~N.~Pusey, and\n H.~Lekkerkerker, Physics World \\textbf{9}, p. 27, April 1996.\n\n\\end{references}\n\n\\begin{figure} %Figure 1\n\\begin{center}\n\\epsfig{file=fig1.ps, width=10cm}\n\\end{center}\n\\caption{\\label{f:system}\nWetting film (thick full line) surrounding two identical homogeneous\nspheres of radius $R$ which are separated by a distance $D$. The whole\nsystem is rotationally symmetric around the $z$ axis which runs\nthrough both centers of mass. The position of the liquid-vapor interface which\nencloses both spheres is described by a function $h(z)$, i.e., in\ncylindrical coordinates the sharp interface is given by\nthe manifold $\\{\\rv(r_{\\perp},\\phi,z) =\n(r_{\\perp}\\cos\\phi,r_{\\perp}\\sin\\phi,z) \\in{\\mathbb\n R}^3|r_{\\perp}=h(z)\\}$. The origin of the coordinate system is in\nthe middle between the two spheres so that their centers are located\nat $z=\\pm D/2$. $a=D-2R$ is the shortest separation between the\nsurfaces of the spheres. Within the so-called sharp-kink\napproximation this interface separates a region of\nconstant liquid number density $\\rho_l$ from the surrounding bulk vapor phase\nof constant number density $\\rho_g$. Close to the surfaces of the\nspheres the repulsive \ninteraction leads to a volume with thickness $d_s$ excluded for the\ncenters of the fluid particles. For sufficiently large values of $D$ the bridgelike\nwetting film configuration shown in (a) breaks up into two disjunct\npieces so that $h(z) = 0$ for a finite interval around $z=0$ (b).}\n\\end{figure}\n\n\\begin{figure} %Figure 2\n\\begin{center}\n\\epsfig{file=fig2.ps, width=7cm, bbllx = 15, bblly = 15,\n bburx = 495, bbury = 840}\n\\end{center}\n\\caption{\\label{f:example1}\nMorphologies of liquidlike wetting layers on two adjacent, identical\nspheres with radius $R=20\\sigma$. The center-of-mass distance between\nthem is $D=50\\sigma$. The pictures show cross-sections through the system\ndefined by the plane $y=0$; the system is rotationally symmetric\naround the $z$ axis (see Fig.~\\ref{f:system}). The thick full lines denote the\nliquid-vapor interface, the \nthin dashed lines the surfaces of the spheres. (a) and (b): layer\nconfiguration with and without liquid bridge, respectively, for the\ntemperature $T^*=k_BT/\\epsilon = 1.3>T_{tt}^*$ and at liquid-vapor\ncoexistence $\\Delta\\mu=0$. Because of its higher free energy the configuration\nwithout bridge is metastable (c.f., Fig.~\\ref{f:btandttt}). These\nconfigurations are characterized by the interaction potential parameters \n$u_3 = 6.283\\epsilon\\sigma^3$, $u_9 =\n0.838\\epsilon\\sigma^9$, and $d_s=\\sigma$. The temperature is above the thin-thick\ntransition temperature $T_{tt}^*\\approx1.271$ for each single sphere. In (c)\nthe interaction \nparameters and $D$ are the same, but the temperature\n$T^*=1.2$ is below the thin-thick transition temperature $T_{tt}^*$, so that the\nwetting layer around a single sphere is thinner than in (a) and\n(b). Also at this temperature the bridge configuration is the stable\none (c.f., Fig.~\\ref{f:btandttt}).} \n\\end{figure}\n\n\\begin{figure} %Figure 3\n\\begin{center}\n\\epsfig{file=fig3.ps, width=7cm, bbllx = 15, bblly = 15,\n bburx = 495, bbury = 840}\n\\end{center}\n\\caption{\\label{f:example2}\nMorphologies of liquidlike wetting layers on two adjacent, identical\nspheres with radius $R=50\\sigma$, $D=120\\sigma$, and for the same\nchoice of interaction parameters as in Fig.~\\ref{f:example1}. The\nthick full lines denote the liquid-vapor interface, the thin dashed lines the\nsurfaces of the spheres. These pictures magnify the\nregion between the spheres. The temperature is $T^*=1.2$, which \nis above the thin-thick transition temperature $T_{tt}^*\\approx1.191$ at\ncoexistence for a single sphere,\nand the pictures differ with respect to the undersaturation:\n$\\Delta\\mu^*=\\Delta\\mu/\\epsilon = 0.05$ in (a), $0.015$ in (b), and\n$0.01$ in (c). Between (a) and (b), at $\\Delta\\mu_{bt}^*\\approx\n0.0235$ the system undergoes the first-order transition \nfrom the state without bridge to the state with bridge, and at\n$\\Delta\\mu_{tt}^*\\approx 0.0103$ between (b) and\n(c) there is a thin-thick transition of the wetting layer around the\nsingle spheres which is the remnant of the prewetting transition on\nthe corresponding flat\nsubstrate (see, c.f., Fig.~\\ref{f:btandttt}). Note that in (b) there\nare six turning points ($\\bullet$) of the \nprofile $h(z)$ whereas in (c) there are only two.}\n\\end{figure}\n\n\\begin{figure} %Figure 4\n\\begin{center}\n\\rotatebox{-90}{\\epsfig{file=fig4.ps, height=13cm, bbllx =\n 35, bblly = 40, bburx = 540, bbury = 800}}\n\\end{center}\n\\caption{\\label{f:btandttt}\nTemperature-undersaturation phase diagram of wetting layer\nconfigurations for two spheres with $R=20\\sigma$ at a \\emph{fixed} distance\n$D=50\\sigma$ ($a=10\\sigma$). The interaction potential parameters are\nthe same as in Fig.~\\ref{f:example1}. The three configurations shown\nin Fig.~\\ref{f:example1}(a)-(c) are located at the respective\nthermodynamic states\n``a'' to ``c'' ($\\blacklozenge$). The line of liquid-vapor coexistence\n$\\Delta\\mu=0$ separates the region where in the bulk the vapor phase\nis stable and the liquid phase is metastable ($-\\Delta\\mu<0$) from the\nregion where the liquid phase is stable and the vapor phase is\nmetastable ($-\\Delta\\mu>0$). The dotted ``metastability line''\n(``ml'') separates the region where the liquid phase in the bulk is\nstill metastable $(-\\Delta\\mu>-\\Delta\\mu_{ml}(T))$ from the region where\nonly the vapor phase is stable in the bulk $(-\\Delta\\mu<-\\Delta\\mu_{ml}(T))$.\nThe liquidlike layer on each individual sphere exhibits a first-order\nthin-thick transition at $-\\Delta\\mu=-\\Delta\\mu_{tt}(T)$ (dashed line\n``tt''). This line intersects the liquid-vapor coexistence line at\n$T_{tt}^*\\approx1.271$ and ends at a critical point\n($\\bullet$) in the vapor phase region; $T_{tt,c}^*\\approx1.275$ and\n$-\\Delta\\mu_{tt,c}^*\\approx-0.0144$. For the present choice of\ninteraction potential parameters, at lower temperatures and larger\nundersaturations $-\\Delta\\mu = \n-\\Delta\\mu_{bt}(T)$ (full line ``bt'') the first-order bridging\ntransitions between the \nconfigurations with bridge $(-\\Delta\\mu>-\\Delta\\mu_{bt}(T))$ and\nwithout bridge $(-\\Delta\\mu<-\\Delta\\mu_{bt}(T))$ occur. This line\nintersects the coexistence line linearly. Within the sharp-kink\napproximation the line of bridging transitions happens to be cut off\nby the ``metastability line''; within a more sophisticated approach\nthe line ``bt'' is expected to end at a critical point, too. The\nlocations of the thin-thick transitions in the \nphase diagram are practically not affected by the presence of the\nbridge. The dashed-double-dotted lines ($-\\cdot\\cdot-$) are metastable\nextensions of the thin-thick and bridging transition lines, respectively. The\ndashed-dotted line ``p'' ($-\\cdot-$) is the prewetting line for the\ncorresponding \\emph{planar} substrate. It joins the liquid-vapor coexistence\nline $\\Delta\\mu=0$ tangentially at the first-order wetting transition\ntemperature $T_w^*\\approx 1.053$ ($\\blacktriangle$) and ends at a\ncritical point ($\\blacksquare$) in the vapor phase region. For a\ndiscussion of the effects of fluctuations on this mean-field phase\ndiagram see the main text.}\n\\end{figure}\n\n\\begin{figure} %Figure 5\n\\begin{center}\n\\epsfig{file=fig5a.ps, width=7cm, bbllx = 20, bblly = 335,\n bburx = 520, bbury = 775}\n\\epsfig{file=fig5b.ps, width=7cm, bbllx = 20, bblly = 335,\n bburx = 520, bbury = 775}\n\\end{center}\n\\caption{\\label{f:eipforexample1}\n(a) Dependence of the grand canonical potential $\\Omega_S$ on the\nseparation $a=D-2R$ and the undersaturation $\\Delta\\mu$ for\nthe same system as in Figs.~\\ref{f:example1}(c) and \\ref{f:btandttt},\ni.e., for $R=20\\sigma$ and \n$T^*=1.2<T_{tt}^*$. The dots indicate the end points of metastable \nbranches. For $a<2d_s$ (with $d_s=\\sigma=0.05R$ here) the excluded volumes\naround the spheres overlap. In the limit $D\\to\\infty$ the\nstable solution is the one without a liquid bridge; in this limit\n$\\Omega_S(D\\to\\infty)=2\\Omega_S^{(1)}$ is twice the free energy of a single sphere\nsurrounded by a wetting layer. At the separation $D_{bt}$ or $a_{bt}$, where the two\nfree energy branches intersect for a given $\\Delta\\mu$, a first-order\nmorphological phase transition between a configuration with a liquid bridge\nand a state without bridge takes place. The equilibrium thickness\nof the homogeneous wetting layer around a single sphere is $l_0\n\\approx1.3\\sigma$, so that $D_{bt}/(R+l_0)\\approx2.39$; the slight deviation\nfrom the prediction of Eqs.~(\\ref{e:Dbt}) and (\\ref{e:lambda}) is due\nto the still rather small size of the spheres. We note that, in contrast to\nthe case shown here, for $T>T_{tt}$\nthe free energy curve corresponding to the solutions without bridge\napproaches its asymptote from \\emph{below}. (b) Same as in (a), showing\nthe excess free energy $\\Omega_E = \\Omega_S - 2\\Omega_S^{(1)}$.\nIn this presentation the results for the solutions without bridge and\nfor different undersaturations $\\Delta\\mu$ collapse\nonto a single line. $\\Omega_E(D\\to\\infty)$ decays as $D^{-6}$.} \n\\end{figure}\n\n\\newpage\n\n\\begin{figure} % Figure 6\n\\begin{center}\n\\epsfig{file=fig6.ps, width=10cm, bbllx = 40, bblly = 335,\n bburx = 520, bbury = 775}\n\\end{center}\n\\caption{\\label{f:eip_largespheres}\nExcess free energy $\\Omega_E=\\Omega_S-2\\Omega_S^{(1)}$ for\n$\\Delta\\mu=0$ in the limit of large \nspheres, i.e., $R\\gg\\sigma$, $\\sigma\\ll a\\approx R$. In this limit the\nexcess free energy branch for the unbridged solution vanishes if\nit is measured in units of $8\\pi R^2$. Off two-phase coexistence, i.e.,\nfor $\\Delta\\mu\\neq0$ the branch for the\nbridged solution is determined only by the contributions $\\Omega_{lg}$\n(Eq.~(\\ref{e:omegalg})) and $\\Omega_{ex}$ (Eq.~(\\ref{e:excess})) to\nthe free energy. At two-phase coexistence $\\Delta\\mu$ and\n$\\Omega_{ex}$ vanish so that $\\Omega_E$ is solely determined by\n$\\Omega_{lg}$. Therefore within the local theory with\n$\\Omega_{lg}^{(loc)}$ (Eq.~(\\ref{e:local})) the bridged solution is a\nminimal area surface, i.e., its mean curvature is zero. Since $a\\gg\nd_s\\approx\\sigma$ the excluded \nvolume at small $a$ disappears from the figure. Therefore, compared\nwith the full curve in Fig.~\\ref{f:eipforexample1}(b) the potential\ncurve here is effectively shifted to smaller values of $a$. Moreover,\nthe actual minimum of the effective interaction potential at small\n$a\\gtrsim\\sigma$ (compare\nFig.~\\ref{f:eipforexample1}), which is due to the influence of the\ncontributions $\\Omega_{ei}$ and $\\Omega_{sl}$, is not visible on this\nscale either. The critical separation\nfor the bridging transition ($\\blacklozenge$) is given by $a_{bt}/R\\approx\n0.32$ (Eqs.~(\\ref{e:Dbt}) and (\\ref{e:lambda})). If the thermodynamic\nstate of the system is driven into the off-coexistence region\n$\\Delta\\mu>0$ the whole excess free energy branch for the bridged\nsolution is shifted upwards (compare Fig.~\\ref{f:eipforexample1}). For any\nfinite value of $\\Delta\\mu$, in the limit $R\\to\\infty$ there is no\nbridging transition anymore (see the main text).}\n\\end{figure}\n\n\\begin{figure} %Figure 7\n\\begin{center}\n\\epsfig{file=fig7.ps, width=10cm, bbllx =\n 40, bblly = 335, bburx = 520, bbury = 775}\n\\end{center}\n\\caption{\\label{f:eipforexample1plusbare}\nExcess free energy\n$\\Omega_E = \\Omega_S-2\\Omega_S^{(1)}$ (dashed lines) and excess total\nfree energy $\\Omega_{E,tot} =\n\\Omega_S-2\\Omega_S^{(1)}+\\Phi$ (full lines)\nfor $\\Delta\\mu=0$. Here $T^*=1.2$ and $R=20\\sigma$ so that the dashed\nlines are identical with \nthe full lines in Fig.~\\ref{f:eipforexample1}(b). The dots indicate\nthe end points of metastable branches. The parameters $\\epsilon_{ss}$\nand $\\sigma_{ss}$ of the pair potential between the particles forming\nthe spheres are chosen such that the condition \n$A_{sf} = \\sqrt{A\\,A_{ss}}$ for the corresponding Hamaker constants is\nsatisfied. Although the wetting-layer induced potential for the\nsolutions without bridge is \\emph{repulsive}, the total interaction\npotential including the bare dispersion potential is\n\\emph{attractive}. For small separations $a$ or $D$ the bare dispersion\npotential dominates. In the limit $D\\to\\infty$, i.e., for\nthe configurations without bridge, $\\Omega_E$ and $\\Omega_{E,tot}$\ndecay as $D^{-6}$ as expected for dispersion interactions.}\n\\end{figure}\n\n\\newpage\n\n\\begin{figure} %Figure 8\n\\begin{center}\n\\epsfig{file=fig8a.ps, width=7cm, bbllx = 20, bblly = 335,\n bburx = 520, bbury = 775}\n\\epsfig{file=fig8b.ps, width=7cm, bbllx = 20, bblly = 335,\n bburx = 520, bbury = 775}\n\\end{center}\n\\caption{\\label{f:indexmatch}\nSame as in Fig.~\\ref{f:eipforexample1} but with $\\Omega_{tot,im} =\n\\Omega_S-\\Omega_{sg}$ (a) and with $\\Omega_{E,im} =\n\\Omega_{tot,im}-2\\Omega_{im}^{(1)}$ (b). We choose again $T^*=1.2$, $R=20\\sigma$,\nand the interaction parameters as in the previous figures. The dots\nindicate the end points of metastable branches. The total interaction\npotential for index-matched spheres and bulk fluid is again\n\\emph{repulsive}: since the temperature is below the thin-thick transition\ntemperature $T_{tt}$ the adjacent spheres dislike the presence of\nadditional liquid in their vicinity and therefore it is energetically advantageous\nto separate them as much as possible. $\\Omega_{E,im}$ for the\nsolutions without bridge is smaller than $\\Omega_E$. However, for the\nbridged solutions, $\\Omega_{E,im}$ and $\\Omega_E$ as well as the\ncorresponding wetting-induced forces are of almost the same size,\nrespectively.}\n\\end{figure}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002133.extracted_bib",
"string": "\\bibitem{hansenmcdonald} See, e.g., J.~P.~Hansen and I.~R.~McDonald,\n \\emph{Theory of Simple Liquids}, 2nd. Ed. (Academic, London, 1986).\n\n\n\\bibitem{asakuraoosawa} S.~Asakura and F.~Oosawa,\n J. Chem. Phys. \\textbf{22}, 1255 (1954).\n\n\n\\bibitem{maoetal} Y.~Mao, M.~E.~Cates, and H.~N.~W.~Lekkerkerker,\n Physica A \\textbf{222}, 10 (1995).\n\n\n\\bibitem{goetzelmannetal} B.~G\\\"otzelmann, R.~Evans, and S.~Dietrich,\n Phys. Rev. E \\textbf{57}, 6785 (1998); B.~G\\\"ot\\-zel\\-mann, R.~Roth,\n S.~Dietrich, M.~Dijkstra, and R.~Evans, Europhys. Lett. \\textbf{47},\n 398 (1999).\n\n\n\\bibitem{rothetal} R.~Roth, B.~G\\\"otzelmann, and S.~Dietrich,\n Phys. Rev. Lett. \\textbf{83}, 448 (1999).\n\n\n\\bibitem{dijkstraetal} M.~Dijkstra, R.~van~Roij, and R.~Evans,\n Phys. Rev. Lett. \\textbf{81}, 2268 (1998); \\emph{ibid} \\textbf{82},\n 117 (1999); Phys. Rev. E \\textbf{59}, 5744 (1999).\n\n\n\\bibitem{eisenriegleretal} T.~W.~Burkhardt and E.~Eisenriegler,\n Phys. Rev. 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Matter \\textbf{4}, 10133 (1992).\n\n\n\\bibitem{wca} J.~D.~Weeks, D.~Chandler, and H.~C.~Andersen,\nJ. Chem. Phys. \\textbf{54}, 5237 (1971); H.~C.~Andersen, J.~D.~Weeks,\nand D.~Chandler, Phys. Rev. A \\textbf{4}, 1597 (1971).\n\n\n\\bibitem{cs} N.~F.~Carnahan and K.~E.~Starling,\nJ. Chem. Phys. \\textbf{51}, 635 (1969).\n\n\n\\bibitem{bauer1} C.~Bauer and S.~Dietrich, Eur. Phys. J. \\textbf{10},\n 767 (1999).\n\n\n\\bibitem{privmanfisher} V.~Privman and M.~E.~Fisher,\n J. Stat. Phys. \\textbf{33}, 385 (1983); J. Appl. Phys. \\textbf{57},\n 3327 (1985).\n\n\n\\bibitem{gelfandlipowsky} M.~P.~Gelfand and R.~Lipowsky, Phys. Rev. B\n \\textbf{36}, 8725 (1987).\n\n\n\\bibitem{hamaker} H.~C.~Hamaker, Physica \\textbf{4}, 1058 (1937).\n\n\n\\bibitem{israelachvili} J.~Israelachvili, \\emph{Intermolecular and\n surface forces}, 2nd ed. (Academic, London, 1991).\n\n\n\\bibitem{vold} M.~J.~Vold, J. Coll. Sci. \\textbf{16}, 1 (1961).\n\n\n\\bibitem{vincentetal} D.~W.~J.~Osmond, B.~Vincent, and\n F.~A.~Waite, J. Colloid Interface Sci. \\textbf{42}, 262 (1973);\n B.~Vincent, J. Colloid Interface Sci. \\textbf{42}, 270 (1973).\n\n\n\\bibitem{kinoshita} M.~Kinoshita, Mol. Phys. \\textbf{94}, 485 (1998);\n and references therein.\n\n\n\\bibitem{puseyvanmegen} P.~N.~Pusey and W.~van~Megen, Nature\n \\textbf{320}, 340 (1986).\n\n\n\\bibitem{poonpuseylekkerkerker} W.~Poon, P.~N.~Pusey, and\n H.~Lekkerkerker, Physics World \\textbf{9}, p. 27, April 1996.\n\n"
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cond-mat0002134
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Nature of the Spin Glass State
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[
{
"author": "Matteo Palassini and A. P. Young"
}
] |
The nature of the spin glass state is investigated by studying changes to the ground state when a weak perturbation is applied to the bulk of the system. We consider short range models in three and four dimensions and the infinite range Sherrington-Kirkpatrick (SK) and Viana-Bray models. Our results for the SK and Viana-Bray models agree with the replica symmetry breaking picture. The data for the short range models fit naturally a picture in which there are large scale excitations which cost a finite energy but whose surface has a fractal dimension, $d_s$, less than the space dimension $d$. We also discuss a possible crossover to other behavior at larger length scales than the sizes studied.
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[
{
"name": "paper7_4.tex",
"string": "%\\documentstyle[prb,aps,epsf,floats,goodfloat]{revtex}\n\\documentstyle[prb,aps,epsfig,floats,goodfloat]{revtex}\n\\newcommand{\\smfrac}[2]{\\mbox{\\small $#1 \\over #2$}}\n\\newcommand{\\bc}{\\begin{center}}\n\\newcommand{\\ec}{\\end{center}}\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\beqn}{\\begin{eqnarray}}\n\\newcommand{\\eeqn}{\\end{eqnarray}}\n%\\newcommand{\\ql}{q_{\\mathrm{link}}}\n\\newcommand{\\ql}{q_{l}}\n\\newcommand{\\ml}{\\mu_{l}}\n\\newcommand{\\av}{_{\\mathrm{av}}}\n\\begin{document}\n\\draft\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse%\n\\endcsname\n\n\\title{\nNature of the Spin Glass State\n}\n\n\\author{Matteo Palassini and A. P. Young}\n\\address{Department of Physics, University of California, Santa Cruz, \nCA 95064}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\nThe nature of the spin glass state is investigated by studying changes to the\nground state when a weak perturbation is applied to the bulk of the system. We\nconsider short range models in three and four dimensions and the\ninfinite range Sherrington-Kirkpatrick (SK) and Viana-Bray models. \nOur results for the SK and Viana-Bray models agree with the replica \nsymmetry breaking picture.\nThe data for the short range models fit naturally a picture in which\nthere are large scale excitations which cost a finite energy but whose\nsurface has a fractal dimension, $d_s$, less than the space dimension $d$. \nWe also discuss a possible crossover to other behavior at\nlarger length scales than the sizes studied. \n\\end{abstract}\n\n\\pacs{PACS numbers: 75.50.Lk, 05.70.Jk, 75.40.Mg, 77.80.Bh}\n%\\vskip 0.3 truein\n]\n\nThe nature of ordering in spin glasses below the transition temperature, $T_c$,\nremains a controversial issue. Two theories have been extensively discussed: the\n``droplet theory'' proposed by Fisher and Huse\\cite{fh} (see also\nRefs.~\\onlinecite{bm,mcmillan,ns}), and the replica symmetry\nbreaking (RSB) theory of Parisi\\cite{parisi,mpv,by}. An important difference\nbetween these theories concerns the number of large-scale, low energy\nexcitations. In the RSB theory, which follows the exact solution of the\ninfinite range SK model, there are excitations which involve turning over a\nfinite fraction of the spins and which cost only a {\\em finite} \nenergy even in the thermodynamic limit. Furthermore, the surface of these\nexcitations is argued\\cite{qlink} to be\nspace filling, i.e. the fractal\ndimension of their surface, $d_s$, is equal to the space dimension, $d$.\nBy contrast, in the droplet theory, the lowest energy excitation which involves\na given spin and which has linear spatial extent $L$ typically costs an energy of order\n$L^\\theta$, where $\\theta$ is a (positive) exponent. Hence, in the\nthermodynamic limit, excitations which flip a finite fraction of the spins cost\nan {\\em infinite} energy. Also, the\n surface of these excitations is not\nspace filling, {\\em i.e.} $d_s < d$.\n\nRecently we\\cite{py1,py2,py3} investigated this issue by looking at how spin\nglass ground states in two and three dimensions change upon changing the\nboundary conditions. Extrapolating from the range of sizes studied to the\nthermodynamic limit, our results suggest that the low energy\nexcitations have $d_s < d$. Similar results were found in two dimensions\nby Middleton\\cite{midd}. In this paper, following a suggestion by\nFisher\\cite{dsf}, we apply a perturbation to the ground states in the {\\em\nbulk} rather than at the surface. The motivation for this is two-fold:\n%\\begin{enumerate}\n%\\item\n(i) We can apply the same method both to models with short range interactions and\nto infinite range models, like the SK model,\n%which have no boundary and for\n%which one cannot therefore change the ``boundary conditions''. \n%We\nand so can verify that the method is able to distinguish\nbetween the RSB picture, which is believed to apply to infinite range models,\nand some other picture which may apply to short range\nmodels.\n%\\item\n(ii) It is possible that there are\nother low energy excitations which are not excited by changing the\nboundary conditions\\cite{kawa,hm}. \n%\\end{enumerate}\n\nWe consider the short-range Ising spin glass in three and four\ndimensions, and, in addition, the SK and Viana-Bray\\cite{vb}\nmodels. The latter is\ninfinite range but with a finite average\ncoordination number $z$, and is expected to show RSB behavior.\nAll these models have a finite\ntransition temperature. \n%In the 3-D case, the exponent $\\theta$ obtained from\n%the magnitude of the change of the ground state energy when\n%the boundary conditions are changed from periodic to anti-periodic is\n%about\\cite{theta-3d} 0.2, whereas in 4-D it is much larger,\n%This number is quite small, which makes it difficult to distinguish\n%between different scenarios. Hence the usefulness of also studying the 4-D\n%model, where $\\theta$ obtained from boundary condition changes is much larger,\n%about\\cite{theta-4d} 0.7.\n\nOur results for the SK and Viana-Bray models\nshow clearly the validity of the RSB picture. \nHowever, for\nthe short range models, our data is consistent with a picture suggested by\nKrzakala and Martin\\cite{km} where there \nare extensive excitations with {\\em finite} energy, i.e. their energy varies\nas $L^{\\theta'}$ with $\\theta' = 0$,\nbut $d_s < d$. In three dimensions, this picture is difficult to\ndifferentiate from the droplet picture where the energy varies as\n$L^\\theta$, because of the small value of $\\theta$ ($\\simeq 0.2$,\nobtained from\nthe magnitude of the change of the ground state energy when\nthe boundary conditions are changed from periodic to anti-periodic\\cite{theta-3d}). \nIt is easier to\ndistinguish the two pictures in 4-D, even though the range of $L$ is less,\nbecause $\\theta$ is much larger\\cite{theta-4d} ($\\simeq 0.7$).\n%An alternative scenario \n%is that the standard droplet\n%theory is asymptotically correct but seen only\n%beyond a certain length scale which increases\n%as $d$ increases and is larger than the sizes studied.\n%This cannot be ruled out.\n\n%Krzakala and Martin\\cite{km} have come to a\n%similar conclusion based on results just in 3-D.\n\nThe Hamiltonian is given by\n\\begin{equation}\n{\\cal H} = -\\sum_{\\langle i,j \\rangle} J_{ij} S_i S_j ,\n\\label{ham}\n\\end{equation}\nwhere, for the short range case, the sites $i$ lie on a \nsimple cubic lattice in dimension $d=3$ or 4 with $N=L^d$ sites \n($L \\le 8$ in 3-D, $L \\le 5$ in 4-D), $S_i=\\pm\n1$, and the $J_{ij}$ are nearest-neighbor interactions chosen from a\nGaussian distribution with zero mean and standard deviation unity. Periodic\nboundary conditions are applied. For the SK\nmodel there are interactions between {\\em all} pairs chosen from a Gaussian\ndistribution of width $1/\\sqrt{N-1}$, where $N \\le 199$. For the Viana-Bray\nmodel each spin is connected with $z=6$ spins on average, chosen randomly,\n%where we took $z=6$,\nthe width of the Gaussian distribution is unity, and\nthe range of sizes is $N \\le 399$.\nTo determine the ground state\n%of the short range and Viana-Bray models\nwe use a hybrid genetic algorithm introduced by Pal\\cite{pal}, as\ndiscussed elsewhere\\cite{py2}.\n%For the SK model we used a procedure of repeated\n%quenching from a random starting configuration to the nearest local minima, and\n%stopping when the lowest energy\n%found did not change for a sufficiently large number of\n%quenches. \n\nLet $S_i^{(0)}$\nbe the spin configuration in the ground state for\na given set of bonds.\nHaving found $S_i^{(0)}$, we\nthen add a perturbation to the Hamiltonian designed to\nincrease the energy of the ground state relative to the other states, and so\npossibly induce a change in the ground state. This perturbation,\nwhich depends upon a positive parameter $\\epsilon$, changes the\ninteractions $J_{ij}$ by an amount proportional to $S_i^{(0)} S_j^{(0)}$, i.e.\n\\begin{equation}\n\\Delta {\\cal H}(\\epsilon) = \\epsilon {1 \\over N_b} \\sum_{\\langle i,j \\rangle}\nS_i^{(0)} S_j^{(0)} S_i S_j,\n\\end{equation}\nwhere \n$N_b$ is the number of bonds in the Hamiltonian.\n%($3N$ for 3-D and the\n%$z=6$ Viana-Bray model, $4N$ for\n%4-D, and $N(N-1)/2$ for the SK model). \nThe energy of the ground state will thus increase exactly by an amount\n$ \\Delta E^{(0)} = \\epsilon .$\nThe energy of any other state, $\\alpha$ say, will increase by the lesser amount\n$ \\Delta E^{(\\alpha)} = \\epsilon\\ \\ql^{(0, \\alpha)},$\nwhere $\\ql^{(0, \\alpha)}$ is the ``link overlap'' between the states\n``0'' and $\\alpha$, defined by\n\\begin{equation}\n\\ql^{(0, \\alpha)} = {1 \\over N_b}\\sum_{\\langle i,j \\rangle} S_i^{(0)} S_j^{(0)} \nS_i^{(\\alpha)} S_j^{(\\alpha)} ,\n\\end{equation}\nin which the sum is over all the $N_b$ pairs where there are interactions.\nNote that the {\\em total} energy of the states is changed by an\namount of order unity.\n\nThe decrease in the energy {\\em difference} between \na low energy excited state and the\n ground state is given by\n\\begin{equation}\n\\delta E^{(\\alpha)} = \\Delta E^{(0)} - \\Delta E^{(\\alpha)} = \n\\epsilon \\ (1 - \\ql^{(0, \\alpha)}) .\n\\label{de}\n\\end{equation}\nIf this exceeds the original difference in energy, $E^{(\\alpha)} - E^{(0)}$,\nfor at least one of the excited states, then the ground state will change \ndue to the perturbation. We denote the new ground state spin\nconfiguration by $ \\tilde{S}_i^{(0)}$, and indicate by\n$\\ql$ and $q$, with no indices, the link- and spin-overlap\n between the new and old ground states.\n\nNext we discuss the expected behavior of $q$ and $\\ql$ for\nthe various models. For the SK\nmodel, it is easy to derive the trivial relation,\n$\\ql = q^2$ (for large $N$).\n%between $\\ql$ and $q$.\nSince RSB theory is expected to be correct, \nthere are some excited states\nwhich cost a finite energy and which have an overlap $q$ less than\nunity. According to Eq.~(\\ref{de}), these have a finite\nprobability of becoming the new ground state. Hence the average value of \n$q$ and $\\ql$ over many samples,\ndenoted by $[\\cdots]\\av$,\nshould tend to a constant less than unity in\nthe thermodynamic limit. \nThis behavior is shown in the inset of Fig.~\\ref{q_sk}. For the\nViana-Bray model, where there is no trivial connection between $q$ and $\\ql$,\nwe show in Fig.~\\ref{q_sk}\ndata for $R = (1-[\\ql]\\av)/(1-[q]\\av)$ for several values of $\\epsilon$.\nThis also appears to saturate. \nWe plot this ratio rather than $[q]\\av$\nor $[q_l]\\av$ for better comparison with the short range case below.\n%Data for $q$, not shown, is qualitatively similar though, in contrast to the\n%SK model, there is no trivial connection between $q$ and $\\ql$.\nFor both models we took\n$\\epsilon$ to be a multiple of\nthe transition temperature (the mean field\napproximation to it, $T_c^{MF} = \\sqrt{z}$,\nfor the Viana-Bray model), so that a perturbation of\ncomparable magnitude was applied in both cases.\n%in which the number of samples simulated was 25000 for the sizes $N=11$, 33,\n%55, 20000 for $N=99$, and 5160 for $N=199$.\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{figure=q_SKVB.eps,width=\\columnwidth}\n\\end{center}\n\\caption{\nData for\n$R = (1-[q_l]\\av)/(1-[q]\\av)$ for\nthe Viana-Bray model with coordination number $z=6$ for several values of\n$\\epsilon$. The curvature is a strong indication that the data tends to a\nnon-zero value for $N \\to \\infty$, as for the SK model. The best fits\nto $a + b/N^c$, shown by the lines, give\n$a= 0.872 \\pm 0.005, 0.883 \\pm 0.01$ and $0.84 \\pm 0.03$\nfor $\\epsilon=\\sqrt{6}, \\epsilon=\\sqrt{6}/2$ and $\\epsilon=\\sqrt{6}/4$\nrespectively.\nInset: \n$1 - [q]\\av$ for the SK model\n%where $q$ is the spin overlap between the original\n%ground state and the perturbed ground state\nwith the strength of the\nperturbation given by $\\epsilon = 1$.\nBecause $\\ql = q^2$, the behavior of $[\\ql]\\av$ is\nvery similar. The data is clearly tending to a constant at large $N$. The solid\nline is the best fit %to the form $1-[q]\\av = a + b / N^c$\nand has\n$a = 0.377 \\pm 0.004$.\n}\n\\label{q_sk}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{figure=q_scaling_3d.eps,width=\\columnwidth}\n\\end{center}\n\\caption{\nA scaling plot of the data for $[q]\\av$ in 3-D according to\nEq.~(\\ref{scaling}). The data collapse is very good with $\\mu = 0.44$.\nThe solid curve is a polynomial fit ($\\chi^2=14.9$, $d.o.f.=13$), constrained to go through the origin,\nomitting the $L=3$ data. The dashed line is the linear term in the fit.\n}\n\\label{q_scaling_3d}\n\\end{figure}\n\nWhat do we expect for the short range models? In the RSB \ntheory, $1-[q]_{av}$ and $1-[q_l]_{av}$ (and hence the\nratio $R$) should saturate to a finite value for large $L$.\nTo derive the prediction of the droplet theory,\nsuppose that the energy to create an excitation of linear dimension, $l$,\nhas a characteristic scale of $l^{\\theta'}$\n(we use $\\theta'$ rather than\n$\\theta$ to allow for the possibility that this exponent is different from the\none found by changing the boundary conditions). Let us assume that large clusters\n($l \\approx L$) dominate and ask for\nthe probability that a large cluster is excited.\nThe energy gained from the perturbation is \n$\\epsilon (1 - \\ql) \\sim \\epsilon / L^{(d - d_s)}$ since $1/L^{(d-d_s)}$ is the\nfraction of the system containing the surface (i.e. the broken bonds) of the\ncluster. Generally this will not be able to overcome the $L^{\\theta'}$ energy\ncost to create the cluster. However, there is a distribution of\ncluster energies and if we make the plausible hypothesis that this\ndistribution has a finite weight at the origin, then the\nprobability that the cluster is excited \nis proportional to\n$1/L^{d - d_s + \\theta'}$. In other words\n\\begin{equation}\n1 - [q]\\av \\sim \\epsilon / L^{\\mu} \\ \\ {\\mathrm where} \\ \\ \n\\mu = \\theta' + d - d_s .\n\\label{psis}\n\\end{equation}\nAs discussed above, $1 - \\ql$ is of order $1/L^{(d-d_s)}$ \n%when a large ($q<1$) cluster\n%is excited, \nand so\n\\begin{equation}\n1 - [\\ql]\\av \\sim \\epsilon / L^{\\ml} \\ \\ {\\mathrm where} \\ \\ \n\\ml = \\theta' + 2(d - d_s) .\n\\label{psil}\n\\end{equation}\nSimilar expressions have been derived by Drossel et al.\\cite{drossel} in\nanother context.\n%to\n%describe the \n%effect of coupling two replicas of the system. \nEqs.~(\\ref{psis}) and\n(\\ref{psil}) are expected to be valid only {\\em asymptotically} \nin the limit $\\epsilon \\to 0$. In order\nto include data for a range of values of $\\epsilon$ we note that the data is\nexpected to scale as \n\\begin{eqnarray}\n1 - [q]\\av & = & F_q(\\epsilon/L^\\mu) , \\nonumber \\\\\n 1 - [\\ql]\\av & = & L^{-(d-d_s)}\nF_{q_{l}}(\\epsilon/L^\\mu) ,\n\\label{scaling}\n\\end{eqnarray}\nwhere the scaling functions $F_q(x)$ and $F_{q_{l}}(x)$\nboth vary linearly for small $x$. Note that the above discussion\napplies also to a picture in which $\\theta'=0$ and $d_s<d$.\n\n%In the RSB picture both $1 - [q]\\av$ and $1 - [\\ql]\\av$\n%tend to a constant which implies that $\\theta' = 0$ and $d_s = d$.\n%In the droplet theory $\\theta' = \\theta > 0$ and $d_s < d$. In the RSB theory,\n%$\\theta' = 0$ but it is not obvious to us what is the prediction for $d-d_s$.\n%For the Viana-Bray model we found that both $1 - [q]\\av$ and $1 - [\\ql]\\av$\n%tend to a constant which would correspond to $d_s = d$ as well as\n%$\\theta' = 0$. However, the Viana-Bray model is not a short range model so it\n%is not obvious to us that this is necessarily the prediction of the RSB theory\n%for short range models.\n%We will see that our results\n%fit best a scenario where $\\theta' = 0$ but $d_s < d$.\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{figure=q_ql_3d.eps,width=\\columnwidth}\n\\end{center}\n\\caption{\nA plot of $R = (1-[q_l]\\av)/(1 - [q]\\av)$, the surface to volume ratio of the\nclusters, in 3D as a function of system size for different different values of \n$\\epsilon$. For clarity the $\\epsilon = \\tau/2$ data is omitted.\nThe dependence on $\\epsilon$ is quite weak, and for\neach value of $\\epsilon$ the data gives a good fit to a straight line with\nslope [equal to $-(d-d_s)$] consistent with Eq.~(\\ref{ds}).\nThe inset shows a scaling plot\nof the data according to Eq.~(\\ref{scaling}) with the same value of $\\mu$ as\nin Fig.~\\ref{q_scaling_3d}. The solid curve is a polynomial fit \n($\\chi^2=26.0$, $d.o.f.=18$). \n}\n\\label{q_ql_3d}\n\\end{figure}\n\nA scaling plot of our results for $1 - [q]\\av$ in 3D\nis shown in\nFig.~\\ref{q_scaling_3d}.\nWe consider a range of $\\epsilon$ from $\\sqrt{6}/4$ to\n$4\\sqrt{6}$ (note that $T_c^{MF}= \\sqrt{6}$) and find that the data collapse\nwell onto the form expected in Eq.~(\\ref{scaling}) with\n$ \\mu = 0.44 \\pm 0.02.$\n\nIt is also convenient to plot the ratio $R$, which\nrepresents the surface to volume ratio of the excited clusters. This has\na rather\nweak dependence on $\\epsilon$ and, as shown in Fig.~\\ref{q_ql_3d},\nthe data for {\\em each}\\/ of the values of $\\epsilon$\nfits well the power law behavior\n$L^{-(d-d_s)}$, expected from Eqs.~(\\ref{psis}) and (\\ref{psil}), with\n$d - d_s$ between 0.40 and 0.41 (the goodness of fit parameter, $Q$,\nis $0.07, 0.03, 0.85, 0.23, 0.10$, in order of increasing $\\epsilon$).\nThe inset to Fig.~\\ref{q_ql_3d} shows that there are \nsmall deviations from the asymptotic behaviour, which can be accounted for\nby a scaling function with the same value of $\\mu$ as in\nFig.~\\ref{q_scaling_3d} and with\n\\begin{equation}\nd - d_s = 0.42 \\pm 0.02 \\ \\ (3D) .\n\\label{ds}\n\\end{equation}\nFrom this value of $\\mu$ and\nEqs.~(\\ref{psis}) and (\\ref{ds})\nwe find\n\\begin{equation}\n\\theta' = 0.02 \\pm 0.03 \\ \\ (3D) .\n\\end{equation}\nIn order to test the RSB prediction, we tried fits of the form \n$ R = a + b/L^c $, which give\n$a = 0.28 \\pm 0.18, 0.01 \\pm 0.14, 0.04 \\pm 0.11$, and $-0.28 \\pm 0.18$\n($Q=0.08,0.01,0.72$, and 0.52)\nfor $\\epsilon/\\tau = 0.25, 0.5, 1$ and $2$. These are consistent with $a=0$\nthough a fairly small positive value, which would imply\n$d_s=d$, cannot be ruled out. For $\\epsilon/\\tau\n=4$ the fit gives a small positive value, $0.18 \\pm 0.07$ ($Q=0.79$),\nbut this is likely too large a value of $\\epsilon$ to be in the \nasymptotic regime for these sizes (see the inset of Fig.~\\ref{q_ql_3d}).\nThe form $R=a+b/L+c/L^2$ also fits reasonably well the data and gives \n$a$ between 0.41 and 0.48 ($Q=0.16,0.03,0.82,0.80,0.16$).\nHowever, for both forms the data are very far from the asymptotic limit $R\\sim a$ \nfor the sizes considered, unlike for the Viana-Bray model \n(compare the main parts of Figs. ~\\ref{q_sk} and\n~\\ref{q_ql_3d}). \n%For example, for $\\epsilon=1$ the correction term \n%$b/L+c/L^2$ is about $70\\%$ of the asymptotic value \n%$a=0.43$ for $L = 8$. \nBy contrast, the deviation from the asymptotic behavior \n$R\\sim L^{-(d-d_s)}$ is quite small (see the inset of\nFig.~\\ref{q_ql_3d}). \n\\begin{figure}\n\\begin{center}\n\\epsfig{figure=qall_4d.eps,width=\\columnwidth}\n\\end{center}\n\\caption{\nThe main figure shows the ratio $R = (1-[q_l]\\av)/(1-[q]\\av)$\nfor different values of $\\epsilon$\nin 4-D. The inset shows corresponding data for $[q]\\av$.\n}\n\\label{q_all_4d}\n\\end{figure}\n\nIn Fig.~\\ref{q_all_4d} we show analogous results\nin 4-D. The calculations were performed for two\ndifferent values $\\epsilon = \\sqrt{8}/4$ and $\\sqrt{8}$ ($= T_c^{MF}$).\n%For these two values, the number of samples was 15000 and 45000 for $L=3$,\n%19000 and 27000 for $L=4$, 4000 and 6400 for $L=5$.\nThe exponents are essentially the same for these two values of the\nperturbation and the fits give\n$ \\mu = 0.26 \\pm 0.04$, $d - d_s = 0.23 \\pm 0.02$ ,\nand so from Eq.~(\\ref{psis}) we get our main results for 4D:\n\\begin{equation}\n\\theta' = 0.03 \\pm 0.05, \\quad d - d_s = 0.23 \\pm 0.02 \\ \\ (4D).\n\\end{equation}\nThe data in Fig.~\\ref{q_all_4d} is {\\em consistent} with the scaling form in\nEq.~(\\ref{scaling}) but the data for the two values of $\\epsilon$ are too\nwidely separated to {\\em demonstrate} scaling.\n\nInterestingly, our results in both 3-D and 4-D are consistent with $\\theta' =\n0$, and, within the error bars, (which are purely statistical) incompatible\nwith the relation $\\theta' = \\theta$, since\n$\\theta \\simeq 0.20$ in\n3-D\\cite{theta-3d,py2} and\n$\\theta \\simeq 0.7$ in 4-D\\cite{theta-4d}. In 3-D, \n$\\theta - \\theta'$ is small, but in 4-D this difference is larger and hence the\nconclusion that $\\theta' \\ne \\theta$ is stronger. However, the conclusion that\n$d-d_s > 0$ is less strong in 4-D because our value for $d-d_s$ is quite small\nand the range of sizes is smaller than in 3-D.\n%Our results do not\n%seem to be compatible with the picture of ``sponge-like'' excitations proposed\n%by Houdayer and Martin\\cite{hm} since, in addition to having $\\theta' = 0$,\n%these also have a space filling surface.\n\nIt would be interesting, in future work, to study the\nnature of these excitations to see\nhow they differ from the\nexcitation of energy $L^\\theta$ (with $\\theta > 0$) induced by boundary\ncondition changes\\cite{py2,theta-3d,theta-4d}.\nIn particular, if their\nvolume is space filling,\n%i.e. that the fractal dimension of their bulk is equal to $d$,\none would \nexpect a non-trivial order parameter distribution, $P(q)$, at finite\ntemperatures. %We should emphasize\n\nTo conclude, an\ninterpretation of our results for short range models\nwhich is natural, in that it fits the data with a minimum number of\nparameters and with small corrections to scaling, is that there\nare large-scale low energy excitations which cost a finite energy, and\nwhose surface has fractal dimension\nless than $d$. This picture differs from the one suggested by \nHoudayer and Martin\\cite{hm}, in which $d_s=d$. \nFurthermore, the results for short range models\nappear quite different from those of the mean-field\nlike Viana-Bray model for samples with a similar coordination number and\na similar number of spins.\nOther scenarios,\nsuch as the droplet theory (with $\\theta^\\prime = \\theta \\ (> 0)$) or\nan RSB picture (where $\\theta^\\prime = 0, d - d_s = 0$), require larger\ncorrections to scaling, but we cannot rule out the possibility of\ncrossover to one of these behaviors at larger sizes.\n%In $d=4$, the data for $\\theta'$ for the sizes studied is incompatible with\n%$\\theta \\ (\\simeq 0.7)$, but since our value for $d-d_s$ ($0.22 \\pm 0.06$,\n%where the error is only statistical) is quite small, our data may be\n%consistent with $d - d_s = 0$.\n\nWe would like to thank D.~S.~Fisher for suggesting this line of enquiry, and\nfor many stimulating comments. We also acknowledge useful discussions and\ncorrespondence with G.~Parisi, E. Marinari,\nO.~Martin, M.~M\\'ezard and J.-P.~Bouchaud. We are grateful to D.~A.~Huse,\nM.~A.~Moore and\nA.~J.~Bray for suggesting the scaling plot in Fig.~\\ref{q_scaling_3d} and\none of the referees for suggesting plotting the ratio $R$.\nThis work was\nsupported by the National Science Foundation under grant DMR 9713977. M.P.\nalso is supported in part by a fellowship of Fondazione Angelo Della Riccia.\nThe numerical calculations were supported by computer time from the\nNational Partnership for Advanced Computational Infrastructure.\n\n\\vspace{-0.5cm}\n\\begin{references}\n\n\\bibitem{fh}\n D.~S.~Fisher and D.~A.~Huse, J. Phys. A. {\\bf 20} L997 (1987); D.~A.~Huse\n and D.~S.~Fisher, J. Phys. A. {\\bf 20} L1005 (1987); D.~S.~Fisher and\n D.~A.~Huse, Phys. Rev. B {\\bf 38} 386 (1988).\n\n\\bibitem{bm}\n A.~J.~Bray and M.~A.~Moore, in {\\em Heidelberg Colloquium on Glassy\n Dynamics and Optimization}\\/, L.~Van~Hemmen and I.~Morgenstern eds.\n (Springer-Verlag, Heidelberg, 1986).\n\n\\bibitem{mcmillan}\n W.~L.~McMillan, J. Phys. C {\\bf 17} 3179 (1984).\n\n\\bibitem{ns}\n C.~M.~Newman and D.~L.~Stein, Phys. Rev. B {\\bf 46}, 973 (1992); Phys. Rev.\n Lett., {\\bf 76} 515 (1996);\n Phys. Rev. E {\\bf 57} 1356 (1998).\n\n\\bibitem{parisi}\n G.~Parisi, Phys. Rev. Lett. {\\bf 43}, 1754 (1979); J. Phys. A {\\bf 13},\n 1101, 1887, L115 (1980; Phys. Rev. Lett. {\\bf 50}, 1946 (1983).\n\n\\bibitem{mpv}\n M.~M\\'ezard, G.~Parisi and M.~A.~Virasoro, {\\em Spin Glass Theory and\n Beyond} (World Scientific, Singapore, 1987).\n\n\\bibitem{by}\n K.~Binder and A.~P.~Young, Rev. Mod. Phys. {\\bf 58} 801 (1986).\n\n\\bibitem{qlink}\n E.~Marinari, G.~Parisi, F.~Ricci-Tersenghi, J.~Ruiz-Lorenzo and F.~Zuliani,\n J. Stat. Phys. {\\bf 98}, 973 (2000).\n\n\\bibitem{py1}\n M.~Palassini and A.~P.~Young, Phys. Rev. {\\bf B60}, R9919, (1999). \n\n\\bibitem{py2}\n M.~Palassini and A.~P.~Young, Phys. Rev. Lett. {\\bf 83}, 5216 (1999).\n\n\\bibitem{py3}\n M.~Palassini and A.~P.~Young, cond-mat/9910278.\n\n\\bibitem{midd}\n A.~A.~Middleton, Phys. Rev. Lett. {\\bf 83}, 1672 (1999).\n\n\\bibitem{dsf}\n D.~S.~Fisher, private communication.\n\n\\bibitem{kawa}\n N.~Kawashima, J. Phys. Soc. Jpn. {\\bf 69}, 987 (2000).\n\n\\bibitem{hm}\n J. Houdayer and O. C. Martin, Europhys. Lett. {\\bf 49}, 794 (2000).\n\n\\bibitem{vb}\n L. Viana and A.~J. Bray, J. Phys. C {\\bf 18}, 3037 (1985).\n\n\\bibitem{theta-3d}\n A.~K.~Hartmann, Phys. Rev. E {\\bf 59}, 84 (1999). \n A.~J.~Bray and M.~A.~Moore, J. Phys. C, {\\bf 17}, L463 (1984);\n W.~L.~McMillan, Phys. Rev. B {\\bf 30}, 476 (1984).\n\n\\bibitem{theta-4d}\n A.~K.~Hartmann, Phys. Rev. E {\\bf 60}, 5135 (1999).\n K.~Hukushima, Phys. Rev. E {\\bf 60}, 3606 (1999).\n\n\\bibitem{km}\n F.~Krzakala and O.~C.~Martin cond-mat/0002055.\n\n\\bibitem{pal}\n K.~F.~Pal, Physica A {\\bf 223}, 283 (1996); K.~F.~Pal, Physica A {\\bf\n 233}, 60 (1996).\n\n\\bibitem{drossel}\n B.~Drossel, H.~Bokil, M.~A.~Moore and A.~J.~Bray, Eur. Phys. J. {\\bf 13}, 369 (2000).\n\n%\\bibitem{finding}\n%Our finding that $[q]\\av \\to 1$ is\n%consistent with a non-trivial $P(q)$\n%because $d_s < d$, so the energy gained\n%by exciting the cluster, which comes from the surface, vanishes as\n%$L^{-(d-d_s)}$. Hence,\n%the {\\em probability} of creating the excitation \n%vanishes with this power of $L$, even though its energy is finite.\n\n\\end{references}\n\n\\end{document}\n\n"
}
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[
{
"name": "cond-mat0002134.extracted_bib",
"string": "\\bibitem{fh}\n D.~S.~Fisher and D.~A.~Huse, J. Phys. A. {\\bf 20} L997 (1987); D.~A.~Huse\n and D.~S.~Fisher, J. Phys. A. {\\bf 20} L1005 (1987); D.~S.~Fisher and\n D.~A.~Huse, Phys. Rev. B {\\bf 38} 386 (1988).\n\n\n\\bibitem{bm}\n A.~J.~Bray and M.~A.~Moore, in {\\em Heidelberg Colloquium on Glassy\n Dynamics and Optimization}\\/, L.~Van~Hemmen and I.~Morgenstern eds.\n (Springer-Verlag, Heidelberg, 1986).\n\n\n\\bibitem{mcmillan}\n W.~L.~McMillan, J. Phys. C {\\bf 17} 3179 (1984).\n\n\n\\bibitem{ns}\n C.~M.~Newman and D.~L.~Stein, Phys. Rev. B {\\bf 46}, 973 (1992); Phys. Rev.\n Lett., {\\bf 76} 515 (1996);\n Phys. Rev. E {\\bf 57} 1356 (1998).\n\n\n\\bibitem{parisi}\n G.~Parisi, Phys. Rev. Lett. {\\bf 43}, 1754 (1979); J. Phys. A {\\bf 13},\n 1101, 1887, L115 (1980; Phys. Rev. Lett. {\\bf 50}, 1946 (1983).\n\n\n\\bibitem{mpv}\n M.~M\\'ezard, G.~Parisi and M.~A.~Virasoro, {\\em Spin Glass Theory and\n Beyond} (World Scientific, Singapore, 1987).\n\n\n\\bibitem{by}\n K.~Binder and A.~P.~Young, Rev. Mod. Phys. {\\bf 58} 801 (1986).\n\n\n\\bibitem{qlink}\n E.~Marinari, G.~Parisi, F.~Ricci-Tersenghi, J.~Ruiz-Lorenzo and F.~Zuliani,\n J. Stat. Phys. {\\bf 98}, 973 (2000).\n\n\n\\bibitem{py1}\n M.~Palassini and A.~P.~Young, Phys. Rev. {\\bf B60}, R9919, (1999). \n\n\n\\bibitem{py2}\n M.~Palassini and A.~P.~Young, Phys. Rev. Lett. {\\bf 83}, 5216 (1999).\n\n\n\\bibitem{py3}\n M.~Palassini and A.~P.~Young, cond-mat/9910278.\n\n\n\\bibitem{midd}\n A.~A.~Middleton, Phys. Rev. Lett. {\\bf 83}, 1672 (1999).\n\n\n\\bibitem{dsf}\n D.~S.~Fisher, private communication.\n\n\n\\bibitem{kawa}\n N.~Kawashima, J. Phys. Soc. Jpn. {\\bf 69}, 987 (2000).\n\n\n\\bibitem{hm}\n J. Houdayer and O. C. Martin, Europhys. Lett. {\\bf 49}, 794 (2000).\n\n\n\\bibitem{vb}\n L. Viana and A.~J. Bray, J. Phys. C {\\bf 18}, 3037 (1985).\n\n\n\\bibitem{theta-3d}\n A.~K.~Hartmann, Phys. Rev. E {\\bf 59}, 84 (1999). \n A.~J.~Bray and M.~A.~Moore, J. Phys. C, {\\bf 17}, L463 (1984);\n W.~L.~McMillan, Phys. Rev. B {\\bf 30}, 476 (1984).\n\n\n\\bibitem{theta-4d}\n A.~K.~Hartmann, Phys. Rev. E {\\bf 60}, 5135 (1999).\n K.~Hukushima, Phys. Rev. E {\\bf 60}, 3606 (1999).\n\n\n\\bibitem{km}\n F.~Krzakala and O.~C.~Martin cond-mat/0002055.\n\n\n\\bibitem{pal}\n K.~F.~Pal, Physica A {\\bf 223}, 283 (1996); K.~F.~Pal, Physica A {\\bf\n 233}, 60 (1996).\n\n\n\\bibitem{drossel}\n B.~Drossel, H.~Bokil, M.~A.~Moore and A.~J.~Bray, Eur. Phys. J. {\\bf 13}, 369 (2000).\n\n%\n\\bibitem{finding}\n%Our finding that $[q]\\av \\to 1$ is\n%consistent with a non-trivial $P(q)$\n%because $d_s < d$, so the energy gained\n%by exciting the cluster, which comes from the surface, vanishes as\n%$L^{-(d-d_s)}$. Hence,\n%the {\\em probability} of creating the excitation \n%vanishes with this power of $L$, even though its energy is finite.\n\n"
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cond-mat0002135
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Non-equilibrium effects in transport through quantum dots
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[
{
"author": "E. Bascones$^1$"
},
{
"author": "C. P. Herrero$^1$"
},
{
"author": "F. Guinea$^1$ and Gerd Sch\\\"on$^{2,3}$"
}
] |
The role of non-equilibrium effects in the conductance through quantum dots is investigated. Associated with single-electron tunneling are shake-up processes and the formation of excitonic-like resonances. They change qualitatively the low temperature properties of the system. We analyze by quantum Monte Carlo methods the renormalization of the effective capacitance and the gate-voltage dependent conductance. Experimental relevance is discussed.
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[
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"name": "version9feb.tex",
"string": "% version 27 January 00\n\\tolerance = 10000\n\\documentstyle[aps,epsf,twocolumn]{revtex}\n\\input epsf.sty\n\\begin{document}\n\\draft\n\\flushbottom\n\\twocolumn[\n\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname\n\n\\title{Non-equilibrium effects in transport through quantum dots}\n\\author{E. Bascones$^1$, C. P. Herrero$^1$,\nF. Guinea$^1$ and Gerd Sch\\\"on$^{2,3}$}\n\\address{\n$^1$Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones\nCient{\\'\\i}ficas,\nCantoblanco, E-28049, Madrid, Spain \\\\\n$^2$Institut f\\\"ur Theoretische Festk\\\"orperphysik, Universit\\\"at\nKarlsruhe, 76128 Karlsruhe, Germany\\\\\n$^3$ Forschungszentrum Karlsruhe, Institut f\\\"ur Nanotechnologie,\n76021 Karlsruhe, Germany}\n\\date{\\today}\n\\maketitle\n\\tightenlines\n\\widetext\n\\advance\\leftskip by 57pt\n\\advance\\rightskip by 57pt\n\n\n\\begin{abstract}\nThe role of non-equilibrium effects in the conductance through \nquantum dots is investigated. Associated with\nsingle-electron tunneling are shake-up processes and \nthe formation of excitonic-like resonances. They change qualitatively the low\ntemperature properties of the system. We analyze by quantum Monte\nCarlo methods the renormalization of the effective capacitance and the\ngate-voltage dependent conductance. Experimental relevance is discussed. \n\\end{abstract}\n\\pacs{PACS numbers: 73.40Gk, 73.23.Hk}\n\n]\n\\narrowtext\n\\tightenlines\n\\section{Introduction.}\nQuantum dots are paradigms to study the transition\nfrom macroscopic to microscopic physics. At present, the role\nof single-electron charging is well understood\\cite{SET}.\nProcesses which otherwise are found in solids at the\nsingle-atom level, such as\nthe Kondo effect, are being currently investigated\\cite{Getal98}. \nOther atomic features,\nlike the existence of high spin ground states have also\nbeen observed\\cite{Oetal98}. The existence of many internal degrees of\nfreedom within the dot leads to a variety of effects reminiscent\nof those found in large molecules. Shake-up processes, associated with\nthe rearrangement of many electronic levels upon the addition\nof one electron, have been reported\\cite{Setal97,Oetal99,Aetal97}.\n%as well as the formation of bonding resonances in double well \n%systems\\cite{Vetal95,Betal98,Oetal98b}.\n\nIn the present work, we will show that internal\nexcitations of the dot lead to non-equilibrium effects which\ncan substantially modify the transport properties.\nIn sufficiently small dots, the addition of a single electron may\ncause significant charge rearrangements\\cite{Setal96}, \n%Resonances associated to shake-up processes of the internal degrees\n%of freedom of the dot were reported in\\cite{Aetal97}. \nand the resulting change in the electrostatic potential of the dot\nmodifies the electronic level structure. \nIn the limit when the level separation is much smaller than other\nrelevant scales, this process leads to an \n^^ ^^ orthogonality catastrophe\"\\cite{A67}, first discussed in\nrelation with the sudden switching of a local potential in\na bulk metal. In addition, an electron tunneling event \nchanging the charge of the dot is associated with\na charge depletion in the leads or in neighboring\ndots. The attraction between the electron in the dot and the\ninduced positive charge leads to the formation of\nan excitonic resonance, similar to the well known\nexcitonic effects in X-ray absorption\\cite{ND69,M91}.\n%A qualitative representation of these processes are shown in\n%fig.[\\ref{fig.sketch}]. \n\nThe relevance of these effects for\ntunneling processes in mesoscopic systems was first discussed\nin refs.~\\cite{UK90,UG91} (see also\\cite{GBC98}). \nThe existence of excitonic-like resonances\nhas indeed been reported in mesoscopic systems\\cite{MVM97}.\nThe formation of bonding resonances has been observed \nin double well systems\\cite{Vetal95}. \nNon-equilibrium effects like those studied here \nhave also been discussed in Ref.~\\cite{ML92} \nwhere transport through a tunnel junction \nvia localized levels due to impurities was analyzed, and have been \nobserved experimentally in Ref.~\\cite{Getal94}.\n% In this case charging effects do not need to be taken into account. \n% An excitonic resonance appears due to the Coulomb interaction \n% between the electron which tunnels to the localized level \n% and the conduction electrons in the leads. \nA different\nway for modifying the conductance of quantum dots through the formation\nof excitonic states has been proposed in\\cite{PAK99}. In that case the\nexciton is a real bound state, while the excitonic mechanism discussed\nhere is a dynamical process. \nIn some devices, the charge rearrangement may\ntake place far from the tunneling region. In this case \nno excitons are formed but the orthogonality catastrophe persists. \nThis process has been discussed in relation to the measurement\nof the charge in a quantum dot by the current through a\nneighboring point contact\\cite{Yetal95}. \n\n\nWe will study the simplest\ndeviations from the standard Coulomb blockade regime. Our analysis\nis valid for quantum dots where the spacing between\nelectronic levels, $ \\Delta \\epsilon$, is much smaller than the\ncharging energy, $E_C$, or the temperature $T$. The non-equilibrium effects\ndiscussed here will be observable if, in addition, the number of\nelectrons in the dot is not too large so that changes in the charge \nstate lead to non-uniform redistributions of the charge. \nThus, we will consider an intermediate situation between the\nKondo regime and orthodox Coulomb blockade (see below for estimates).\nThe processes that we consider will be present in double-dot\ndevices, but, for definiteness, we study here a single dot.\n\nThe main features discussed above can be described by a generalization of\nthe dissipative quantum rotor model\\cite{UG91}, which has been\nstudied widely in connection with conventional Coulomb blockade\nprocesses\\cite{SZ90}. We present\na detailed numerical analysis of the generalized model,\nalong similar lines as previous work by some of us\non the dissipative quantum rotor\\cite{HSZ99}.\n\nThe paper is organized as follows:\nIn the next section, we show how to\nestimate the parameters which characterize non-equilibrium effects.\nThe model is reviewed in section III, with emphasis on details\nneeded for the subsequent calculations. The numerical method is\npresented in section IV.\nIn Secs. V and VI, we give results\nfor the renormalized capacitance and conductance.\nIn Sec. VII we discuss some possible experimental evidence of the effects studied here.We close with some conclusions.\n\n\n\\section{Non-equilibrium effects.}\n\\subsection{Inhomogeneous charge redistribution.}\nThe standard Coulomb blockade model assumes that, upon a\nchange in the charge state of the dot,\nthe electronic levels within \na quantum dot are rigidly shifted by the charging energy, $E_C=e^2/2C$ with $C$ the capacitance of the dot . \nDeviations from this assumption\nhave been studied by means of an expansion in terms of $g^{-1}$, \nthe inverse dimensionless\nconductance, $g \\sim k_{\\rm F} l$, where $k_{\\rm F}$ is the Fermi wave-vector,\nand $l$ is the mean free path\\cite{BMM97}. It is also assumed that\n$k_{\\rm F}$ is small compared to the inverse Thomas-Fermi screening length,\n$k_{\\rm TF} = \\sqrt{4 \\pi e^2 N ( \\epsilon_{\\rm F} )/ \\epsilon_0}$.\nTo lowest order beyond standard Coulomb blockade effects, the change\nin the charge state of the dot \nleads to an inhomogeneous potential, and induces a \nterm in the Hamiltonian, which can be written as\\cite{BMM97,G00}:\n\\begin{equation}\n{\\cal H}_{\\rm int} = ( Q - Q_{\\rm offset} )\n\\int \\psi^{\\dag} ( {\\bf \\vec{r}} ) U ( {\\bf \\vec{r}} ) \n\\psi ( {\\bf \\vec{r}} ) d^d {\\bf r} \\; .\n\\label{potential}\n\\end{equation}\nHere $Q_{\\rm offset}$ denotes offset charges in the\nenvironment, the operator $Q$ measures\nthe total electronic charge in the dot,\nand $\\psi^\\dag ( \\bf{\\vec{r}} )$ creates an electron \nat position $\\bf{\\vec{r}}$.\nThe potential $U({\\bf \\vec{r}} )$ modifies \nthe constant shift of the energy levels\nof the dot assumed in the standard Coulomb blockade model. \nIt appears due to the restricted geometry of the dot. \nAfter a charge tunnels into the dot, a pile-up of electrons\nat the surface of the dot is induced. As a result there is a net attraction of\nelectrons towards the surface, besides a constant shift given by\n$e^2 / 2C$. In general, the potential $U$ in eq.~(\\ref{potential}) can be\nobtained from the Hartree approximation (in the Thomas-Fermi limit)\nfor dots and leads of arbitrary shape. For a\nspherical dot of radius $R$\nthe potential $U ( \\bf{\\vec{r}} )$ has the simple form\\cite{BMM97}: \n\\begin{equation}\nU ( {\\bf \\vec{r}} ) = - \\frac{e^2 e^{- k_{FT} ( R - r )}}\n{\\epsilon_0 k_{\\rm TF} R^2} + K \n\\end{equation}\nand, for a two-dimensional circular dot:\n\\begin{equation}\nU ( {\\bf \\vec{r}} ) = - \\frac{e^2}{2 \\epsilon_0 k_{\\rm TF} R\n\\sqrt{R^2 - r^2}} + K \\; .\n\\end{equation} \n\nK is a constant which ensures $\\langle U({\\bf \\vec{r}})\\rangle=0$.\nNon-equilibrium effects arise because\nthe potential in eq.~(\\ref{potential}) is time dependent,\nas it changes upon the addition of electrons to the dot.\nHamiltonians with terms such as eq.~(\\ref{potential})\nwere first discussed in\\cite{UG91} (see next section). \nNote that the potential is\nlocalized in the surface region, where the tunneling electron is\nsupposed to land. Finally, the potential is attractive, leading \nto the localization of the new electron near the surface, giving\nrise to excitonic effects.\n\nOther non-equilibrium effects can arise \nif the charge of the dot induces inhomogeneous potentials in other\nmetallic regions of the device. In this case, the only effect\nexpected is the orthogonality catastrophe (see below), due to the shake-up of\nthe electrons both in the dot and in the other regions. We take this\npossibility into account in the analysis in the\nfollowing sections.\n\n\n\\subsection{Effective tunneling density of states.}\n\nIn the absence of non-equilibrium effects, \nthe conductance of a junction between\nthe dot and the leads is \n\\begin{equation}\ng = \\frac{2 e^2}{h} \\sum_{\\rm channels} | t_i |^2 \nN_{i,{\\rm lead}} ( E_{\\rm F} ) N_{i,{\\rm dot}} ( E_{\\rm F} ) \\; ,\n\\label{Bardeen}\n\\end{equation}\nwhere the summation is over the channels, \n$t_i$ is the hopping matrix element through channel $i$,\n$N(E_{\\rm F})$ is the density of states at the Fermi level,\nand we use the standard theory of\ntunneling in the weak transmission limit\\cite{BMS83}.\nEq. (\\ref{Bardeen}) implicitly assumes a constant density\nof states, as appropriate for a metallic contact.\n\nThe non-equilibrium effects to be considered can be taken into account\nthrough a modification of the effective tunneling density of\nstates\\cite{UG91,DRG98}. In this case the electron propagators in\neq.~(\\ref{Bardeen}) are the non-equilibrium ones, \nin an analogous way to the modifications\nrequired in the study of X-ray absorption spectra of core levels in\nmetals\\cite{ND69}, or tunneling between Luttinger \nliquids\\cite{KF92,MG93,SK97}.\nAs in those problems, we can distinguish two cases:\n\ni) The analogue of the X-ray absorption process: The charging of the dot\nleads to an effective potential which modifies the electronic levels.\nAt the same time, an electronic state localized in a region within the\nrange of the \npotential is filled. The interaction between the electron\nin this state and the induced\npotential must be taken into account (the excitonic effects, in the\nlanguage of the Mahan-Nozi\\`eres-de Dominicis theory).\n\nii) The analogue of X-ray photoemission: The charging process leads\nto a potential which modifies the electronic levels. The tunneling\nelectron appears in a region outside the range of this potential.\nOnly the orthogonality effect caused by the potential needs to be\nincluded.\n\nTaking into account the distinctions between these two possibilities,\nthe effective (non-equilibrium) \ndensity of states in the lead and the dot becomes\n(omitting the channel index, $i$):\n%\\begin{eqnarray}\n%D_{\\rm eff} ( \\omega ) &= &\n\\begin{eqnarray} \nD_{\\rm eff} ( \\omega )& = &\\int_0^\\omega d \\omega' N_{\\rm dot}^{\\rm empty} (\n\\omega' )N_{\\rm lead}^{\\rm occ} ( \\omega - \\omega' ) \\nonumber \\\\ \n& \\propto & |\\omega |^{1-\\epsilon} \n\\label{effdosnew}\n\\end{eqnarray}\nwith $\\epsilon$ given by \n\\begin{eqnarray}\n\\epsilon = \\left\\{ \\begin{array}{ll} \n \\sum_{j=1,2} 2 \\frac{\\delta_j}{\\pi} - \\left(\n\\frac{\\delta_j}{\\pi} \\right)^2 & \n\\rm{\\left( excitonic \\right.} \\\\ &\\rm{\\left. resonance \\right)} \\\\\n -\\sum_{j=1,2} \\left(\n\\frac{\\delta_j}{\\pi} \\right)^2 & \n\\rm{\\left( orthogonality \\right.} \\\\ & \\rm{\\left.\ncatastrophe \\right)} \\end{array} \\right.\n\\label{effdos1}\n\\end{eqnarray} \n%&\\left\\{ \\begin{array}{cc}\n%| \\omega |^{1 + \\sum_{j=1,2} - 2 \\frac{\\delta_j}{\\pi} + \\left(\n%\\frac{\\delta_j}{\\pi} \\right)^2} &\n%\\rm{\\left( excitonic \\right.} \\\\ &\\rm{\\left. resonance \\right)} \\\\\n%| \\omega |^{1 + \\sum_{j=1,2} \\left(\n%\\frac{\\delta_j}{\\pi} \\right)^2} \n%&\\rm{\\left( orthogonality \\right.} \\\\ & \\rm{\\left.\n%catastrophe \\right)} \\end{array} \\right.\n%\\label{effdos1}\n%\\end{eqnarray}\nHere $\\delta_j$ is the phase shift induced by the \nnew electrostatic potential in the lead states ($j=1$) or in the\ndot states ($j=2$). \nThe exponent is positive, $\\epsilon > 0$, if excitonic effects prevail, while\n$\\epsilon < 0$ if the leading process is the orthogonality catastrophe.\n%It is straightforward to extend this analysis to a double dot system.\n\nWe can get an accurate estimate for $\\epsilon$ in the simple cases\nof a spherical or circular quantum dot decoupled from other metallic\nregions discussed in\\cite{BMM97}. We assume that tunneling\ntakes place through a single channel, and the contact is\nof linear dimensions $\\propto k_{\\rm F}^{-1}$.\nIn Born approximation the effective phase shift becomes:\n\\begin{equation}\n\\delta \\approx N ( \\epsilon_{\\rm F} ) \\int_{\\Omega} U ( {\\bf \\vec{r}} )\nd^d {\\bf r} \\approx \\left\\{ \\begin{array}{ll}\n\\frac{e^2 N ( \\epsilon_{\\rm F} )}{\\epsilon_0 k_{FT} k_{\\rm F}^3 R^2}\n&\\rm{spherical \\, \\, dot} \\\\ \n\\frac{e^2 N ( \\epsilon_{\\rm F} )}{\\epsilon_0 k_{FT} k_{\\rm F} R} \n&\\rm{circular \\, \\, dot} \\\\ \\end{array} \\right.\n\\end{equation} \nwhere $\\Omega$ is the region where tunneling processes to the leads\nare non negligible, typically\nof dimensions comparable to $k_F^{-1}$.\nNote that, for a very elongated dot (d=1), the phase shift will\nnot depend on its linear size. As mentioned earlier, the leads\ncan modify significantly these estimates. The tunneling electron can \nbe attracted to the image potential that it induces, enhancing\nthe excitonic effects ($\\epsilon > 0$). On the other hand, shake-up\nprocesses in metallic regions decoupled from the tunneling processes \nwill increase the orthogonality catastrophe, without contributing\nto the formation of the excitonic resonance at the Fermi energy.\n\n\\section{The model.}\nThe shake-up processes mentioned in the preceding section\nare described by the Hamiltonian\\cite{UG91}:\n\\begin{eqnarray}\n{\\cal H} &= &{\\cal H}_Q + {\\cal H}_{\\rm R} + {\\cal H}_{\\rm L} +\n{\\cal H}_{\\rm T} + {\\cal H}_{\\rm int} \\nonumber \\\\\n{\\cal H}_Q &= &\\frac{( Q - Q_{\\rm offset} )^2}{2 C} \\nonumber \\\\\n{\\cal H}_i &= &\\sum_k \\epsilon_{k,i} c^\\dag_{k,i} c_{k,i}\n\\;\\; , \\;\\; i = {\\rm L,R} \\nonumber \\\\\n{\\cal H}_{\\rm T} &= &t e^{i \\phi} \\sum_{k,k'} c^\\dag_{k,{\\rm R}}\nc_{k',{\\rm L}} + h. c. \n\\nonumber \\\\ \n{\\cal H}_{\\rm int} &= &( Q - Q_{\\rm offset} )\n \\sum_{k,k'} \\left( V^R_{k,k'} c^\\dag_{k,{\\rm R}} c_{k',{\\rm R}}\n- V^L_{k,k'} c^\\dag_{k,{\\rm L}} c_{k',{\\rm L}} \\right) \\nonumber \\\\\n\\label{Hamiltonian}\n\\end{eqnarray}\n Here $\\left[ \\phi , Q \\right] = i e$. The Hamiltonian separates the junction degrees of freedom into\na collective mode, the charge $Q$, and the electron degrees of freedom\nof the electrodes and the dot. This separation is standard\nin analyzing electron liquids, where collective charge oscillations\n(the plasmons) are treated separately from the low-energy electron-hole\nexcitations. In our case, this implies that only those states\nwith energies lower than the charging energy are to be included\nin ${\\cal H}_i$, ${\\cal H}_T$ and ${\\cal H}_{int}$\nin eq.(\\ref{Hamiltonian}). Higher electronic states\ncontribute to the dynamics of the charge, described by\n${\\cal H}_Q$. The Hamiltonian, eq.(\\ref{Hamiltonian}), suffices\nto describe transport processes at voltages and temperatures\nsmaller than the charging energy.\n\nIn the following, we will express the offset charge $Q_{\\rm offset}$,\nintroduced in eq.~(\\ref{potential}),\nby the dimensionless parameter $n_e = Q_{offset} / e$.\nBy $V_{k,k'}$ we denote the \nmatrix elements of $U ( {\\bf \\vec{r}} )$ in the basis of the\neigenfunctions near the Fermi level. We allowed that inhomogeneous\npotentials can be generated on both sides of the junction. We assume that tunneling can take place through several channels (index channel has been omitted). The transmission through each channel should be small for perturbation theory to apply.\n\n\nThe electrical relaxation associated with the tunneling process takes place\nin two stages. In the first, the tunneling electron is screened by the \nexcitation of plasmons, forming the screened Coulomb potential. The\ntime scale for this process is of the order of the inverse plasma\nfrequency. Next, the screened Coulomb \npotential excites electron-hole pairs. As the electrons at the Fermi level have a much \nlonger response time, they feel this change as a sudden and local perturbation. \n\nWe will restrict ourselves to the regime where the level spacing is\nsmall, $\\Delta\\epsilon \\ll T,E_c$. \nUsing standard techniques\\cite{SZ90,AES82}, we can integrate out the\nelectron-hole pairs \nand describe the system in terms of the phase $\\phi$ and charge $Q$ \nonly. \nThis procedure leads to retarded interactions which\nare long-range in time, as the electron-hole pairs have a\ncontinuous spectrum \ndown to zero energy. It is best to describe the resulting model\nwithin a path-integral formalism. Because of the non-equilibrium effects \nthe effective action is\na generalization of that derived for tunnel junctions\\cite{BMS83,AES82}\n\\begin{eqnarray}\n{\\cal S}[\\phi] &=& \\int_0^{\\beta} d \\tau \\frac{1}{4 E_C} \n\\left( \\frac{\\partial \\phi}{\\partial \\tau} \\right)^2 + \\nonumber \\\\\n& &\\alpha \\int_0^{\\beta} d \\tau \\int_0^{\\beta} d \\tau' \n E_C^{\\epsilon} \\left( \\frac{\\pi}{\\beta} \\right)^{2-\\epsilon}\n\\frac { 1 - \\cos [ \\phi ( \\tau ) - \\phi ( \\tau' ) ] }\n{\\sin^{2 - \\epsilon} [ \\pi ( \\tau - \\tau' ) / \\beta ]} \\; . \\nonumber \\\\\n\\label{action}\n\\end{eqnarray}\nIt describes the low energy processes below\nan upper cutoff of order of the unscreened charging energy,\n$E_C$.\nThe parameter \n$\\alpha \\propto t^2 N_R ( \\epsilon_{\\rm F} ) N_L ( \\epsilon_{\\rm F} )$\nis a measure of the high temperature conductance, \n$g_0$, in units of $e^2 / h$:\n\\begin{equation}\n\\alpha = \\frac{g_0}\n{4 \\pi^2 ( e^2 / h )} \\; .\n\\end{equation}\nNote that the definition of the action, eq.~(\\ref{action}), does not\nallow us to study temperatures much higher than $E_C$.\nThe kernel which describes the retarded\ninteraction is given by the effective\ntunneling density of states, eq. (\\ref{effdosnew}).\nThe value of $\\epsilon$ is the anomalous exponent\nin the tunneling density of states, given in eq. (\\ref{effdosnew}).\n\nThe action (\\ref{action}) has been studied extensively for\n$\\epsilon = 0$\\cite{SZ90,HSZ99,FSZ95,WEG97}, describing charging\neffects in the single-electron transistor in the usual limit where\nthe electrodes are assumed to be in equilibrium. If\n$\\epsilon > 0$, the model \nhas a non-trivial phase transition\\cite{K77,GS86,SG97}. In this case, for\n$\\alpha > \\alpha_{crit} \\sim 2 / ( \\pi^2 \\epsilon )$, the system \ndevelops long-range order when $T = 1/\\beta \\rightarrow 0$, leading to\nphase coherence and a diverging conductance.\n\n\n\\section{Computational method}\n\nFor a given offset charge $n_e$, the grand partition \nfunction can be written in terms of the phase $\\phi$, as\na path integral \\cite{SZ90}:\n\\begin{equation}\n Z(n_e) = \\sum_{m=-\\infty}^{\\infty} \\exp(2 \\pi i m n_e) \\, \n\t \\int {\\cal D} \\phi \\, \\exp \\left( - S[\\phi] \\right)\n \\hspace{.2cm} ,\n \\label{part1}\n\\end{equation}\nwhere $m$ is the winding number of $\\phi$,\nand the paths $\\phi(\\tau)$ satisfy in sector $m$ the\n boundary condition\n$\\phi(\\beta) = \\phi(0) + 2 \\pi m$.\n\nThe effective action and partition function can be rewritten\n in terms of the phase fluctuations \n$ \\theta(\\tau) = \\phi(\\tau) - 2 \\pi m \\tau / \\beta$,\n with boundary condition\n$\\theta(\\beta) = \\theta(0)$, in the form\n\\begin{equation}\nZ = \\sum_{m=-\\infty}^{\\infty} \n \\exp(2 \\pi i m n_e) \\, I_m(\\alpha,\\epsilon,\\beta) \\hspace{.2cm} .\n \\label{part2}\n\\end{equation}\nThe coefficients $I_m(\\alpha,\\epsilon,\\beta) = \\int {\\cal D} \\theta \n \\, \\exp \\left( - S_m[\\theta ] \\right) $\n are to be evaluated with the effective action\n$ S_m[\\theta(\\tau)] = S[\\theta(\\tau) + 2 \\pi m \\tau / \\beta] $. \nThey depend on the winding number\n$m$, the temperature, and the dimensionless parameters $\\alpha$\nand $\\epsilon$, but are independent of the offset charge $n_e$.\nThis means that the problem reduces, from a computational point of view,\n to the calculation of the relative values of\n$I_m(\\alpha,\\epsilon,\\beta)$, which can be obtained from \nMonte Carlo (MC) simulations apart from an overall normalization\nconstant \\cite{HSZ99,WEG97}.\nThe partition function is even and periodic with respect to $n_e$,\n$Z(n_e) = Z(-n_e) = Z(n_e + 1)$, and therefore\none can restrict the analysis to the range $0 \\le n_e \\le 0.5$.\n\n\nThe MC simulations have been carried out by the usual\ndiscretization of the quantum\npaths into $N$ (Trotter number) imaginary-time slices \\cite{S93}.\nIn order to keep roughly the same precision in the calculated\n quantities, as the temperature is reduced, the number of time-slices\n$N$ has to increase as $1 / T$.\n We have found that a value $N = 4 \\beta E_C$ is sufficient to\n reach convergence of $I_m$.\n Therefore, the imaginary-time slice employed in the discretization of the\n paths is $\\Delta \\tau = \\beta / N = 1 / (4 E_C) $. \n\nWhen discretizing the paths $\\phi(\\tau)$ into $N$ points,\n it is important to treat correctly the \n$| \\tau - \\tau' | \\rightarrow 0$ divergence that appears in\n the tunneling term $S_t[\\phi]$ of the effective action \n[second term on the r.h.s. of Eq.\\,(\\ref{action})].\nThis divergence can be handled as follows. \n In the discretization procedure, the double integral\nin $S_t[\\phi]$ translates into a sum extended\nto $N^2$ two-dimensional plaquettes, each one with area\n$(\\Delta \\tau)^2$. The above-mentioned divergence appears\nin the $N$ ``diagonal'' terms ($\\tau = \\tau'$) and can be dealt\nwith by approximating the integrand close to $\\tau = \\tau'$ by \n$ E_C^{\\epsilon} |\\tau - \\tau'|^{\\epsilon} (d \\phi / d \\tau)^2 / 2$.\nThus, by integrating this expression over the ``diagonal'' plaquette\n$(i,i)$, with $1 \\le i \\le N$, one finds that its contribution\nto $S_t[\\phi]$ is given by\n\\begin{equation}\n \\Delta S_t (\\tau_i,\\tau_i) = 2 \\, E_C^{\\epsilon}\n \\, \\frac{\\alpha}{\\epsilon + 1} \n \\left( \\frac {\\Delta \\tau}{2} \\right)^{\\epsilon + 2}\n \\left( \\frac {d \\phi}{d \\tau} \\right)^2_{\\tau = \\tau_i}\n \\hspace{.2cm} ,\n \\label{aprox2}\n\\end{equation} \nwhich is regular for $\\epsilon \\neq -1$. The error introduced by\nthis replacement in the discretization procedure is of the same\norder as that introduced by the usual discretization of the\n``non-diagonal'' terms. We have checked that the results\nof our Monte Carlo simulations obtained by using this procedure \nconverge with the Trotter number $N$.\n\nThe partition function in Eq.(\\ref{part2}) has been sampled by\nthe classical Metropolis method \\cite{BH88}\nfor temperatures down to $k_{\\rm B} T = E_C / 200$.\nA simulation run proceeds via successive MC steps.\nIn each step, all path-coordinates (imaginary-time slices) \nare updated. For each set of parameters ($\\alpha, \\epsilon, T$),\nthe maximum distance allowed\nfor random moves was fixed in order to obtain an acceptance ratio\nof about $50 \\%$. Then, we chose a starting configuration for \nthe MC runs after system equilibration during about $3 \\times10^4$ MC\nsteps. Finally, ensemble-averaged values for the quantities\nof interest were calculated from samples of $\\sim 1 \\times10^5$\nquantum paths.\nMore details on this kind of MC simulations can be found \nelsewhere \\cite{HSZ99}.\n\n\n\\section{Renormalization of the capacitance.}\n\nWe will study the capacitance renormalization for tunneling conductance \n$\\alpha > 0$ by calculating the effective charging energy\n$ E_C^*(T) = e^2 / 2 C^*(T)$, which can be obtained\nas a second derivative of the free energy\n$F = - k_{\\rm B} T \\ln Z$:\n\\begin{equation}\n E_C^*(T) = \\frac{1}{2} \\left. \n \\frac{\\partial^2 F}{\\partial n_e^2} \\right|_{n_e=0}\n \\hspace{.2cm} .\n \\label{ec1}\n\\end{equation}\t\nAt high temperatures, the free energy $F(n_e)$ depends weakly on\n$n_e$, and the curvature [i.e., $E_C^*(T)$] approaches zero.\nAt low $T$, and for weak tunneling ($\\alpha \\ll 1$), it coincides\nwith the usual charging energy $E_C$.\nBy using Eqs.\\,(\\ref{part1}) and (\\ref{ec1}) \nthis renormalized charging energy can readily be expressed as\n\\begin{equation}\nE_C^*(T) =\n2 \\pi^2 k_{\\rm B} T \\langle m^2 \\rangle_{n_e=0} \\;,\n \\label{ec2}\n\\end{equation}\nwhere $\\langle m^2 \\rangle_{n_e=0}$ is the second moment of\n the coefficients $I_m$.\n\nThe correlation function in imaginary time $G(\\tau)$, that\nwill be used below to calculate the conductance, can be\ncalculated from the MC simulations as\n$G(\\tau) = \\langle \\cos[\\phi(\\tau) - \\phi(0)] \\rangle$.\nThis means in our context:\n\\begin{figure}\n\\epsfxsize=\\hsize\n\\centerline{\\epsfbox{eca25.eps}} %\\vspace{-1.5cm}\n\\caption{Effective charging energy of the single\nelectron transistor for different values of $\\epsilon$ and\n$\\alpha = 0.25$, as a function of temperature.}\n\\label{fig:eca25}\n\\end{figure} \n\\begin{eqnarray}\n G(\\tau) & = & \\frac{1}{Z} \\sum_{m=-\\infty}^{\\infty} \\exp(2 \\pi i m n_e) \n \\, \\times \\nonumber \\\\ \n & & \\int {\\cal D} \\phi \\, e^{- S[\\phi]} \\cos[\\phi(\\tau) - \\phi(0)]\n \\hspace{.2cm} .\n \\label{gtau}\n\\end{eqnarray}\n\nA number of features, directly related to the free energy\nof the model given by action (\\ref{action}), are reasonably well\nunderstood for $\\epsilon = 0$. The effective charge induced by an\n arbitrary offset charge, $n_e$, has been extensively analyzed.\n A number of analytical schemes\ngive consistent results in the weak coupling ($\\alpha \\ll 1$) \nregime\\cite{GG98,KS98}. These calculations have been extended\nto the strong coupling limit by numerical methods\\cite{HSZ99}.\n\nIn fig.~(\\ref{fig:eca25}), we present results for the effective\ncharging energy $E^*_C$\nas a function of temperature, and for different values of $\\epsilon$.\nThe value of $E^*_C$ is enhanced for $\\epsilon < 0$, where the orthogonality\ncatastrophe dominates the physics. A positive $\\epsilon$ reduces the\neffective charging energy, and, beyond some critical\nvalue (see discussion below), $E^*_C$\nscales towards zero as the temperature is decreased, showing\nnon-monotonic behavior.\nThe same trend can be appreciated in fig.~(\\ref{fig:ect02}), where the\neffective charging energy at low temperatures is plotted as a function\nof $\\alpha$. \nRenormalization group arguments\\cite{K77,GS86} show that\n$E_C^* ( \\alpha , \\epsilon )$ \nshould go to zero for $\\alpha > \\alpha_{crit} ( \\epsilon )$.\nWe have checked the consistency of this prediction with\nthe numerical results by fitting\nthe values of $E^*_C ( \\alpha )$ at low temperatures by the expression\nexpected from the scaling analysis near the transition\\cite{K77}:\n\\begin{figure}\n\\epsfxsize=\\hsize\n\\centerline{\\epsfbox{ect02.eps}} %\\vspace{-1.5cm}\n\\caption{Effective charging energy of the single\nelectron transistor for different values of $\\epsilon$, as\na function of $\\alpha$ for $T / E_C = 0.02$.}\n\\label{fig:ect02}\n\\end{figure} \n\\begin{equation}\nE_C^* ( \\alpha , \\epsilon ) \n= \\left[ 1 - \\frac{\\alpha}{\\alpha_{crit} ( \\epsilon )} \\right]^{\n\\frac{1}{\\epsilon}}\n\\label{alphacrit}\n\\end{equation}\nIn this expression we use $\\alpha_{crit} ( \\epsilon )$ as the only adjustable\nparameter.\nThe results are shown in fig.~(\\ref{fig:alphacrit}).\nNote that the same equation (\\ref{alphacrit}) can be used to fit\nthe results for $\\epsilon < 0$, if one uses a negative value\nfor $\\alpha_{crit}$. There is no phase transition, however,\nfor $\\epsilon < 0$. \n\\begin{figure}\n\\epsfxsize=\\hsize\n\\centerline{\\epsfbox{alphacrit.eps}} %\\vspace{-1.5cm}\n\\caption{Critical line determined by the RG calculation: $\\alpha_{crit}=2/(\\pi^{2}\\epsilon)$ and values calculated fitting\nthe numerical data for $E^*_C ( \\alpha , \\epsilon )$ to the \nexpression (\\ref{alphacrit}).}\n\\label{fig:alphacrit}\n\\end{figure}\nThe results show that the present calculations are very accurate even for\nrelatively large values of $\\alpha$, where $E^*_C$ converges at very\nlow temperatures. \n\n\\section{Evaluation of the conductance.}\n\\subsection{$\\epsilon = 0$}\nThe conductance of the single-electron\ntransistor is notoriously more difficult to\ncalculate than standard thermodynamic\naverages. It cannot be derived in a simple fashion from the partition\nfunction, and requires the analytical continuation of the response\nfunctions from imaginary to real times or frequencies\\cite{GLT22}.\nHence, there are no comprehensive results valid for the whole\nrange of values of $\\alpha, T/E_C$ and $n_e$. For $\\epsilon = 0$, \nthe conductance $g(T)$ can be written as\\cite{ZS91,IZ92,G99}:\n\\begin{equation}\ng ( T ) = 2 g_0 \\beta \\int_0^\\infty \\frac{d \\omega}{2 \\pi}\n\\frac{\\omega S ( \\omega )}{e^{\\beta \\omega} - 1}\n\\label{conductance}\n\\end{equation}\nwhere $g_0$ is the normal state conductance,\nand $S ( \\omega )$ is related to the correlation function\nin imaginary time\\cite{W93}:\n\\begin{eqnarray}\nG ( \\tau ) &= \n&\\langle e^{i \\phi ( \\tau )} e^{-i \\phi ( 0 )} \\rangle \\nonumber \\\\ \n&= &\\frac{1}{2 \\pi} \\int_0^\\infty d \\omega \\,\n \\frac{e^{(\\beta - \\tau ) \\omega} + \n e^{\\tau \\omega}}{e^{\\beta \\omega} - 1} A ( \\omega )\n\\end{eqnarray}\nand\n\\begin{equation}\nA ( \\omega ) = ( 1 - e^{- \\beta \\omega} ) S ( \\omega )\n\\end{equation}\nIn the previous expressions, the charging energy is the natural cutoff\nfor the energy integrals.\n\nAt high temperatures, $\\beta E_C \\sim 1$, we can expand in eq.\n(\\ref{conductance}):\n\\begin{equation}\ng ( T ) \\approx 2 g_0 \\beta \\int_0^{\\infty} \\frac{d \\omega}{2 \\pi}\n\\left[ \\frac{1}{\\beta} - \\frac{\\beta \\omega^2}{24} + \\frac{7 \\beta^3\n\\omega^4}{5760} + ... \\right] \\frac{e^{(\\beta \\omega )/2} A ( \\omega )}\n{e^{\\beta \\omega} - 1}\n\\end{equation}\nso that:\n\\begin{equation}\ng ( T ) \\approx g_0 \\left[ G \\left( \\frac{\\beta}{2} \\right)\n- \\frac{\\beta^2}{24} G'' \\left( \\frac{\\beta}{2} \\right)\n+ \\frac{7 \\beta^4}{5760} G^{iv} \\left( \\frac{\\beta}{2} \\right)\n+ ... \\right]\n\\end{equation}\n\nAt low temperatures, $\\beta E_C \\gg 1$, the conductance is dominated\nby the low energy behavior of $A ( \\omega )$ or, alternatively,\n$S ( \\omega )$. To lowest order, we expect an expansion\nof the form:\n\\begin{equation}\nS ( \\omega ) \\approx 2 \\pi \\delta ( \\omega - \\omega_0 ) + A | \\omega | + ...\n\\end{equation}\nwhere $\\omega_0$ is an energy of the order of the renormalized\ncharging energy (or zero at resonance, $n_e \\approx 1 / 2$), and\n$A$ is a constant which describes cotunneling processes\\cite{AN90}.\nInserting this expression in eq.~(\\ref{conductance}), we obtain:\n\\begin{equation}\ng ( T ) \\approx g_0 \\frac{2\\beta \\omega_0}{ e^{\\beta \\omega_0} - 1} + g_0\n\\frac{A}{\\pi \\beta^2} \\int_0^{\\infty} dx \\frac{x^2}{e^x -1} + ...\n\\label{glowt}\n\\end{equation}\nwhile, on the other hand:\n\\begin{equation}\nG \\left( \\frac{\\beta}{2} \\right) \\approx 2e^{-\\frac{1}{2} \\beta\n\\omega_0} + \\frac{A}{\\pi} \\left( \\frac{2}{\\beta} \\right)^2 + ...\n\\label{Glowt}\n\\end{equation}\nIf $\\omega_0 = 0$, both $g$ and $g_0 G ( \\beta / 2 )$ have the\nsame limit, as $T \\rightarrow 0$. When $\\omega_0 \\ne 0$ the leading\nterm goes as $T^2$ (cotunneling) in both cases, with prefactors \nequal to $2.404 A / \\pi$ and $4 A / \\pi$, respectively.\n\n From the above discussion of the relation between the high- and \nlow-temperature behavior of $g ( T )$ and $G ( \\beta / 2 )$, we find\nthat the interpolation formula\n\\begin{equation}\ng ( T ) \\approx g_0 G \\left( \\frac{\\beta}{2} \\right)\n\\label{inter}\n\\end{equation}\nshould give a reasonable approximation over the entire range\nof parameters (note that the above discussion is independent\nof the values of $\\alpha$ and $n_e$). Eq. (\\ref{inter})\nis consistent with the the main\nphysical features expected both in the high and low temperature limits,\nat and away from resonance. The advantage of using\n$G ( \\beta / 2 )$ is that it can be computed, to a high\ndegree of accuracy, by standard Monte Carlo techniques, as\nit does not require to continue the results to real times.\nA similar approximation, used to avoid inaccurate analytical\ncontinuations has been applied for bulk systems in Ref.~\\cite{RTMS92}.\n\n\n\nWe show the adequacy of the approximation,\neq.~(\\ref{inter}), by plotting the conductances\nestimated in this way, as a function of the bias charge $n_e$,\nin fig.~(\\ref{fig:conductance}).\n\\begin{figure}\n\\epsfxsize=\\hsize\n\\centerline{\\epsfbox{resone0.eps}} %\\vspace{-1.5cm}\n\\caption{Conductance of the single-electron transistor ($\\epsilon=0$),\nas a function of the dimensionless bias charge $n_e$, for several \nvalues of the coupling $\\alpha$, and two different temperatures.}\n\\label{fig:conductance}\n\\end{figure}\n\\begin{figure}\n\\epsfxsize=\\hsize\n\\centerline{\\epsfbox{maxmine0.eps}} %\\vspace{-1.5cm}\n\\caption{Maximum and minimum values of\nthe conductance of the single-electron transistor ($\\epsilon=0$), as\na function of temperature, for different values of the \ncoupling $\\alpha$.} \n\\label{fig:maxmine0}\n\\end{figure}\nThe minimum ($n_e = 0$) and maximum conductances ($n_e = 1/2$)\nfor different values of $\\alpha$ and temperatures are shown\nin fig.~(\\ref{fig:maxmine0}). \n\\subsection{$\\epsilon \\ne 0$}\nWe now extend the previous approximation to the conductance,\neq.~(\\ref{inter}) to the case $\\epsilon \\ne 0$. The main modification\nin eq.~(\\ref{conductance}) is that a factor $\\omega / ( 1 - e^{-\\beta\n\\omega} )$ within the integral has to be replaced by the\neffective tunneling density of states, $D_{\\rm eff}$, given, at zero\ntemperature, by eq.~(\\ref{effdosnew}). At finite temperatures,\nthe corresponding expression is approximately\n\\begin{equation}\nD_{\\rm eff} ( \\omega ) \\propto \\hbox{max} [ T ( T / E_C )^{- \\epsilon} , \\omega \n( \\omega / E_C )^{- \\epsilon} ] \\; .\n\\end{equation}\n The relevant range in the integrand in\neq.~(\\ref{conductance}) is from $\\omega = 0$ to $\\omega \\approx T$.\nIn the following, we will factor the $\\epsilon$ dependent\npart of the effective density of states, and we write the generalization\nof eq.~(\\ref{effdosnew}) to finite temperatures as:\n\\begin{equation}\nD_{\\rm eff} ( \\omega ) = \\frac{\\omega}{1 - e^{-\\beta \\omega}}\nD_{res} ( \\epsilon , \\omega )\n\\end{equation} \nFinally, when inserting this expression into eq.~(\\ref{conductance}),\nwe make the approximation:\n\\begin{equation}\nD_{res} ( \\epsilon , \\omega ) \\approx \\left( \\frac{T}{E_C} \n\\right)^{-\\epsilon}\n\\end{equation}\nWith this approximation, we can perform the same analysis in the high and\nlow temperature regimes as before, to obtain the interpolation\nformula:\n\\begin{equation}\ng ( T ) \\approx g_0 \\left( \\frac{T}{E_C} \\right)^{-\\epsilon}\nG \\left( \\frac{\\beta}{2} \\right)\n\\end{equation}\nThis expression includes again the relevant physical processes at high\nand low temperatures.\n\nResults for the maximum and minimum values of the conductances,\nfor different values of $\\epsilon$, are presented in fig.~(\\ref{fig:exciton}).\nIn the non phase-coherent regime, at very low temperatures, $T \\ll E^*_C$, the conductance away \nfrom resonance should vary as $g \\propto T^{2 - 2 \\epsilon}$.\nExactly at resonance, $n_e = 1/2$, the conductance\ndiverges as $T^{-\\epsilon}$. \nThe most interesting result is the divergence of the conductance,\nat low temperatures, for $\\epsilon = 0.5$, where the excitonic effects\nare strong enough to drive the system to the phase-coherent phase.\n\n\\begin{figure}\n\\epsfxsize=\\hsize\n\\centerline{\\epsfbox{exciton.eps}} %\\vspace{-1.5cm}\n\\caption{Maximum and minimum values of\nthe conductance of the single-electron transistor,\nfor different values of $\\epsilon$.}\n\\label{fig:exciton}\n\\end{figure}\nThe full conductance, as a function of $n_e$, is shown in \nfig.~(\\ref{fig:exciton1}), for $\\alpha = 0.25$ and $\\epsilon = 0.5$.\nAs mentioned above, for these parameters the system is\nalready in the phase coherent regime.\nThe conductance behaves in a way similar to that in the usual case\n($\\epsilon = 0$), and a peaked structure develops.\nThe absolute magnitude, however, increases as\nthe temperature is lowered. Note that the effective charging energy\nis finite as $T \\rightarrow 0$ (see fig.~(\\ref{fig:eca25})).\nIt is interesting to note that, in this phase with complete\nsuppression of Coulomb blockade effects at low temperatures\n(high values of $\\epsilon$ and high conductances), the\npeak structure appears only for an intermediate range\nof temperatures, and it is washed out at very low temperatures.\n\\begin{figure}\n\\epsfxsize=\\hsize\n\\centerline{\\epsfbox{resa25e5.eps}} %\\vspace{-1.5cm}\n\\caption{Conductance, as a function of $n_e$ and temperature,\nof a single-electron transistor with $\\alpha = 0.25$ and $\\epsilon = 0.5$.}\n\\label{fig:exciton1}\n\\end{figure}\nIn fig.~(\\ref{fig:exciton2}) we show the conductance as\na function of $\\epsilon$ for a fixed temperature. It is evident the increase of the \nconductance as $\\epsilon$ increases. \n\\begin{figure}\n\\epsfxsize=\\hsize\n%\\centerline{\\epsfbox{exciton2.eps}} %\\vspace{-1.5cm}\n\\centerline{\\epsfbox{resa25t03.eps}} \n\\caption{Conductance, as a function of $n_e$ ,\nof a single-electron transistor with $\\alpha = 0.25$ and different\nvalues of $\\epsilon$ ($T = 0.03 E_C$).}\n\\label{fig:exciton2}\n\\end{figure}\n\n\n\\section{Discussion.} \nIn the following, we discuss\nsome experimental evidence which can be explained within \nthe model discussed here. \n\nIt has been pointed out\\cite{BG99} that the correlations between\nthe conductances for neighboring charge states of a quantum dot\\cite{Cetal97}\nare too weak to be explained using standard methods for\ndisordered, non-interacting systems. The experiments reported\nin\\cite{Cetal97} are in the cotunneling regime. In the\npresence of non-equilibrium processes, we expect a behavior\nof the type $g \\propto T^{2 - 2 \\epsilon}$. \nNote that $\\epsilon$ is determined by phase shifts, which depend on\nmicroscopic details of the contacts. Thus, it can be expected to vary with\nthe charge state of the dot, and to lead to large differences in\nthe conductances of neighboring valleys. \n\nIt has been shown that the temperature dependence of the conductance quantum dots,\naway from resonances,\ncan be opposite to that expected in a system exhibiting\nCoulomb blockade\\cite{Metal99}. This effect could not be\nattributed to Kondo physics, as the data do not show an even-odd\nalternation. The reported behavior can be explained within our\nmodel, assuming that the value of $\\epsilon$ is sufficiently large,\nand dependent on the charge state of the dot.\n\n\nIn ref.\\cite{Fetal98} the inelastic contribution to the conductance \nin a double dot sytem, where the electronic states in the two\ndots are separated by an energy $\\epsilon$ is measured.\nThe result is approximately given by\n$I_{\\rm inel} ( \\epsilon ) \\propto \\epsilon^{\\lambda}$, \nwhere $\\lambda$ is a negative constant of\norder unity. \nTaking into account only one\nelectronic state within each dot, the problem can be reduced\nto that of a dissipative, biased two level system. \nThe inelastic conductance reflects the nature of the low\nenergy excitations coupled to the two level system. The observed power\nlaw decay implies that the spectral strength of the coupling, that is\nthe function $J ( \\omega )$ in the standard literature\\cite{W93}, should\nbe ohmic, $J ( \\omega ) \\propto | \\omega |$.\nThis has led to the proposal that the excitations coupled to\nthe charges in the double dot system are piezoelectric phonons\\cite{Fetal98,BK99}.\nIt is interesting to note that the excitation of electron-hole\npairs leads also to an ohmic spectral function.\nThus, at sufficiently low energies,\nan orthogonality catastrophe due to electron-hole pairs\nshows a behavior\nindistinguishable from that arising from piezoelectric phonons.\nThe contributions from the two types of excitations can be distinguished\nat the natural cutoff scale for phonons, which\nis the energy of a phonon whose wavelength is of the order of the \ndimensions of the device. On the other hand, the simplest \nprediction for the expected behavior of the current induced by\nthe emission of the electron-hole pairs is:\n\\begin{equation}\nI ( \\epsilon ) \n\\propto K \\frac{1}{1 - e^{- \\beta \\sqrt{\\Delta_0^2 + \\epsilon_b^2}}}\n\\frac{\\Delta_0^2}{\\sqrt{\\Delta_0^2 + \\epsilon_b^2}}\n\\label{rate}\n\\end{equation}\nwhere $\\Delta_0$ is the tunneling element between the two dots,\n$\\epsilon_b$ is the bias, and $K$ is the coupling constant\n(referred to as $\\epsilon$ in other sections of this paper).\nThis expression gives the absorption rate of a dissipative two\nlevel system in the weak coupling regime, $K \\ll 1$\\cite{W93}.\nThe natural cutoff for electron-hole pairs is bounded by the charging energy\nof the system.\nIt would be interesting to disentangle the relative contributions\nof electron-hole pairs and piezoelectric phonons to the inelastic\ncurrent.\n\nThe photo-induced conductance in a double dot system has also been\nmeasured\\cite{Betal98b}. The analysis of the contribution of inelastic processes\ndue to electron hole pairs\nto the measured conductance proceeds in the same way as in the interpretation\nof the previous experiment. Let us suppose that a photon of\nenergy $\\omega_{ph}$ excites\nan electron within one dot. Assuming that\nthe coupling to the environment is weak,\nthe rate at which the electron tunnels to the second dot\nby losing an energy $\\epsilon$ is given by\neq.(\\ref{rate}). The experiments in\\cite{Betal98b}\nsuggest that the number of states within each dot are discrete.\nThen, the induced conductance should show a series of peaks, related\nto resonant photon absorption within one dot. The height of each peak\nis determined by the decay rate to lower excited states in the other dot,\nand it can be written as a sum of terms with the dependence\ngiven in eq.(\\ref{rate}), where $\\epsilon$\nis the energy difference between the initial and final states.\nThe envelope of\nthe spectrum should look like a power law, in qualitative agreement\nwith the experiments. \n\n\nIt is interesting to note\nthat bunching of energy levels in quantum dots have been reported\\cite{ZAPW97}.\nThe separation between peaks defines the charging energy, which, according\nto the experiments, vanishes for certain charge states. This behavior\ncan be explained if the excitonic effects drive the quantum dot\nbeyond the transition, and charging effects are totally suppressed. \nThis mechanism can also play some role in the observed\ntransitions in granular wires\\cite{DRG98,Hetal96}.\n\\section{Conclusions.}\nWe have analyzed the effects of non-equilibrium transients\nafter a tunneling process on the conductance of quantum dots.\nThey are related to the change in the electrostatic potential \nof the dot upon the addition of a single electron. These effects can enhance or suppress the Coulomb blockade.\nThe most striking effect arise\nfrom the formation of an exciton-like resonance at the\nFermi level after the charging process and lead to the complete suppression of the Coulomb blockade and a diverging conductance\nat low temperatures. It appears for sufficiently large values of the conductance, $\\alpha$ , and the non-equilibrium phase shifts which define $\\epsilon$ in our model. 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{
"name": "cond-mat0002135.extracted_bib",
"string": "\\bibitem{SET} see several articles in\n{\\it Single Charge Tunneling}, H. Grabert and M. H. Devoret eds.\n(Plenum Press, New York, 1992).\n\n\\bibitem{Getal98} \nD. Goldhaber-Gordon, H. Shtrikman, D. Mahalu,\nD. Abusch-Magder, U. Meirav and M. A. Kastner, Nature {\\bf 391}, 156 (1998).\nS. M. Cronenwett, T. H. Oosterkamp and L. P. Kouwenhoven,\nScience {\\bf 281}, 540 (1998).\n\n\n\\bibitem{Oetal98}\nT. H. Oosterkamp, S. F. Godijn, M. J. Uilenreef, Y. V. Nazarov,\nN. C. van der Vaart and L. P. Kouwenhoven, Phys. Rev. Lett.\n{\\bf 80}, 4951 (1998).\n\n\n\\bibitem{Setal97}\nD. R. Stewart, D. Sprinzak, C. M. Marcus, C. I. Duru\\\"oz and\nJ. S. Harris Jr., Science {\\bf 278}, 1784 (1997).\nS. R. Patel, D. R. Stewart, C. M. Marcus, M. G\\\"okcedag, Y. Alhassid,\nA. D. Stone, C. I. Duru\\\"oz and J. S. Harris Jr., Phys. Rev. Lett.\n{\\bf 81}, 5900 (1998).\n\n\n\\bibitem{Oetal99}\nT. H. Oosterkamp, J. W. Janssen, L. P. Kouwenhoven, D. G. Austing,\nT. Honda and S. Tarucha, Phys. 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(World Scientific, Singapore, 1990).\n\n\n\\bibitem{UG91}\nM. Ueda and F. Guinea, Zeits. f\\\"ur Phys. B {\\bf 85}, 413 (1991).\n\n\n\\bibitem{GBC98}\nF. Guinea, E. Bascones and M. J. Calder\\'on in\n{\\it Lectures on the Physics of Highly Correlated Electrons},\nF. Mancini ed., AIP press (New York), 1998.\n\n\\bibitem{MVM97}\nM. Matters, J. J. Versluys and J. E. Mooij, Phys. Rev. Lett.\n{\\bf 78}, 2469 (1997). \n\n\\bibitem{Vetal95}\nN. C. van der Waart, S. F. Godijn, Y. V. Nazarov, C. J. P. M.\nHarmans, J. E. Mooij, L. W. Molenkamp and C. T. Foxon, Phys. Rev. Lett.\n {\\bf 74}, 4702 (1995).\nR. H. Blick, D. Pfannkuche, R. J. Haug, K. v. Klitzing, and K. Eberl,\nPhys. Rev. Lett. {\\bf 80}, 4032 (1998).\nT. H. Oosterkamp, T. Fujisawa, \nW.G. van der Wiel, K. Ishibashi, R.V. Hijman, S. Tarucha\nand L.P. Kouwenhoven, Nature {\\bf 395}, 873 (1998).\n\n\n\n\\bibitem{ML92}\nK. A. Matveev and A. I. Larkin, Phys. Rev. B {\\bf 46}, 15337 (1992).\n\n\n\\bibitem{Getal94}\nA. K. Geim, P. C. Main, N. La Scala Jr., \nL. Eaves, T. J. Foster, P. H. Beton, J. W. Sakai, \nF. W. Sheard, M. Henini, G. Hill, and M. A. Pate, \nPhys. Rev. Lett. {\\bf 72}, 2061 (1994). \n\n\n\\bibitem{PAK99}\nM. Pustilnik, Y. Avishai and K. Kikoin, preprint \n(cond-mat/9908004), and Physica B, to be published. \n\n\n\\bibitem{Yetal95}\nA. Yacoby, M. Heiblum, D. Mahalu and H. Shtrikman,\nPhys. Rev. Lett. {\\bf 74}, 4047 (1995). \nY. Levinson, Europhys. Lett. {\\bf 39}, 299 (1997).\nI. L. Aleiner, N. S. Wingreen, and Y. Meir, Phys. Rev. Lett.\n{\\bf 79}, 3740 (1997).\n\n% \n\\bibitem{Wetal95}\n% F. Waugh, M. J. Berry, D. J. Mar, R. M. Westervelt, K. L. Campman\n% and A. C. Gossard, Phys. Rev. Lett. {\\bf 75}, 705 (1995).\n\n\n\\bibitem{SZ90}\nG. Sch\\\"on and A. D. Zaikin, Phys. Rep. {\\bf 198}, 238 (1990).\n\n\\bibitem{HSZ99}\nC. P. Herrero, G. Sch\\\"on and A. D. Zaikin, Phys. Rev. B {\\bf 59}, 5728 (1999).\n\n\\bibitem{BMM97}\nYa. M. Blanter, A. D. Mirlin and B. A. Muzykantskii,\nPhys. Rev. Lett. {\\bf 78}, 2449 (1997).\n\n\\bibitem{G00}\nFor a review see L. I. Glazman, {\\bf Single electron tunneling}, to be published in a special issue of Journal of Low Temperature Physics . \n\n\\bibitem{BMS83}\nE. Ben-Jacob, E. Mottola and G. Sch\\\"on, Phys. Rev. Lett.\n{\\bf 51}, 2064 (1983). \n\n\n\\bibitem{DRG98}\nS. Drewes, S. R. Renn and F. Guinea,\nPhys. Rev. Lett. {\\bf 80}, 1046 (1998). \n\n\n\n\\bibitem{KF92}\nC. Kane and M. P. A. Fisher, Phys. Rev. B {\\bf 46}, 15233 (1992).\n\n\\bibitem{MG93}\nK. A. Matveev and L. I. Glazman, Phys. Rev. Lett. {\\bf 70},\n990 (1993).\n\n\\bibitem{SK97}\nM. Sassetti and B. Kramer, Phys. Rev. B {\\bf 55}, 9306 (1997). \n\n\n\\bibitem{AES82}\nV. Ambegaokar, U. Eckern and G. Sch\\\"on, \n\tPhys.\\ Rev.\\ Lett. {\\bf 48}, 1745 (1982).\n\n\n\n\n\\bibitem{FSZ95}\nG. Falci, G. Sch\\\"on, and G.T. Zimanyi,\n \tPhys.\\ Rev.\\ Lett. {\\bf 74}, 3257 (1995).\n\n\\bibitem{WEG97}\nX. Wang, R. Egger, and H. Grabert, Europhys. Lett. {\\bf 38}, 545 (1997).\n\n\\bibitem{K77}\nJ. M. Kosterlitz, Phys. Rev. Lett. {\\bf 37}, 1577 (1977).\n\n\\bibitem{GS86}\nF. Guinea and G. Sch\\\"on, J. Low Temp. Phys. {\\bf 69}, 219 (1986).\n\n\\bibitem{SG97}\nT. Strohm and F. Guinea, Nucl. Phys. B {\\bf 487}, 795 (1997).\n\n\\bibitem{S93}\n{\\it Quantum Monte Carlo Methods in Condensed Matter Physics},\n edited by M. Suzuki (World Scientific, Singapore, 1993).\n\n\\bibitem{BH88}\nK. Binder and D.W. Heermann, {\\it Monte Carlo Simulation in\n Statistical Physics} (Springer, Berlin, 1988).\n\n\\bibitem{GG98}\nG. G\\\"oppert, H. Grabert, N. Prokof'ev, and B. V. Svistunov, \nPhys. Rev. Lett. {\\bf 81}, 2324 (1998).\n\n\\bibitem{KS98}\nJ. K\\\"onig and H. Schoeller, Phys. Rev. Lett. {\\bf 81}, 3511 (1998).\n\n\\bibitem{GLT22}\nG.G\\\"oppert, B. Hupper and H. Grabert, to be published in Physica B. \n\n\\bibitem{ZS91}\nW. Zwerger and M. Scharpf, Z. Phys. B {\\bf 85}, 421 (1991).\n\n\\bibitem{IZ92}\nG. Ingold and A. V. Nazarov in ref. [1].\n\n\\bibitem{G99}\nG. G\\\"oppert and H. Grabert, C.R.Acad Sci. 327, 885 (1999). G. G\\\"oppert and H. Grabert, cond-mat/9910237 \n\n\\bibitem{W93}\nU. Weiss, {\\it Quantum dissipative systems} (World Scientific,\nSingapore, 1993). \n\n\\bibitem{AN90}\nD. V. Averin and Yu. V. Nazarov, Phys. Rev. Lett. {\\bf 65},\n2446 (1990).\n\n\\bibitem{RTMS92}\nM. Randeria, N. Trivedi, A. Moreo and R. T. Scalettar,\nPhys. Rev. Lett. {\\bf 69}, 2001 (1992).\nN. Trivedi and M. Randeria, Phys. Rev. Lett. {\\bf 75}, 312 (1995).\n\n\\bibitem{BG99}\nR. Baltin and Y. Gefen, Phys. Rev. Lett. {bf 83}, 5094 (1999).\n\n\\bibitem{Cetal97}\nS. M. Cronnenwett, S. R. Patel, C. M. Marcus, K. Campman and\nA. C. Gossard, Phys. Rev. Lett. {\\bf 79}, 2312 (1997).\n\n\\bibitem{Metal99}\nS. M. Maurer, S. R. Patel, C. M. Marcus, C. I. Duru\\\"oz and S. J. Harris,\nPhys. Rev. Lett. {\\bf 83}, 1403 (1999).\n\n\\bibitem{Fetal98}\nT. Fujisawa, T.H. Oosterkamp, W.G. van der Wiel, B. Broer, R. Aguado, \nS. Tarucha, and L.P. Kouwenhoven, Science {\\bf 282}, 932 (1998).\n\n\\bibitem{BK99}\nT. Brandes and B. Kramer, Phys. Rev. Lett. {\\bf 83}, 3021 (1999).\n\n\\bibitem{Betal98b}\nR. H. Blick, D. W. van der Weide, R. J. Haug and K. Eberl,\nPhys. Rev. Lett. {\\bf 81}, 689 (1998).\n\n\\bibitem{ZAPW97}\nN. V. Zhitenev, A. C. Ashoori, L. N. Pfeiffer and K. W. West,\nPhys. Rev. Lett. {\\bf 79}, 2308 (1997).\n\n\\bibitem{Hetal96}\nA. V. Herzog, P. Xiong, F. Sharifi and R. C. Dynes, Phys. Rev.\nLett. {\\bf 76}, 668 (1996). \n"
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cond-mat0002136
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Thermodynamic properties \protect\\ of the periodic nonuniform spin-$\frac{1}{2}$ isotropic $XY$ chains \protect\\ in a transverse field
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[
{
"author": "Oleg Derzhko$^{\\dagger,\\ddagger}$"
},
{
"author": "Johannes Richter$^{\\star}$ and Oles' Zaburannyi$^{\\dagger}$"
},
{
"author": "{\\em {1 Svientsitskii St., L'viv-11, 290011, Ukraine}}"
},
{
"author": "{\\em {12 Drahomanov St., L'viv-5, 290005, Ukraine}}"
},
{
"author": "{\\em $^{\\star}${Institut f\\\"{u}r Theoretische Physik"
},
{
"author": "Universit\\\"{a}t Magdeburg}}"
},
{
"author": "{\\em {P.O. Box 4120, D-39016 Magdeburg, Germany}}"
}
] |
Using the Jordan-Wigner transformation and the continued-fraction method we calculate exactly the density of states and thermodynamic quantities of the periodic nonuniform spin-$\frac{1}{2}$ isotropic $XY$ chain in a transverse field. We discuss in detail the changes in the behaviour of the thermodynamic quantities caused by regular nonuniformity. The exact consideration of thermodynamics is extended including a random Lorentzian transverse field. The presented results are used to study the Peierls instability in a quantum spin chain. In particular, we examine the influence of a non-random/random field on the spin-Peierls instability with respect to dimerization.
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[
{
"name": "lra07.tex",
"string": "\\section{Introduction}\n\nThe study of regularly nonuniform spin models\nis an attracting problem of statistical mechanics.\nBesides of its general academic importance\nthe development of magnetic materials in recent years\nmakes the study of nonuniform spin models particularly interesting \nfor experimental application.\nIn order to achieve a progress in understanding\nthe generic features generated by periodic nonuniformity\nit is desirable to examine simple models\nthat may be investigated without making any approximation.\nAmong possible candidates of such systems one can mention\nthe spin-$\\frac{1}{2}$ $XY$ model in one dimension.\nThe uniform one-dimensional spin-$\\frac{1}{2}$ $XY$ model \nin a transverse field \nwas introduced by Lieb, Schultz and Mattis.$^{1}$\nThese authors noticed\nthat a lot of statistical mechanics calculations\nfor such a spin model can be performed exactly\nsince\nit can be rewritten\nas a model of noninteracting spinless fermions by means\nof the Jordan-Wigner transformation.\nThe nonuniform version of the \ntransverse spin-$\\frac{1}{2}$ $XY$ chain\nalso can be mapped onto a chain of free spinless fermions,\nhowever, with an on-site energy and hopping integrals\nthat vary from site to site. \nEspecially attractive is the case of isotropic \nspin coupling\nwith regularly alternating exchange integrals and transverse fields\nsince after fermionization one faces a model for which a lot\nof work has been done.\nOne should mention here the results for the tight-binding Hamiltonian\nof periodically modulated chains$^{2,3}$\nand the\nspinless Falicov-Kimball\nchain.$^{4,5}$\n\nOne of the goals of the present paper\nis to give a magnetic interpretation of\nthose results derived for\nthe one-dimensional tight-binding spinless fermions.\nExploiting the\ncontinued-fraction\napproach developed in the mentioned papers$^{2-5}$\nwe shall be able to calculate exactly the one-fermion Green functions\nand therefore to obtain\nthe thermodynamic quantities for the periodic nonuniform\nspin-$\\frac{1}{2}$ isotropic $XY$ chain in a transverse field.\nWe shall treat few examples of\nthe periodic nonuniform spin-$\\frac{1}{2}$ isotropic $XY$ chain\nin a transverse field\nin order to reveal\nthe changes in the thermodynamic properties\ninduced by periodic nonuniformity.\nThe model of the considered regularly nonuniform magnetic chain allows\neven a natural extension to include additional disorder remaining the model\nexactly solvable.\nNamely, one can assume the transverse fields to be random\nindependent Lorentzian variables\nwith regularly alternating mean values and widths of distribution.\nTo derive exactly the random-averaged density of states for such a model\none should at first average a set of equations for\nthe Green functions using contour integrals.$^{6-11}$\nAs a result\none comes to a set of equations\nsimilar to that for the periodic nonuniform non-random case.\n\nIt should be noted that the periodic nonuniform\nspin-$\\frac{1}{2}$\nisotropic $XY$ chain was considered\nin several papers$^{12-25}$\ndealing mainly with \nthe adiabatic treatment of \nthe spin-Peierls instability.\nHowever, those papers were focussed\nmostly on the influence of the structural degrees of freedom\nupon the magnetic ones,\nrather than on\nthe exhaustive analysis of\nthe properties of a magnetic chain\nwith regularly alternating exchange couplings.\nAnother closely related study concerns the\nspin-$\\frac{1}{2}$ isotropic $XY$ model\non a one-dimensional superlattice$^{26}$.\nThe treatment reported in Ref. 26,\nhowever, was restricted to the magnon spectrum.\nA related study of\nthe spin-$\\frac{1}{2}$ isotropic $XY$ chain in a transverse\nfield with two kinds of coupling constant aimed on examining the\ncondition for appearance of an energy gap was reported in Ref. 27.\nFinally, let us note that the periodic nonuniform chain can be viewed\nas the uniform chain with a crystalline unit cell containing\nseveral sites of the initial lattice\n(as a matter of fact such a point of view\nwas adopted, for example, in Refs. 12, 13)\nand thus\nthe standard methods elaborated for such complex crystals may be\nexploited.\nHowever, we prefer to treat periodically nonuniform chains\nsince from such a viewpoint\nan elegant continued-fraction approach\nimmediately arises\nthat seems to be a natural and\nconvenient language for describing such compounds.\n\nThe present paper is a more extensive version of the results\nbriefly reported in Refs. 28, 29\ncontaining more details on the calculation and more applications.\nWe show that the presented below \nmethod based on continued-fraction representation \nfor the one-fermion diagonal Green functions immediately reproduces \nthe results for the spin-$\\frac{1}{2}$ transverse isotropic $XY$ chains \nhaving periods 2 and 3 and in contrast to other approaches easily yields the \nresults for larger periods (e.g., 4 and 12). \nIt should be stressed that the elaborated approach is a systematic method \nthat permits to consider in the same fashion the regularly nonuniform chains\nwith randomness that cannot be done within the frames of the approaches \nexploited in Refs. 12-27. We present a comprehensive study of the \nthermodynamic properties (density of states, gap in the energy spectrum,\nentropy, specific heat, magnetization, susceptibility) of regularly \nalternating chains having periods 2, 3, 4, and 12\ndiscussing in detail the dependences of the energy gap and\nthe low-temperature transverse\nmagnetization on the transverse field and comparing the latter dependence \nwith the corresponding\none for the classical spin chain. We underline a \npossibility of a nonzero transverse magnetization at the zero average \ntransverse field in the periodic nonuniform chain \nowing to a regular\nnonuniformity. This material constitutes Section 2. In Section 3 we \ndemonstrate the influence of a diagonal disorder on the effects caused by \nperiodic \nnonuniformity assuming the transverse fields to be independent random \nLorentzian variables. For simplicity we restrict ourselves by the case\nof a \nchain having period 2. The results obtained for the spin-$\\frac{1}{2}$ \ntransverse isotropic $XY$\nchains with period 2 are applied for an analysis\nof the spin-Peierls instability\nin the adiabatic limit\nwith respect to dimerization \nin the presence of a non-random/random (Lorentzian) transverse field \n(Section 4).\nWe find how\nthe (random) transverse field\ninfluences the dependence\nof the (averaged) total energy\non the dimerization parameter\ntracing a suppression of dimerization by\nthe non-random (random) field.\n\n\\section{Periodic nonuniform spin-$\\frac{1}{2}$\n\tisotropic $XY$ chain in a transverse field}\n\nLet us consider a cyclic\nnonuniform isotropic $XY$ chain of $N$\n(eventually $N\\to\\infty$) spins $s=\\frac{1}{2}$\nin a transverse field. The Hamiltonian of the system reads\n\\begin{eqnarray}\nH=\\sum_{n=1}^{N}\\Omega_ns_n^z\n+2\\sum_{n=1}^{N}I_n\\left(s^x_ns^x_{n+1}+s^y_ns^y_{n+1}\\right)\n\\nonumber\\\\\n=\\sum_{n=1}^{N}\\Omega_n\\left(s_n^+s_n^--\\frac{1}{2}\\right)\n+\\sum_{n=1}^{N}I_n\\left(s^+_ns^-_{n+1}+s^-_ns^+_{n+1}\\right),\n\\;\\;\\;\ns^{\\alpha}_{n+N}=s^{\\alpha}_{n}.\n\\end{eqnarray}\nHere $\\Omega_n$ is the transverse field at site $n$\nand $2I_n$ is the\nexchange coupling between the sites $n$ and $n+1$.\nLet us note that \n$s^z=\\sum_{n=1}^Ns_n^z$\ncommutes with the Hamiltonian $H$ (1)\nand hence\nthe eigenfunctions of $H$ can be classified according to \neigenvalues of \n$s^z$.\nMoreover, at $\\Omega_n=0$\nthe ground state of $H$ corresponds to $s^z=0$.\nAfter making use of the Jordan-Wigner transformation\none comes to a cyclic chain of spinless fermions\ngoverned by the Hamiltonian\n\\begin{eqnarray}\nH=\\sum_{n=1}^{N}\\Omega_n\\left(c_n^+c_n-\\frac{1}{2}\\right)\n+\\sum_{n=1}^{N}I_n\\left(c^+_nc_{n+1}-c_nc^+_{n+1}\\right).\n\\end{eqnarray}\nThe so-called boundary term is not\nessential for calculation of thermodynamic functions$^{30}$\nand has been omitted.\nWe shall discuss the most general case, i.e. \nassuming that\nboth transverse fields\nand exchange couplings vary from site to site.\nNote, that \nin the particular case when\nthe transverse field is uniform\none recognizes in Eq. (2) the Hamiltonian of the system\nconsidered in Ref. 2.\nIn addition, in another limiting case after substitution\n$\\Omega_n\\rightarrow Uw_n,$\n$I_n\\rightarrow -t$\nEq. (2) transforms into the\nHamiltonian of a one-dimensional spinless Falicov-Kimball model\nin the notations used in Refs. 4, 5.\n\nLet us introduce the temperature double-time Green functions\n\\linebreak\n$G_{nm}^{\\mp}(t)\n=\\mp{\\mbox{i}}\\theta(\\pm t)\n\\langle\\left\\{c_n(t), c_m^+(0)\\right\\}\\rangle,$\n$G_{nm}^{\\mp}(t)=\\left(1/2\\pi\\right)\n\\int_{-\\infty}^{\\infty}{\\mbox{d}}\\omega\n\\exp\\left(-{\\mbox{i}}\\omega t\\right)\nG_{nm}^{\\mp}(\\omega\\pm{\\mbox{i}}\\epsilon),$\n$\\epsilon\\rightarrow+0,$\nwhere the angular brackets denote the thermodynamic average.\nConsider further the set of equations\nof motion\nfor\n$G_{nm}^{\\mp}\\equiv G_{nm}^{\\mp}(\\omega\\pm{\\mbox{i}}\\epsilon)$\n\\begin{eqnarray}\n\\left(\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_n\\right)G_{nm}^{\\mp}\n-I_{n-1}G_{n-1,m}^{\\mp}\n-I_{n}G_{n+1,m}^{\\mp}\n=\\delta_{nm}.\n\\end{eqnarray}\nOur task is to evaluate the diagonal Green functions\n$G_{nn}^{\\mp}$,\nthe imaginary part of which gives the density of states\n$\\rho(\\omega)$,\n\\begin{eqnarray}\n\\rho(\\omega)\n=\\mp\\frac{1}{\\pi N}\n\\sum_{n=1}^N{\\mbox{Im}}G_{nn}^{\\mp},\n\\end{eqnarray}\nthat on its part, yields the thermodynamic properties of\nthe spin model (1).\n\nIt is a simple matter to obtain\nfrom Eq. (3) the following representation for $G_{nn}^{\\mp}$\n\\begin{eqnarray}\nG_{nn}^{\\mp}\n=\\frac{1}\n{\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_n-\\Delta^-_n-\\Delta^+_n},\n\\nonumber\\\\\n\\Delta^-_n=\\frac{I_{n-1}^2}\n{\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_{n-1}-\n\\frac{I_{n-2}^2}\n{\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_{n-2}-_{\\ddots}}},\n\\nonumber\\\\\n\\Delta^+_n=\\frac{I_{n}^2}\n{\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_{n+1}-\n\\frac{I_{n+1}^2}\n{\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_{n+2}-_{\\ddots}}}.\n\\end{eqnarray}\nEquations (4), (5) are extremely useful\nfor examining thermodynamic properties\nof the {\\em periodic} nonuniform\nspin-$\\frac{1}{2}$ isotropic $XY$ chain in a transverse field,\nsince the evaluation of periodic continued fractions$^{31}$ emerging in \n(5)\nis quite simple\nand reduces to solving quadratic equations.\n\nIt should be noted here that the continued-fraction representation\nof the one-particle Green functions has been widely used for\ntight-binding electrons over the last two decades.\nAs an example let us refer\nhere to the papers of Haydock, Heine and Kelly$^{32,33}$\nand the review articles.$^{34,35}$\nHowever, those studies were aimed mainly on\ngetting the electronic band structure of non-translationally invariant\nsystems (alternatively to the band theory)\nstarting from the local environment of atom\nand in practice were connected with an appropriate approximative\ntermination\nof continued fractions.\nIn what follows we shall use the exact values of continued fractions\n(since they are periodic)\nto reveal the effects of regular nonuniformity on\nthe magnon band structure.\n\nConsider at first a uniform chain\n$\\Omega_0I\\Omega_0I\\ldots\\;$.\nIn this case one comes to\na periodic continued fraction having a period 1\n\\begin{eqnarray}\n\\Delta^-_n\n=\\Delta^+_n\n=\\Delta\n=\\frac{I^2}\n{\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0-\n\\frac{I^2}\n{\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0-_{\\ddots}}}\n=\\frac{I^2}\n{\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0 -\\Delta}.\n\\end{eqnarray}\nThe quadratic equation for $\\Delta$ (6) can be solved with\n\\begin{eqnarray}\n\\Delta\n=\\left\\{\\frac{1}{2}\n\\left[\n\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0\n+\\sqrt{\\left(\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0\\right)^2-4I^2}\n\\right],\\;\n\\frac{1}{2}\n\\left[\n\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0\n-\\sqrt{\\left(\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0\\right)^2-4I^2}\n\\right]\\right\\}\n\\end{eqnarray}\nand therefore\n$\\rho(\\omega)$\naccording to (4), (5) becomes\n\\begin{eqnarray}\n\\rho(\\omega)\n=\\left\\{\n\\begin{array}{ll}\n\\frac{1}{\\pi}\n\\frac{1}{\\sqrt{4I^2-\\left(\\omega-\\Omega_0\\right)^2}},\n& {\\mbox{if}}\\;\\;\\;4I^2-(\\omega-\\Omega_0)^2>0,\n\\\\\n0,\n& {\\mbox{otherwise.}}\n\\end{array}\n\\right.\n\\end{eqnarray}\nThe self-consistent equation for the continued fraction (6)\nintroduces a spurious root.\nHowever, the false solution is eliminated\nrequiring $\\rho(\\omega)$ to be not negative.\nLet us emphasize the attractive features of\nthe continued-fraction approach\nreminding how\n$\\rho(\\omega)$ (8)\ncan be obtained within the frames\nof the standard technique.\nUsually one substitutes into Eq. (3)\n$G_{nm}^{\\mp}=(1/N)\\sum_{\\kappa}\n\\exp[{\\mbox{i}}(n-m)\\kappa]G_{\\kappa}^{\\mp}$\nto obtain\n$G_{\\kappa}^{\\mp}\n=1/(\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0-2I\\cos\\kappa)$\nand then evaluates the integral\n$G_{nn}^{\\mp}\n=(1/2\\pi)\\int_{-\\pi}^{\\pi}{\\mbox{d}}\\kappa\n/(\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0-2I\\cos\\kappa)$\nusing, for example, contour integrals\nto get\n$G_{nn}^{\\mp}=1\n/\\sqrt{\\left(\\omega\\pm{\\mbox{i}}\\epsilon-\\Omega_0\\right)^2-4I^2}$\nand therefore\nthe density of states (8).\n\nThe advantages of the continued-fraction approach\nbecome clear while treating the periodic nonuniform chains.\nWe shall demonstrate this in some \ndetail for regularly modulated chains with\nperiods of modulation of 2, 3 and 4.\n\n(i)\nConsider a regular alternating chain\n$\\Omega_1I_1\\Omega_2I_2\\Omega_1I_1\\Omega_2I_2\\ldots\\;$.\nIn this case\nperiodic continued fractions having a period 2 emerge.\nSolving similar quadratic equations as (6)\nfor\n$\\Delta_n^-,$\n$\\Delta_n^+,$\n$\\Delta_{n+1}^-,$\n$\\Delta_{n+1}^+$\none obtains as a result the Green functions\n$G_{nn}^{\\mp},$\n$G_{n+1,n+1}^{\\mp}$\nand therefore the density of states\n$\\rho(\\omega)$\n\\begin{eqnarray}\n\\rho(\\omega)\n=\\left\\{\n\\begin{array}{ll}\n\\frac{1}{2\\pi}\n\\frac{\\mid 2\\omega-\\Omega_1-\\Omega_2\\mid}\n{\\sqrt{{\\cal{B}}(\\omega)}},\n& {\\mbox{if}}\\;\\;\\;{\\cal{B}}(\\omega)>0,\n\\\\\n0,\n& {\\mbox{otherwise;}}\n\\end{array}\n\\right.\n\\nonumber\\\\\n{\\cal{B}}(\\omega)\n=4I_1^2I_2^2\n-\\left[\n\\left(\\omega-\\Omega_1\\right)\n\\left(\\omega-\\Omega_2\\right)\n-I_1^2-I_2^2\n\\right]^2\n\\nonumber\\\\\n=-\\left(\\omega-b_1\\right)\n\\left(\\omega-b_2\\right)\n\\left(\\omega-b_3\\right)\n\\left(\\omega-b_4\\right).\n\\end{eqnarray}\nHere\n$b_1\\ge b_2\\ge b_3\\ge b_4$\ndenote the four roots of the equation\n${\\cal{B}}(\\omega)=0$, namely\n\\begin{eqnarray}\n\\left\\{b_i\\right\\}\n=\n\\left\\{\n\\frac{1}{2}\n\\left(\n\\Omega_1+\\Omega_2\n\\right)\n\\pm{\\sf{b}}_1,\n\\;\\;\\;\n\\frac{1}{2}\n\\left(\n\\Omega_1+\\Omega_2\n\\right)\n\\pm{\\sf{b}}_2\n\\right\\}\n\\end{eqnarray}\nwith\n${\\sf{b}}_1=\\frac{1}{2}\\sqrt{\\left(\\Omega_1-\\Omega_2\\right)^2\n+4\\left(\\vert I_1\\vert+\\vert I_2\\vert\\right)^2}$,\n${\\sf{b}}_2=\\frac{1}{2}\\sqrt{\\left(\\Omega_1-\\Omega_2\\right)^2\n+4\\left(\\vert I_1\\vert-\\vert I_2\\vert\\right)^2}$.\nSolving the inequality\n${\\cal{B}}(\\omega)>0$\none can write the density of states\n$\\rho(\\omega)$ (9) in the explicit form\n\\begin{eqnarray}\n\\rho(\\omega)\n=\\left\\{\n\\begin{array}{ll}\n0,\n& {\\mbox{if}}\\;\\;\\;\\omega<b_4,\\;\\;\\;b_3<\\omega<b_2,\\;\\;\\;b_1<\\omega,\n\\\\\n\\frac{1}{2\\pi}\n\\frac{\\mid 2\\omega-\\Omega_1-\\Omega_2\\mid}\n{\\sqrt{{\\cal{B}}(\\omega)}},\n& {\\mbox{if}}\\;\\;\\;b_4<\\omega<b_3,\\;\\;\\;b_2<\\omega<b_1.\n\\end{array}\n\\right.\n\\end{eqnarray}\nThe result for\nthe uniform chain (8)\nis contained\nin the density of states (11), (10), (9)\nas a partial case\nwhen $\\Omega_1=\\Omega_2=\\Omega_0$,\n$I_1=I_2=I$.\n\n(ii) Next we consider the regularly modulated chain\n$\\Omega_1I_1\\Omega_2I_2\\Omega_3I_3\\Omega_1I_1\\Omega_2I_2\\Omega_3I_3\\ldots\\;$.\nIn this case \none generates\nthe corresponding periodic continued fractions of period 3.\nGoing along the lines as described above one gets\n\\begin{eqnarray}\n\\rho(\\omega)\n=\\left\\{\n\\begin{array}{ll}\n\\frac{1}{3\\pi}\n\\frac{\\mid\nI_1^2+I_2^2+I_3^2\n-\\left(\\omega-\\Omega_1\\right)\\left(\\omega-\\Omega_2\\right)\n-\\left(\\omega-\\Omega_1\\right)\\left(\\omega-\\Omega_3\\right)\n-\\left(\\omega-\\Omega_2\\right)\\left(\\omega-\\Omega_3\\right)\n\\mid}\n{\\sqrt{{\\cal{C}}(\\omega)}},\n& {\\mbox{if}}\\;\\;\\;{\\cal{C}}(\\omega)>0,\n\\\\\n0,\n& {\\mbox{otherwise;}}\n\\end{array}\n\\right.\n\\nonumber\\\\\n{\\cal{C}}(\\omega)\n=4I_1^2I_2^2I_3^2\n-\\left[\nI_1^2\\left(\\omega-\\Omega_3\\right)\n+I_2^2\\left(\\omega-\\Omega_1\\right)\n+I_3^2\\left(\\omega-\\Omega_2\\right)\n-\\left(\\omega-\\Omega_1\\right)\n\\left(\\omega-\\Omega_2\\right)\n\\left(\\omega-\\Omega_3\\right)\n\\right]^2\n\\nonumber\\\\\n=-\\prod_{j=1}^6\\left(\\omega-c_j\\right),\n\\end{eqnarray}\nwhere $c_j$ are the six roots of the equation ${\\cal{C}}(\\omega)=0$.\nTo find them one must solve two qubic equations\nthat follow from Eq. (12).\n\n(iii) Finally, let us\nconsider the regularly modulated chain\n$\\Omega_1I_1\\Omega_2I_2\\Omega_3I_3\\Omega_4I_4\n\\Omega_1I_1\\Omega_2I_2\\Omega_3I_3\\Omega_4I_4\\ldots\\;$.\nIn this case one gets\nperiodic continued fractions with period 4.\nThe density of states for such a chain is given by\n\\begin{eqnarray}\n\\rho(\\omega)\n=\\left\\{\n\\begin{array}{ll}\n\\frac{1}{4\\pi}\n\\frac{\\mid{\\cal{W}}(\\omega)\\mid}\n{\\sqrt{{\\cal{D}}(\\omega)}},\n& {\\mbox{if}}\\;\\;\\;{\\cal{D}}(\\omega)>0,\n\\\\\n0,\n& {\\mbox{otherwise;}}\n\\end{array}\n\\right.\n\\nonumber\\\\\n{\\cal{W}}(\\omega)\n=I_1^2(2\\omega-\\Omega_3-\\Omega_4)\n+I_2^2(2\\omega-\\Omega_1-\\Omega_4)\n+I_3^2(2\\omega-\\Omega_1-\\Omega_2)\n+I_4^2(2\\omega-\\Omega_2-\\Omega_3)\n\\nonumber\\\\\n-(\\omega-\\Omega_1)(\\omega-\\Omega_2)(\\omega-\\Omega_3)\n-(\\omega-\\Omega_1)(\\omega-\\Omega_2)(\\omega-\\Omega_4)\n\\nonumber\\\\\n-(\\omega-\\Omega_1)(\\omega-\\Omega_3)(\\omega-\\Omega_4)\n-(\\omega-\\Omega_2)(\\omega-\\Omega_3)(\\omega-\\Omega_4),\n\\nonumber\\\\\n{\\cal{D}}(\\omega)\n=4I_1^2I_2^2I_3^2I_4^2\n-\\left[\n\\left(\\omega-\\Omega_1\\right)\n\\left(\\omega-\\Omega_2\\right)\n\\left(\\omega-\\Omega_3\\right)\n\\left(\\omega-\\Omega_4\\right)\n\\right.\n\\nonumber\\\\\n\\left.\n-I_1^2\\left(\\omega-\\Omega_3\\right)\\left(\\omega-\\Omega_4\\right)\n-I_2^2\\left(\\omega-\\Omega_1\\right)\\left(\\omega-\\Omega_4\\right)\n\\right.\n\\nonumber\\\\\n\\left.\n-I_3^2\\left(\\omega-\\Omega_1\\right)\\left(\\omega-\\Omega_2\\right)\n-I_4^2\\left(\\omega-\\Omega_2\\right)\\left(\\omega-\\Omega_3\\right)\n\\right.\n\\nonumber\\\\\n\\left.\n+I_1^2I_3^2+I_2^2I_4^2\n\\right]^2\n=-\\prod_{j=1}^8\\left(\\omega-d_j\\right),\n\\end{eqnarray}\nwhere $d_j$ are the eight roots of the equation\n${\\cal{D}}(\\omega)=0$.\nTo find them one must solve two equations of 4th order\nthat follow from Eq. (13).\nLet us note that all\n$d_j$\n(as well as all $c_j$)\nare real\nsince they can be viewed as eigenvalues of symmetric matrices.$^2$\n\nThere are no principal difficulties in proceeding the\nanalytic\ncalculations of $\\rho(\\omega)$\nfor larger periods,\nexcept the fact that they become more cumbersome.\nAll\nGreen functions\nrequired for getting the density of states $\\rho(\\omega)$ (4)\nare calculated by solving quadratic equations,\nhowever, further analysis of the band structure is becoming\nmore complicated.\nThis analysis, however, can be easily implemented on a computer\nand the results for the chains having period 12 presented below\nwere obtained in such a manner.\n\nLet us discuss the\nresults for the density of states\nfor the considered periodic\nnonuniform chains.\nThe main consequence of introducing the nonuniformity\nis a splitting of the initial magnon band\ninto several subbands\n(compare (8) and (9) - (13)).\nThe edges of the subbands are determined by the roots of equations\n${\\cal{B}}(\\omega)=0$,\n${\\cal{C}}(\\omega)=0$,\n${\\cal{D}}(\\omega)=0$, etc..\n$\\rho(\\omega)$ is positive inside the subbands,\ntends to infinity inversely proportionally to the square root of\n$\\omega-\\omega_e$\nwhen $\\omega$ approaches the subbands edges $\\omega_e$,\nand is equal to zero outside the subbands.\nThe number of subbands does not exceed the period of the chain.\nAt special (symmetric) values of the Hamiltonian parameters the roots of\nthe equation\nthat determines the subband edges may become multiple\nand the zeros in the denominator and the numerator in the expression for\n$\\rho(\\omega)$\nmay cancel each other.\nAs a result due to an increase of symmetry one may observe a smaller\nnumber of subbands.\nThis `mechanism' is easily traced, for example,\nin formulas (9) - (11)\nif putting $\\Omega_1=\\Omega_2$, $\\vert I_1\\vert=\\vert I_2\\vert$.\nThe described magnon band structure\ncan be seen in\nFigs. 1 and 2a, 3 and 4a, and 5 and 6a\nwhere we show\n$\\rho(\\omega)$ for\na few particular periodic nonuniform chains\nhaving periods 2, 3, and 12, respectively.\nThe splitting caused by periodic nonuniformity\nin fact\nis not surprising.\nThe periodic nonuniform chain\nis simply another viewpoint on\nthe uniform chain with a crystalline unit cell\ncontaining several sites.\nOn the other hand,\nit is generally known that\none may expect several subbands for a crystal having several\natoms per unit cell.$^{36}$\n\nFurther\none can easily calculate the widths of energy gaps\nin the magnon spectrum\nthat appear due to nonuniformity.\nFor example, for a chain having a period 2 one finds\n\\begin{eqnarray}\nb_2-b_3\n=\\sqrt{(\\Omega_1-\\Omega_2)^2+4(\\vert I_1\\vert-\\vert I_2\\vert)^2}.\n\\end{eqnarray}\nThis quantity is connected with a gap\nbetween the ground state energy and the first excited state energy\nof the spin chain.\nAs an example consider the chain \n$\\Omega_0I_1\\Omega_0I_2\n\\Omega_0I_1\\Omega_0I_2\\ldots\\;$.\nThe edges of the upper magnon subband are given by\n$\\Omega_0+\\vert I_1\\vert+\\vert I_2\\vert$\nand\n$\\Omega_0+\\vert\\vert I_1\\vert-\\vert I_2\\vert\\vert$,\nwhereas the edges of the lower magnon subband are given by\n$\\Omega_0-\\vert\\vert I_1\\vert-\\vert I_2\\vert\\vert$,\nand\n$\\Omega_0-\\vert I_1\\vert-\\vert I_2\\vert$.\nAt $\\Omega_0=0$ the ground state (for which $s^z=0$)\ncorresponds to the filled lower subband \nand the empty upper subband \nand the energy spectrum exhibits a gap \n$\\Delta(0)=\\vert\\vert I_1\\vert-\\vert I_2\\vert\\vert$\n(this is the energy required to create a hole in the lower subband ---\nthe first excited state of the spin chain\n(with $s^z\\ne 0$)).\nWith increasing of $\\Omega_0$ \nthe gap\n$\\Delta$ decreases as\n$\\Delta(\\Omega_0)=\\vert\\vert I_1\\vert-\\vert I_2\\vert\\vert-\\Omega_0$\nand becomes zero at\n$\\Omega_0=\\vert\\vert I_1\\vert-\\vert I_2\\vert\\vert$.\nWith further increasing of $\\Omega_0$ the gap remains equal to zero up to \nthe value of the transverse field \n$\\Omega_0=\\vert I_1\\vert+\\vert I_2\\vert$\nafter which the gap opens and increases as\n$\\Delta(\\Omega_0)=\\Omega_0-\\vert I_1\\vert-\\vert I_2\\vert$\n(the ground state for the transverse field larger than \n$\\vert I_1\\vert+\\vert I_2\\vert$\ncorresponds to the empty subbands and\nthe written\n$\\Delta(\\Omega_0)$ is the energy\nrequired to create a particle in the vicinity of the lower edge of the \nlower subband). For chains with larger periods one finds more complicated \nbehaviour of the energy gap $\\Delta$ with varying of the field $\\Omega_0$\n(see Fig. 7a and Figs. 7b - 7d).\n\nThe splitting of the magnon band into subbands caused by nonuniformity\nhas interesting consequences for thermodynamic properties.\nThe entropy, specific heat, transverse magnetization and\nstatic transverse linear susceptibility are determined through the density\nof states\naccording to the following formulas\n\\begin{eqnarray}\ns=\\int\\limits_{-\\infty}^{\\infty}\n{\\mbox{d}}E\\rho(E)\n\\left[\\ln\\left(2\\cosh\\frac{E}{2kT}\\right)-\n\\frac{E}{2kT}\\tanh \\frac{E}{2kT}\\right],\n\\\\\nc=\\int\\limits_{-\\infty}^{\\infty}\n{\\mbox{d}}E\\rho(E)\n\\left(\n\\frac{\\frac{E}{2kT}}\n{\\cosh\\frac{E}{2kT}}\n\\right)^2,\n\\\\\nm_z=-\\frac{1}{2}\\int\\limits_{-\\infty}^{\\infty}\n{\\mbox{d}}E\\rho(E)\\tanh\\frac{E}{2kT},\n\\\\\n\\chi_{zz}=-\\frac{1}{kT}\\int\\limits_{-\\infty}^{\\infty}\n{\\mbox{d}}E\\rho(E)\\frac{1}{4\\cosh^2\\frac{E}{2kT}}.\n\\end{eqnarray}\n\nApparently the most spectacular changes caused by regular nonuniformity\nare observed in the dependence of transverse magnetization (17)\non transverse field at low temperatures\n(Figs. 2b, 4b, 6b).\nSince for $T\\rightarrow 0$,\n$\\tanh\\frac{E}{2kT}$ tends either to $-1$ if $E<0$,\nor to $1$ if $E>0$,\none immediately finds\ndue to the splitting of the magnon band into \nsubbands\nthat the low-temperature dependence of\n$m_z$ versus $\\Omega_0$\nmust be composed of sharply increasing parts\n(they appear when $E=0$ moves with increasing of $\\Omega_0$\nfrom the bottom to the top of each subband)\nseparated by horizontal parts\n(they appear when $E=0$ moves with increasing of $\\Omega_0$\ninside the gaps).\nThe number of plateaus is determined by the number of subbands.\nIt should be emphasized here that a study of magnetization plateaus for \nquantum spin chains is a hot topic at the present time.$^{37}$\nHowever in such studies usually more general spin chains are attacked\nwhich cannot be treated within the frames of the described approach. For \nexample, the spin-$\\frac{1}{2}$ $XXX$ chain can be mapped onto the chain of \n{\\it {interacting}} \nspinless fermions with the intersite interaction of \nthe same order as the hopping integral and hence the results derived \nrigorously for noninteracting fermions cannot be immediately extended for \nthis more complicated spin chain.\n\nIt is interesting to note that the appearance of plateaus in the dependence\nof transverse magnetization on transverse field at $T=0$\nfor the regularly nonuniform isotropic $XY$ chains\nessentially differs in the quantum and classical\ncases.\nThe Hamiltonian of the classical nonuniform\nisotropic $XY$ chain in a transverse field reads\n\\begin{eqnarray}\nH=\\sum_{n=1}^N\\Omega_ns\\cos\\theta_n\n+2\\sum_{n=1}^NI_ns^2\\cos(\\phi_n-\\phi_{n+1})\n\\sin\\theta_n\n\\sin\\theta_{n+1}\n\\end{eqnarray}\nthat\nimmediately yields the ansatz for the ground state energy\nin the uniform case\n\\begin{eqnarray}\nE_0\n=N\\Omega_0 s\\cos\\theta-2 N\\vert I\\vert s^2\\sin^2\\theta\n=N\\Omega_0 m_z+2 N\\vert I\\vert\\left(m_z^2-s^2\\right)\n\\end{eqnarray}\nwhere the ground state transverse magnetization\n$m_z=s\\cos\\theta$ has been introduced.\nMinimizing $E_0$ with respect to\n$\\cos\\theta$ one finds\nthat\nfor $s=\\frac{1}{2}$\nthe quantity\n$-m_z$ increases as $\\frac{1}{2}\\frac{\\Omega_0}{2\\vert I\\vert}$\nwhile $\\Omega_0$ increases from $0$ to\n$2\\vert I\\vert$ and\n$-m_z=\\frac{1}{2}$\nwith further increase of $\\Omega_0$.\nUsing numerical calculations for finite chains\n(the number of spins $N$ is a multiple of 12)\nwith periods 2, 3, 12\nwe found that\nthe detailed profiles for the quantum and classical chains\nare different,\nalthough the values of the transverse field\nat which a saturation of the transverse\nmagnetization occurs are the same.\nThough one could argue that the magnetization plateaus are connected with\nthe quantum nature of the spins we found\nfor special parameter sets even in the classical chain\nplateaus\nin the dependence $-m_z$ versus $\\Omega_0$\n(compare\ndashed curves\nin Figs. 8a, 8b and\nin Figs. 2b, 4b).\nFor instance,\nthe well pronounced plateau shown by the dashed line in Fig. 8b occurs\nat the same height as in the quantum case. \nThe corresponding classical state\nis a state \n$\\downarrow \\uparrow \\uparrow \\downarrow \\uparrow \\uparrow \n\\downarrow \\uparrow \\uparrow \\downarrow \\uparrow \\uparrow \\ldots $ \nwhere\nthe arrows symbolize classical spins pointing either \nin $-z$- or $+z$-direction.\nAn\nevident difference between the quantum and classical case is connected with\nthe slope of the $m_z(\\Omega_0)$ curve at $T=0$. \nThe slope remains finite in the\nclassical case but becomes infinite approaching the plateaus in the quantum\ncase. The infinite slope in the quantum case is clearly a\nconsequence of the singularities in the density of states.\n\nOne of the interesting magnetic properties of the periodic nonuniform\nspin-$\\frac{1}{2}$ isotropic $XY$ chain is the possibility\nof the existence of a non-zero transverse magnetization\n$m_z$ at zero average transverse field\n($\\sum_{n=1}^N\\Omega_n=0$). For illustration we consider as\nan example a chain having\nthe period 4 and the parameters $\\Omega_1=\\Omega_3=0$,\n$\\Omega_2=-\\Omega_4<0$,\n$\\vert I_1\\vert =\\vert I_2\\vert >0$,\n$\\vert I_3\\vert =\\vert I_4\\vert =0$.\nAt site $n+1$ we have the transverse field\n$\\Omega_2<0$ surrounded on the left and right side by the strong\ncouplings \n$\\vert I_1\\vert =\\vert I_2\\vert$. \nAt site $n+3$ we have the\ntransverse field $-\\Omega_2 >0$ surrounded by the weak couplings\n$\\vert I_3\\vert =\\vert I_4\\vert =0$.\nOne may expect that the local transverse\nmagnetization at site $n+1$ has\na smaller value and opposite direction\nwith respect to that quantity at site $n+3$\nand therefore a non-zero total transverse magnetization\nat zero average transverse field may be expected.\nConsider the described chain in more detail.\nFrom Eq. (13) for the above set of parameters\nit follows that\n\\begin{eqnarray}\n\\rho(\\omega)\n=\\lambda_1\\delta\\left(\\omega-\n\\frac{\\Omega_2-\\sqrt{\\Omega_2^2+8I_1^2}}{2}\\right)\n+\\lambda_2\\delta\\left(\\omega\\right)\n+\\lambda_3\\delta\\left(\\omega-\n\\frac{\\Omega_2+\\sqrt{\\Omega_2^2+8I_1^2}}{2}\\right)\n+\\lambda_4\\delta\\left(\\omega+\\Omega_2\\right)\n\\end{eqnarray}\nand the coefficients $\\lambda_j$ may be found comparing (21) and (13)\nin the vicinity of\n$\\frac{\\Omega_2-\\sqrt{\\Omega_2^2+8I_1^2}}{2}$,\n$0$,\n$\\frac{\\Omega_2+\\sqrt{\\Omega_2^2+8I_1^2}}{2}$\nand\n$-\\Omega_2$.\nAs a result one gets $\\lambda_j=\\frac{1}{4}$\n(see Fig. 9a).\nNow transverse magnetization (17) at $T=0$ is\n$m_z=-\\frac{1}{8}\\ne 0$\nalthough $\\sum_{n=1}^N\\Omega_n=0$\n(solid curves in Figs. 9b, 9c;\nin the latter figure the solid curve especially clearly shows that\n$-m_z=\\frac{1}{8}$).\nIf $\\vert I_3\\vert=\\vert I_4\\vert\\ne 0$ \nthe magnon subbands look as in Fig. 9a\nand at $T=0$ one has $m_z=0$ (Figs. 9b, 9c).\nHowever, such a position of the subbands\nprovides an interesting temperature dependence\nof $m_z$\nat $\\sum_{n=1}^N\\Omega_n=0$\n(dashed and dotted curves in Fig. 9c)\nreminding the `order from disorder' phenomenon,$^{38-40}$\ni.e. increasing of order with increasing temperature.\n\nLet us turn to other thermodynamic quantities.\nEvery infinite slope in the dependence\n$m_z$ versus $\\Omega_0$ at $T=0$\ninduces a singularity in the dependence\n$\\chi_{zz}$ versus $\\Omega_0$ at $T=0$.\nHowever, there is no need to plot this dependence.\nSince $1/4kT\\cosh^2\\frac{E}{2kT}$\ntends to $\\delta(E)$ as $T\\to0$\none gets from (18) that at $T=0$\n$\\;$\n$-\\chi_{zz}=\\rho(0)$.\nThe latter dependence as a matter of fact can be seen\nin Figs. 2a, 4a, 6a.\nThe changes\nin the temperature dependences of entropy and specific heat\ndue to nonuniformity\nwhich are\ndisplayed in Figs. 2c, 4c, 6c\nand 2d, 4d, 6d\ncan be understood while bearing in mind\nthe behaviour of integrands in (15), (16)\nthat are products of\nthe functions with evident dependences on the temperature\nand the density of states.\nNote that as a result of the magnon band splitting\nthe temperature dependence of the specific heat may\nexhibit a two-peak structure (Fig. 2d)\nor even a more complicated behaviour (solid curve in Fig. 4d).\nFinally we look at $\\chi_{zz}$. As mentioned above\nat $T=0$ we have\n$\\;$\n$-\\chi_{zz}=\\rho(0)$.\nAnalysing the density of states depicted in Figs. 2a, 4a, 6a\none finds that nonuniformity may either suppress or enhance\nthe initial (that is at $\\Omega_0=0$)\nstatic transverse linear susceptibility\n$-\\chi_{zz}$ at $T=0$\nshown in\nFigs. 2f, 4f, 6f.\n\n\\section{Periodic nonuniform spin-$\\frac{1}{2}$\nisotropic $XY$ chain\nin a random Lo\\-rentzian transverse field}\n\nIn this Section we consider a generalization of model (1) including \nadditional\nrandomness in the transverse fields. We assume\nthe transverse fields to be independent random variables\neach with a Lorentzian probability distribution\n\\begin{eqnarray}\np(\\Omega_n)=\\frac{1}{\\pi}\n\\frac{\\Gamma_n}\n{\\left(\\Omega_{0n}-\\Omega_n\\right)^2+\\Gamma_n^2}.\n\\end{eqnarray}\nHere $\\Omega_{0n}$ is the mean value of the transverse field at site\n$n$ and $\\Gamma_n$ is the width of its distribution.\nWe are interested in the random-averaged density of states\n$\\overline{\\rho(\\omega)}$\nthat follows from the random-averaged diagonal Green functions\n$\\overline{G_{nn}^{\\mp}}$ according to Eq. (4).\nRepeating the arguments presented in Refs. 6-11\none gets the following set of equations for the random-averaged\nGreen functions\n\\begin{eqnarray}\n\\left(\\omega\\pm{\\mbox{i}}\\Gamma_n\n-\\Omega_{0n}\\right)\n\\overline{G_{nm}^{\\mp}}\n-I_{n-1}\\overline{G_{n-1,m}^{\\mp}}\n-I_{n}\\overline{G_{n+1,m}^{\\mp}}\n=\\delta_{nm}\n\\end{eqnarray}\nthat immediately yields\n\\begin{eqnarray}\n\\overline{G_{nn}^{\\mp}}\n=\\frac{1}\n{\\omega\\pm{\\mbox{i}}\\Gamma_n\n-\\Omega_{0n}-\\Delta^-_n-\\Delta^+_n},\n\\nonumber\\\\\n\\Delta^-_n=\\frac{I_{n-1}^2}\n{\\omega\\pm{\\mbox{i}}\\Gamma_{n-1}\n-\\Omega_{0,n-1}-\n\\frac{I_{n-2}^2}\n{\\omega\\pm{\\mbox{i}}\\Gamma_{n-2}\n-\\Omega_{0,n-2}-_{\\ddots}}},\n\\nonumber\\\\\n\\Delta^+_n=\\frac{I_{n}^2}\n{\\omega\\pm{\\mbox{i}}\\Gamma_{n+1}\n-\\Omega_{0,n+1}-\n\\frac{I_{n+1}^2}\n{\\omega\\pm{\\mbox{i}}\\Gamma_{n+2}\n-\\Omega_{0,n+2}-_{\\ddots}}}.\n\\end{eqnarray}\nIn case\n$\\Omega_{0n},$\n$\\Gamma_n,$\n$I_n$\nvary regularly from site to site\none again comes to the periodic\ncontinued fractions.\nThey can be calculated as solutions of the corresponding\nquadratic equations.\nThus one gets rigorously the random-averaged Green functions\nand therefore the random-averaged density of states.\nFor example, for a regular random chain\n$\\Omega_{01}\\Gamma_1I_1\\Omega_{02}\\Gamma_2I_2\n\\Omega_{01}\\Gamma_1I_1\\Omega_{02}\\Gamma_2I_2\n\\ldots$\none finds\n\\begin{eqnarray}\n\\overline{\\rho(\\omega)}\n=\\frac{1}{2\\pi}\\frac{\\vert{\\cal{Y}}(\\omega)\\vert}\n{{\\cal{B}}(\\omega)};\n\\nonumber\\\\\n{\\cal{Y}}(\\omega)\n=(\\Gamma_1+\\Gamma_2)\n\\sqrt{\\frac{{\\cal{B}}(\\omega)+{\\cal{B}}^{\\prime}(\\omega)}{2}}\n-{\\mbox{sgn}}{\\cal{B}}^{\\prime\\prime}(\\omega)\n(2\\omega-\\Omega_{01}-\\Omega_{02})\n\\sqrt{\\frac{{\\cal{B}}(\\omega)-{\\cal{B}}^{\\prime}(\\omega)}{2}},\n\\nonumber\\\\\n{\\cal{B}}(\\omega)\n=\\sqrt{\\left({\\cal{B}}^{\\prime}(\\omega)\\right)^2\n+\\left({\\cal{B}}^{\\prime\\prime}(\\omega)\\right)^2},\n\\nonumber\\\\\n{\\cal{B}}^{\\prime}(\\omega)\n=\\left[\n(\\omega-\\Omega_{01})\n(\\omega-\\Omega_{02})\n-\\Gamma_1\\Gamma_2-I_1^2-I_2^2\n\\right]^2\n-\n\\left[\n(\\omega-\\Omega_{01})\\Gamma_2\n+\n(\\omega-\\Omega_{02})\\Gamma_1\n\\right]^2\n-4I_1^2I_2^2,\n\\nonumber\\\\\n{\\cal{B}}^{\\prime\\prime}(\\omega)\n=2\\left[\n(\\omega-\\Omega_{01})\n(\\omega-\\Omega_{02})\n-\\Gamma_1\\Gamma_2-I_1^2-I_2^2\n\\right]\n\\left[\n(\\omega-\\Omega_{01})\\Gamma_2\n+\n(\\omega-\\Omega_{02})\\Gamma_1\n\\right].\n\\end{eqnarray}\nThe random-averaged density of states (25)\ntransforms into (9)\nif $\\Gamma_1=\\Gamma_2=0$,\nand into the result reported in Ref. 9,\n$\\overline{\\rho(\\omega)}\n=\\mp(1/\\pi){\\mbox{Im}}\n1/\\sqrt{(\\omega\\pm{\\mbox{i}}\\Gamma-\\Omega_0)^2-4I^2},$\nif\n$\\Omega_{01}=\\Omega_{02}=\\Omega_{0},$\n$\\Gamma_1=\\Gamma_2=\\Gamma,$\n$I_1=I_2=I.$\n\nLet us discuss the effects of\nthe considered diagonal Lorentzian disorder.\nThe main effect of the randomness\nis smearing out the band structure.\nHowever, one can see a difference\nin smoothed magnon subbands for the uniform disorder\n(when $\\Gamma_1=\\Gamma_2$) (see Fig. 10a)\nand the nonuniform disorder\n(when $\\Gamma_1\\ne\\Gamma_2$) (see Fig. 11a).\nNamely, in the former case both subbands are smeared out in the same way,\nwhereas in the latter case,\nthe subbands are smeared out differently and,\nat least for small strengths of disorder, in one subband the peaks at\nthe band edges persist.\nThis circumstance in the latter case induces an interesting \nstep-like behaviour of the low-temperature\ntransverse magnetization as a function of transverse field.\nNamely, as can be seen in Fig. 11b\nthe disorder smooths only one step\nin contrast to Fig. 10b in which both steps are smeared out.\nThe difference in the influence of the uniform and nonuniform\ndisorders on other thermodynamic quantities\ncan be seen in Figs. 10c - 10f and 11c - 11f."
},
{
"name": "lrb07.tex",
"string": "\\section{Periodic nonuniform spin-$\\frac{1}{2}$ isotropic $XY$ chains\n\tand spin-Peierls instability}\n\nIn this Section we want to demonstrate that the results\nfor the density of states of the \nperiodic nonuniform \nspin-$\\frac{1}{2}$ isotropic $XY$ chains\nobtained\nwithin\nthe continued-fraction approach may be of use\nfor the study of the spin-Peierls instability in these chains\nin adiabatic limit.\nThe discovery of existence of the spin-Peierls transition in\nthe inorganic compound CuGeO$_3$$^{41,42}$\nhas stimulated much research work in this field.\nIn particular, the influence of an external field\nor randomness attracts much interest both from experimental and\ntheoretical viewpoints (see e.g. Refs. 42-49).\n\nLet us start from the non-random case.\nIn order to examine \nthe instability \nof the spin chain \nwith respect to dimerization\none must calculate the ground state energy per site\nof the regularly alternating chain\n$\\Omega_1I_1\\Omega_2I_2\\Omega_1I_1\\Omega_2I_2\\ldots$\n(see Eqs. (9) - (11))\n\\begin{eqnarray}\ne_0=\n-\\frac{1}{2}\\int\\limits_{-\\infty}^{\\infty}\n{\\mbox{d}}E\\rho(E)\\vert E\\vert\n=-\\frac{1}{2\\pi}\n\\int\\limits_{-{\\sf{b}}_1}^{-{\\sf{b}}_2}\n{\\mbox{d}}E\\frac{\\vert E\\vert\n\\left(\n\\vert E-\\hat{\\Omega}\\vert+\\vert E+\\hat{\\Omega}\\vert\n\\right)}\n{\\sqrt{-\\left(E^2-{\\sf{b}}_1^2\\right)\n\\left(E^2-{\\sf{b}}_2^2\\right)}}\n\\end{eqnarray}\nwhere $\\hat{\\Omega}=(\\Omega_1+\\Omega_2)/2$.\nDepending on the value of $\\hat{\\Omega}$\nformula (26) can be rewritten as follows\n\\begin{eqnarray}\ne_0=\n-\\frac{1}{\\pi}\n\\int\\limits_{-{\\sf{b}}_1}^{-{\\sf{b}}_2}\n{\\mbox{d}}E\\frac{\\vert\\hat{\\Omega}\\vert \\vert E\\vert}\n{\\sqrt{-\\left(E^2-{\\sf{b}}_1^2\\right)\n\\left(E^2-{\\sf{b}}_2^2\\right)}}\n\\end{eqnarray}\nif ${\\sf{b}}_1\\le \\vert\\hat{\\Omega}\\vert$,\n\\begin{eqnarray}\ne_0=\n-\\frac{1}{\\pi}\n\\int\\limits^{-\\vert\\hat{\\Omega}\\vert}_{-{\\sf{b}}_1}\n{\\mbox{d}}E\\frac{E^2}\n{\\sqrt{-\\left(E^2-{\\sf{b}}_1^2\\right)\n\\left(E^2-{\\sf{b}}_2^2\\right)}}\n-\n\\frac{1}{\\pi}\n\\int\\limits_{-\\vert\\hat{\\Omega}\\vert}^{-{\\sf{b}}_2}\n{\\mbox{d}}E\\frac{\\vert\\hat{\\Omega}\\vert \\vert E\\vert}\n{\\sqrt{-\\left(E^2-{\\sf{b}}_1^2\\right)\n\\left(E^2-{\\sf{b}}_2^2\\right)}}\n\\end{eqnarray}\nif ${\\sf{b}}_2 \\le \\vert\\hat{\\Omega}\\vert<{\\sf{b}}_1$, and\n\\begin{eqnarray}\ne_0=\n-\\frac{1}{\\pi}\n\\int\\limits_{-{\\sf{b}}_1}^{-{\\sf{b}}_2}\n{\\mbox{d}}E\\frac{E^2}\n{\\sqrt{-\\left(E^2-{\\sf{b}}_1^2\\right)\n\\left(E^2-{\\sf{b}}_2^2\\right)}}\n\\end{eqnarray}\nif $\\vert\\hat{\\Omega}\\vert<{\\sf{b}}_2$.\nIntroducing a new variable $\\varphi$ by the relation\n$E=-\\sqrt{{\\sf{b}}_1^2-({\\sf{b}}_1^2-{\\sf{b}}_2^2)\\sin^2\\varphi}$\none gets the following final expression for the ground state energy\n\\begin{eqnarray}\ne_0=\n-\\frac{1}{\\pi}\n\\left[\n{\\sf{b}}_1\n{\\mbox{E}}\n\\left(\n\\psi,\n\\frac{{\\sf{b}}_1^2-{\\sf{b}}_2^2}{{\\sf{b}}_1^2}\n\\right)\n+\n\\vert\\hat{\\Omega}\\vert\\left(\\frac{\\pi}{2}-\\psi\\right)\n\\right]\n\\end{eqnarray}\nwhere\n${\\mbox{E}}(\\psi,a^2)=\\int_0^{\\psi}{\\mbox{d}}\\varphi\n\\sqrt{1-a^2\\sin^2\\varphi}$\nis the elliptic integral of the second kind$^{50}$\nand\n\\begin{eqnarray}\n\\psi=\n\\left\\{\n\\begin{array}{ll}\n0,\n& {\\mbox{if}}\\;\\;\\;{\\sf{b}}_1\\le \\vert\\hat{\\Omega}\\vert,\\\\\n{\\mbox{arcsin}}\n\\sqrt{\\frac{{\\sf{b}}_1^2-\\hat{\\Omega}^2}\n{{\\sf{b}}_1^2-{\\sf{b}}_2^2}},\n& {\\mbox{if}}\\;\\;\\;{\\sf{b}}_2\\le \\vert\\hat{\\Omega}\\vert<{\\sf{b}}_1,\\\\\n\\frac{\\pi}{2},\n& {\\mbox{if}}\\;\\;\\;\\vert\\hat{\\Omega}\\vert<{\\sf{b}}_2.\n\\end{array}\n\\right.\n\\end{eqnarray}\n\nThe result obtained by Pincus$^{13}$\nfollows from (30), (31) if\n$\\Omega_1=\\Omega_2=0$.\nHowever,\nthe described approach permits to get the ground state\nenergy (or the Helmholtz free energy) for\nmore complicated regular nonuniformities\n(e.g., for chains with regularly alternating non-random\nor random (Lorentzian) transverse fields).\nTo demonstrate this let us consider at first\nthe spin-Peierls instability\nwith respect to dimerization\nin the presence of a non-random transverse field.\nWe introduce dimerization parameter $\\delta$ and assume in (30), (31)\n$\\vert I_1\\vert=\\vert I\\vert(1+\\delta)$,\n$\\vert I_2\\vert=\\vert I\\vert(1-\\delta)$,\n$0\\le \\delta\\le 1$.\nTaking into account that the elastic energy per site is $\\alpha\\delta^2$\none must seek the minimum of the total energy \n${\\cal{E}}(\\delta)=e_0(\\delta)+\\alpha\\delta^2$\nas a function of $\\delta$. For\n${\\cal{E}}(\\delta)$ we find\n\\begin{eqnarray}\n{\\cal{E}}(\\delta)\n=-\\frac{\\sqrt{(\\Omega_1-\\Omega_2)^2+16 I^2}}\n{2\\pi}\n{\\mbox{E}}\n\\left(\n\\psi,\n\\frac{4 I^2(1-\\delta^2)}\n{\\frac{1}{4}(\\Omega_1-\\Omega_2)^2+4 I^2}\n\\right)\n-\\vert\\Omega_1+\\Omega_2\\vert\n\\left(\n\\frac{1}{4}-\\frac{\\psi}{2\\pi}\n\\right)\n+\\alpha\\delta^2\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\psi=\n\\left\\{\n\\begin{array}{ll}\n0,\n& {\\mbox{if}}\\;\\;\\;\n\\sqrt{(\\Omega_1-\\Omega_2)^2+16 I^2}\n\\le \\vert\\Omega_1+\\Omega_2\\vert,\\\\\n{\\mbox{arcsin}}\n\\sqrt{\\frac{4 I^2-\\Omega_1\\Omega_2}{4 I^2(1-\\delta^2)}},\n& {\\mbox{if}}\\;\\;\\;\n\\sqrt{(\\Omega_1-\\Omega_2)^2+16 I^2\\delta^2}\n\\le \\vert\\Omega_1+\\Omega_2\\vert\n<\\sqrt{(\\Omega_1-\\Omega_2)^2+16 I^2},\\\\\n\\frac{\\pi}{2},\n& {\\mbox{if}}\\;\\;\\;\n\\vert\\Omega_1+\\Omega_2\\vert\n<\\sqrt{(\\Omega_1-\\Omega_2)^2+16 I^2\\delta^2} .\n\\end{array}\n\\right.\n\\end{eqnarray}\nEqs. (32), (33) in the limit of uniform field\n$\\Omega_1=\\Omega_2$\ncoincide with the result reported in Ref. 16.\nFor strong fields\n$\\vert\\Omega_1+\\Omega_2\\vert\\ge\n\\sqrt{(\\Omega_1-\\Omega_2)^2+16 I^2}$\none finds that \n${\\cal{E}}(\\delta)=-\\frac{1}{4}\\vert\\Omega_1+\\Omega_2\\vert+\\alpha\\delta^2$\nand\nthe equation\n$\\frac{\\partial{\\cal{E}}(\\delta)}{\\partial\\delta}=0$\nhas \nonly the zero solution\n$\\delta^{\\star}=0$\n(no dimerization in strong enough fields),\nwhereas for weaker fields besides the zero solution\nthere may be a non-zero one\n$\\delta^{\\star}\\ne 0$\ncoming from the equation\n\\begin{eqnarray}\n\\alpha\n=\\frac{\\sqrt{(\\Omega_1-\\Omega_2)^2+16 I^2}}\n{4\\pi(1-\\delta^2)}\n\\left[\n{\\mbox{F}}\n\\left(\n\\psi,\n\\frac{4 I^2(1-\\delta^2)}\n{\\frac{1}{4}(\\Omega_1-\\Omega_2)^2+4 I^2}\n\\right)\n-\n{\\mbox{E}}\n\\left(\n\\psi,\n\\frac{4 I^2(1-\\delta^2)}\n{\\frac{1}{4}(\\Omega_1-\\Omega_2)^2+4 I^2}\n\\right)\n\\right]\n\\end{eqnarray}\nwhere\n${\\mbox{F}}(\\psi,a^2)=\\int_0^{\\psi}{\\mbox{d}}\\varphi\n/\\sqrt{1-a^2\\sin^2\\varphi}$\nis the elliptic integral of the first kind.$^{50}$\n\nIn the following discussion of results we choose a\nuniform transverse field \n$\\Omega_1=\\Omega_2=\\Omega_0$,\n$\\vert \\Omega_0 \\vert <2\\vert I\\vert$.\nTo give a guide for further reading this paragraph\nwe summarize the main results valid for sufficiently hard\nlattices (having $\\alpha>\\frac{\\vert I\\vert}{4}$).\n(i)\nFor zero field we have a minimum of the total energy\n${\\cal{E}}(\\delta)$\nat a nonzero value of the dimerization parameter\n$\\delta^{\\star}\\ne 0$.\n(ii)\nFor finite but small fields\n${\\cal{E}}(\\delta)$\nstill exhibits one minimum\nat $\\delta^{\\star}\\ne 0$\nthe position of which remains unchanged.\n(iii)\nWhen the field achieves a certain characteristic value\n$\\Omega_{0a}$\na second local minimum appears at $\\delta^{\\star}=0$.\nThe two minima at\n$\\delta^{\\star}=0$\nand\n$\\delta^{\\star}\\ne 0$\nare separated by a maximum.\n(iv)\nAt a second characteristic field\n$\\Omega_{0b}$\nboth minima at\n$\\delta^{\\star}=0$\nand\n$\\delta^{\\star}\\ne 0$\nhave the same depth.\n(v) Further increasing $\\Omega_0$\nthe minimum at\n$\\delta^{\\star}=0$\nbecomes the global one and at a certain characteristic\nfield\n$\\Omega_{0c}$\nthe minimum at\n$\\delta^{\\star}\\ne 0$\nabruptly disappears.\nThe scenario\ndescribed in (i) -- (v) is typical for a\nfirst order transition\ncharacterized by the order parameter $\\delta^{\\star}$\nand driven by the transverse field $\\Omega_0$.\nNow we illustrate it in a more detail.\n\nIn Fig. 12 we show for different values\nof $\\alpha$\nhow the dependence of\n${\\cal{E}}(\\delta)-{\\cal{E}}(0)$\non\nthe dimerization parameter\nvaries with\nthe\nstrength of\nthe field\n$\\Omega_0$.\nAs it follows from Eqs. (32), (33)\n(and can be also\nseen in Fig. 12 where, however,\nthe difference\n${\\cal{E}}(\\delta)-{\\cal{E}}(0)$\nis depicted)\nthe total energy ${\\cal{E}}(\\delta)$\nat sufficiently large values of $\\delta$\n($\\delta\\ge\\frac{\\vert\\Omega_0\\vert}{2\\vert I\\vert}$,\n${\\cal{E}}(\\delta)-{\\cal{E}}(0)$ \nat the value\n$\\frac{\\vert\\Omega_0\\vert}{2\\vert I\\vert}$\nis denoted by dark circles in Fig. 12)\nbecomes independent\nof the field.\nIn Fig. 13 we plot the solution of Eqs. (34), (33)\nfor different lattices (i.e. different values of $\\alpha$)\nin the presence of the field.\nAs a matter of fact we calculated r.h.s. of Eq. (34)\nvarying $\\delta$ from 0 to 1 and\nfinding in such a way for what $\\alpha$ this value of $\\delta^{\\star}$\nrealizes.\nNote that solutions of Eqs. (34), (33)\n$\\delta^{\\star}$\nwhich are smaller than\n$\\frac{\\vert\\Omega_0\\vert}{2\\vert I\\vert}$\nrealize a maximum of the total energy,\nwhereas solutions\n$\\delta^{\\star}$\nwhich are larger than\n$\\frac{\\vert\\Omega_0\\vert}{2\\vert I\\vert}$\nrealize a minimum.\nThis can be seen, for example, for a lattice with\n$\\alpha=0.4$ in Figs. 12c and 13b, 13c:\nat $\\Omega_0=0.1$\nthe total energy\n${\\cal{E}}(\\delta)$ exhibits two minima at\n$\\delta^{\\star}=0$\nand\n$\\delta^{\\star}\\ne 0$\nseparated by a maximum at intermediate value of\n$\\delta^{\\star}$;\nat $\\Omega_0=0.2$\nthe total energy\n${\\cal{E}}(\\delta)$ exhibits only a minimum at\n$\\delta^{\\star}=0$.\nFrom Figs. 12, 13\nand Eqs. (34), (33)\none concludes\nthat for soft lattices having\n$\\alpha<\\frac{\\vert I\\vert}{4}$\nthere is no solution of Eqs. (34), (33)\nfulfilling the presupposition $\\delta^{\\star}\\le 1$.\nSuch lattices are excluded from further consideration.\nFor other lattices \nthe solution of Eqs. (34), (33)\n$\\delta^{\\star}\\ne 0$\nexisting for zero transverse field\ndoes not feel the presence of a small field,\nhowever, abruptly vanishes at a certain value of the\ntransverse field.\nMoreover,\nfor soft lattices one needs larger fields than for hard lattices\nfor a disappearance of the solution of Eqs. (34), (33)\n(compare Figs. 13b - 13f with Fig. 13a).\nThus, in the case of hard lattices\neven small transverse fields may destroy the dimerization.\nAs it is seen e.g.\nfor\na lattice with $\\alpha=0.2$ (Figs. 12, 13)\nabove a certain\ncharacteristic\nvalue of the transverse field\n$\\Omega_{0a}$\n(for which Eqs. (34), (33) has the solution $\\delta^{\\star}=0$)\n($\\Omega_{0a}\\approx 0.2$)\n${\\cal{E}}(\\delta)$ starts to exhibit in addition to the global\nminimum at $\\delta^{\\star}\\ne 0$,\na local one at $\\delta^{\\star}=0$,\ntwo minima are separated by a maximum at the intermediate value of\nthe dimerization parameter.\nWith increasing of $\\Omega_0$ the depths of the minima at first become\nequal\n(when $\\Omega_0$ has a characteristic value\n$\\Omega_{0b}$)\nand then the minima at $\\delta^{\\star}=0$ becomes a global one.\nThe latter minima remains the only one at\n$\\Omega_0$ having a characteristic value\n$\\Omega_{0c}$\n(for which Eqs. (34), (33) has the\nsolution $\\delta^{\\star}=\\frac{\\vert\\Omega_0\\vert}{2\\vert I\\vert}$)\n($\\Omega_{0c}\\approx 0.5$)\nthat manifests a complete suppression of the dimerization by the field.\nIn Fig. 14 we show different regions in the plane transverse field\n$\\Omega_0$ -- lattice parameter $\\alpha$\nin which\n${\\cal{E}}(\\delta),$ \n$0\\le\\delta\\le 1$\nexhibits one minimum at\n$\\delta^{\\star}\\ne 0$ (region A),\ntwo minima at\n$\\delta^{\\star}=0$ and $\\delta^{\\star}\\ne 0$\nseparated by a maximum\n(regions B$_1$ and B$_2$;\nin the region B$_1$ the minimum at $\\delta^{\\star}\\ne 0$ is deeper,\nwhereas in the region B$_2$ the minimum at $\\delta^{\\star}=0$ is deeper),\none minimum at\n$\\delta^{\\star}=0$ (region C).\nTo find the line that separates B$_1$ and B$_2$ \none must find for a given $\\Omega_0$ such a $\\delta^{\\star}$ \nat which ${\\cal{E}}(\\delta)-{\\cal{E}}(0)$ (32), (33) with \n$\\alpha$ given by the r.h.s. of Eq. (34), (33) equals to zero,\nand then to evaluate the r.h.s. of Eq. (34) at the sought $\\delta^{\\star}$. \nCrossing the phase diagram by a vertical line corresponding\nto a certain lattice (e.g. with $\\alpha=0.2$ in Fig. 14)\none obtains the field at which the first order transition\nbetween the dimerized and uniform phases occur\n($\\Omega_{0b}$ in Fig. 14)\nand the width of hysteresis\n(determined by\n$\\Omega_{0a}$ and $\\Omega_{0c}$\nin Fig. 14).\n\nNext we consider the influence of\na random Lorentzian transverse field\non the spin-Peierls instability with respect to\ndimerization. For that we calculate the difference\nin random-averaged total energy\n(to avoid non-physical infinities\ndue to the Lorentzian probability distribution)\n\\begin{eqnarray}\n\\overline{{\\cal{E}}(\\delta)}\n-\\overline{{\\cal{E}}(0)}\n=-\\frac{1}{2}\\int\\limits_{-\\infty}^{\\infty}\n{\\mbox{d}}E\n\\left(\\overline{\\rho_{\\delta}(E)}\n-\\overline{\\rho_0(E)}\\right)\n\\vert E\\vert\n+\\alpha\\delta^2\n\\end{eqnarray}\nwith $\\overline{\\rho_{\\delta}(E)}$ given by Eq. (26)\nwhere\n$\\vert I_1\\vert=\\vert I\\vert(1+\\delta)$,\n$\\vert I_2\\vert=\\vert I\\vert(1-\\delta)$.\nLet us start from the case\n$\\Omega_{01}=\\Omega_{02}=0$,\n$\\Gamma_1=\\Gamma_2=\\Gamma$\ngeneralizing in such a way\nthe consideration\nfor the zero transverse field by assuming the latter to be random\n(Lorentzian) with the zero mean value.\nAs can be seen in Fig. 15\nthe randomness leads\nto a continuous decrease of the non-zero value of\ndimerization parameter\nat which\nthe random-averaged total energy\nexhibits minimum. At\nsufficiently large strengths of disorder $\\Gamma$\nthe minimum of\nthe random-averaged total energy\noccurs already at\nthe zero\ndimerization parameter, i.e.\nrandomness acts against dimerization and may suppress it completely for\nsufficiently large strength of disorder.\nConsidering the equation\n\\begin{eqnarray}\n\\frac{\\partial\\overline{{\\cal{E}}(\\delta)}}{\\partial\\delta}\n=-\\frac{1}{2}\n\\int\\limits_{-\\infty}^{\\infty}\n{\\mbox{d}}E\n\\frac{\\partial\\overline{\\rho_{\\delta}(E)}}{\\partial\\delta}\n\\vert E\\vert\n+2\\alpha\\delta=0\n\\end{eqnarray}\none can find its solution\n$\\delta^{\\star}$ for different $\\Gamma$\n(see Fig. 16).\nFrom Fig. 16 one sees\nthat in the case of hard lattices even small disorder may destroy the\ndimerization.\nIn Fig. 17 we depicted different regions in the plane\nstrength of disorder $\\Gamma$ -- lattice parameter $\\alpha$\nin which\n$\\overline{{\\cal{E}}(\\delta)}-\\overline{{\\cal{E}}(0)}$,\n$0\\le \\delta\\le 1$\nexhibits one minimum at $\\delta^{\\star}\\ne 0$\n(region A) or\none minimum at $\\delta^{\\star}=0$\n(region C). \nThe boundary curve between the regions C and A is obtained by calculating \n$\\alpha$ from (36) with varying $\\Gamma$ for fixed $\\delta=0$.\nThus, the random field with zero mean value suppresses dimerization\nwith increasing the strength of disorder,\nhowever the dimerization parameter\n$\\delta^{\\star}$ vanishes according to a second order phase transition\nscenario in contrast to the previous case.\n\nFinally we consider the case of random field with non-zero average\nvalue, i.e.,\n$\\Omega_{01}=\\Omega_{02}=\\Omega_0\\ne 0$,\n$\\Gamma_1=\\Gamma_2=\\Gamma$.\nFor small strengths of randomness\n$\\Gamma$\nthe above discussed scenario of one or two minimum in\n$\\overline{{\\cal{E}}(\\delta)}$\nin dependence of the value of the field remains valid.\nA switching on randomness for a system being in the region A\nat $\\Gamma=0$ (Fig. 14)\nleads to continuous decreasing of $\\delta^{\\star}\\ne 0$\nto zero.\nFor a system being in the regions B$_1$ or B$_2$\nan increasing of randomness usually leads \nat first to\na continuous decrease of\n$\\delta^{\\star}\\ne 0$ with a decrease of the depth of that minimum\nand then to an abrupt disappearance of\n$\\delta^{\\star}\\ne 0$ above a certain strength of disorder. \nWe also observed another influence of small randomness for a system \nbeing in the region B$_1$, namely, an increasing of randomness \nleads at first to a disappearance of the minimum at $\\delta^{\\star}=0$\nthat appears again for larger strength of disorder.\nThe details can be traced\nin Fig. 18\nwhere we plotted the dependence\n$\\overline{{\\cal{E}}(\\delta)}-\\overline{{\\cal{E}}(0)}$\nvs $\\delta$ for different $\\Gamma$\nconsidering two mean values of the random transverse field\n$\\Omega_0=0.1$\nand\n$\\Omega_0=0.3$\nand in Fig. 19 where we illustrated\nthe vanishing and appearance of the minimum at \n$\\delta^{\\star}=0$ with increase of randomness.\nBoth the one minimum profile (solid curve in Fig. 18b)\nand the two minima profile (solid curves in Figs. 18c, 18e)\nof that dependence existing in the non-random case\n$\\Gamma=0$\nare finally destroyed by increasing disorder.\nThe phase diagrams in\nthe $\\Gamma$ -- $\\alpha$ plane\nfor the two mentioned values of $\\Omega_0$\nare shown in\nFig. 20.\n\nClosing this Section,\nwe want to make some comments concerning\nthe conclusions on spin-Peierls\ninstability that can be drawn using exact results for thermodynamic\nquantities of regularly nonuniform spin-$\\frac{1}{2}$ isotropic $XY$\nchain in a transverse field.\nAlthough the described basic\npicture of a first order phase transition in\na uniform field\nseems to be qualitatively correct\nwe should keep in mind\nthat an increasing of\nfield at low temperature leads to a transition from dimerized to\nincommensurate phase. This fact was observed experimentally and analysed\ntheoretically mainly for the models of CuGeO$_3$ in a number of\npapers.$^{42,51-54}$\nClearly, the simple ansatz for the lattice distortion\n$\\delta_1\\delta_2\\delta_1\\delta_2\\ldots\\;$,\n$\\delta_1+\\delta_2=0$ permitted us to compare the ground state energies\nonly for dimerized and uniform phases. To detect a transition from the\ndimerized to the incommensurate phase with increasing of field one may\nanalyse the ground state energy of a chain having larger period, say 12.\nThe presence of randomness requires even more complicated lattice\ndistortions to be examined and the continued-fraction approach for\nrigorous study of thermodynamics of the regularly alternating\nspin-$\\frac{1}{2}$ isotropic $XY$ chain in a transverse field provides\nsome possibilities to perform such an analysis.\nWe must also keep in mind that the known spin-Peierls compounds\nare described by\nthe spin-$\\frac{1}{2}$ isotropic Heisenberg chain rather than\n$XY$ chain, however, one may expect that the basic features\nof the studied phenomenon\nshould be similar\nfor both quantum spin models.\n\n\\section{Summary}\n\nTo summarize, we have studied\nrigorously the magnon density of states and the thermodynamics\nof the periodic nonuniform\nspin-$\\frac{1}{2}$ isotropic $XY$ chain\nin non-random/random (Lorentzian) transverse field.\nWe have exploited the Jordan-Wigner transformation, the temperature \ndouble-time Green functions and the continued fractions. The Green \nfunctions approach seems to be the most convenient tool for a study of \nthermodynamics of the considered spin chains since it permits to examine \nsuch models\nwith regular nonuniformity or some type of randomness or both. \nRegular\nnonuniformity leads to a splitting of the magnon band into subbands\nthat in its turn leads to\nsome spectacular changes in the behaviour of \nthe gap in the energy spectrum and the \nthermodynamic quantities.\nIn particular, \nthe low-temperature dependence\nof the transverse magnetization on the transverse field is composed of \nsharply\nincreasing parts separated by plateaus, the temperature dependence\nof specific heat may exhibit a well pronounced two-peak structure,\nthe temperature dependence of the initial\ntransverse linear susceptibility may be enhanced or suppressed.\nRegularly nonuniform spin-$\\frac{1}{2}$ isotropic $XY$ chain\nmay exhibit a non-zero transverse magnetization\nat the zero average transverse field.\nThe regularly alternating Lorentzian disorder in the transverse field\nmay in specific manner influence the thermodynamic quantities leading, for\ninstance,\nto a smearing out of only one `step'\nin the step-like dependence of the\ntransverse magnetization versus the transverse field\nat $T=0$.\nThe derived results for the \n(random-averaged) ground state energy permit to analyse\nthe effects of external non-random/random field\non the spin-Peierls instability.\nBoth, magnetic field as well as randomness may destroy\nthe dimerization\nas the analysis of the (random-averaged) total energy manifests.\n\nThe presented treatment\nof the regularly periodic spin-$\\frac{1}{2}$ isotropic $XY$ chains\nis restricted to the density of states and therefore only to thermodynamics.\nIt will be interesting to\nstudy the effects of periodic nonuniformity\non spin correlations and their dynamics\nespecially for a model of spin-Peierls instability.\nSome work for the\ndynamic $zz$ spin correlations for such models has been done \nin Ref. 16.\nAnother interesting problem concerns\nthe treatment of\nthe periodic nonuniform spin-$\\frac{1}{2}$ transverse $XY$ chains\nwith an\nanisotropic exchange coupling\n(and in particular the extremely anisotropic case, i.e.\nthe spin-$\\frac{1}{2}$ transverse Ising chain).\nSome results for \nthermodynamics of\nsuch regularly nonuniform chains\nhaving period 2\nwere obtained\nin Refs. 14, 17, 23.\nTheir relation to the spin-Peierls instability\nseems to be an intriguing issue.\n\n\\section*{Acknowledgments}\n\nThe present study was partly supported\nby the DFG (projects 436 UKR 17/20/98 and Ri 615/6-1).\nO. 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B {\\bf{7,}} 67 (1999).\n\\end{thebibliography}\n\n\\clearpage\n%\\vspace{10mm}\n\n\\noindent\n{\\bf List of figure captions}\n\n\\vspace{1.25cm}\n\nFIG. 1.\nMagnon band structure for periodic chains\n$\\Omega_1I_1\\Omega_2I_2\\Omega_1I_1\\Omega_2I_2\\ldots,$\n$\\Omega_j=\\Omega_0+\\Omega_j^{\\prime}$;\nthe shadowed areas correspond to the allowed magnon energies.\na) $\\Omega_1^{\\prime}+\\Omega_2^{\\prime}=2$,\n$\\vert I_1\\vert=\\vert I_2\\vert=0.5$;\nb) $\\Omega_1^{\\prime}+\\Omega_2^{\\prime}=2$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.25$;\nc) $\\Omega_1^{\\prime}=\\Omega_2^{\\prime}=1$,\n$\\vert I_1\\vert+\\vert I_2\\vert=1$;\nd) $\\Omega_1^{\\prime}=1.5$, $\\Omega_2^{\\prime}=0.5$,\n$\\vert I_1\\vert+\\vert I_2\\vert=1$.\nThe horizontal lines single out the following particular chains:\n$\\Omega_1^{\\prime}=\\Omega_2^{\\prime}=1$,\n$\\vert I_1\\vert=\\vert I_2\\vert=0.5$ (dotted curves),\n$\\Omega_1^{\\prime}=2$, $\\Omega_2^{\\prime}=0$,\n$\\vert I_1\\vert=\\vert I_2\\vert=0.5$ (dashed curve),\n$\\Omega_1^{\\prime}=\\Omega_2^{\\prime}=1$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.25$ (dashed-dotted curves),\n$\\Omega_1^{\\prime}=1.5$, $\\Omega_2^{\\prime}=0.5$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.25$ (solid curves).\n\n\\vspace{1.25cm}\n\nFIG. 2.\nThe density of states (a),\nthe dependence of the transverse magnetization\non transverse field at $T=0$ (b),\nthe temperature dependence of the entropy (c),\nspecific heat (d),\ntransverse magnetization (e),\nand static linear transverse susceptibility (f) at $\\Omega_0=0$\nfor\nperiodic chains $\\Omega_1I_1\\Omega_2I_2\\Omega_1I_1\\Omega_2I_2\\ldots,$\n$\\Omega_j=\\Omega_0+\\Omega_j^{\\prime}$.\nThe dotted curves correspond to the uniform case\n$\\Omega_1^{\\prime}=\\Omega_2^{\\prime}=1$,\n$\\vert I_1\\vert=\\vert I_2\\vert=0.5$,\nthe dashed curves correspond to the case\n$\\Omega_1^{\\prime}=2$, $\\Omega_2^{\\prime}=0$,\n$\\vert I_1\\vert=\\vert I_2\\vert=0.5$,\nthe dashed-dotted curves correspond to the case\n$\\Omega_1^{\\prime}=\\Omega_2^{\\prime}=1$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.25$,\nand the solid curves correspond to the case\n$\\Omega_1^{\\prime}=1.5$, $\\Omega_2^{\\prime}=0.5$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.25$.\n\n\\vspace{1.25cm}\n\nFIG. 3.\nMagnon band structure for periodic chains\n$\\Omega_1I_1\\Omega_2I_2\\Omega_3I_3\\Omega_1I_1\\Omega_2I_2\\Omega_3I_3\\ldots,$\n$\\Omega_j=\\Omega_0+\\Omega_j^{\\prime}$;\nthe shadowed areas correspond to the allowed magnon energies.\na) $\\Omega_1^{\\prime}+\\Omega_2^{\\prime}+\\Omega_3^{\\prime}=3$,\n$\\Omega_1^{\\prime}-2\\Omega_2^{\\prime}+\\Omega_3^{\\prime}=0$,\n$\\vert I_1\\vert=\\vert I_2\\vert=\\vert I_3\\vert=0.5$;\nb) $\\Omega_1^{\\prime}+\\Omega_2^{\\prime}+\\Omega_3^{\\prime}=3$,\n$\\Omega_1^{\\prime}-2\\Omega_2^{\\prime}+\\Omega_3^{\\prime}=1.5$,\n$\\vert I_1\\vert=\\vert I_2\\vert=\\vert I_3\\vert=0.5$;\nc) $\\Omega_1^{\\prime}+\\Omega_2^{\\prime}+\\Omega_3^{\\prime}=3$,\n$\\Omega_1^{\\prime}-2\\Omega_2^{\\prime}+\\Omega_3^{\\prime}=0$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.5$, $\\vert I_3\\vert=0.25$;\nd) $\\Omega_1^{\\prime}+\\Omega_2^{\\prime}+\\Omega_3^{\\prime}=3$,\n$\\Omega_1^{\\prime}-2\\Omega_2^{\\prime}+\\Omega_3^{\\prime}=1.5$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.5$, $\\vert I_3\\vert=0.25$;\ne) $\\Omega_1^{\\prime}=\\Omega_2^{\\prime}=\\Omega_3^{\\prime}=1$,\n$\\vert I_1\\vert+\\vert I_2\\vert+\\vert I_3\\vert=1.5$,\n$\\vert I_1\\vert-2\\vert I_2\\vert+\\vert I_3\\vert=0$;\nf) $\\Omega_1^{\\prime}=\\Omega_2^{\\prime}=\\Omega_3^{\\prime}=1$,\n$\\vert I_1\\vert+\\vert I_2\\vert+\\vert I_3\\vert=1.5$,\n$\\vert I_1\\vert-2\\vert I_2\\vert+\\vert I_3\\vert=0.75$;\ng) $\\Omega_1^{\\prime}=1.5$, $\\Omega_2^{\\prime}=1$, $\\Omega_3^{\\prime}=0.5$,\n$\\vert I_1\\vert+\\vert I_2\\vert+\\vert I_3\\vert=1.5$,\n$\\vert I_1\\vert-2\\vert I_2\\vert+\\vert I_3\\vert=0$;\nh) $\\Omega_1^{\\prime}=1.5$, $\\Omega_2^{\\prime}=1$, $\\Omega_3^{\\prime}=0.5$,\n$\\vert I_1\\vert+\\vert I_2\\vert+\\vert I_3\\vert=1.5$,\n$\\vert I_1\\vert-2\\vert I_2\\vert+\\vert I_3\\vert=0.75$.\nThe horizontal lines single out the following particular chains:\n$\\Omega_1^{\\prime}=\\Omega_2^{\\prime}=\\Omega_3^{\\prime}=1$,\n$\\vert I_1\\vert=\\vert I_2\\vert=\\vert I_3\\vert=0.5$ (dotted curves),\n$\\Omega_1^{\\prime}=2.5$, $\\Omega_2^{\\prime}=0.5$, $\\Omega_3^{\\prime}=0$,\n$\\vert I_1\\vert=\\vert I_2\\vert=\\vert I_3\\vert=0.5$ (dashed curve),\n$\\Omega_1^{\\prime}=1$, $\\Omega_2^{\\prime}=0.5$, $\\Omega_3^{\\prime}=1.5$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.5$, $\\vert I_2\\vert=0.25$\n(dashed-dotted curve),\n$\\Omega_1^{\\prime}=1.5$, $\\Omega_2^{\\prime}=1$, $\\Omega_3^{\\prime}=0.5$,\n$\\vert I_1\\vert=1$, $\\vert I_2\\vert=\\vert I_3\\vert=0.25$\n(solid curve).\n\n\\vspace{1.25cm}\n\nFIG. 4.\nThe same as in Fig. 2 for periodic chains\n$\\Omega_1I_1\\Omega_2I_2\\Omega_3I_3\\Omega_1I_1\\Omega_2I_2\\Omega_3I_3\\ldots,$\n$\\Omega_j=\\Omega_0+\\Omega_j^{\\prime}.$\nThe dotted, dashed, dashed-dotted, and solid curves correspond\nto the cases pointed out in the capture to Fig. 3.\n\n\\vspace{1.25cm}\n\nFIG. 5.\nThe same as in Figs. 1, 3 for periodic chains\nhaving a period 12,\n$\\Omega_1I_1\\ldots\\Omega_{12}I_{12}\n\\Omega_1I_1\\ldots\\Omega_{12}I_{12}\\ldots,$\n$\\Omega_1=\\Omega_2=\\ldots=\\Omega_6,$\n$\\Omega_7=\\Omega_8=\\ldots=\\Omega_{12},$\n$I_1=I_2=\\ldots=I_6,$\n$I_7=I_8=\\ldots=I_{12},$\n$\\Omega_j=\\Omega_0+\\Omega_j^{\\prime}$.\na) $\\Omega_1^{\\prime}+\\Omega_7^{\\prime}=2$,\n$\\vert I_1\\vert=\\vert I_7\\vert=0.5$;\nb) $\\Omega_1^{\\prime}+\\Omega_7^{\\prime}=2$,\n$\\vert I_1\\vert=0.75$, $\\vert I_7\\vert=0.25$;\nc) $\\Omega_1^{\\prime}=\\Omega_7^{\\prime}=1$,\n$\\vert I_1\\vert+\\vert I_7\\vert=1$;\nd) $\\Omega_1^{\\prime}=1.5$, $\\Omega_7^{\\prime}=0.5$,\n$\\vert I_1\\vert+\\vert I_7\\vert=1$.\nThe horizontal lines single out the following particular chains:\n$\\Omega_1^{\\prime}=\\Omega_7^{\\prime}=1$,\n$\\vert I_1\\vert=\\vert I_7\\vert=0.5$ (dotted curves),\n$\\Omega_1^{\\prime}=2$, $\\Omega_7^{\\prime}=0$,\n$\\vert I_1\\vert=\\vert I_7\\vert=0.5$ (dashed curve),\n$\\Omega_1^{\\prime}=\\Omega_7^{\\prime}=1$,\n$\\vert I_1\\vert=0.75$, $\\vert I_7\\vert=0.25$ (dashed-dotted curves),\n$\\Omega_1^{\\prime}=1.5$, $\\Omega_7^{\\prime}=0.5$,\n$\\vert I_1\\vert=0.75$, $\\vert I_7\\vert=0.25$ (solid curves).\n\n\\vspace{1.25cm}\n\nFIG. 6.\nThe same as in Figs. 2, 4 for the chains singled out in Fig. 5.\n\n\\vspace{1.25cm}\n\nFIG. 7.\nThe dependence of the energy gap\n$\\Delta$\nbetween the ground state and the first \nexcited state on transverse field\n$\\Omega_0$\nfor certain regularly nonuniform\nchains. a) The chain \n$\\Omega_0I_1\\Omega_0I_2\\Omega_0I_1\\Omega_0I_2\\ldots\\;$,\n$\\vert I_1\\vert=0.75$,\n$\\vert I_2\\vert=0.25$;\nb) - d) the chains having periods 2, 3, and 12, respectively,\nwith the notations as in Figs. 2, 4, 6.\n\n\\vspace{1.25cm}\n\nFIG. 8.\nThe dependence of the transverse magnetization on the transverse field\n$\\Omega_0$\nat\n$T=0$\nfor classical periodic nonuniform isotropic $XY$ chains in a\ntransverse field.\na) Chains having a period 2\n($\\Omega^{\\prime}_1=2$, $\\Omega^{\\prime}_2=0$,\n$\\vert I_1\\vert=\\vert I_2\\vert=0.5$ (dashed curve),\n$\\Omega^{\\prime}_1=\\Omega^{\\prime}_2=1$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.25$ (dashed-dotted curve),\n$\\Omega^{\\prime}_1=1.5$, $\\Omega^{\\prime}_2=0.5$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.25$ (solid curve));\nb) chains having a period 3\n($\\Omega^{\\prime}_1=2.5$, $\\Omega^{\\prime}_2=0.5$, $\\Omega^{\\prime}_3=0$,\n$\\vert I_1\\vert=\\vert I_2\\vert=\\vert I_3\\vert=0.5$ (dashed curve),\n$\\Omega^{\\prime}_1=1$, $\\Omega^{\\prime}_2=0.5$, $\\Omega^{\\prime}_3=1.5$,\n$\\vert I_1\\vert=0.75$,\n$\\vert I_2\\vert=0.5$,\n$\\vert I_3\\vert=0.25$ (dashed-dotted curve),\n$\\Omega^{\\prime}_1=1.5$, $\\Omega^{\\prime}_2=1$, $\\Omega^{\\prime}_3=0.5$,\n$\\vert I_1\\vert=1$,\n$\\vert I_2\\vert=\\vert I_3\\vert=0.25$ (solid curve));\nc) chains having a period 12\n($\\Omega^{\\prime}_1=2$, $\\Omega^{\\prime}_7=0$,\n$\\vert I_1\\vert=\\vert I_7\\vert=0.5$ (dashed curve),\n$\\Omega^{\\prime}_1=\\Omega^{\\prime}_7=1$,\n$\\vert I_1\\vert=0.75$, $\\vert I_7\\vert=0.25$ (dashed-dotted curve),\n$\\Omega^{\\prime}_1=1.5$, $\\Omega^{\\prime}_7=0.5$,\n$\\vert I_1\\vert=0.75$, $\\vert I_7\\vert=0.25$ (solid curve)).\n\n\\vspace{1.25cm}\n\nFIG. 9.\nIllustration of the existence of a non-zero transverse magnetization\nat the zero average transverse field\nin a chain having period 4.\n$\\Omega^{\\prime}_1=\\Omega^{\\prime}_3=0$,\n$\\Omega^{\\prime}_2=-\\Omega^{\\prime}_4=-1$,\n$\\vert I_1\\vert=\\vert I_2\\vert=0.5$,\n$\\vert I_3\\vert=\\vert I_4\\vert=0$ (solid curves),\n$\\vert I_3\\vert=\\vert I_4\\vert=0.05$ (dashed curves),\n$\\vert I_3\\vert=\\vert I_4\\vert=0.25$ (dotted curves).\n\n\\vspace{1.25cm}\n\nFIG. 10.\nThe random-averaged density of states (a),\nthe dependence of the transverse magnetization\non transverse field at $T=0$ (b),\nthe temperature dependence of the entropy (c),\nspecific heat (d),\ntransverse magnetization (e),\nand static linear transverse susceptibility (f) at $\\Omega_0=0$\nfor periodic chains\n\\linebreak\n$\\Omega_{01}\\Gamma_1I_1\\Omega_{02}\\Gamma_2I_2\n\\Omega_{01}\\Gamma_1I_1\\Omega_{02}\\Gamma_2I_2\\ldots,$\n$\\Omega_{0j}=\\Omega_0+\\Omega_j^{\\prime}$,\n$\\Omega_1^{\\prime}=1.5$, $\\Omega_2^{\\prime}=0.5$,\n$\\vert I_1\\vert=0.75$, $\\vert I_2\\vert=0.25$\nfor the case of uniform disorder $\\Gamma_1=\\Gamma_2=\\Gamma$.\nThe solid curves correspond to the non-random case $\\Gamma=0$;\nthe long-dashed curves correspond to $\\Gamma=0.1$;\nthe short-dashed curves correspond to $\\Gamma=0.25$;\nthe dotted curves correspond to $\\Gamma=0.5$.\n\n\\vspace{1.25cm}\n\nFIG. 11.\nThe same as in Fig. 10 for nonuniform disorder\n$\\Gamma_1\\ne 0$, $\\Gamma_2=0$.\nThe solid curves correspond to the non-random case $\\Gamma_1=0$;\nthe long-dashed curves correspond to $\\Gamma_1=0.1$;\nthe short-dashed curves correspond to $\\Gamma_1=0.25$;\nthe dotted curves correspond to $\\Gamma_1=0.5$.\n\n\\vspace{1.25cm}\n\nFIG. 12.\nChange of\nthe total energy\n${\\cal{E}}(\\delta)-{\\cal{E}}(0)$\nas a function of the dimerization parameter $\\delta$\nin the presence of the uniform transverse field;\n$\\vert I\\vert=0.5$;\na) $\\alpha=0$,\nb) $\\alpha=0.2$,\nc) $\\alpha=0.4$;\n$\\Omega_0=0$ (solid curves),\n$\\Omega_0=0.1$ (dashed-dotted-dotted curves),\n$\\Omega_0=0.2$ (dashed-dotted curves),\n$\\Omega_0=0.3$ (dashed curves),\n$\\Omega_0=0.4$ (dotted curves).\n\n\\vspace{1.25cm}\n\nFIG. 13.\nDimerization parameter $\\delta^{\\star}$\nas a function of $\\alpha$\nin the presence of a uniform transverse field $\\Omega_0$;\n$\\vert I\\vert=0.5$;\n$\\Omega_0=0$ (a),\n$\\Omega_0=0.1$ (b),\n$\\Omega_0=0.2$ (c),\n$\\Omega_0=0.3$ (d),\n$\\Omega_0=0.4$ (e),\n$\\Omega_0=0.5$ (f).\nThe solid curves show the solution of Eqs. (34), (33) corresponding to a\nminimum of the total energy;\nthe dashed curve in (a) corresponds to the \ndependence $\\delta^{\\star}$ versus $\\alpha$\nvalid for hard lattices \nthat was\nobtained in Ref. 13;\nthe dashed curves in (b) - (f) \nshow the solution of Eqs. (34), (33)\ncorresponding to a maximum\nof the total energy.\n\n\\vspace{1.25cm}\n\nFIG. 14.\nDifferent types of solution for\nthe dimerization parameter $\\delta^{\\star}$\n($0\\le \\delta^{\\star}\\le 1$)\nin the plane\n$\\Omega_0$ -- $\\alpha$;\n$\\vert I\\vert=0.5$.\nRegion A:\n${\\cal{E}}(\\delta)$ has one minimum at\n$\\delta^{\\star}\\ne 0$,\nregions B$_1$, B$_2$:\n${\\cal{E}}(\\delta)$ has two minima at\n$\\delta^{\\star}=0$ (favourable in B$_2$)\nand\n$\\delta^{\\star}\\ne 0$ (favourable in B$_1$)\nseparated by\na maximum,\nmoreover,\nthe depths of the minima\nat the line\nthat separates B$_1$ and B$_2$\nare the same;\nregion C:\n${\\cal{E}}(\\delta)$ has one minimum at\n$\\delta^{\\star}=0$.\n\n\\vspace{1.25cm}\n\nFIG. 15.\nChange of the\nrandom-averaged total energy\nas a function of the dimerization parameter\nin the presence of \na uniform random Lorentzian transverse field with zero mean\nvalue;\n$\\vert I\\vert=0.5$,\n$\\Gamma_1=\\Gamma_2=\\Gamma=0$ (solid curves),\n$\\Gamma=0.02$ (dashed-dotted curves),\n$\\Gamma=0.1$ (dashed curves),\n$\\Gamma=0.5$ (dotted curves);\na) $\\alpha=0$,\nb) $\\alpha=0.2$,\nc) $\\alpha=0.4$.\n\n\\vspace{1.25cm}\n\nFIG. 16.\nThe solution of Eq. (36) as a function of $\\alpha$\nin the presence of disorder;\n$\\vert I\\vert=0.5$,\n$\\Omega_{01}=\\Omega_{02}=0$,\n$\\Gamma_1=\\Gamma_2=\\Gamma=0$ (solid curves),\n$\\Gamma=0.02$ (dashed-dotted curves),\n$\\Gamma=0.1$ (dashed curves),\n$\\Gamma=0.5$ (dotted curves).\n\n\\vspace{1.25cm}\n\nFIG. 17.\nDifferent types of solution for the dimerization parameter $\\delta^{\\star}$\nin the plane $\\Gamma$ -- $\\alpha$;\n$\\vert I\\vert=0.5$,\n$\\Omega_0=0$.\nRegion A:\n$\\overline{{\\cal{E}}(\\delta)}\n-\\overline{{\\cal{E}}(0)}$\nhas one minimum at $\\delta^{\\star}\\ne 0$,\nregion C:\n$\\overline{{\\cal{E}}(\\delta)}\n-\\overline{{\\cal{E}}(0)}$\nhas one minimum at $\\delta^{\\star}=0$.\n\n\\vspace{1.25cm}\n\nFIG. 18.\nChange of the\nrandom-averaged total energy\nas a function of the dimerization parameter\nin the presence of \nthe uniform random Lorentzian transverse field with a non-zero mean\nvalue\n$\\Omega_0=0.1$ (a, b, c)\nand\n$\\Omega_0=0.3$ (d, e, f);\n$\\vert I\\vert=0.5$,\n$\\Gamma_1=\\Gamma_2=\\Gamma=0$ (solid curves),\n$\\Gamma=0.02$ (dashed-dotted curves),\n$\\Gamma=0.1$ (dashed curves),\n$\\Gamma=0.5$ (dotted curves);\n$\\alpha=0$ (a, d),\n$\\alpha=0.2$ (b, e),\n$\\alpha=0.4$ (c, f).\n\n\\vspace{1.25cm}\n\nFIG. 19.\nChange of $\\overline{{\\cal{E}}(\\delta)}-\\overline{{\\cal{E}}(0)}$ \nas a function of $\\delta$ in the presence of the uniform random Lorentzian \ntransverse field with \n$\\Omega_0=0.3$, \n$\\Gamma=0.01$ (solid curves),\n$\\Gamma=0.1$ (dashed-dotted curves),\n$\\Gamma=0.2$ (dashed curves),\n$\\Gamma=0.3$ (dotted curves);\n$\\vert I\\vert=0.5$, $\\alpha=0.15$.\n\n\\vspace{1.25cm}\n\nFIG. 20.\nDifferent types of solution for the dimerization parameter\n$\\delta^{\\star}$ in the plane\nin the plane $\\Gamma$ -- $\\alpha$;\n$\\vert I\\vert=0.5$,\n$\\Omega_0=0.1$ (a),\n$\\Omega_0=0.3$ (b).\nRegion A:\n$\\overline{{\\cal{E}}(\\delta)}\n-\\overline{{\\cal{E}}(0)}$\nhas one minimum at $\\delta^{\\star}\\ne 0$,\nregion B$_1$:\n$\\overline{{\\cal{E}}(\\delta)}\n-\\overline{{\\cal{E}}(0)}$\nhas two minima at\n$\\delta\\ne 0$\nand\n$\\delta=0$\nand the first one is favourable,\nregion B$_2$:\n$\\overline{{\\cal{E}}(\\delta)}\n-\\overline{{\\cal{E}}(0)}$\nhas two minima at\n$\\delta\\ne 0$\nand\n$\\delta=0$\nand the second one is favourable,\nregion C:\n$\\overline{{\\cal{E}}(\\delta)}\n-\\overline{{\\cal{E}}(0)}$\nhas one minimum at $\\delta^{\\star}=0$.\n\n%\\end{document}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g01.eps}\n%\\end{center}\n%\\vspace{-4mm}\n\\vspace{15mm}\n\\caption{FIGURE 1.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g02.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 2.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g03.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 3.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g04.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 4.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g05.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 5.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g06.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 6.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g07.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 7.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g08.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 8.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g09.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 9.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g10.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 10.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=120mm\n\\epsfbox{g11.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 11.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=150mm\n\\epsfbox{g12.eps}\n%\\end{center}\n\\vspace{-30mm}\n\\caption{FIGURE 12.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=140mm\n\\epsfbox{g13.eps}\n%\\end{center}\n\\vspace{15mm}\n\\caption{FIGURE 13.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=200mm\n\\epsfbox{g14.eps}\n%\\end{center}\n\\vspace{-60mm}\n\\caption{FIGURE 14.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=150mm\n\\epsfbox{g15.eps}\n%\\end{center}\n\\vspace{-30mm}\n\\caption{FIGURE 15.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\vspace{15mm}\n\\epsfxsize=180mm\n\\epsfbox{g16.eps}\n%\\end{center}\n\\vspace{-35mm}\n\\caption{FIGURE 16.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=200mm\n\\epsfbox{g17.eps}\n%\\end{center}\n\\vspace{-60mm}\n\\caption{FIGURE 17.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=150mm\n\\epsfbox{g18.eps}\n%\\end{center}\n\\vspace{-30mm}\n\\caption{FIGURE 18.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=150mm\n\\epsfbox{g19.eps}\n%\\end{center}\n\\vspace{-60mm}\n\\caption{FIGURE 19.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n%\\begin{center}\n\\epsfxsize=180mm\n\\epsfbox{g20.eps}\n%\\end{center}\n\\vspace{-60mm}\n\\caption{FIGURE 20.}\n\\end{figure}\n\n\n\\end{document}"
},
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"name": "lrm07.tex",
"string": "%\\documentstyle[12pt,epsf]{article}\n\\documentstyle[epsf]{article}\n\\oddsidemargin -5mm\n\\topmargin -20mm\n\\textwidth 170mm\n\\textheight 245mm\n\n\\title{Thermodynamic properties\n \\protect\\\\\n of the periodic nonuniform spin-$\\frac{1}{2}$ isotropic $XY$ chains\n \\protect\\\\\n in a transverse field}\n\\author{Oleg Derzhko$^{\\dagger,\\ddagger}$,\n\tJohannes Richter$^{\\star}$\n and\n Oles' Zaburannyi$^{\\dagger}$\\\\\n\\small {\\em $^{\\dagger}${Institute for Condensed Matter Physics}}\\\\\n\\small {\\em {1 Svientsitskii St., L'viv-11, 290011, Ukraine}}\\\\\n\\small {\\em $^{\\ddagger}${Chair of Theoretical Physics,\n Ivan Franko State University of L'viv}}\\\\\n\\small {\\em {12 Drahomanov St., L'viv-5, 290005, Ukraine}}\\\\\n\\small {\\em $^{\\star}${Institut f\\\"{u}r Theoretische Physik,\n\t\t\t Universit\\\"{a}t Magdeburg}}\\\\\n\\small {\\em {P.O. Box 4120, D-39016 Magdeburg, Germany}}}\n\n\\date{\\today}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nUsing the Jordan-Wigner transformation and the continued-fraction method\nwe calculate exactly the density of states\nand thermodynamic quantities\nof the periodic nonuniform\nspin-$\\frac{1}{2}$ isotropic $XY$ chain\nin a transverse field.\nWe discuss in detail the changes\nin the behaviour of the thermodynamic quantities\ncaused by regular nonuniformity.\nThe exact consideration of thermodynamics is extended\nincluding a random Lorentzian transverse field.\nThe presented results are used to study\nthe Peierls instability\nin a quantum spin chain.\nIn particular, we examine the influence of a non-random/random field \non the spin-Peierls instability with respect to dimerization.\n\\end{abstract}\n\n\\vspace{1cm}\n\n\\noindent\n{\\bf {PACS numbers:}}\n75.10.-b\n\n\\vspace{1cm}\n\n\\noindent\n{\\bf {Keywords:}}\nSpin-$\\frac{1}{2}$ $XY$ chain;\nPeriodic nonuniformity;\nDiagonal Lorentzian disorder;\nDensity of states;\nThermodynamics;\nSpin-Peierls dimerization\\\\\n\n\\vspace{1mm}\n\n\\noindent\n{\\bf Postal addresses:}\\\\\n\n\\vspace{2mm}\n\n\\noindent\n{\\em\nDr. Oleg Derzhko (corresponding author)\\\\\nOles' Zaburannyi\\\\\nInstitute for Condensed Matter Physics\\\\\n1 Svientsitskii St., L'viv-11, 290011, Ukraine\\\\\nTel: (0322) 42 74 39\\\\\nFax: (0322) 76 19 78\\\\\nE-mail: derzhko@icmp.lviv.ua\\\\\n\n\\vspace{1mm}\n\n\\noindent\nProf. Johannes Richter\\\\\nInstitut f\\\"{u}r Theoretische Physik, Universit\\\"{a}t Magdeburg\\\\\nP.O. Box 4120, D-39016 Magdeburg, Germany\\\\\nTel: (0049) 391 671 8841\\\\\nFax: (0049) 391 671 1217\\\\\nE-mail: Johannes.Richter@Physik.Uni-Magdeburg.DE\n\n\\clearpage\n\n\\renewcommand\\baselinestretch {1.25}\n\\large\\normalsize\n\n\\input{lra07.tex}\n\n\\input{lrb07.tex}\n\n\\end{document}"
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"name": "cond-mat0002136.extracted_bib",
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JETP {\\bf 26,} 687 (1968).\n\\bibitem{p}\nP. Pincus,\nSolid State Commun. {\\bf 9,} 1971 (1971).\n\\bibitem{capel}\nJ. H. H. Perk, H. W. Capel, M. J. Zuilhof, and Th. J. Siskens,\nPhysica A {\\bf 81,} 319 (1975).\n\\bibitem{lt}\nR. A. T. Lima and C. Tsallis,\nPhys. Rev. B {\\bf 27,} 6896 (1983).\n\\bibitem{taylormuller}\nJ. H. Taylor and G. M\\\"{u}ller,\nPhysica A {\\bf{130,}} 1 (1985)\n(and references therein).\n\\bibitem{oy}\nK. Okamoto and K. Yasumura,\nJ. Phys. Soc. Jpn. {\\bf 59,} 993 (1990)\n(and references therein).\n\\bibitem{o1}\nK. Okamoto,\nJ. Phys. Soc. Jpn. {\\bf 59,} 4286 (1990).\n\\bibitem{zvyagin1}\nA. A. Zvyagin,\nPhys. Lett. A {\\bf{158,}} 333 (1991).\n\\bibitem{zvyagin2}\nA. A. Zvyagin,\nFiz. Niz. Temp. (Kharkiv) {\\bf{18,}} 788 (1992)\n(in Russian).\n\\bibitem{o2}\nK. Okamoto,\nSolid State Commun. {\\bf 83,} 1039 (1992).\n\\bibitem{so}\nY. Saika and K. Okamoto, cond-mat/9510114.\n\\bibitem{f}\nA. Fujii, cond-mat/9707137.\n\\bibitem{bm}\nS. M. Bhattacharjee and S. Mukherji,\nJ. Phys. A {\\bf 31,} L695 (1998).\n\\bibitem{si}\nS. Sil,\nJ. Phys.: Condens. Matter {\\bf 10,} 8851 (1998).\n\\bibitem{gl}\nL. L. Gon\\c{c}alves and J. P. de Lima,\nJ. Magn. Magn. Mater. {\\bf 140-144,} 1606 (1995).\n\\bibitem{sa}\nS. Sasaki,\nPhys. Rev. E {\\bf 53,} 168 (1996).\n\\bibitem{od}\nO. Derzhko, \nFiz. Niz. Temp. (Kharkiv) {\\bf 25,} 575 (1999);\nLow Temp. Phys. {\\bf 25,} 426 (1999).\n\\bibitem{drzaburannyi}\nO. Derzhko, J. Richter, and O. Zaburannyi,\ncond-mat/9909251, submitted to Phys. Lett. A.\n\\bibitem{sm}\nTh. J. Siskens and P. Mazur,\nPhysica A {\\bf 71,} 560 (1974).\n\\bibitem{jt}\nW. B. Jones and W. J. Thron,\n{\\em Continued Fractions.\nAnalytic Theory and Applications}\n(Addison-Wesley Publishing Company,\nLondon, Amsterdam, Don Mills, Ontario, Sydney, Tokyo,\n1980).\n\\bibitem{hhk1}\nR. Haydock, V. Heine, and M. J. Kelly,\nJ. Phys. C {\\bf 5,} 2845 (1972).\n\\bibitem{hhk2}\nR. Haydock, V. Heine, and M. J. Kelly,\nJ. Phys. C {\\bf 8,} 2591 (1975).\n\\bibitem{h}\nR. Haydock,\nSolid State Physics {\\bf 35,} 215 (1980).\n\\bibitem{k}\nM. J. Kelly,\nSolid State Physics {\\bf 35,} 295 (1980).\n\\bibitem{d}\nA. S. Davydov,\n{\\em Tjeorija tvjerdogo tjela}\n(Nauka, Moskwa, 1976) (in Russian).\n\\bibitem{oya}\nM. Oshikawa, M. Yamanaka, and I. Affleck,\nPhys. Rev. Lett. {\\bf{78,}} 1984 (1997).\n\\bibitem{villain}\nJ. Villain, R. Bidaux, J.-P. Carton, and R. Conte,\nJ. Physique {\\bf{41,}} 1263 (1980).\n\\bibitem{shender}\nE. F. Shender,\nSov. Phys. JETP {\\bf{56,}} 178 (1982).\n\\bibitem{richterkruger}\nJ. Richter, S. E. Kr\\\"{u}ger, A. Voigt, and C. Gros,\nEurophys. Lett. {\\bf{28,}} 363 (1994).\n\\bibitem{htu}\nM. Hase, I. Terasaki, and K. Uchinokura,\nPhys. Rev. Lett. {\\bf{70,}} 3651 (1993).\n\\bibitem{boucherregnault}\nFor a review on CuGeO$_3$ see:\nJ. P. Boucher and L. P. Regnault,\nJ. Phys. I {\\bf{6,}} 1939 (1996).\n\\bibitem{42}\nM. Hase, I. Terasaki, K. Uchinokura, M. Tokunaga, N. Miura,\nand H. Obara,\nPhys. Rev. B {\\bf{48,}} 9616 (1993).\n\\bibitem{43}\nK. Hirota, M. Hase, J. Akimitsu, T. Masuda, K. Uchinokura,\nand G. Shirane,\nJ. Phys. Soc. Jpn {\\bf{67,}} 645 (1998).\n\\bibitem{44}\nV. N. Glazkov, A. I. Smirnov, O. A. Petrenko, D. M$^{\\mbox{c}}$K. Paul,\nA. G. Vetkin, and R. M. Eremina,\nJ. Phys.: Condens. Matter {\\bf{10,}} 7879 (1998).\n\\bibitem{45}\nH. Fukuyama, T. Tanimoto, and M. Saito,\nJ. Phys. Soc. Jpn. {\\bf{65,}} 1182 (1996).\n\\bibitem{46}\nH. Yoshioka and Y. Suzumura,\nJ. Phys. Soc. Jpn. {\\bf{66,}} 3962 (1997).\n\\bibitem{47}\nM. Mostovoy, D. Khomskii, and J.Knoester,\nPhys. Rev. B {\\bf{58,}} 8190 (1998).\n\\bibitem{48}\nM. Fabrizio, R. M\\'{e}lin, and J. Souletie,\ncond-mat/9807093.\n\\bibitem{as}\n{\\em Handbook of mathematical functions with formulas,\ngraphs and mathematical tables,}\nedited by M. Abramovitz and I. A. Stegun\n(National Bureau of Standards, 1964).\n\\bibitem{50}\nM. C. Cross,\nPhys. Rev. B {\\bf{20,}} 4606 (1979).\n\\bibitem{51}\nJ. Mertsching and H. J. Fishbeck,\nPhysica Status Solidi B {\\bf{103,}} 783 (1981).\n\\bibitem{52}\nG. S. Uhrig, F. Sch\\\"{o}nfeld, and J. P. Boucher,\nEurophys. Lett. {\\bf{41,}} 431 (1998).\n\\bibitem{53}\nG. S. Uhrig, F. Sch\\\"{o}nfeld, M. Laukamp, and E. Dagotto,\nEur. Phys. J. B {\\bf{7,}} 67 (1999).\n\\end{thebibliography}"
}
] |
cond-mat0002137
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Vortex state in a doped Mott insulator
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[
{
"author": "M. Franz and Z. Te\\v{s}anovi\\'c"
}
] |
\address{~ \parbox{14cm}{\medskip We analyze the recent vortex core spectroscopy data on cuprate superconductors and discuss what can be learned from them about the nature of the ground state in these compounds. We argue that the data are inconsistent with the assumption of a simple metallic ground state and exhibit characteristics of a doped Mott insulator. A theory of the vortex core in such a doped Mott insulator is developed based on the U(1) gauge field slave boson model. In the limit of vanishing gauge field stiffness such theory predicts two types of singly quantized vortices: an insulating ``holon'' vortex in the underdoped and metallic ``spinon'' vortex in the overdoped region of the phase diagram. We argue that the holon vortex exhibits a pseudogap excitation spectrum in its core qualitatively consistent with the existing experimental data on Bi$_2$Sr$_2$CaCu$_2$O$_8$. As a test of this theory we propose that spinon vortex with metallic core might be observed in the heavily overdoped samples. }} %
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[
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"name": "m3.tex",
"string": "%\\documentstyle[aps,prl,preprint]{revtex}\n\\documentstyle[aps,floats,epsf,twocolumn]{revtex}\n\n\n\\begin{document} \\draft\n\n\\title{Vortex state in a doped Mott insulator}\n\n\\author{M. Franz and Z. Te\\v{s}anovi\\'c}\n\\address{Department of Physics and Astronomy, Johns Hopkins University,\nBaltimore, MD 21218\n\\\\ {\\rm(\\today)}\n}\n%\\maketitle\n%\n%\\begin{abstract}\n\\address{~\n\\parbox{14cm}{\\rm\n\\medskip\nWe analyze the recent vortex core spectroscopy data on cuprate\nsuperconductors and discuss what can be learned from them about the nature \nof the ground state in these compounds. We argue that the data are\ninconsistent with the assumption of a simple metallic \nground state and exhibit characteristics of a doped Mott insulator. \nA theory of the vortex core in such a doped Mott insulator is \ndeveloped based on the U(1) gauge field slave boson model. In the limit of\nvanishing gauge field stiffness such theory predicts two types of singly \nquantized vortices: an insulating ``holon'' vortex in the underdoped \nand metallic ``spinon'' vortex in the overdoped region of the phase diagram.\nWe argue that the holon vortex exhibits a pseudogap excitation spectrum in \nits core qualitatively consistent with the existing experimental data on \nBi$_2$Sr$_2$CaCu$_2$O$_8$. As a test of this theory we propose that spinon \nvortex with metallic core might be observed in the heavily overdoped samples. \n}}\n%\\end{abstract}\n\\maketitle\n\n%\\pacs{74.60.-w,74.60.Ec,74.72.-h}\n\n%\n\\narrowtext\n\n\\section{Introduction}\nNature of the ground state as a function of doping remains one of the recurring\nunresolved issues \nin the theory of high-$T_c$ cuprate superconductors. The problem is\npartly due to formidable difficulties related to the theoretical description of\ndoped Mott insulators and partly due to experimental hurdles in accessing\nthe normal state properties in the $T\\to 0$ limit because of the intervening\nsuperconducting order. Probes that suppress superconductivity and \nreveal the properties of the underlying ground state are therefore of \nconsiderable value. So far only pulsed\nmagnetic fields\\cite{ando1} in excess of $H_{c2}$ and impurity doping\nbeyond the critical concentration\\cite{lemberger1} have been used\ntowards this goal. Here we argue that the vortex core\nspectroscopy performed using scanning tunneling microscope (STM) \ncan provide new insights into the nature of the ground state in cuprates.\nWe analyze the existing experimental data\\cite{maggio1,renner1,pan1,pan2} \nand conclude that they imply strongly correlated ``normal'' ground state,\npresumably derivable from a doped Mott insulator. We then \ndevelop a theoretical framework for the problem of \ntunneling in the vortex state of such a doped Mott insulator.\n\nIn the vortex core the superconducting order parameter is locally suppressed \nto zero and the region within a coherence length $\\xi$ from its center can\nbe to the first approximation thought of as normal. Spectroscopy of the\nvortex core therefore provides information on the normal state electronic\nexcitation spectrum in the $T\\to 0$ limit. More accurately, the core \nspectroscopy reflects the spectrum in the spatially non-uniform situation\nwhere the order parameter amplitude rapidly varies in response to the\nsingularity in the phase imposed by the external magnetic field. In order to\nextract useful information regarding the underlying ground state\nfrom such measurements a detailed understanding of the vortex core \nphysics is necessary. So far the problem has been addressed using the weak \ncoupling approach based on the \nBogoliubov-de Gennes theory generalized to the $d$-wave symmetry of \nthe order parameter\\cite{soininen1,wang1,franz1,kita1}, and semiclassical\ncalculations\\cite{volovik1,maki1,ichioka1}.\nThe early theoretical debate focused on the existence\nor absence of the vortex core bound states \n\\cite{maki1,franz2,himeda1}. This debate, now resolved in favor\nof absence of any bound states in pure $d_{x^-y^2}$ state\n\\cite{franz1,kita1,resende1}, has somewhat eclipsed the possibly\nmore important issues related to the nature of the ground state \nin cuprates.\n\nThe body of work based on mean field, weak coupling calculations\n\\cite{wang1,franz1,kita1,ichioka1} yields results for the\nlocal density of states in the vortex core which exhibit two generic features:\n(i) the coherence peaks (occurring at $E=\\pm\\Delta_0$ in the bulk) are \nsuppressed,\nwith the spectral weight transferred to a (ii) broad featureless peak \ncentered around the zero energy. \n%Both features are easily understood on \n%physical grounds. The smearing of the coherence peaks results from the gap\n%not being sharp on the length-scale $\\xi$ in the core and the broad peak is a\n%remnant the bound states that would exist in the $s$-wave \n%vortex\\cite{caroli1} strongly hybridized into the continuum states \n%present in a $d$-wave superconductor. \nHere we wish to emphasize the heretofore little appreciated fact that \nthese features are {\\em qualitatively inconsistent}\nwith the existing experimental data on cuprate superconductors. STM\nspectroscopy on Bi$_2$Sr$_2$CaCu$_2$O$_8$\n(BSCCO) at 4.2K indicates a ``pseudogap'' spectrum in the vortex core with the \nspectral weight from the coherence peaks at $\\pm\\Delta_0\\simeq 40$meV\ntransferred to {\\em high energies}, and no peak whatsoever\naround $E=0$\\cite{renner1}. Recent high resolution \ndata on the same compound\\cite{pan2}\nconfirmed these findings down to 200mK and found evidence for weak bound\nstates at $\\pm7$meV. Experiments on YBa$_2$Cu$_3$O$_7$ (YBCO)\\cite{maggio1}\nalso indicate low energy bound states, but are somewhat more difficult to\ninterpret because of the high zero-bias conductance of unknown origin\nappearing even in the absence of magnetic field. \n\nThe fundamental discrepancy between the theoretical predictions and \nthe experimental findings strongly suggests that models based on a simple\nweak coupling theory break down in the vortex core. The pseudogap observed \nin the core hints that the underlying ground state revealed\nby local suppression of the superconducting order parameter is \na doped Mott insulator and not a conventional metal. Taking into account \nthe effects of strong correlations appears to be necessary to\nconsistently describe the physics of the vortex core. \nConversely, studying the vortex core physics could provide information\nessential for understanding the nature of the underlying ground state \nin cuprates. \n\nThe first step in this direction was taken by Arovas {\\em et al.}\n\\cite{arovas1} who proposed that within the framework of the SO(5) theory\n\\cite{zhang1} vortex cores could become antiferromagnetic (AF). \nThey found that such AF cores can be stabilized\nat low $T$ but only in the close vicinity of the bulk AF phase. In contrast,\nexperimentally the pseudogap in the core is found to persist \ninto the overdoped \nregion\\cite{renner1}. More recently microscopic calculations within the same \nmodel\\cite{andersen1} revealed electronic excitations in such AF cores with \nbehavior roughly resembling the experimental data. Quantitatively, \nhowever, these spectra exhibit asymmetric\nshifts in the coherence peaks (related to the fact that spin gap in the AF \ncore is no longer tied to the Fermi level) not observed experimentally. These\ndiscrepancies suggest that generically cores will not exhibit the true AF \norder. Finally, these previous approaches are still of the \nHartree-Fock-Bogoliubov type and cannot be expected to properly capture the\neffects of strong correlations.\n\nHere we consider a model for the vortex core based on a version of \nthe U(1) gauge field slave boson theory formulated recently by \nLee\\cite{dhlee1}. Originally proposed by Anderson\\cite{anderson1} \nthe slave boson theory was formulated \nto describe strongly correlated electrons in the CuO$_2$ planes of the \nhigh-$T_c$ cuprates. Various versions of\nthis theory have been extensively discussed in the \nliterature \\cite{baskaran1,ruckenstein1,kotliar1,affleck1,lee1}. \nInterest in spin-charge separated \nsystems revived recently\\cite{wen1,lee2,balents1,dhlee1} due to the\nrealization that it provides a natural description of the pseudogap phenomenon\nobserved in the underdoped cuprates. The common ingredient in these theories\nis ``splintering'' of the electron into quasiparticles carrying its spin\nand charge degrees of freedom. Within the theories based on Hubbard and \n$t$-$J$ models this splintering is formally implemented by the\ndecomposition of the electron creation operator\n%\n\\begin{equation}\nc^\\dagger_{i\\sigma}=f^\\dagger_{i\\sigma} b_i\n\\label{}\n\\end{equation}\n%\ninto a fermionic spinon $f_{i\\sigma}$ and bosonic holon $b_i$. The local\nconstraint of the single occupancy $b^\\dagger_ib_i+\nf^\\dagger_{i\\sigma}f_{i\\sigma}=1$ is enforced by a fluctuating U(1) gauge \nfield ${\\bf a}$. The mean field phase diagram is known to contain\nfour phases distinguished \nby the formation of spinon pairs, $\\Delta_{ij}=\\langle\\epsilon_{\\sigma\\sigma'}\nf^\\dagger_{i\\sigma}f^\\dagger_{j\\sigma'}\\rangle$, and Bose-Einstein condensation\nof the individual holons $b=\\langle b_i\\rangle$\\cite{lee1}, and is illustrated\n in Figure (\\ref{fig1}).\n%\n\\begin{figure}[t]\n\\epsfxsize=8.5cm\n\\epsffile{fig1.ps}\n\\caption[]{Schematic phase diagram of the system with spin-charge\nseparation in the doping$-$temperature plane, as applied to cuprate\nsuperconductors. }\n\\label{fig1}\n\\end{figure}\n%\n\nThe effects of magnetic field on such spin-charge separated system is\nmost conveniently studied in the framework of an effective Ginzburg-Landau \n(GL) theory for the condensate fields $\\Delta$ and $b$. The corresponding \neffective action can be constructed\\cite{sachdev1,lee3}\nbased on the requirements of local gauge invariance with respect to \nthe physical electromagnetic vector potential ${\\bf A}$ and the internal\ngauge field ${\\bf a}$:\n%\n\\begin{eqnarray}\nf_{\\rm GL}&=&|(\\nabla-2i{\\bf a})\\Delta|^2 +{r_\\Delta}|\\Delta|^2 +\n{1\\over 2}{u_\\Delta}|\\Delta|^4\\nonumber \\\\\n&+&|(\\nabla-i{\\bf a}-ie{\\bf A})b|^2 +r_b|b|^2 +{1\\over 2} u_b|b|^4 \n +v|\\Delta|^2|b|^2 \\nonumber \\\\\n&+& {1\\over 8\\pi}(\\nabla\\times{\\bf A})^2 +f_{\\rm gauge}.\n\\label{fgl1}\n\\end{eqnarray}\n%\nThe factor of 2 in the spinon gradient term reflects the fact that {\\em pairs}\nof spinons were assumed to condense. $f_{\\rm gauge}$ describes the dynamics\nof the internal gauge field ${\\bf a}$. We note that unlike the physical \nelectromagnetic field ${\\bf A}$ the gauge field ${\\bf a}$ has no independent dynamics\nin the underlying microscopic model since it serves only to enforce a \nconstraint. Sachdev\\cite{sachdev1} and Nagaosa and Lee\\cite{lee3} assumed\nthat upon integrating out the microscopic degrees of freedom a term\n%\n\\begin{equation}\nf_{\\rm gauge}={\\sigma\\over2}(\\nabla\\times{\\bf a})^2\n\\label{fgauge}\n\\end{equation}\n%\nis generated in the free energy. They then analyzed vortex solutions of\nthe free energy (\\ref{fgl1}) and came to the conclusion that two types of\nvortices are permissible: a ``holon vortex'' with the singularity in the \n$b$ field and a ``spinon vortex'' with the singularity in the\n $\\Delta$ field. Because holons\ncarry electric charge $e$ the holon vortex is threaded by electronic flux\nquantum $hc/e$, i.e. twice the conventional \nsuperconducting flux quantum $\\Phi_0=hc/2e$.\nSpinons on the other hand condense in pairs, and the spinon vortex therefore\ncarries flux $\\Phi_0$.\nStability analysis then implies that spinon vortex will be stable over the\nmost of the superconducting phase diagram, while the \n$hc/e$ holon vortex can be stabilized\nonly in the close vicinity of the phase boundary on the underdoped side\n\\cite{sachdev1,lee3}. \nThis is a direct consequence of the fact that singly quantized vortices are \nalways energetically favorable\\cite{abrikosov1,fetter1}.\n\nAs far as the electronic excitations are concerned, the spinon vortex is\nvirtually indistinguishable from the vortex in a conventional weak coupling\nmean field\ntheory: the spin gap $\\Delta$, which gives rise to the gap in the electron\nspectrum, vanishes in the core. Consequently, the vortex state based on the\nresults of Sachdev-Nagaosa-Lee (SNL) theory\\cite{sachdev1,lee3} does not\nexhibit the pseudogap in the core and suffers \nfrom the same discrepancy with the experimental data as the weak coupling\ntheories\\cite{soininen1,wang1,franz1,kita1} based on the conventional Fermi \nliquid description. Moreover, no evidence exists at present\nfor stable doubly quantized holon vortices predicted by SNL. What is\nneeded to account for the experimental data is a {\\em singly quantized\nholon vortex} stable over the large portion of the superconducting phase in the\nphase diagram of Figure \\ref{fig1}. In the core of such a holon vortex the spin\ngap $\\Delta$ remains finite and leads naturally\nto the pseudogap excitation spectrum. In what follows we show that under\ncertain conditions the free energy (\\ref{fgl1}) permits precisely such \nsolution. \n\nThe results of the SNL theory are predicated upon the assumption that the \n``stiffness'' $\\sigma$ of the gauge field is relatively large and that singular\nconfigurations in which $\\nabla\\times{\\bf a}$ contains a full flux quantum\nthrough an elementary plaquette are prohibited. Consider now a precisely \nopposite physical situation, allowing unconstrained fluctuations in \n${\\bf a}$. This amounts\nto the assumption that the $f_{\\rm gauge}$ term (\\ref{fgauge})\ncan be neglected in (\\ref{fgl1}), i.e. $\\sigma\\to 0$. Physically this\ncorresponds to the ``extreme type-I'' limit of the GL ``superconductor''\n(\\ref{fgl1}) with respect to fluctuations in ${\\bf a}$. Based on Elitzur's \ntheorem\\cite{elitzur1} Nayak\\cite{nayak1} recently argued that\nthe exact local U(1) symmetry of the model cannot be broken, implying absence\nof the phase stiffness term (\\ref{fgauge}) at all energy scales. Our assumption\ntherefore appears reasonable and in Section III. we shall give a more thorough\ndiscussion of the significance of the $f_{\\rm gauge}$ term for the vortex\nsolutions of interest here. For the time being we shall assume that \n$f_{\\rm gauge}$ can be neglected and explore physical consequences of the\nresulting theory. \n\n$f_{\\rm GL}$ given by Eq.\\ (\\ref{fgl1}) is quadratic in ${\\bf a}$ and with \nthe $\\nabla\\times{\\bf a}$ term absent the gauge\nfluctuations can be trivially integrated out. Within the closely related \nmicroscopic model this procedure has been recently implemented by \nLee\\cite{dhlee1}. The resulting effective free energy density reads\n%\n\\begin{eqnarray}\nf&=& f_{\\rm amp} +{\\rho_\\Delta^2\\rho_b^2\\over 4\\rho_\\Delta^2+\\rho_b^2}\n(\\nabla\\phi-2\\nabla\\theta+2e{\\bf A})^2 \\nonumber \\\\\n&+&{1\\over 8\\pi}(\\nabla\\times{\\bf A})^2,\n\\label{feff}\n\\end{eqnarray}\n%\nwhere we have set $\\Delta=\\rho_\\Delta e^{i\\phi}$, $b=\\rho_b e^{i\\theta}$, and \n%\n\\begin{eqnarray}\nf_{\\rm amp}&=&(\\nabla\\rho_\\Delta)^2 +{r_\\Delta}\\rho_\\Delta^2 +\n{1\\over 2}{u_\\Delta}\\rho_\\Delta^4\\nonumber \\\\\n&+&(\\nabla\\rho_b)^2 +r_b\\rho_b^2 +{1\\over 2} u_b\\rho_b^4 \n +v\\rho_\\Delta^2\\rho_b^2\n\\label{famp}\n\\end{eqnarray}\n%\nis the amplitude piece. The most important feature of the effective free\nenergy (\\ref{feff}) is that it no longer depends on the individual phases\n$\\phi$ and $\\theta$ but only on their particular combination\n%\n\\begin{equation}\n\\Omega=\\phi-2\\theta.\n\\label{omega}\n\\end{equation}\n%\nSince the physical superconducting order parameter $\\Psi=\\Delta^*b^2=\n\\rho_\\Delta\\rho_b^2 e^{-i(\\phi-2\\theta)}$ it is reasonable \nto identify $\\Omega$ with the phase of a {\\em Cooper pair}. Physically, the\nunconstrained fluctuations of the gauge field in Eq.\\ (\\ref{fgl1}) resulted in\npartial restoration of the original electronic degrees of freedom in Eq.\\ \n(\\ref{feff}). In the underlying microscopic model this means that on long\nlength scales spinons and holons are always confined, in agreement with\nElitzur's theorem \\cite{elitzur1,nayak1}. On lengthscales shorter than\nthe confinement length, such as inside the vortex core, spinons and holons\ncan still appear locally decoupled. In the present effective theory \nthis aspect is reflected by two amplitude degrees of freedom\npresent in (\\ref{feff}). More detailed discussion of these issues is given in \nRefs.\\ \\cite{dhlee1,nayak1}. \n\nWe have thus arrived at an effective theory of a spin-charge separated\nsystem containing one phase degree\nof freedom $\\Omega$ and two amplitudes, $\\rho_\\Delta$ and $\\rho_b$. Deep in the\nsuperconducting phase,\nwhere both amplitudes are finite, the physics of (\\ref{feff}) will be very\nsimilar to that of a conventional GL theory. In the situations where\nthe superconducting order parameter $\\Psi$ is strongly suppressed, such as\nin the vortex core, near an impurity or a wall, the new theory has an extra\ndegree of richness, associated with the fact that it is sufficient (and\ngenerally preferred by the energetics) when only {\\em one}\n of the two amplitudes \nis suppressed. Since the two amplitudes play very different roles in the \nelectronic excitation spectrum, the effective theory (\\ref{feff})\nwill lead to a number of nontrivial effects.\n\nTo illustrate this consider what will happen in the core of a superconducting\nvortex. Under the\ninfluence of the magnetic field the phase $\\Omega$ will develop a singularity\nsuch that $\\nabla\\Omega \\sim 1/r$ close to the vortex center. For \nthe free energy to remain finite the amplitude prefactor in the second\nterm of Eq.\\ (\\ref{feff}) must vanish for $r\\to 0$. \nThis is analogous to $|\\Psi|$ vanishing\nin the core of a conventional vortex. In the present case, \nhowever, it is sufficient when the product $\\rho_\\Delta\\rho_b$ vanishes. Since \nsuppressing any of the two amplitudes costs condensation energy, in general\nonly one amplitude will be driven to zero. Which of the two is suppressed\nwill be determined by the energetics of the amplitude term (\\ref{famp}).\nOn general grounds we expect that the state in the vortex core will be\nthe same as the corresponding bulk ``normal'' state obtained by raising \ntemperature above $T_c$. Thus, very crudely, we expect that holon vortex \nwill be stable in the underdoped while the spinon vortex will be stable in the\noverdoped region of the phase diagram Figure \\ref{fig1}. \n\nAn important point by which our approach differs from the SNL theory\nis that in the present theory {\\em both} types of vortices carry the \n{\\em same} superconducting flux quantum $\\Phi_0$ and thus compete\non equal footing. This is a direct consequence of our assumption of\nthe vanishing phase stiffness $\\sigma$. \n\nIn what follows we study in detail the vortex solutions of the free energy\n(\\ref{feff}). Our main objective is to obtain the precise estimates for \nthe energy of the two types of vortices as a function of temperature and \ndoping and deduce the corresponding phase diagram for the state inside\nthe vortex core. We show that for generic parameters in (\\ref{feff})\nthe singly quantized holon vortex with a pseudogap spectrum in the core can \nbe stabilized over a large portion of the superconducting phase, as\nrequired by the experimental constraints discussed above.\n\n\n\n\\section{Solution for a single vortex}\n\n\\subsection{General considerations}\n\nIn order to provide a more quantitative discussion we now adopt some \nassumptions\nabout the coefficients entering the free energy (\\ref{feff}). We assume that \n%\n\\begin{equation}\nr_i=\\alpha_i(T-T_i), \\ \\ i=b,\\Delta,\n\\label{ri}\n\\end{equation}\n% \nwhere $T_i$ are corresponding ``bare'' critical temperatures, which we\nassume depend on doping concentration $x$ in the following way:\n %\n\\begin{equation}\nT_\\Delta = T_0(2x_m-x), \\ \\ \\\nT_b = T_0x.\n\\label{tc}\n\\end{equation}\n%\nHere $x_m$ denotes the optimal doping and $T_0$ sets the overall temperature\nscale. We furthermore assume that $u_i$ and $v$ are all positive and\nindependent of doping and temperature. It is easy to see that such choice \nof parameters qualitatively reproduces the bulk phase diagram of cuprates\nin the $x$-$T$ plane shown in Figure \\ref{fig1}. \nThe effect of the $v$-term is to suppress $T_c$ from \nits bare value away from the optimal doping. In real systems fluctuations\nwill lead to additional suppression of $T_c$ which we do not consider here. \n\nIn the absence of perturbations the bulk values of the amplitudes are \ngiven by \n%\n\\begin{eqnarray}\n\\bar{\\rho}_\\Delta^2 &=& -({r_\\Delta} u_b-r_bv)/D, \\nonumber \\\\\n\\bar{\\rho}_b^2 &=& -(r_b{u_\\Delta} -{r_\\Delta} v)/D,\n\\label{rho}\n\\end{eqnarray}\n%\nwith $D=u_b{u_\\Delta}-v^2$. \nIn analogy with conventional GL theories we may define coherence\nlengths for the two amplitudes\\cite{sachdev1}\n%\n\\begin{eqnarray}\n\\xi_\\Delta^{-2} &=& -({r_\\Delta}-r_bv/u_b), \\nonumber \\\\\n\\xi_b^{-2} &=& -(r_b-{r_\\Delta} v/{u_\\Delta}), \n\\label{xi}\n\\end{eqnarray}\n%\none of which always diverges at $T_c$ as $(T-T_c)^{-1/2}$. \n\nMinimization of the free energy (\\ref{feff}) with respect to the vector \npotential ${\\bf A}$ yields an equation \n%\n\\begin{equation}\n\\nabla\\times\\nabla\\times{\\bf A}=e\\rho_s(\\nabla\\Omega-2e{\\bf A}),\n\\label{lona}\n\\end{equation}\n%\nwhere \n%\n\\begin{equation}\n\\rho_s={4\\rho_\\Delta^2\\rho_b^2\\over 4\\rho_\\Delta^2+\\rho_b^2}\n\\label{rhos}\n\\end{equation}\n%\nis the effective superfluid density. The term in brackets can be \nidentified as twice the conventional superfluid velocity \n%\n$${\\bf v}_s={1\\over 2}\\nabla\\Omega-e{\\bf A}.$$ \n%\nMaking use of the Ampere's law\n$4\\pi{\\bf j}=\\nabla\\times{\\bf B}$ we see that Eq.\\ (\\ref{lona}) specifies\nthe supercurrent in terms superfluid density and velocity:\n${\\bf j}=2e\\rho_s{\\bf v}_s$. Minimization of (\\ref{feff}) with respect\nto $\\Omega$ then implies $\\nabla\\cdot{\\bf j}=0$; the supercurrent is \nconserved.\n\nMinimizing the free energy (\\ref{feff}) with respect to the amplitudes results\nin the pair of coupled GL equations:\n%\n\\begin{mathletters}\n\\label{gl:all}\n\\begin{equation}\n-\\nabla^2\\rho_\\Delta + {r_\\Delta}\\rho_\\Delta + {u_\\Delta}\\rho_\\Delta^3 + v\\rho_b^2\\rho_\\Delta\n+{4\\rho_\\Delta^2\\rho_b^2\\over (4\\rho_\\Delta^2+\\rho_b^2)^2}{\\bf v}_s^2 = 0, \\label{gl:a}\n\\end{equation}\n\\begin{equation}\n-\\nabla^2\\rho_b + r_b\\rho_b + u_b\\rho_b^3 + v\\rho_\\Delta^2\\rho_b\n+{16\\rho_\\Delta^2\\rho_b^2\\over (4\\rho_\\Delta^2+\\rho_b^2)^2}{\\bf v}_s^2 = 0. \\label{gl:b}\n\\end{equation}\n\\end{mathletters}\nWe are interested in the behavior of the amplitudes in the vicinity of the \nvortex center. In this region, for a strongly type-II superconductor, we\nmay neglect the vector potential {{\\bf A}} \nin the superfluid velocity ${\\bf v}_s$. In\na singly quantized vortex $\\Omega$ winds by $2\\pi$ around the origin \nleading to a singularity\nof the form ${\\bf v}_s\\simeq{1\\over 2}\\nabla\\Omega =\\hat\\varphi/2r$. \nFirst, for the {\\em holon} vortex we assume that\n$\\rho_b$ vanishes in the core as some power $\\rho_b(r)\\sim r^\\nu$ and \n$\\rho_\\Delta(r)\\approx\\bar\\rho_\\Delta$ \nremains approximately constant. Eq.\\ (\\ref{gl:b}) then becomes\n%\n\\begin{equation}\n({1\\over 4}-\\nu^2)r^{\\nu-2} + (r_b+v\\bar\\rho_\\Delta^2)r^\\nu + \nu_b\\bar\\rho_b^2r^{3\\nu}=0,\n\\label{glcore}\n\\end{equation}\n%\nwhere we have neglected $\\rho_b^2(r)$ compared to $4\\bar\\rho_\\Delta^2$\nin the denominator of the last term in Eq.\\ (\\ref{gl:b}). The most singular\nterm in Eq.\\ (\\ref{glcore}) is the first one and we must demand that the\ncoefficient of $r^{\\nu-2}$ vanishes. This implies $\\nu={1\\over 2}$. The\nasymptotic short distance behavior of the holon amplitude therefore\ncan be written as\n%\n\\begin{equation}\n\\rho_b(r)\\simeq c_b\\bar\\rho_b \\left(r\\over\\xi_b\\right)^{1/2},\n\\label{rhobcore}\n\\end{equation}\n%\nwhere $c_b$ is a constant of order unity which may be determined \nby the full integration of Eqs.\\ (\\ref{gl:all}).\nSimilar analysis of Eq.\\ (\\ref{gl:a}) in the vicinity of the {\\em spinon} \nvortex yields \n%\n\\begin{equation}\n\\rho_\\Delta(r)\\simeq c_\\Delta\\bar\\rho_\\Delta \\left(r\\over\\xi_d\\right), \n\\label{rhodcore}\n\\end{equation}\n%\nwith $\\rho_b$ approximately constant. \n\nWe notice the different power laws\nin the holon and spinon results. Operationally this difference arises from\ndifferent numerical prefactors of the respective superfluid velocity terms \nin Eqs.\\ (\\ref{gl:all}). Physically, the unusual $r$ dependence of the\nholon amplitude in the core reflects the fact that the field $b$ describes\na condensate of single holons, each carrying charge $e$. Superconducting \nvortex with the flux quantum $\\Phi_0$ represents a magnetic \n``half-flux'' for the \nholon field which results in non-analytic behavior of $\\rho_b(r)$ at the \norigin. Singly quantized holon vortex is therefore a peculiar object and we\nshall discuss it more fully in Section III. Here we note that the \nphysical superconducting order parameter amplitude $|\\Psi|=\\rho_\\Delta\\rho_b^2$\nremains analytic in the core of both the spinon and the holon\nvortex.\n\n\n\\subsection{Holon vs. spinon vortex: the phase diagram}\n\nWe are now in the position to estimate the energies of the two types \nof vortices and deduce the phase diagram for the ``normal'' state in the \nvortex core. To this end we consider a single isolated vortex centered\nat the origin.\nThe total vortex line energy can be divided into electromagnetic \nand core contributions\\cite{fetter1}.\nThe electromagnetic contribution \nconsists of the energy of the supercurrents and the magnetic \nfield outside the core region. It may be estimated \nby assuming that the amplitudes $\\rho_\\Delta$ and $\\rho_b$ \nhave reached their bulk values $\\bar\\rho_\\Delta$ and $\\bar\\rho_b$ respectively.\nTaking curl of Eq.\\ (\\ref{lona}) and noting\nthat $\\nabla\\times\\nabla\\Omega=2\\pi\\delta({\\bf r})$ for a singly quantized vortex\nwe obtain the London equation for the magnetic field \n${\\bf B}=\\nabla\\times{\\bf A}$ of the form\n%\n\\begin{equation}\nB-\\lambda^2\\nabla^2B=\\Phi_0\\delta({\\bf r})\n\\label{lon}\n\\end{equation}\n%\nwhere \n%\n\\begin{equation}\n\\lambda^{-2}=8\\pi e^2 \n{4\\bar\\rho_\\Delta^2\\bar\\rho_b^2\\over 4\\bar\\rho_\\Delta^2+\\bar\\rho_b^2}.\n\\label{lam}\n\\end{equation}\n%\nhas the meaning of the London penetration depth for the effective GL theory\n(\\ref{feff}). Aside from the unusual form of $\\lambda$, Eq.\\ (\\ref{lon})\nis identical to the conventional London equation.\nThe corresponding electromagnetic energy is therefore \nthe same for both types of vortices\nand can be calculated in the usual manner\\cite{abrikosov1,fetter1,sachdev1} \nobtaining \n%\n\\begin{equation}\nE_{\\rm EM}\\simeq\\left({\\Phi_0\\over 4\\pi\\lambda}\\right)^2 \\ln\\kappa,\n\\label{em}\n\\end{equation}\n%\nwith $\\kappa=\\lambda/\\max(\\xi_\\Delta,\\xi_b)$ being the generalized GL ratio. \n\nTo estimate the core contribution to the vortex line energy we assume that \none of the amplitudes is suppressed to zero in the core\n%\n\\begin{equation}\n\\rho_i(r)=0, \\ \\ \\ r<\\xi_i,\n\\label{rhor}\n\\end{equation}\n%\nwhile the other one stays constant and equal to its bulk value. This is a\nvery \ncrude approximation which we justify below by an exact numerical computation.\nWith these assumptions, the core energy is \n%\n\\begin{equation}\nE_{\\rm core}^{(i)}\\simeq\\left({\\Phi_0\\over 4\\pi\\lambda_i}\\right)^2, \n\\label{ecore}\n\\end{equation}\n%\nwhere $i=\\Delta,b$ for spinon and holon vortex respectively and \n%\n\\begin{equation}\n\\lambda^{-2}_i=8\\pi e^2 \\bar\\rho_i^2.\n\\label{lami}\n\\end{equation}\n%\nSuch a crude approximation overestimates the core energy. A more accurate\nanalysis\\cite{abrikosov1,fetter1}, which we do not pursue here,\nallows for a more realistic variation of $\\rho_i(r)$ in the core and\nindicates that the\nvalue of $E_{\\rm core}^{(i)}$ has the same form as Eq.\\ (\\ref{ecore})\nmultiplied by a numerical factor $c_1\\approx 0.5$\\cite{hu1,alama1}. \nThus, the total energy\nof the vortex line can be written as\n%\n\\begin{equation}\nE^{(i)}=\\left({\\Phi_0\\over 4\\pi\\lambda}\\right)^2\\ln\\kappa\n+c_1\\left({\\Phi_0\\over 4\\pi\\lambda_i}\\right)^2, \n\\label{evortex}\n\\end{equation}\n%\nwhere again $i=\\Delta,b$ for spinon and holon vortex respectively.\nEq.\\ (\\ref{evortex}) parallels the Abrikosov expression for the vortex line \nenergy in a conventional GL theory\\cite{abrikosov1} \nwhere $\\lambda$ and $\\lambda_i$ are identical and equal to the ordinary \nLondon penetration depth.\n%\n\\begin{figure}[t]\n\\epsfxsize=8.5cm\n\\epsffile{fig2.ps}\n\\caption[]{Vortex core phase diagram for GL parameters chosen as follows:\n$\\alpha_\\Delta=0.13$, $\\alpha_b=0.10$, $T_0=200$K, $x_m=0.2$,\n$u_\\Delta=u_b=1.0$ and $v=0.5$.\nDashed line marks the phase boundary $T_g(x)$ obtained from \nEq.\\ (\\ref{tg}) while the solid circles correspond to the \nnumerical calculation with the same parameters. }\n\\label{fig2}\n\\end{figure}\n%\n\nIn the vortex state described by the free energy (\\ref{feff}) the vortex with\nlower energy $E^{(i)}$ will be stabilized. Eq.\\ (\\ref{evortex}) implies that \nthe difference in energy between the two types of vortices comes primarily\nfrom the\ncore contribution, as expected on the basis of the physical argument presented\nabove. Condition $\\lambda_\\Delta=\\lambda_b$ marks the \ntransition point between the two solutions. For fixed GL parameters $T_0$, \n$x_m$, $\\alpha_i$, $u_i$ and $v$ this defines a transition line in the $x$-$T$ \nplane. According to (\\ref{lami}) the equation for this line is \n%\n\\begin{equation}\n\\bar\\rho_\\Delta(x,T)=\\bar\\rho_b(x,T).\n\\label{tran}\n\\end{equation}\n%\nUsing Eqs.\\ (\\ref{ri}-\\ref{rho}) one can obtain an explicit expression \nfor the transition temperature $T_g$ between two types of vortices as\na function of doping\n%\n\\begin{equation}\nT_g(x)=T_0\\left[{2x_m-x\\over 1-\\beta}+{x\\over 1-\\beta^{-1}}\\right],\n\\label{tg}\n\\end{equation}\n% \nwith \n%\n\\begin{equation}\n\\beta={\\alpha_b(u_\\Delta+v)\\over \\alpha_\\Delta(u_b+v)}.\n\\label{beta}\n\\end{equation}\n%\nEq.\\ (\\ref{tg}) describes a straight line in the $x$-$T$ plane, originating\nat $[x_m,T_0x_m]$, i.e. maximal $T_c$ at optimum doping, and terminating at \n$[2x_m/(1+\\beta),0]$. Generically, we expect that parameters $\\alpha_i$ and\n$u_i$ will be comparable in magnitude for the holon and spinon channels.\nParameter $\\beta$ defined in Eq.\\ (\\ref{beta}) will therefore be of order\nunity. The typical situation for $\\beta=0.77$ \nis illustrated in Figure \\ref{fig2}.\nMore generally the quartic coefficients $u_i$ and $v$ could exhibit weak\ndoping and temperature dependences leading to a curvature in the phase \nboundary. \n\nThe appealing feature of the present theory is that parameter \n$\\beta$ may vary from compound to compound. Thus, the experimental fact\nthat in BSCCO the pseudogap in the core persists\ninto the overdoped region is easily accounted for in the present theory. \nIt would be interesting to see if the transition\nfrom holon to spinon vortex as a function of doping could be experimentally \nobserved. A good candidate for such observation would be LSCO, \nwhere the transport measurements in pulsed magnetic fields\\cite{ando1}\nestablished a metal-insulator transition around optimal doping, i.e.\\\n $\\beta\\approx 1$. The current\ntheory predicts a holon vortex with the pseudogap spectrum in the\nunderdoped (insulating) region and spinon vortex with conventional metallic\nspectrum on the overdoped side.\n\n\n\\subsection{Numerical results}\n\nIn order to put the above analytical estimates on firmer ground we now pursue\nnumerical computation of the vortex line energy. For simplicity we\nconsider the strongly type-II situation $(\\kappa\\gg1)$ where the vector\npotential term in ${\\bf v}_s$ can be neglected to an excellent approximation,\nas long as we focus on the behavior close to the core. \nWe are then faced with the task of numerically minimizing\nthe free energy (\\ref{feff}) with respect to the two cylindrically\nsymmetric amplitudes $\\rho_\\Delta(r)$ and $\\rho_b(r)$. As noted by Sachdev\n\\cite{sachdev1} direct numerical minimization of the free energy (\\ref{feff})\nprovides a more robust solution than the numerical\nintegration of the coupled differential equations (\\ref{gl:all}). \n\nWe discretize the free energy functional (\\ref{feff}) on a disk of a \nradius $R\\gg\\xi_i$ in the radial coordinate $r$ with up to $N= 2000$\nspatial points. \nWe then employ the Polak-Ribiere variant of the Conjugate Gradient Method\n\\cite{numrec} to minimize this discretized functional with respect to \n$\\rho_\\Delta(r_j)$ and $\\rho_b(r_j)$, initialized to suitable single vortex trial \nfunctions. The procedure converges very rapidly and the results are \ninsensitive to the detailed shape of the trial functions as long as they \nsaturate to the correct bulk values outside the vortex core.\n\nTypical results of our numerical computations are displayed in Figure\n(\\ref{fig3}) and are in complete agreement with the analytical\nconsiderations of the preceding subsections. Note in particular that \n$\\rho_b(r)$ in the holon vortex vanishes with infinite slope, consistent\nwith Eq.\\ (\\ref{rhobcore}). Plotting $\\rho_b^2(r)$ confirms that the \nexponent is indeed $1/2$. In the spinon vortex\n$\\rho_\\Delta(r)$ is seen to vanish linearly as expected on the basis of\nEq.\\ (\\ref{rhodcore}). The nonvanishing order parameter is slightly \nelevated in the core reflecting the effective ``repulsion'' between the\ntwo amplitudes contained in the $v$-term of the free energy. The results\nfor the spinon vortex are consistent with those of Ref.\\ \\cite{sachdev1}.\n%\n\\begin{figure}[t]\n\\epsfxsize=8.5cm\n\\epsffile{fig3.ps}\n\\caption[]{Order parameter amplitudes near a single isolated vortex for\nGL parameters specified in Figure \\ref{fig2}.\nThe holon vortex is plotted for $T=0$ and\n$x=0.22$ (implying coherence lengths $\\xi_\\Delta=0.63$ and $\\xi_b=0.70$),\nwhile the spinon vortex is plotted for $T=0$ and $x=0.24$ (implying\n$\\xi_\\Delta=0.75$ and $\\xi_b=0.60$).}\n\\label{fig3}\n\\end{figure}\n% \n\nWe explored a number of other parameter configurations and obtained \nsimilar results.\nWe find that Eq.\\ (\\ref{tran}) is a good predictor of the \ntransition line between the holon and spinon vortex, although the precise\nnumerical value of the transition temperature $T_g$ for given $x$ tends\nto deviate slightly from the value predicted by Eq.\\ (\\ref{tg}). This is\nillustrated in Figure (\\ref{fig2}) where we compare the vortex core\nphase diagrams obtained numerically and from Eq.\\ (\\ref{tg}). Interestingly,\nthe deviation always tends to enlarge the holon vortex sector of the phase\ndiagram at the expense of the spinon vortex sector. This is presumably \nbecause the sharper $\\sim\\sqrt{r}$ suppression of the holon order parameter\nin the core costs less condensation energy. \n\n\n\n\\section{Gauge fluctuations and the spectral properties in the core}\n\nTheory of the vortex core based on the effective action (\\ref{feff})\nappears to yield results consistent with the STM data on cuprates\n\\cite{renner1,pan2} in that it implies stable holon vortex solution over\nthe large portion of the superconducting phase diagram. The state inside the \ncore of such a holon vortex is characterized by vanishing amplitude of the\nholon condensate field,\n$|b|=0$, and a finite spin gap $|\\Delta|\\approx\\Delta_{\\rm bulk}$. This is the \nsame state as in the pseudogap region above $T_c$. One would thus expect\nthe electronic spectrum in the core to be similar to that found in the normal \nstate of the underdoped cuprates, in agreement with the \ndata\\cite{renner1,pan2}. The holon vortex\nwith this property carries conventional superconducting flux quantum \n$\\Phi_0$, in accord with experiment. This general agreement between theory\nand experiment would suggest that the effective action (\\ref{feff}) \nprovides the sought for phenomenological description of the vortex core \nphysics in cuprates.\nIn what follows we amplify our argumentation that it is also\ntenable in a broader theoretical context in that it naturally\nfollows from the U(1) slave boson models extensively studied in the \nclassic and more recent high-$T_c$ literature. We then provide a more detailed\ndiscussion of the vortex core spectra and propose an explanation for the\nexperimentally observed core bound states.\n\n\n\\subsection{Significance of the $f_{\\rm gauge}$ term} \n\nDerivation of the effective action (\\ref{feff}) from the more general U(1)\naction (\\ref{fgl1}) hinges on our assumption that the stiffness $\\sigma$ \nof the \ngauge field ${\\bf a}$ is low and that the $f_{\\rm gauge}$ term (\\ref{fgauge})\ncan be neglected. Assumption of large $\\sigma$ by SNL leads to very different\nvortex solutions\\cite{sachdev1,lee3} which appear inconsistent with the recent\nexperimental data. We first expand on our discussion as to why is \n$f_{\\rm gauge}$ term important and then we argue why it may be permissible\nto neglect it in the realistic models of cuprates. \n\nTo facilitate the discussion let us rewrite Eq.\\ (\\ref{fgl1}) by resolving\nthe complex matter fields into amplitude and phase components:\n%\n\\begin{eqnarray}\nf_{\\rm GL}&=& f_{\\rm amp} +\\rho_\\Delta^2(\\nabla\\phi -2{\\bf a})^2\n+\\rho_b^2(\\nabla\\theta-{\\bf a}-e{\\bf A})^2 \\nonumber \\\\\n&+&{1\\over 8\\pi}(\\nabla\\times{\\bf A})^2 + \n{\\sigma\\over2}(\\nabla\\times{\\bf a})^2,\n\\label{fgl2}\n\\end{eqnarray}\n%\nwith $f_{\\rm amp}$ specified by Eq.\\ (\\ref{famp}). Now consider situation\nin which the sample is subjected to uniform magnetic field \n${\\bf B}=\\nabla\\times{\\bf A}$. Two scenarios (discussed previously by SNL) \nappear possible. In the first, the internal gauge field develops no net \nflux, $\\langle\\nabla\\times{\\bf a}\\rangle=0$, and the holon phase \n$\\theta$ develops \nsingularities in response to ${\\bf A}$ such that \n%\n$$\\nabla\\times\\nabla\\theta=2\\pi\\sum_j\\delta({\\bf r}-{\\bf r}_j),$$ \n%\nwhere ${\\bf r}_j$ denotes the vortex positions. The holon amplitude $\\rho_b$\nis driven to zero at ${\\bf r}_j$, essentially to prevent the free energy from\ndiverging due to the singularity in the phase gradient.\nSince holons carry charge $e$, each vortex is threaded by \nflux $hc/e$, i.e.\\ twice the superconducting flux quantum $\\Phi_0=hc/2e$. \nThis solution represents the doubly quantized holon vortex lattice, considered\nby SNL. \n\nIn the second scenario\n${\\bf a}$ develops a net flux such that ${\\bf a}\\approx -e{\\bf A}$, which screens out the\n${\\bf A}$ field in the holon term but produces a net flux $-2e{\\bf A}$ in the \nspinon term. In response to this flux, spinon phase $\\phi$ develops \nsingularities such that \n%\n$$\\nabla\\times\\nabla\\phi=2\\pi\\sum_j\\delta\n({\\bf r}-\\tilde{\\bf r}_j),$$\n% \ncorresponding to the spinon vortex lattice. $\\tilde{\\bf r}_j$ denotes vortex\npositions which will be different from ${\\bf r}_j$ since at the fixed field\n$B$ there will be twice as many spinon vortices as holon vortices. \n(Spinon vortices carry conventional superconducting quantum of flux $\\Phi_0$.) \nIn this case $\\rho_\\Delta$ is driven to zero at the vortex centers.\nIn this scenario \none pays a penalty for nucleating the net flux in $\\nabla\\times{\\bf a}$ due\nto last term in Eq.\\ (\\ref{fgl2}). This energy cost can be estimated as\n%\n\\begin{equation}\nE_\\sigma\\simeq 8\\pi\\sigma e^2 \\left({\\Phi_0\\over 4\\pi\\lambda}\\right)^2\n\\label{esig}\n\\end{equation}\n%\nper vortex. Stiffness $\\sigma$ must be small enough so that $E_\\sigma$ is\nsmall compared to the vortex energy (\\ref{evortex}). Taking the dominant \n$E_{\\rm EM}$ term and neglecting $\\ln\\kappa$ this implies that\n%\n\\begin{equation}\n\\sigma\\ll {1\\over 8\\pi e^2},\n\\label{sig}\n\\end{equation}\n%\nwhich is the same condition as considered in Ref.\\ \\cite{sachdev1}. \n\n\nNow consider a {\\em third} scenario in which a {\\em singly \nquantized} holon vortex emerges. As a starting point consider the spinon \nvortex solution just described. In the underdoped regime the amplitude piece\n$f_{\\rm amp}$ would favor suppressing the holon amplitude in the core \ninstead of the spinon amplitude but according to our previous considerations\nthis would ordinarily require formation of a doubly quantized vortex whose \nmagnetic energy is too large. However, if the gauge field stiffness $\\sigma$\nis sufficiently small, the system could lower its free energy by\nsetting up singularities in ${\\bf a}$ which would precisely cancel the \nsingularities in $\\nabla\\phi$ and shift them to the holon term. \nTo arrive at this situation imagine\ncontracting the initially uniform flux $\\nabla\\times{\\bf a}$ so that it\nbecomes localized in the individual vortex core regions. Taking this \nprocedure to the extreme, i.e. taking the limit $\\sigma\\to 0$,\nthe gauge field will form ``flux spikes'' of the form \n%\n\\begin{equation}\n2(\\nabla\\times{\\bf a})\n=-\\nabla\\times\\nabla\\phi=-2\\pi\\sum_j\\delta({\\bf r}-\\tilde{\\bf r}_j),\n\\label{sing}\n\\end{equation}\n%\ncompletely localized at the vortex centers. \nGauge field of this form indeed completely cancels the singularities in the \nspinon\nphase gradient in Eq.\\ (\\ref{fgl2}) and $\\rho_\\Delta$ is no longer forced\nto vanish in the core. The singularities now appear in the holon term,\nbut they stem from ${\\bf a}$ rather that $\\nabla\\theta$ which remains \nnonsingular. Consequently, $\\rho_b$ is forced to vanish in the vortex cores. \nBy construction the vortices are located at $\\tilde{\\bf r}_j$ and \nare therefore singly quantized.\nThis is the singly quantized holon vortex \ndiscussed in the framework of the free energy (\\ref{feff}). Based on the\nabove discussion the singly quantized holon vortex can be thought of as a\ncomposite object formed by attaching half quantum $(h/2)$ of the \nfictitious gauge flux $\\nabla\\times{\\bf a}$ to the spinon vortex. Within\nthe full compact U(1) theory this is essentially equivalent to the Z$_2$\nvortex discussed by Wen\\cite{wen2} in the framework of topological orders\nin spin liquids. \n\nIn the framework of the free energy (\\ref{fgl2}) one pays a penalty for such \na singular solution due to the gauge stiffness term. In the present \ncontinuum model\nthis penalty per single vortex is actually infinite, since according to \nEq.\\ (\\ref{sing}) it involves a \nspatial integral over $[\\delta({\\bf r}-\\tilde{\\bf r}_j)]^2$. Thus, in the\ncontinuum model\nthe singular solutions of this type are prohibited. In reality, however,\nwe have to recall that our effective action (\\ref{fgl1}) descended from\na microscopic \nlattice model for spinons and holons in which the gauge field ${\\bf a}$ lives\non the nearest neighbor bonds of the ionic lattice. The ionic lattice constant\n$d$ therefore provides a natural short distance cutoff and the delta\nfunction in Eq.\\ (\\ref{sing}) should be interpreted as a flux quantum $\\Phi_0$\npiercing an elementary plaquette of the lattice. The energy cost per vortex\nthus becomes finite and is given by \n%\n\\begin{equation}\nE'_\\sigma\\simeq {\\sigma e^2\\over 2} \\left({\\Phi_0\\over d}\\right)^2.\n\\label{esigp}\n\\end{equation}\n%\nAgain, for the solution to be stable, $E'_\\sigma$ must be negligible compared\nto the vortex energy (\\ref{evortex}). This implies \n%\n\\begin{equation}\n\\sigma\\ll {1\\over 8\\pi^2e^2} \\left({d\\over \\lambda}\\right)^2,\n\\label{sigp}\n\\end{equation}\n%\nwhich is a much more stringent condition than (\\ref{sig}) since in cuprates\n$d\\ll\\lambda$. \n\nWhen condition (\\ref{sigp}) is satisfied it is permissible to neglect \nthe $f_{\\rm gauge}$ term in the effective action (\\ref{fgl1}) and\nit becomes fully equivalent to (\\ref{feff}) as far as the vortex solutions are \nconcerned. \nEq.\\ (\\ref{sigp}) gives the precise meaning to the requirement of the weak \nstiffness of the gauge field loosely stated when deriving the effective\naction (\\ref{feff}).\n\n\n\n\\subsection{Microscopic considerations}\n\nAs mentioned in the introduction, the gauge field ${\\bf a}$ has no dynamics\nin the original U(1) microscopic model, as it only serves\nto enforce a constraint on spinons and holons. The stiffness term \n(\\ref{fgauge}) in the effective\ntheory was assumed to arise in the process of integrating out the\nmicroscopic degrees of freedom\\cite{sachdev1,lee3}. While such term is\ncertainly permitted by symmetry, assessing its strength\n$\\sigma$ is a nontrivial issue since even deep in the superconducting phase\nneither holons nor spinons are truly gapped. Thus, in general, integrating\nout these degrees of freedom may lead to singular and nonlocal interactions\nbetween the condensate and the gauge fields. To our knowledge the procedure\nhas not been explicitly performed for the U(1) model and the precise form or\nmagnitude of the gauge stiffness term is unknown. General \nconsiderations\\cite{nayak1} suggest\nthat the gauge stiffness term is negligible in the class \nof models with exact local U(1) symmetry connecting the phases of holons \nand spinons.\n\nConsider now an intermediate representation of the problem where only\nhigh energy microscopic degrees of freedom have been integrated out. In the \npresence of a cutoff this is a well defined procedure even for gapless\nexcitations, as explicitly shown by Kwon and Dorsey\\cite{kwon1} for a \nsimple BCS model. The corresponding effective Lagrangian density of the \npresent U(1) model can be written as\n %\n\\begin{eqnarray}\n{\\cal L}_{\\rm eff}&=&\n{\\kappa_\\Delta^\\mu\\over 2}(\\partial_\\mu\\phi-2a_\\mu)^2\n+{\\kappa_b^\\mu\\over 2}(\\partial_\\mu\\theta-a_\\mu-eA_\\mu)^2\n-f_{\\rm amp} \\nonumber \\\\\n&+&(\\partial_\\mu\\phi-2a_\\mu)J_{\\rm sp}^\\mu\n+(\\partial_\\mu\\theta-a_\\mu-eA_\\mu)J_h^\\mu \\nonumber \\\\ \n&+&{\\cal L}_{\\rm sp}[\\psi_{\\rm sp},\\psi_{\\rm sp}^\\dagger;\\rho_\\Delta]\n+{\\cal L}_h[\\psi_h,\\psi_h^\\dagger;\\rho_b] +{\\cal L}_{\\rm EM}[A_\\mu].\n\\label{leff}\n\\end{eqnarray}\n%\nThe Greek index $\\mu$ runs over time and two spatial dimensions, \n$\\kappa_i^0$ are compressibilities of the holon and spinon condensates,\nwhile \n%\n\\begin{equation}\n\\kappa_i^j=-2(\\rho_i)^2,\\ \\ \\ i=\\Delta, b, \\ \\ \\ j=1,2, \n\\label{kappa}\n\\end{equation}\n%\nare the\nrespective phase stiffnesses. $J_{\\rm sp}^\\mu$ and $J_h^\\mu$ are spinon and\nholon three currents respectively and ${\\cal L}_{\\rm sp}$ and \n${\\cal L}_h$ are the low energy effective Lagrangians for \nthe fermionic spinon field $\\psi_{\\rm sp}$ and bosonic holon field $\\psi_h$. \n${\\cal L}_{\\rm EM}$ is the Maxwell Lagrangian for the physical electromagnetic\nfield. Thus, ${\\cal L}_{\\rm eff}$ describes\nan effective low energy theory of spinons and holons coupled to their \nrespective collective modes and a fluctuating U(1) gauge field. Similar theory \nhas been recently considered by Lee\\cite{dhlee1}. \n \nThe precise form of the microscopic Lagrangians ${\\cal L}_{\\rm sp}$ and \n${\\cal L}_h$ is not important for our discussion. The salient feature which \nwe exploit here is that only the amplitude of the respective condensate field \nenters into ${\\cal L}_{\\rm sp}$ and ${\\cal L}_h$. Coupling to the phases and\nthe gauge field is contained entirely in the Doppler shift terms [second line\nof Eq.\\ (\\ref{leff})]. Such form of the coupling is largely dictated by the \nrequirements of the gauge invariance and the particular form Eq.\\ (\\ref{leff})\ncan be explicitly derived by gauging away the respective phase factors\nfrom the $\\psi$ fields\\cite{balents1,kwon1}. \n\nThe gauge field $a_\\mu$ enters the effective Lagrangian (\\ref{leff}) only via\ntwo gauge invariant terms: $(\\partial_\\mu\\phi-2a_\\mu)$ and \n$(\\partial_\\mu\\theta-a_\\mu-eA_\\mu)$, which may be interpreted as the three \nvelocities of the spinon and holon condensates respectively. \nFurthermore, the only coupling between\nholons and spinons arises from $a_\\mu$. Therefore, if we now proceed to \nintegrate out the remaining microscopic degrees of freedom from \n${\\cal L}_{\\rm eff}$, the two \nvelocity terms will not mix. This consideration suggests that upon \nintegrating out all of the microscopic degrees of freedom, the resulting\ngauge stiffness term will be of the form\n%\n\\begin{eqnarray}\nf'_{\\rm gauge}&=&{\\sigma_\\Delta\\over2}[\\nabla\\times(2{\\bf a}-\\nabla\\phi)]^2\n\\nonumber \\\\\n&+&{\\sigma_b\\over2}[\\nabla\\times({\\bf a}+e{\\bf A}-\\nabla\\theta)]^2.\n\\label{fgaugep}\n\\end{eqnarray}\n%\nClearly, such term is permitted by the gauge symmetry. Furthermore, we note \nthat for smooth (i.e.\\ vortex free) configurations of phases the gradient\nterms will contribute nothing and we recover the gauge term considered\nin Ref.\\ \\cite{lee3}. \n\nIn the presence of a vortex in $\\phi$ or $\\theta$ the $f'_{\\rm gauge}$\nterm will contribute formally divergent energy. Regularizing\nthis on the lattice, as discussed above Eq.\\ (\\ref{esigp}),\nthis energy will become finite and can be interpreted simply as the\nenergy of the spinon or holon vortex core states, which have been integrated \nout. In the microscopic theory (\\ref{leff}) such energy would arise upon\nsolving the relevant fermionic or bosonic vortex problem. \n\nWe stress that, as concluded\nin the preceding subsection, the main theoretical obstacle to the \nformation of a singly quantized holon vortex in the original SNL theory \nwas the appearance of a formally\ndivergent contribution in the $f_{\\rm gauge}$ term (\\ref{fgauge}). The \nargument above suggests that $f_{\\rm gauge}$ in Eq.\\ (\\ref{fgl1})\n should be replaced by Eq.\\ \n(\\ref{fgaugep}), in which such formally divergent contribution appears for\n{\\em arbitrary} vortex configuration and upon regularization has a simple\nphysical interpretation in terms of the energy of the vortex core states. \nUsage of the physically\nmotivated term (\\ref{fgaugep}) in place of (\\ref{fgauge}) therefore \nremoves the bias against the singly quantized holon vortex solution, which \nappears to be realized in real materials. With (\\ref{fgaugep}) any bias \nbetween the holon and spinon vortex solutions can result only from the \ndifference between the two stiffness constants $\\sigma_\\Delta$ and \n$\\sigma_b$. It is reasonable on physical grounds to assume that\nconstants $\\sigma_\\Delta$ and $\\sigma_b$ are of the similar magnitudes. \nFurthermore, on the basis of Ref.\\ \\cite{nayak1} we expect these constants\nto be negligibly small in the physically relevant models. \nConsequently we\nexpect that neglecting the $f_{\\rm gauge}$ term as in our derivation\nof effective action (\\ref{feff}) will result in accurate determination\nof the phase diagram for the state in the vortex core.\n\n\n\\subsection{Vortex core states}\n\nThe phenomenological theory based on the effective action (\\ref{feff}) \ndoes not allow us to address the interesting question of the nature of \nthe fermionic states in the vortex core. To do this we need to consider \nthe microscopic Lagrangian density\n(\\ref{leff}). While the fully self consistent calculation is likely \nto be prohibitively difficult, one can obtain qualitative insights by first\nsolving the GL theory (\\ref{feff}) as described in Sec.\\ II, and then\nusing the order parameters $\\rho_\\Delta$ and $\\rho_b$ \nas an input to the fermionic and bosonic sectors \nof the theory specified by Eq.\\ (\\ref{leff}). The work on a detailed\nsolution of this type is in progress. Here we wish to point out some \ninteresting features of such a theory and argue that it may indeed exhibit\nstructure in the low energy spectral density similar to that found \nexperimentally\\cite{maggio1,pan2}.\n\nIt is instructive to integrate out the gauge fluctuations from the \nLagrangian (\\ref{leff}) as first discussed by Lee\\cite{dhlee1}. Since\n${\\cal L}_{\\rm eff}$ is quadratic in $a_\\mu$ the integration can be\nexplicitly performed resulting in the Lagrangian of the form\n%\n\\begin{eqnarray}\n{\\cal L}'_{\\rm eff}&=&\n{1\\over 2}K_\\mu(v_s^\\mu)^2 -f_{\\rm amp} + {\\cal L}_{\\rm EM}\\nonumber \\\\\n&-& {2\\kappa_b^\\mu\\over 4\\kappa_\\Delta^\\mu+\\kappa_b^\\mu}\n(v_s^\\mu J_{\\rm sp}^\\mu)\n+{4\\kappa_\\Delta^\\mu\\over 4\\kappa_\\Delta^\\mu+\\kappa_b^\\mu}\n(v_s^\\mu J_h^\\mu)\\nonumber \\\\\n&+&{\\cal L}_{\\rm sp} +{\\cal L}_h\n-{1\\over 2}{1\\over 4\\kappa_\\Delta^\\mu+\\kappa_b^\\mu}(2J_{\\rm sp}^\\mu+J_h^\\mu)^2,\n\\label{leffi}\n\\end{eqnarray}\n%\nwhere $K_\\mu=4\\kappa_\\Delta^\\mu\\kappa_b^\\mu/(4\\kappa_\\Delta^\\mu+\\kappa_b^\\mu)$\nand \n%\n\\begin{equation}\nv_s^\\mu=(\\partial_\\mu\\theta-{1\\over2}\\partial_\\mu\\phi-eA_\\mu)\n\\label{vs}\n\\end{equation}\n%\nis\nthe physical superfluid velocity. The first line reproduces the GL effective\naction (\\ref{feff}) for the condensate fields, \nthe second line describes the Doppler shift coupling \nof the superfluid velocity to the microscopic currents, and the third line\ncontains spinon and holon pieces with additional current-current interactions\ngenerated by the gauge fluctuations\\cite{dhlee1}. \n\n\nWe now discuss the physical implications of Eq.\\ (\\ref{leffi}) for the two \ntypes of vortices. We focus\non the static solutions (i.e. we ignore the time dependences of various \nquantities, e.g.\\ taking $v_s^0=0$) of ${\\cal L}'_{\\rm eff}$ in the \npresence of a single isolated vortex. We are interested in the local\nspectral function of a physical electron. This is\ngiven by a convolution in the energy variable of the spinon and holon \nspectral functions. According\nto the analysis presented in Ref.\\ \\cite{lee2}, at low temperatures \nthe electron spectral function will be essentially equal to the spinon\nspectral function. Convolution with the\nholon spectral function which is dominated by the sharp coherent peak due to \nthe condensate merely leads\nto a small broadening of the order $T$. In the following we therefore focus\non the behavior of spinons in the vicinity of the two types of vortices. \n\n\nBy inspecting Eq.\\ (\\ref{leffi}) it is easy to see that the excitations\ninside the {\\em spinon vortex} will be qualitatively \nvery similar to those found in the conventional vortex described by the weak\ncoupling $d$-wave BCS theory\\cite{wang1,franz1,kita1}. In particular according \nto Eq. (\\ref{rhodcore}) we have $\\kappa_\\Delta\\sim r^2$,\nand $\\kappa_b\\sim$ const in the core. Recalling furthermore that \n$|{\\bf v}_s|\\sim 1/r$ we observe that the spinon current ${\\bf J}_{\\rm sp}$ \nis coupled to a term that diverges as $1/r$ in the core (just as in a \nconventional vortex), while the holon current ${\\bf J}_h$ is coupled to a \nnonsingular term. Thus, one may conclude that holons remain essentially\nunperturbed by the phase singularity in the spinon vortex while the spinons \nobey the essentially conventional Bogoliubov-de Gennes equations for a \n$d$-wave vortex. \n\nIn the {\\em holon vortex} the situation is quite different. According to \nEq. (\\ref{rhobcore}) we have $\\kappa_b\\sim r$\nand $\\kappa_\\Delta\\sim$ const in the core. The spinon current \n${\\bf J}_{\\rm sp}$ is now\ncoupled to a nonsingular term ($1/r$ divergence in $v_s$ is canceled by \n$\\kappa_b\\sim r$).\nTherefore, there will be no topological perturbation in the spinon sector\nand we expect the spinon wavefunctions to be essentially unperturbed by \nthe diverging superfluid velocity. Spinon spectral density in the core \nshould be qualitatively similar to that far outside the core. This is our basis\nfor expecting a pseudogap-like spectrum in the core of a holon vortex. \n\nWe now address the possible origin of the experimentally observed vortex\ncore states\\cite{maggio1,pan2} within the present scenario for a holon vortex.\nTo this end consider the effect of the last term in Eq.\\ (\\ref{leffi}) which \nwe ignored so far. Upon expanding the binomial \nthe temporal component is seen to contain a density-density \ninteraction of the form $J_{\\rm sp}^0 J_h^0$ where $J_h^0$ is the local\ndensity of {\\em uncondensed} holons. Since the holon order parameter\nvanishes in the core and the electric neutrality dictates that \nthe total density of holons must be approximately constant in space, we \nexpect that uncondesed holon density will behave roughly as\n%\n\\[\nJ_h^0(r)=\\bar\\rho_b-\\rho_b(r);\n\\]\n%\n$J_h^0(r)$ will have a spike in the core of a holon vortex.\nInsofar as $J_h^0(r)$ can be viewed as a static potential acting on\nspinons, the uncondensed holons in the vortex core can be thought of as \ncreating a scattering potential, akin to an \nimpurity embedded in a $d$-wave superconductor. \nIn fact, formally the spinon problem is identical to the problem of a \nfermionic quasiparticle in a $d$-wave superconductor\nin zero field in the presence of a localized impurity potential. It is known\nthat such problem exhibits a pair of marginally bound impurity \nstates\\cite{balatsky1}\nat low energies which result in sharp resonances in the spectral density\ninside the gap. Such states have been extensively studied theoretically\n\\cite{balatsky2,flatte1,shnirman1,atkinson1} and their existence was \nrecently confirmed experimentally by Pan {\\em et al.}\n\\cite{pan1}. We propose here that, within the formalism of Eq.\\ (\\ref{leffi}),\nthe same mechanism could give rise to the low energy quasiparticle states \nin the core of a holon vortex. Such structure, if indeed confirmed by a\nmicroscopic calculation, could explain the spectral features observed\nexperimentally in the vortex cores of cuprate superconductors\n\\cite{maggio1,pan2}.\n\n\n\n\\section{Conclusions}\n\nScanning tunneling spectroscopy of the vortex cores affords a unique \nopportunity for probing the underlying ``normal'' ground state in cuprate \nsuperconductors. The existing experimental data\non YBCO and BSCCO strongly suggest that conventional mean field weak coupling\ntheories \\cite{soininen1,wang1,franz1,kita1,volovik1,maki1,ichioka1}\nfail to describe the physics of the vortex core. \nOur main objective was to develop a theoretical framework for understanding \nthese spectra and the nature of the strongly\ncorrelated electronic system which emerges once the superconducting \norder is suppressed. We have shown that phenomenological model (\\ref{fgl1})\nbased on a variant of the U(1) gauge field slave boson theory\n\\cite{dhlee1} contains the right physics, provided that the gauge field \nstiffness\nis vanishingly small. The latter assumption is consistent with the \ngeneral arguments involving local gauge symmetry\\cite{nayak1}. In such a \ntheory the gauge field can be explicitly integrated out, resulting in the \neffective action (\\ref{feff}) which contains one phase degree of freedom \nrepresenting the phase of a Cooper pair and two amplitude degrees of\nfreedom representing the holon and spinon condensates. \n\nAnalysis of the effective theory (\\ref{feff}) in the presence of a \nmagnetic field\nestablishes existence of two types of vortices, spinon and holon, with \ncontrasting spectral properties in their core regions.\nOur holon vortex is singly quantized and therefore differs in a profound way\nfrom the doubly quantized holon vortex discussed by SNL\\cite{sachdev1,lee3}.\nAs indicated in Figure \\ref{fig2} such a singly quantized \nholon vortex is expected to be \nstable over the large portion of the phase diagram on the underdoped side. \nQuasiparticle spectrum in the core of a holon vortex is predicted to exhibit \na ``pseudogap'', similar to that found in the underdoped normal region \nabove $T_c$. This is consistent with the data of Renner {\\em et al.}\n\\cite{renner1} who pointed out a remarkable similarity between the vortex\ncore and the normal state spectra in BSCCO. Spinon vortex, on the other hand, \nshould be virtually indistinguishable from the conventional $d$-wave BCS\nvortex and is expected to occur on the overdoped side of the phase diagram. \nTransition from the insulating holon vortex to the metallic spinon vortex\nas a function of doping is a concrete testable prediction of the present \ntheory. \n\nPhenomenological theory based on the effective \naction (\\ref{feff}) does not permit \nexplicit evaluation of the electronic spectral function. To this end\nwe have considered the corresponding microscopic theory \n(\\ref{leffi}) and concluded that\nholon vortex will indeed exhibit a pseudogap like spectrum. Such qualitative\nanalysis furthermore suggests a plausible mechanism for the sharp vortex \ncore states observed in YBCO\\cite{maggio1} and BSCCO\n\\cite{pan2}. We stress that conventional mean field weak coupling theories \nyield neither pseudogap nor the core states.\nIn the core of a holon vortex such states will arise\nas a result of spinons scattering off of the locally uncondensed holons,\nin a manner analogous to the quasiparticle resonant states in the vicinity\nof an impurity in a $d$-wave superconductor\n\\cite{balatsky1,balatsky2,flatte1,shnirman1,atkinson1}.\nThe latter conclusion is somewhat speculative and must be \nconfirmed by explicitly solving the fermionic sector of the microscopic\ntheory (\\ref{leffi}). \n\nOn a broader theoretical front the importance of the vortex core spectroscopy\nas a window to the normal state in the $T\\to 0$ limit lies in its potential\nto discriminate between various microscopic theories of cuprates. \nIt is reasonable to assume that the observed pseudogap in the \nvortex core reflects the same physics as the pseudogap observed \nin the normal state.\nThis means that the mechanism responsible for the pseudogap must be operative \non extremely short lengthscales, of order of several lattice spacings.\nThe U(1) slave boson theory considered in this work apparently satisfies \nthis requirement. Obtaining the correct vortex core spectral functions could\nserve as an interesting test for other theoretical approaches describing \nthe physics of the underdoped cuprates\\cite{lee2,pines1,levin1}.\n\nIt will be of interest to explore \nthe implications of the effective theories (\\ref{feff}) and (\\ref{leffi}) \nin other physical situations. Of special interest are situations where \nthe holon condensate amplitude is suppressed, locally\nor globally, giving rise to ``normal'' transport properties (vanishing \nsuperfluid density) but quasiparticle excitations that are characteristic \nof a superconducting state. These include\nthe spectra in the vicinity of an impurity, \ntwin boundary or a sample edge. In the latter case one might hope to observe \na signature of the zero bias tunneling peak anomaly\n(normally seen for certain geometries deep in the superconducting phase in \nthe optimally doped cuprates) even above $T_c$ in the underdoped samples. \n\n\\acknowledgments\nThe authors are indebted to A. J. Berlinsky, \nJ. C. Davis, \\O. Fischer, C. Kallin, D.-H. Lee, P. A. Lee,\nS.-H. Pan, C. Renner, J. Ye and S.C. Zhang\nfor helpful discussions. This research was \nsupported in part by NSF grant DMR-9415549 and by Aspen Center for Physics\nwhere part of the work was done. \n\n\\vskip 10pt\n{\\em Note added in proof.} After submission of this manuscript we learned\nabout complementary microscopic treatments of the spin-charge separated\nstate in the \nvortex core within U(1) \\cite{han1} and SU(2) \\cite{wen1} slave boson \ntheories. 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{
"name": "cond-mat0002137.extracted_bib",
"string": "\\bibitem{ando1} G. S. Boebinger, Y. Ando, A. Passer, T. Kimura,\nM. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa and S. Uchida,\n \\prl {\\bf 77}, 5417 (1996).\n\n\\bibitem{lemberger1} K. Karpinska, M. Z. Cieplak, S. Guha, A. Malinowski,\nT. Skoskiewicz, W. Plesiewicz, M. Berkowski, B. Boyce, T. R. Lemberger and\nP. Lindenfeld, \\prl {\\bf 84} 155 (2000).\n\n\\bibitem{maggio1} I. Maggio-Aprile, Ch. Renner, A. Erb, E. Walker, and \n\\O. Fischer, \\prl {\\bf 75}, 2754 (1995).\n\n\\bibitem{renner1} Ch. Renner, B. Revaz, K. Kadowaki, I. Maggio-Aprile,\n and \\O. Fischer, \\prl {\\bf 80}, 3606 (1998).\n\n\\bibitem{pan1} E. W. Hudson, S. H. Pan, A. K. Gupta, K.-W. Ng and \nJ. C. Davis, {\\em Science} {\\bf 285}, 88 (1999).\n\n\\bibitem{pan2} S. H. Pan, E. W. Hudson, A. K. Gupta, K.-W. Ng,\nH. Eisaki, S. Uchida and J. C. Davis,\\prl {\\bf 85}, 1536 (2000).\n\n\\bibitem{soininen1} P.I. Soininen, C. Kallin, and A.J. Berlinsky, \\prb{\\bf 50},\n13883 (1994).\n\n\\bibitem{wang1} Y. Wang and A. H. MacDonald, \\prb {\\bf 52}, R3876 (1995).\n\n\\bibitem{franz1} M. Franz and Z. Te\\v{s}anovi\\'c, \\prl{\\bf 60}, 4763\n (1998).\n\n\\bibitem{kita1} K. Yasui and T. Kita, \\prl {\\bf 83}, 4168 (1999).\n\n\\bibitem{volovik1} G.\\ E.\\ Volovik, JETP Lett.{\\bf 58}, 469 (1993).\n\n\\bibitem{maki1} Y. Morita, M. Kohmoto, and K. Maki, \\prl {\\bf 78}, 4841\n(1997).\n\n\\bibitem{ichioka1} M. Ichioka, N. Hayashi, N. Enomoto, and K. Machida, \n\\prb {\\bf 53}, 15316 (1996).\n\n\\bibitem{franz2} M. Franz and M. Ichioka, \\prl {\\bf 79}, 4513 (1997).\n\n\\bibitem{himeda1} A. Himeda, M. Ogata, Y. Tanaka, and S. Kashiwaya,\nJ.\\ Phys.\\ Soc.\\ Japan {\\bf 66}, 3367 (1997).\n\n\\bibitem{resende1} X. R. Resende and A. H. MacDonald, Bull.\\ Am.\\ Phys.\\ Soc.\\\n{\\bf 43}, 573 (1998).\n\n\\bibitem{arovas1} D. P. Arovas, A. J. Berlinsky, C. Kallin, and\nS.-C. Zhang, \\prl {\\bf 79}, 2871 (1997).\n\n\\bibitem{zhang1} S.-C. 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Wen,\n \\prb {\\bf 57}, 6003 (1998).\n\n\\bibitem{balents1} L. Balents, M. P. A. Fisher and C. Nayak, Int.\\ J.\\ Mod.\\\nPhys.\\ B {\\bf 12}, 1033 (1998); \\prb {\\bf 60}, 1654 (1999). \n\n\\bibitem{sachdev1} S. Sachdev, \\prb {\\bf 45}, 389 (1992).\n\n\\bibitem{lee3} N. Nagaosa and P. A. Lee, \\prb {\\bf 45}, 966 (1992).\n\n\\bibitem{abrikosov1} A.A. Abrikosov, Zh.\\ Eksp.\\ Teor.\\ Fiz.\\ {\\bf 32}, 1442 \n(1957) [Sov.\\ Phys.\\ JETP {\\bf 5}, 1174 (1957)].\n\n\\bibitem{fetter1} A. L. Fetter and P. C. Hohenberg, in \n{\\em Superconductivity}, edited by R. D. Parks (Marcel Dekker, New York 1969).\n\n\\bibitem{elitzur1} S. Elitzur, \\prd {\\bf 12}, 3978 (1975).\n\n\\bibitem{nayak1} C. Nayak, \\prl {\\bf 85}, 178 (2000).\n\n\\bibitem{hu1} C.-R. Hu, \\prb {\\bf 6} 1756 (1972).\n\n\\bibitem{alama1} S. Alama, A. J. Berlinsky, L. Bronsard, and T. Giorgi, \n\\prb {\\bf 60}, 6901 (1999). \n\n\\bibitem{numrec} W. H. Press, B. P. Flannery, S. A. Teukolsky, and \nW. T. 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Levin, \\prl {\\bf 81}, 4708\n(1998); \\prb {\\bf 59}, 7083 (1999).\n\n\\bibitem{han1} J. H. Han and D.-H. Lee, \\prl{\\bf 85}, 1100 (2000).\n\n\\bibitem{wen1} P. A. Lee and X.-G. Wen, cond-mat/0008419\n\n\\bibitem{senthil1} T. Senthil and M.P.A. Fisher, \\prb {\\bf 62}, 7850 (2000);\ncond-mat/0006481.\n\n%\n\\bibitem{} , \\prl {\\bf }, ().\n"
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cond-mat0002138
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Modelling an Imperfect Market
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[
{
"author": "Raul Donangelo"
},
{
"author": "\\footnote{Permanent address: Instituto de F\\'\\i sica"
},
{
"author": "Universidade Federal do Rio de Janeiro"
},
{
"author": "Cidade Universit{\\'a}ria, CP 68528, 21945--970 Rio de Janeiro--RJ, Brazil} Alex Hansen"
},
{
"author": "Blegdamsvej 17"
},
{
"author": "DK-2100 Copenhagen {\\O}, Denmark} and Sergio R.\\ Souza$^*$"
}
] |
We propose a simple market model where agents trade different types of products with each other by using money, relying only on local information. Value fluctuations of single products, combined with the condition of maximum profit in transactions, readily lead to persistent fluctuations in the wealth of individual agents.
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[
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"name": "paper5.tex",
"string": "\\documentstyle[prl,aps,preprint]{revtex}\n\\begin{document}\n%------------------------------------------------------------------------------\n\\title{Modelling an Imperfect Market}\n\\author{Raul Donangelo,\\footnote{Permanent address: Instituto de F\\'\\i sica,\nUniversidade Federal do Rio de Janeiro, Cidade Universit{\\'a}ria, CP 68528,\n21945--970 Rio de Janeiro--RJ, Brazil} \nAlex Hansen,\\footnote{Permanent address: Department of Physics, Norwegian \nUniversity of Science and Technology, NTNU, N--7034 Trondheim, Norway}\nKim Sneppen \\footnote{Permanent address: NORDITA, Blegdamsvej 17, DK-2100 \nCopenhagen {\\O}, Denmark}\nand Sergio R.\\ Souza$^*$}\n\\address{International Centre for Condensed Matter Physics,\nUniversidade de Bras{\\'\\i}lia, CP 04513, 70919--970 Bras{\\'\\i}lia--DF,\nBrazil}\n\\address{and}\n\\address{NORDITA, Blegdamsvej 17, DK--2100 Copenhagen {\\O}, Denmark}\n\\date{\\today}\n\\maketitle\n%--------------------------------------------------------------------\n\\begin{abstract} \nWe propose a simple market model where agents trade\ndifferent types of products with each other by using money,\nrelying only on local information.\nValue fluctuations of single products, combined with the condition\nof maximum profit in transactions, readily lead to persistent\nfluctuations in the wealth of individual agents.\n\\end{abstract} \n%--------------------------------------------------------------------\n\\section{Introduction}\n\\label{intro}\n%--------------------------------------------------------------------\nFinancial markets usually consist in trades of commodities and currencies. \nHowever, one can easily find cases in other types of\nhuman endeavour that parallel the activities observed\nin financial markets.\nFor example, a politician may select the standpoints in his/her\nplatform in exchange for votes in the coming election\\cite{D57}.\nAlso, scientists may trade ideas in order to generate citations.\nHowever, the easiest quantifiable marketplace is the financial one,\nwhere the existence of money allows a direct measure of value.\n\nThere have been several proposals to model such markets.\nA recent review was presented by Farmer \\cite{F99}.\nBak, Paczuski and Shubik \\cite{BPS97} have proposed a model where they\nconsider the price fluctuations of a single product traded by many agents, \nall of which use the same strategy. \nIn that model fat tails and anomalous Hurst exponents appear when global \ncorrelation between agents are introduced.\nAn alternative model, presented by Lux and Marchesi \\cite{LM99},\nis driven by exogenous fluctuations in ``fundamental'' values of\na single good, with induced non-gaussian fluctuations in price\nassessment of this good arising from switching of strategies \n(trend followers and fundamentalists) by the agents. \n\nBelow we give a few considerations that had led us in the development\nof a model where the dynamics arises solely from the interaction of \nagents trying to exchange several types of goods.\n\nFinancial markets exhibit a dynamical behavior that, even in the absence\nof production, allow people to become either wealthy or poor.\nIf in these markets there were some sort of equilibrium, e.g.\\ due to\ncomplete rationality of the players, this would not be possible.\nIn order to create wealth or bankruptcy, people have to outsmart each other.\nThis means that each trader attempts to buy as cheaply and sell as\nexpensively as possible.\nThis demands that an agent should find a seller which sells at a low price,\nand later another one that is willing to buy the same product at a higher one.\nThus different agents price differently the same product.\nIn this work we demonstrate that it is possible to devise a simple model\nfor such a non-equilibrium market.\nWe call it the Fat Cat (FC) model, as it functions on greed\n(each agent buys to optimize his own assets)\nand creates a market with large fluctuations, i.e.\\ fat tails. \nIn Sect.\\ \\ref{model} we present this model and show examples of its\ndynamical evolution. \nWe show that it leads to an ever fluctuating market.\nIn the following Sect. \\ref{fluctuations}, the time series generated by this\nmodel is analyzed and it is shown that it exhibits \npersistency in the fluctuations of the wealth of individual agents. \nIn the final section we summarize the work and present suggestions\nfor generalizing the model to let individual agents evolve their\nindividual trading strategies.\n%--------------------------------------------------------------------\n\\section{Description of the FC model}\n\\label{model}\n%--------------------------------------------------------------------\nConsider a market with $N_{ag}$ agents, \neach having initially a stock of $N_{un}$ \nunits of products selected among $N_{pr}$ different types of products.\nIn a previous work \\cite{DS99} we have shown that, for the case of agents \nhaving memory of past transactions, such a market spontaneously selects\none of the products as the most adequate as a means of exchange. \nThe product so chosen acts as money in the sense that it is accepted even when\nthe agent does not need it, because through the memory of past requests of\nproducts the agent knows it is in high demand, and therefore it will be \nuseful to have it to trade for other products.\nThe selection of a product as an accepted means of exchange is not indefinite:\nafter some time another one substitutes it as the favorite currency. \nThe time scale for these currency substitutions is large when measured in \nnumber of exchanges between individual agents.\n \nIn the present model we consider that each of the agents initially has\n$N_{mon}$ units of money.\nAccording to the discussion in the preceding paragraph, in principle money\ncould be viewed as one of the products, but here we consider it a separate\nentity in order to quantify prices.\nThe $N_{un}$ products given initially to each agent are selected\nrandomly.\nLater, during the time evolution of the system, each agent $i$ \nhas, at each time step, an amount of money $M (i)\\,,i=1,\\dots,N_{ag}$, and a \nstock of the different products $j$, $S (i,j)$, where $j=1,...,N_{pr}$.\nSince the model uses money as means of exchange, agents assign different\nprices to the different products in their possessions. \nThe prices of the different items in the stock of agent $i$, $P(i,j)$, are\ntaken initially to be integers uniformly distributed in the interval $[1,5]$. \nWe have verified that the evolution of the system does \nnot depend on this particular choice.\n\nHow do we picture such a market?\nWe may imagine antique collectors trying to buy objects directly from each\nother, using their own estimates for the prices of the different stock. \nWhen two agents meet, one of them, the buyer, checks the seller's price list,\nand compares it with his own price list. We have chosen the antique collector\nmarket as an example because \nfew other markets show spatial price fluctuations at \nsuch a high level.\nThe decision to buy or not, and the changes in the value of the agents'\nproducts are given by some strategy, which for now assume is the same for\nall agents in the system.\nAmong all the products that the buyer considers to be possible buys (having\na price set by the seller which is lower than the one he would sell the same\nproduct for), he will single out the best one, and will then attempt to buy it.\nBut, if the buyer finds no products that he considers as good buys,\nthe seller will consider that he has overpriced his goods and will as a\nconsequence tend to lower his prices.\nAt the same time, the buyer will think that his price estimate was too low, \nand as a consequence raise his price estimate.\n\nThis is the basis for our computer simulation for such a market place.\nIn it, we assume that at each time step the following procedure takes place:\n\\begin{enumerate}\n\\item Buyer ($b$) and seller ($s$) are selected at random among the $N_{ag}$\nagents.\nIf the seller has no products to offer, then another seller is chosen.\\\\\n\n\\item The buyer selects the product $j$ in the seller's stock which maximizes\n$P(b,j)-P(s,j)$ (i.e. his profit).\nThe corresponding $j$, we call $j_{bb}$ (best buy).\\\\\n\n\\item If the buyer does not have enough money, \n(i.e.\\ if $M(b) < P(b,j_{bb})$), we go back to the first step, choosing\na new pair of agents.\\\\\n\n\\item If the buyer has enough money we proceed.\\\\\nIf $P(s,j_{bb}) < P(b,j_{bb})$, the transaction is performed at the\nseller's price. This means that we adjust:\n$S(b,j_{bb})\\rightarrow S(b,j_{bb})+1$, $S(s,j_{bb})\\rightarrow S(s,j_{bb})-1$,\n\n$M(b)\\rightarrow M(b)-P(s,j_{bb})$, $M(s)\\rightarrow M(s)+P(s,j_{bb})$.\\\\\n\n\\item If $P(s,j_{bb})\\ge P(b,j_{bb})$, the transaction is not performed.\\\\\nIn this case, the seller lowers his price by one unit,\n\n$P(s,j_{bb})\\rightarrow \\max(P(s,j_{bb})-1,0)$,\n\nand the buyer raises his price by one unit,\n\n$P(b,j_{bb})\\rightarrow P(b,j_{bb})+1$.\\\\\n\\end{enumerate}\n\nWe see that, according to these rules, buyer and seller decide on a\ntransaction based only on their local information, i.e. their\nestimates of the prices for the different products they possess.\nThese prices are always non-negative integers.\nAlso note that since, as defined in step 3 above, the price offered by the\nbuyer cannot be higher than the amount of money he has, we are not\nallowing for the agents to get in debt. \nFurther, the model tends to equilibrize large price differences, according\nto step 5, but induces price differences when buyer and seller agree on\nthe price of the most tradeable product.\nThis non-equilibrizing step is essential to induce dynamics in a model like\nthe present one, where all agents follow precisely the same strategy.\nWithout it the system would freeze into a state where all agents agree on all\nprices. \n\nThe rules given above are just one possible set of rules for transactions.\nWe have found other sets that lead to a behavior qualitatively similar\nto the one shown below for the present rules.\n\\footnote{A quote from Marx is appropriate here: ``These are my principles. \nIf you do not like them, I have others\" \\cite{M34}.}\nWe now show that, under this set of local rules, the distribution of wealth\norganizes itself into a dynamically stable pattern, and the same phenomenon\ntakes place with the prices.\n\nOne should emphasize that there is not an accepted market value for the\nproducts.\nIndeed, due to the price adjustments performed in unsuccessful encounters, \nthe prices never reach equilibrium, and different agents may assign\ndifferent prices for the same product.\nIn Fig.\\ \\ref{fig1} we illustrate this point, showing the price assigned\nby two different agents to the same product, as a function of time.\nTime is defined here in terms of the number of encounters between agents,\nand one time unit is the average time between events where a given agent\nacts as a buyer.\nWe note that during a considerable fraction of the time there is a relatively\nlarge difference between the prices assigned by the agents.\nThis shows that there is a margin for making profit in such a market,\ni.e.\\ arbitrage is possible. We have verified that\nthe average market price of a good\nfluctuates with a Hurst exponent of $\\sim 0.5$.\n\nIn Fig.\\ \\ref{fig2} we show, for the same time interval, an example of the\nevolution of key quantities in the model associated to Agent 1 in\nFig.\\ \\ref{fig1}.\nThe total wealth of an agent $i$ is the amount of money plus the \nvalue of all goods in the agent's possession:\n\\begin{equation}\n\\label{eq1}\nw(i) = M(i) + G(i)\\;.\n\\end{equation}\nHere the value of product $j$ is defined as the average of what all\nagents consider its value to be:\n\\begin{equation}\n\\label{eq2}\nP_{ave} (j) = \\frac{1}{N_{ag}} \\sum_{i=1}^{N_{ag}} P(i,j)\\;,\n\\end{equation}\nand the value of all agent $i$'s goods $G(i)$ is then defined as\n\\begin{equation}\n\\label{eq3}\nG(i)=\\sum_j S(i,j) P_{ave} (j)\\;.\n\\end{equation}\n\nWe note that there are considerable fluctuations in the wealth of this\nagent. The study of these fluctuations is essential to the understanding\nof the properties of the model, and we develop this in the following section.\n\n%--------------------------------------------------------------------\n\\section{Fluctuations in the FC model}\n\\label{fluctuations}\n%--------------------------------------------------------------------\n\nIn order to quantify the fluctuations in wealth, we show in Fig.\\ \\ref{fig3}\nthe RMS fluctuations of the wealth of a selected agent as function of time. \nThe figure illustrates that the wealth fluctuations can be characterized\nby a Hurst exponent \\cite{F88} $H\\approx 0.7 $. \nWe have examined variants of both model parameters and rules\nto check the stability of this result.\nWe found it to be stable, as long as one keeps greed in the model.\nFor example, if one reduces the number of product types to only \n$N_{pr}=2$ (keeping $N_{ag}=100$, $N_{mon}=500$ and $N_{un}=100$ initially)\nthe Hurst exponent remains unchanged although the scaling regime shrinks.\nSimilarly setting $N_{un}=2$ (with $N_{pr}=100$\n$N_{ag}=100$ and $N_{mon}=500$) resembling antique dealing where\neach agents owns a few of many possible products, also\nlets the Hurst exponent unchanged.\nOn the other hand, if greed is removed from the model, e.g.\\ \nthe buyer selects a product at random from the sellers store,\nwithout consideration to the profit margin, the Hurst exponent\ndrops to $0.5$, signaling that no correlations develop in such a case.\n\nThus, the optimization of product selection expressed by step 2 in the\nprocedure is closely related to the persistent fluctuations seen in our model.\nWe have checked that other optimization procedures, as e.g.\\ selecting the\ncheapest product or the product the buyer has the least of in stock also give\nsimilar persistent fluctuations.\nOppositely, random selection reflect an economy where different products do\nnot interact significantly with each other, and where our market of $N_{pr}$\ndifferent types of products nearly decouples into $N_{pr}$ different markets.\nWith random selection, the only interaction between products is indirect;\nit appears due to the constraint of the agents' money take only non-negative\nvalues. Overall, the random strategy gives a less \nfluctuating market where agents agree more on prices.\nGreed indeed makes our model world both richer and more interesting \n(which is {\\it not\\/} to say {\\it better.\\/}).\n\nWe now try to quantify these wealth fluctuations. Fig.\\ \\ref{fig4}\ndisplays the changes in the value of $w$ for several time intervals $\\Delta t$.\nThe three curves are histograms of wealth changes for respectively \n$\\Delta t=10$, $\\Delta t=100$ and $\\Delta t=1000$.\nOne observes fairly broad distributions with a tendency to asymmetry\nin having bigger probability for large losses than for large gains.\nSimilar skewness is seen in real stock market data.\nFurthermore, the tails are outside the Gaussian regime \\cite{Plerou99}. \nIn the upper panel of Fig.\\ \\ref{fig5}, this is investigated further by\nplotting the histograms as function of the logarithmic changes\nin $w$, $r_{\\Delta t}=\\log_2 w(t+\\Delta_t)-\\log_2 w(t)$ ({\\sl log-returns}),\nfor the same three time intervals.\nIn that figure we have collapsed the curves onto each other by\nrescaling them using the Hurst exponent $H=0.69$, consistent with \nthe one found in Fig.\\ \\ref{fig3}. For the two short time intervals\nthe collapse is nearly perfect, even in the non-Gaussian fat tails.\nFor larger time intervals the distribution changes from a steep power \nlaw or stretched exponential, to an exponential, and finally becomes\nGaussian for very large intervals (not shown as it is very narrow on the\nscale of this figure). \nWe think it is interesting to notice that our model is consistent with\nthe empirical observation that in real markets the probability \nfor large negative fluctuations is larger than that for large positive ones.\n\nIn the lower panel of Fig. \\ref{fig5} we examine in details the fluctuations\nfor $\\Delta t=1$, and compare the fat tails with truncated power law decays\n$P(r) \\sim 1/r^4 \\cdot \\exp(-|r|/R)$ which for such small time intervals\nis nearly symmetrical.\nThe $1/r^4$ is consistent with the fat tail observed on 5-minute interval\ntrading of stocks \\cite{Plerou99}.\nThe cut off size $R=0.8$\ncorresponds to cut offs when price changes are about\na factor 2 from the original price, a regime which is not addresses in the\nshort time trading analysis of \\cite{Plerou99}.\nWe stress that our analysis of fat tails includes a wide distribution\nof wealth, thus large relative changes of wealth \nare presumably mostly associated to poor agents.\nThus the seemingly good fit to fat tails observed\nfor stock market fluctuations in the 1000 largest US companies\n\\cite{Plerou99} may be coincidental.\n\n\\section{Summary and Discussion}\n\\label{theend}\n\nThe appearance of fat tails\n\\cite{M63,E95,Plerou99} and Hurst exponents\\cite{E95,M91} \nlarger than 0.5 in the\ndistribution of monetary value appears to be a characteristic of\nreal markets.\nThe present model is, as it was the case with \nthe previous version \\cite{DS99}, qualitatively consistent with these features. \nWe stress that here, as for the previous\nmodel, we are not including any development of strategy by the agents that\nmight force the emergence of cooperativity \\cite{Z97,PB99}. \nA more important difference to game theoretic models is that the minority game,\nas well as the evolving Boolean network of Paczuski, Bassler and Corral\n\\cite{PB99} evolves on basis on a global reward function.\nWith the present model we would like to open for models which evolve\nwith imperfect information, \npreferably in a form which allows direct\ncomparison with financial data.\nThe present model does this, and in a setting where there are many products\nand thus possibilities for making arbitrage along different ``coordinates''.\n\nCompared to models with fat tails or persistency arising from \nboom-burst cycles, as the trend enhancing model\nof Delong \\cite{Delong90} or the trend following model of Lux and Marchesi\n\\cite{LM99}, the present model discuss \nanomalous scaling in a market where no agent has a precise knowledge of the \nglobal or average value of a product.\nThere is only local optimization of utility (estimated market value) and all\ntrades are done locally without the effects of a global information pool.\nThis imperfect information gives a possibility for arbitrage and opens for \na dynamic and evolving market. \n\nThe model we propose is for a market composed of agents, goods and money. \nWe have demonstrated that such a market easily shows persistent\nfluctuations of wealth, and seen that this persistency is closely related \nto having a market with several products which influence each others� trading.\nAs seen in Fig.\\ \\ref{fig2}, wealth increase of an agent is associated\nwith active trading of few products. This may be understood as\nfollows: when the number of options a seller presents is small,\nit is hard for the buyer to find a bargain, so if a transaction is\nperformed it will probably be at a good price for the seller.\n\nThe simpler model of Ref.~\\cite{DS99}, had persistency in the fluctuations in\nthe demand for different products, whereas, as mentioned above, the persistence\nhere is in the fluctuations in the wealth of the different agents.\nIt is interesting to mention that the Hurst exponents in the two models take\nvery similar values, and this for a wide variety of the respective parameters.\nThere is a kind of duality in that in the simpler model a product increased\nin value when it was held by relatively few agents, whereas in the present\nmodel the agents increase their wealth by specializing to few products.\n\nThe setup proposed here with agents and products with individual local\nprices allows for also different individual strategies of the agents. \nFor example both the selection of greed versus random strategy under \nstep 2, and the particular adjustment of values defined under step 4-5\ncould be defined differently from agent to agent.\nOne may accordingly have different strategies for different agents, and each\nagent could change its strategy in order to improve his performance. The\nevolution of these strategies would then become an inherent part of the\ndynamics. This opens for evolution of strategies as part of the \nfinancial market, and will be discussed in a separate publication\\cite{DHSS}.\n\\vspace{0.5cm}\n\nWe thank F.A.\\ Oliveira and H.N.\\ Nazareno for warm\nhospitality and the I.C.C.M.P.\\ \nfor support during our stay in Bras\\'i lia. \nR.D., A.H.\\ and S.R.S\\ thank NORDITA for support during our stay in \nCopenhagen. R.D.\\ and S.R.S.\\ thank MCT/FINEP/CNPq (PRONEX) under \ncontract number 41.96.0886.00 for partial financial support.\n%--------------------------------------------------------------------\n% BIBLIOGRAPHY\n% --------------------------------------------------------------------\n\\begin{thebibliography}{10}\n\n\\bibitem{D57} \nA.\\ Downs, {\\sl An Economic Theory of Democracy\\/} (Harper\nBros., New York, 1957).\n\n\\bibitem{F99}\nJ.D. Farmer, Comp.\\ in Science and Eng.\\ {\\bf 1}, (6), 26 (1999). \n\n\\bibitem{BPS97}\nP.\\ Bak, M.\\ Paczuski and M.\\ Shubik, Physica A {\\bf 246}, 430 (1997).\n\n\\bibitem{LM99}\nT.\\ Lux and M.\\ Marchesi, Nature {\\bf 397} 498 (1999).\n\n\\bibitem{DS99} R.\\ Donangelo and K.\\ Sneppen, Physica A, {\\bf 276}, 572 (2000).\n\n\\bibitem{M34} G.\\ Marx, unpublished.\n\n\\bibitem{F88} J.\\ Feder, {\\sl Fractals\\/} (Plenum Press, New York, 1988).\n\n\\bibitem{M63}\nB.\\ Mandelbrot, Journal of Business of the University of Chicago\n{\\bf 36}, 307 (1963).\n\n\\bibitem{E95}\nC.J.G.\\ Evertz, Fractals {\\bf 3}, 609 (1995).\n\n\\bibitem{Plerou99}\nV.\\ Plerou, P.\\ Gopikrishnan, L.A.N.\\ Amaral, M.\\ Meyer and H.E.\\ Stanley, \nCond-mat/9907161 (1999).\n\n\\bibitem{M91}\nR.N.\\ Mantegna, Physica A {\\bf 179} (1991) 232.\n\n\\bibitem{Z97}\nD.\\ Challet and Y.-C.\\ Zhang, Physica A {\\bf 246}, 407 (1997);\nY.-C.\\ Zhang, Europhys. News {\\bf 29}, 51 (1998).\n\n\\bibitem{PB99}\nM.\\ Paczuski, K.\\ Bassler and A. Corral, Cond-mat/9905082.\n\n\\bibitem{DHSS}\nR.\\ Donangelo, A.\\ Hansen, K.\\ Sneppen and S.R.\\ Souza, in preparation.\n\n\\bibitem{Delong90}\nJ.B.\\ Delong, J.\\ Finance {\\bf 45} 379 (1990).\n\\end{thebibliography}\n\n% --------------------------------------------------------------------\n% FIGURE CAPTIONS\n% --------------------------------------------------------------------\n\\begin{figure}\n\\caption{Selling prices of a specific product by two different agents, as a function\nof time. The simulation was performed for the following parameter values:\n$N_{ag}=100$, $N_{pr}=100$, $N_{mon}=500$ and $N_{un}=100$, which apply to the\ncalculations presented in all figures in this work.}\n\\label{fig1} \n\\end{figure}\n\n\\begin{figure}\n\\caption{Amount of money $M$, number of different products and combined wealth\n$w=M+G$, held by an agent as a function of time.\nSee text for further\ndetails.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\caption{ RMS fluctuations of the wealth of a given agent as function of time\ninterval $\\Delta t$. In order to guide the eye we also plot the power\nfunction $\\Delta t^{0.7}$.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Probability of having changes in wealth\n$\\Delta w=w(t+\\Delta t)-w(t)$ as a function of their size,\nfor the three different time steps\n$\\Delta t =10$ (full line),\n$\\Delta t = 100$ (long dashed line), and\n$\\Delta t = 1000$ (short dashed line).}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}\n\\caption{a) Similar to Fig.~4 except that wealth fluctuation is measured here\nin units of $\\log_2 \\left[ w(t+\\Delta t)/w(t) \\right] / \\Delta t^H$,\nwhere $\\Delta t$ takes the same values as above, and we took for the Hurst\nexponent the value $H = 0.69$.\nb) Fits to the probability of having a wealth loss (gain) as a function\nof the log-returns $r=\\log_2 \\left[ w(t+\\Delta t)/w(t) \\right]$, for\nthe case $\\Delta t=1$. The fit \n$P(r) = 5/(r^2+0.03)^2 \\cdot \\exp(-|r|/0.4)$\nhave asymptotic expressions for large $|r|$ of the form\n$P(r) \\sim 1/r^4 \\cdot \\exp(-|r|/R)$, with $R=0.4$.}\n\\label{fig5}\n\\end{figure}\n% --------------------------------------------------------------------\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002138.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{D57} \nA.\\ Downs, {\\sl An Economic Theory of Democracy\\/} (Harper\nBros., New York, 1957).\n\n\\bibitem{F99}\nJ.D. Farmer, Comp.\\ in Science and Eng.\\ {\\bf 1}, (6), 26 (1999). \n\n\\bibitem{BPS97}\nP.\\ Bak, M.\\ Paczuski and M.\\ Shubik, Physica A {\\bf 246}, 430 (1997).\n\n\\bibitem{LM99}\nT.\\ Lux and M.\\ Marchesi, Nature {\\bf 397} 498 (1999).\n\n\\bibitem{DS99} R.\\ Donangelo and K.\\ Sneppen, Physica A, {\\bf 276}, 572 (2000).\n\n\\bibitem{M34} G.\\ Marx, unpublished.\n\n\\bibitem{F88} J.\\ Feder, {\\sl Fractals\\/} (Plenum Press, New York, 1988).\n\n\\bibitem{M63}\nB.\\ Mandelbrot, Journal of Business of the University of Chicago\n{\\bf 36}, 307 (1963).\n\n\\bibitem{E95}\nC.J.G.\\ Evertz, Fractals {\\bf 3}, 609 (1995).\n\n\\bibitem{Plerou99}\nV.\\ Plerou, P.\\ Gopikrishnan, L.A.N.\\ Amaral, M.\\ Meyer and H.E.\\ Stanley, \nCond-mat/9907161 (1999).\n\n\\bibitem{M91}\nR.N.\\ Mantegna, Physica A {\\bf 179} (1991) 232.\n\n\\bibitem{Z97}\nD.\\ Challet and Y.-C.\\ Zhang, Physica A {\\bf 246}, 407 (1997);\nY.-C.\\ Zhang, Europhys. News {\\bf 29}, 51 (1998).\n\n\\bibitem{PB99}\nM.\\ Paczuski, K.\\ Bassler and A. Corral, Cond-mat/9905082.\n\n\\bibitem{DHSS}\nR.\\ Donangelo, A.\\ Hansen, K.\\ Sneppen and S.R.\\ Souza, in preparation.\n\n\\bibitem{Delong90}\nJ.B.\\ Delong, J.\\ Finance {\\bf 45} 379 (1990).\n\\end{thebibliography}"
}
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cond-mat0002139
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Fluctuating spin $g$-tensor in small metal grains
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[
{
"author": "P. W. Brouwer$^{a}$"
},
{
"author": "X. Waintal$^{a}$"
},
{
"author": "and B. I. Halperin$^{b}$"
}
] |
[
{
"name": "gtens.tex",
"string": "\\documentstyle[aps,prl,epsf,twocolumn,floats]{revtex}\n\n\\begin{document}\n\\draft\n\\title{Fluctuating spin $g$-tensor in small metal grains}\n\n\\author{P. W. Brouwer$^{a}$, X. Waintal$^{a}$, and B. I. Halperin$^{b}$}\n\\address{$^{a}$Laboratory of Atomic and Solid State Physics,\nCornell University, Ithaca, NY 14853-2501\\\\\n$^{b}$Lyman Laboratory of Physics, Harvard University, Cambridge MA \n02138\\\\\n{\\rm \\today}\n\\medskip \\\\ \\parbox{14cm}{\\rm\nIn the presence of spin-orbit scattering, the splitting of an energy\nlevel $\\varepsilon_{\\mu}$ in a generic small metal grain due to the\nZeeman coupling to a magnetic field $\\vec B$ depends on the direction\nof $\\vec B$, as a result of mesoscopic fluctuations. The anisotropy is\ndescribed by the eigenvalues $g_j^2$ ($j=1,2,3$) of a tensor ${\\cal\nG}$, corresponding to the (squares of) $g$-factors along three\nprincipal axes. We consider the statistical distribution of ${\\cal G}$\nand find that the anisotropy is enhanced by eigenvalue repulsion between\nthe $g_{j}$.\n\\medskip\\\\\n%\nPACS numbers: 71.24.+q, 71.70.Ej}}\n\n% 71. Electronic structure (see also 73.20 Surface \n% and interface electron states)\n% 71.24.+q Electronic structure of clusters and nanoparticles\n% 71.70.Ej Spin-orbit coupling, Zeeman and Stark splitting, \n% Jahn-Teller effect\n\n\\maketitle\n\nWith the advance of nanoparticle technology, it has become possible to\nresolve individual energy levels for electrons in ultrasmall metal\ngrains. Recent experiments addressed their Zeeman\nsplitting under the application of a magnetic field $\\vec B$\n\\cite{Ralph,Davidovic,Salinas}. The splitting of a level\n$\\varepsilon_{\\mu}$ is described by a $g$-factor, $\\delta\n\\varepsilon_{\\mu} = \\pm \\case{1}{2} \\mu_B g B_z$, where $\\mu_B$ is the Bohr\nmagneton. A free electron has $g = 2$, but in\nsmall metal grains the effective $g$-factor may be reduced as a result\nof spin-orbit scattering \\cite{Halperin}. \nIn order to study this reduction, Salinas\n{\\rm et al.} \\cite{Salinas} have doped Al grains (which do not have\nsignificant spin-orbit scattering) with Au (which has). For small\nconcentrations of Au, the effective $g$-factor was seen to drop\nfrom 2 to around 0.7. Even lower values $g \\sim 0.3$ were reported in \nexperiments on Au grains \\cite{Davidovic}.\n\nFor disordered systems with spin-orbit scattering, the splitting of\na level $\\varepsilon_{\\mu}$ does not only depend on the magnitude of\nthe magnetic field $\\vec B$, but also on its direction. Hence, an\nanalysis in terms of a ``$g$-tensor'' is more appropriate\n\\cite{Slichter}. To be precise, \nthe Zeeman field splits the Kramers' doublet $\\varepsilon_{\\mu} \\to\n\\varepsilon_{\\mu} \\pm \\delta \\varepsilon_{\\mu}$ with\n\\begin{eqnarray}\n\\delta \\varepsilon_{\\mu}^2 = (\\mu_B/2)^2 \\vec B \\cdot {\\cal G_{\\mu}} \n \\cdot \\vec B,\n \\label{eq:deltaE}\n\\end{eqnarray}\nwhere ${\\cal G}_{\\mu}$ is a $3 \\times 3$ tensor. In the absence of\nspin-orbit scattering, the tensor ${\\cal G}_{\\mu}$ is isotropic,\n$({\\cal G_{\\mu}})_{ij} = 4 \\delta_{ij}$. The effect of spin-orbit\nscattering on ${\\cal G}_{\\mu}$ is threefold: It\nleads to a decrease of the typical magnitude of ${\\cal G}_{\\mu}$, it\nmakes the tensor structure of ${\\cal G}_{\\mu}$ important (i.e., it\nintroduces an anisotropic response to the magnetic field $\\vec B$),\nand it causes ${\\cal G}_{\\mu}$ to be different for each level\n$\\varepsilon_{\\mu}$. Hence ${\\cal G}_{\\mu}$ becomes a fluctuating\nquantity, and it is\nimportant to know its statistical distribution. The latter problem\nwas addressed in a recent paper by Matveev et al.\\ \\cite{Matveev},\nhowever without considering the tensor structure of ${\\cal\nG}_{\\mu}$. The anisotropy of the $g$-tensor is a measurable quantity and\nwe here consider the distribution of the entire tensor ${\\cal G}_{\\mu}$.\nThe distribution $P({\\cal G}_{\\mu})$ is defined with respect to an ensemble \nof small metal grains of roughly equal size. The same distribution applies\nto the fluctuations of ${\\cal G}_{\\mu}$ as a function of the level \n$\\varepsilon_{\\mu}$ in the same grain.\n\nIn general, ${\\cal G}_{\\mu}$ has a contribution ${\\cal\nG}_{\\mu}^{\\rm spin}$ from the magnetic moment of electron spins, and a\ncontribution ${\\cal G}_{\\mu}^{\\rm orb}$ for the orbital angular moment of\nthe state $|\\psi_{\\mu}\\rangle$. In Ref.\\ \\onlinecite{Matveev}, the\ntypical sizes of both contributions were estimated as ${\\cal G}^{\\rm\nspin} \\sim \\tau_{\\rm so} \\Delta$ and ${\\cal G}^{\\rm orb} \\sim \\ell/L$,\nwhere $\\tau_{\\rm so}$ is the mean spin-orbit scattering time, $L$ is\nthe grain size, $\\Delta \\propto L^{-3}$ is the mean level spacing, and\n$\\ell \\ll L$ is the elastic mean free path.\nWe restrict ourselves to the spin\ncontribution ${\\cal G}^{\\rm spin}$, which should be dominant for small\ngrain sizes \\cite{Matveev}, provided $\\tau_{\\rm so}$ does not\ndepend on system size, as should be the case for the experiments of\nRef.\\ \\onlinecite{Salinas}. \nWhen orbital contributions are important, the anisotropy of ${\\cal G}$\nwill be affected by the shape of the grain. In that case, our main\nconclusions apply only to a roughly spherical grain.\nAs the typical magnitude of ${\\cal G}$\n(we drop the superscript ``spin'' and the subscript $\\mu$ if there\nis no ambiguity)\ndepends on the microscopic parameters $\\tau_{\\rm so}$ and $\\Delta$,\nwhich are in most cases not known accurately, we choose to have\nthe typical magnitude of ${\\cal G}$ serve as an external parameter in\nour theory. \n\nWe first present our main results. \nWith a suitable choice of the coordinate axes (``principal\naxes''), the tensor ${\\cal G}$ can be diagonalized. Writing its\neigenvalues as $g_j^2$ and denoting the components of the magnetic\nfield along the principal axes by $B_j$, $j=1,2,3$, Eq.\\ (\\ref{eq:deltaE})\ntakes a particularly simple form,\n\\begin{equation}\n \\delta \\varepsilon_{\\mu}^2 = \\case{1}{4} \\mu_B^2 (\n g_1^2 B_1^2 + g_2^2 B_2^2 + g_3^3 B_3^2). \\label{eq:deltaE2}\n\\end{equation}\nWe refer to the numbers $g_1$, $g_2$, and $g_3$ as principal\n$g$-factors. For a generic metal grain of a cubic material, \nrotational\nsymmetry implies that, for a given level $\\varepsilon_{\\mu}$, the\npositioning of the principal axes is entirely random in space,\nas long as they are mutually orthogonal. Hence, it remains to\nstudy the distribution $P(g_1,g_2,g_3)$ of the principal $g$-factors\n$g_1$, $g_2$, and $g_3$ for the level $\\varepsilon_{\\mu}$. Our main\nresult is, that for sufficiently strong spin-orbit scattering,\n$P(g_1,g_2,g_3)$ is given by the distribution\n\\begin{equation}\n P(g_1,g_2,g_3) \\propto \\prod_{i<j} |g_i^2 - g_j^2| \\prod_{i} \n e^{-3 g_i^2/2 \\langle g^2 \\rangle}, \\label{eq:PgGSE0}\n\\end{equation} \nwhere $g^2 = \\case{1}{3}(g_1^2 + g_2^2 + g_3^2)$ is the average of \n$(2 \\delta \\varepsilon_{\\mu}/ \\mu_B B)^2$ over all directions of $\\vec B$\nand $\\langle g^2 \\rangle$ is its average over the ensemble of grains.\nIn random matrix theory \\cite{Mehta}, this distribution is known as\nthe Laguerre ensemble. \nWithout loss of generality we may assume\nthat $g_1^2 \\le g_2^2 \\le g_3^2$. Figure \\ref{fig:3} shows the\naverages $\\langle g_j^2 \\rangle$ and a realization of the principal\n$g$-factors $g_1$, $g_2$, and $g_3$ for a specific sample, as a\nfunction of a parameter $\\lambda \\sim (\\tau_{\\rm so} \\Delta)^{-1/2}$\nmeasuring the strength of the spin-orbit scattering. (A formal\ndefinition of $\\lambda$ in a random-matrix model will be\ngiven below.) {}From the figure, one readily observes that,\ntypically, the three principal $g$-factors differ by a factor\n$2$--$3$. This implies that, in spite of the average rotational\nsymmetry of the grains, the response of a given level\n$\\varepsilon_{\\mu}$ to an applied magnetic field is highly\nanisotropic because of mesoscopic fluctuations. \nThe mathematical origin of this effect is the ``level repulsion''\nfactor $|g_i^2 - g_j^2|$ in the probability distribution\n(\\ref{eq:PgGSE0}), which signifies that, to a certain extent, ${\\cal G}_{\\mu}$\ncan be viewed a as a ``random matrix''.\n\n\\begin{figure}\n\\vglue -0.5cm\n\\epsfxsize=0.99\\hsize\n%\\hspace{0.1\\hsize}\n\\epsffile{fig1.eps}\n\n\\caption{\\label{fig:3} Average of the squares of principal \n$g$-factors versus spin-orbit scattering strength\n$\\lambda$, obtained from numerical simulation of the random matrix\nmodel (\\protect\\ref{eq:HSA}) with $N=100$. Inset: \n$g_1$, $g_2$, and $g_3$ for a\nspecific realization. We have included the sign of $g_1$; see the\ndiscussion below Eq.\\ (\\protect\\ref{eq:ga}).}\n\\end{figure}\n\nLet us now turn to a more detailed discussion of our results. Without\nmagnetic field, the Hamiltonian ${\\cal H}$ of the grain is invariant under\ntime-reversal, so that all eigenstates come in doublets\n$|\\psi_{\\mu}\\rangle$ and $|{\\cal T} \\psi_{\\mu}\\rangle$, where ${\\cal\nT} \\psi = i \\sigma_2 \\psi^{*}$ is the time-reversal operator. To study\nthe splitting of the doublets by a magnetic field, we add a term $\n\\mu_B \\vec B \\cdot \\vec \\sigma$ to ${\\cal H}$, $\\vec \\sigma =\n(\\sigma_1,\\sigma_2,\\sigma_3)$ being the vector of Pauli matrices.\nFrom degenerate perturbation theory we find that a level\n$\\varepsilon_{\\mu}$ is split into $\\varepsilon_{\\mu} \\pm \\delta\n\\varepsilon_{\\mu}$, with $\\delta \\varepsilon_{\\mu}$ of the form\n(\\ref{eq:deltaE}). For the real symmetric $3 \\times 3$ matrix\n${\\cal G}_{\\mu}$ one has\n\\begin{equation}\n {\\cal G}_{\\mu} = G_{\\mu}^{\\rm T} G_{\\mu},\n\\end{equation}\nwhere $G_{\\mu}$ is a real $3 \\times 3$ matrix with elements\n\\begin{eqnarray}\n (G_{\\mu})_{1j} + i (G_{\\mu})_{2j} &=&\n - 2 \\langle {\\cal T} \\psi_{\\mu} | \\sigma_j | \\psi_{\\mu} \\rangle\n \\nonumber \\\\\n (G_{\\mu})_{3j} &=&\n 2\\langle \\psi_{\\mu} | \\sigma_{j} | \\psi_{\\mu} \\rangle,\n \\label{eq:gpsi} \n\\end{eqnarray}\nWe use random-matrix theory (RMT) to compute the distribution \nof ${\\cal G}_{\\mu}$. In RMT, the microscopic Hamiltonian ${\\cal H}$ is\nreplaced by a $2N \\times 2N$ random hermitian matrix $H$, where\nat the end of the calculation the limit $N \\to \\infty$ is taken. \n(The factor $2$ accounts for spin.) \nThe wavefunction $\\psi_{\\mu}(\\vec r)$\nis replaced by an $N$-component spinor eigenvector $\\psi_{\\mu n}$ of\n$H$, where $n$ is a vector index. \nTo study the effect of spin-orbit scattering, we\ntake $H$ of the form\n\\begin{mathletters} \\label{eq:HSA}\n\\begin{equation}\n H(\\lambda) = S \\otimes \\openone_2 + i {\\lambda \\over \\sqrt{4N}} \n \\sum_{j} A_j \\otimes \\sigma_j, \n\\end{equation}\nwhere $S$ ($A_j$) is a real symmetric (antisymmetric) $N \\times N$\nmatrix with the Gaussian distribution\n\\begin{eqnarray}\n P(S) &\\propto& e^{- (\\pi^2/4 N \\Delta^2)\\, {\\rm tr}\\, S^{\\rm T} S}, \n \\label{eq:distr}\\\\\n P(A_j) &\\propto& e^{- (\\pi^2/4 N \\Delta^2)\\, {\\rm tr}\\, A_j^{\\rm T} A_j},\\ \n \\ j=1,2,3. \\nonumber\n\\end{eqnarray}\n\\end{mathletters}\nThe Hamiltonian $H(\\lambda)$ is similar to the Pandey-Mehta\nHamiltonian used to describe the effect of time-reversal symmetry\nbreaking in a system of spinless particles \\cite{Pandey}. In Eq.\\\n(\\ref{eq:distr}), $\\Delta$ is the average spacing between the Kramers\ndoublets near $\\varepsilon=0$. \nThe amount of spin-orbit scattering is measured by the\nparameter $\\lambda \\sim (\\tau_{\\rm so} \\Delta)^{-1/2}$ \\cite{Halperin}. \nThe case $\\lambda=0$ corresponds to the absence\nof spin-orbit scattering, when $H = S$ is a member of the Gaussian\nOrthogonal Ensemble (GOE) of random matrix theory. The case\n$\\lambda=(4N)^{1/2}$ corresponds to the case of strong spin-orbit\nscattering, when $H$ is a member of the Gaussian Symplectic Ensemble\n(GSE). The ensemble of Hamiltonians $H(\\lambda)$ corresponds to a\ncrossover from the GOE to the GSE. Similar crossovers were studied\npreviously in the literature, in particular for the cases GOE--GUE\nand GSE--GUE (GUE is Gaussian Unitary Ensemble) \n\\cite{Pandey,French,Sommers,Falko,VanLangen}.\n\nThe distribution of the tensor ${\\cal G}_{\\mu}$ for an\neigenvalue $\\varepsilon_{\\mu}$ of the matrix $H(\\lambda)$ is\nrelated to the statistics of eigenvectors of $H(\\lambda)$ in this\ncrossover ensemble. To deal with the twofold degeneracy of the\neigenvalue $\\varepsilon_{\\mu}$, we combine the two $N$-component spinor \neigenvectors\n$\\psi_{\\mu}$ and ${\\cal T}\\psi_{\\mu}$ into a single $N$-component vector\nof quaternions $\\bar \\psi = (\\psi,{\\cal T}\\psi)$ \\cite{Mehta,quaternion}.\nThe quaternion vector $\\bar \\psi$ can be parameterized as,\n\\begin{equation}\n \\bar \\psi = \\sum_{k=0}^{3} \\alpha_k u_k \\otimes \\phi_k, \\label{eq:barpsiphi}\n\\end{equation}\nwhere the $u_k$ are quaternion numbers with $\\mbox{tr}\\, u_k^{\\dagger}\nu_l = 2 \\delta_{kl}$ (``quaternion phase factors''), the $\\phi_k$ are\n$N$-component real orthonormal vectors, and the $\\alpha_k$ are positive\nnumbers such that $\\sum_{k} \\alpha_k^2 = 1$\n($k,l=0,1,2,3$). A eigenvector in the GOE corresponds to $\\alpha_0 = 1$,\n$\\alpha_1 = \\alpha_2 = \\alpha_3 = 0$, while an eigenvector in the GSE\nhas typically $\\alpha_0 \\approx \\alpha_1 \\approx \\alpha_2 \\approx \\alpha_3\n\\approx \\case{1}{2}$. A similar parameterization has been applied to the\nGOE--GUE crossover \\cite{French}. \nOrthogonal invariance of the distributions of $S$ and $A_j$,\ntogether with the freedom to choose the overall quaternion phase of\n$\\bar \\psi$, give a distribution of the $u_k$ and $\\phi_k$\nthat is as random as possible, provided the above mentioned\northogonality constraints are obeyed. Hence, all nontrivial information \nabout the eigenvector statistics is encoded in the numbers $\\alpha_k$. \nSubstitution of the parameterization (\\ref{eq:barpsiphi}) into Eq.\\ \n(\\ref{eq:gpsi}) yields\n\\begin{eqnarray}\n% g_{j} &=& 2(\\alpha_0^2 - \\alpha_1^2 - \\alpha_2^2 - \\alpha_3^2) +\n% 4 \\alpha_j^2, \\ \\ j=1,2,3. \\label{eq:ga}\n g_{1} &=& 2(\\alpha_0^2 + \\alpha_1^2 - \\alpha_2^2 - \\alpha_3^2), \\nonumber \\\\\n g_{2} &=& 2(\\alpha_0^2 - \\alpha_1^2 + \\alpha_2^2 - \\alpha_3^2), \n \\label{eq:ga}\\\\\n g_{3} &=& 2(\\alpha_0^2 - \\alpha_1^2 - \\alpha_2^2 + \\alpha_3^2). \\nonumber\n\\end{eqnarray}\nWhile the squares $\\alpha_k^2$ ($k=0,1,2,3$) are all positive, the\nprincipal $g$-factors as given by Eq.\\ (\\ref{eq:ga}) can also be\nnegative. Permutations of the $\\alpha_k$ alter the signs of the\nindividual $g_j$, but not of their product $g_1 g_2 g_3$. [The product\n$g_1 g_2 g_3 = \\det G$ also follows from Eq.\\ (\\ref{eq:gpsi}); one \nverifies that it does not change when $|\\psi\\rangle$ is\nreplaced by a linear combination of $|\\psi\\rangle$ and $|{\\cal\nT}\\psi\\rangle$.] Without loss of generality, we may assume that\n$g_1^2 \\le g_2^2 \\le g_3^2$, and that $g_2$ and $g_3$ are\npositive. Then equation (\\ref{eq:ga}) provides the constraint\n$g_2 + g_3 \\le 2 + g_1$, which poses a bound on the occurrence of \nnegative values for the product $g_1 g_2 g_3$.\nWe conclude that all information on the eigenvector statistics in the\nGOE--GSE crossover is encoded in the magnitudes of $g_1$, $g_2$, and\n$g_3$ and the sign of their product. Since for the\nlevel splitting $\\delta \\varepsilon_{\\mu}(\\vec B)$ only the squares\n$g_j^2$ are of relevance, we disregard the sign of $g_1 g_2 g_3$ in the\nremainder of the paper. The sign of $g_1 g_2 g_3$ may be determined\nin principle, however, by a spin-resonance experiment \\cite{spinres}.\n\nIn order to calculate the distribution $P(g_1,g_2,g_3)$ one has, in\nprinciple, to carry out the same program as was done in Refs.\\\n\\onlinecite{Sommers,Falko} for the GOE--GUE crossover. However, it\nturns out that in the present case the calculation is considerably\nmore complicated. This can already be seen from the mere observation\nthat the wavefunction statistics in the GOE--GSE crossover is governed by\nthree variables $g_1$, $g_2$, and $g_3$, whereas in the case of\nthe GOE--GUE crossover only one variable was needed \n\\cite{Sommers,Falko,VanLangen}. In the\nfield-theoretic language of Ref.\\ \\onlinecite{Falko}, one has to use a\nnonlinear sigma model of $16 \\times 16$ supermatrices, instead of the\nusual $8 \\times 8$ for the GOE--GUE crossover \\cite{Klaus}. Here we\nrefrain from such a truly heroic enterprise. Instead we focus on\nthe regimes of strong and weak spin-orbit coupling, and study the\nintermediate regime by means of numerical simulations of the model\n(\\ref{eq:HSA}).\n\nBefore we address the case of strong spin-orbit scattering \n$\\lambda \\gg 1$ in the crossover Hamiltonian, we first consider\nthe GSE, corresponding to $\\lambda^2 = 4N$. In the GSE, the\nwavefunction $\\psi$ is a vector of independently Gaussian distributed\ncomplex numbers. Then, one easily verifies that, for large $N$, the\nelements of the matrix $G$ of Eq.\\ (\\ref{eq:gpsi}) are real random\nvariables, independently distributed, with a Gaussian distribution of\nzero mean and variance $2/N$. Hence $G$ is a random real matrix with\ndistribution\n\\begin{equation}\n P(G) \\propto \\exp(-N {\\rm tr}\\, G^{\\rm T} G/4). \\label{eq:GProb}\n\\end{equation}\nThe principal $g$-factors are the eigenvalues $g_{j}^2$ of the product\n${\\cal G} = G^{\\rm T} G$. The distribution of the eigenvalues of such\na matrix product is known in literature \\cite{Brezin}. It is given by\nEq.\\ (\\ref{eq:PgGSE0}) with $\\langle g^2 \\rangle = 6/N$.\n\nLet us now turn to the Hamiltonian $H(\\lambda)$ for large $\\lambda \\gg 1$,\nbut still $\\lambda \\ll N^{1/2}$. In that case, spin-rotation invariance is\nbroken globally (so that a wavefunction as a whole does not have a\nwell-defined spin), but not locally; on short length scales, the\nparticle keeps a well-defined spin. We then argue that, in the random\nmatrix language, one may think of the quaternion wavevector $\\bar \\psi$\nas consisting of $\\sim \\lambda^2 \\gg 1$ components,\neach with a well-defined spin (or ``quaternion phase''), but with\nuncorrelated spins for each component. The distribution of ${\\cal G}$ is\nthen given by the distribution for the GSE with $N$\nreplaced by a number $\\sim \\lambda^2$ \\cite{phaserigidity}.\nWe have found that the precise\ncorrespondence is $N \\to 2 \\lambda^2$, by estimating the\nexponential term in the exact distribution, along the lines of Ref.\\\n\\onlinecite{Sommers,phaserigidity}. \nIn order to verify this statement we have numerically generated\nrandom matrices of the form (\\ref{eq:HSA}). The comparison with the\nGSE distribution with $N$ replaced by $2 \\lambda^2$ is excellent, see\nFig.\\ \\ref{fig:1}.\n\n\\begin{figure}\n\\epsfxsize=0.89\\hsize\n%\\hspace{0.1\\hsize}\n\\epsffile{fig2.eps}\n\\caption{\\label{fig:1} Distribution of the orientationally averaged\n$g$-factor $g^2 = (g_1^2 + g_2^2 + g_3^2)/3$ (upper left) and of the\nratios $r_{12} = |g_{1}/g_{2}|$ (circles) and $r_{23} = |g_{2}/g_{3}|$ \n(squares, main figure). The solid curves are computed from the theory\n(\\protect\\ref{eq:Pr12}), the data points are numerical simulations of\nthe random matrix model (\\protect\\ref{eq:HSA}) \nwith $N=200$ and $\\lambda = 7.7$. \nThe slight discrepancy between theory and simulations for $r_{12}$\nis a finite-$N$ effect; good agreement is obtained with the\nGSE distribution with $N=200$ (dotted curve).\nThe lower inset shows $\\langle g^2 \\rangle$ vs. $1/N$ for\n$\\lambda=4.3$ (diamonds), $6.2$ (squares), and $8.1$ \n(open circles), together with\nthe theoretical prediction $\\langle g^2 \\rangle = 3\\lambda^{-2}$ for \n$N \\to \\infty$ (closed circles).}\n\\end{figure}\n\nIn order to further analyze $P({\\cal G})$ for strong spin-orbit scattering, \nwe introduce the orientationally \naveraged $g$-factor,\n\\begin{equation}\n g^2 = \\case{1}{3}(g_1^2 + g_2^2 + g_3^2) \n = \\left \\langle { (2\\delta \\varepsilon_{\\mu} / \\mu_B |B|)^2} \n\\right \\rangle_{\\Omega}, \n\\end{equation}\nwhere the brackets $\\langle \\ldots \\rangle_{\\Omega}$ indicate an\naverage over all directions of the magnetic field. Further, we\nintroduce the ratios $r_{12} = |g_1/g_2|$ and $r_{23} = |g_2/g_3|$ to\ncharacterize the anisotropy of ${\\cal G}$.\nChanging variables in Eq.\\ (\\ref{eq:PgGSE0}), we find that\n$P(g,r_{12},r_{23})$ reads\n\\begin{eqnarray}\n P \n &\\propto& % \\nonumber\n {r_{23}^3 (1-r_{23}^2) (1-r_{23}^2 r_{12}^2)\n (1-r_{12}^2) \\over (1 + r_{23}^2 + r_{23}^2 r_{12}^2)^{9/2}}\\, \n g^8 e^{-9 g^2/2 \\langle g^2 \\rangle}.\\! \\label{eq:Pr12}\n\\end{eqnarray}\nNote that the distribution of $r_{12}$ and $r_{23}$ does not depend on\n$\\langle g^2 \\rangle$ (provided the spin-orbit scattering is\nsufficiently strong). The ``$g$-factor'' $g_z$ for a magnetic field in\nthe $z$-direction (which is a random direction with respect to the\nprincipal axes) is given by $g_z = ({\\cal G}_{zz})^{1/2}$. Its\ndistribution follows from Eq.\\ (\\ref{eq:GProb}) as $P(g_z) \\propto\ng_z^2 \\exp(-3 g_z^2/2\\langle g^2 \\rangle)$, in agreement with Ref.\\\n\\onlinecite{Matveev}.\n\nThe case of weak spin-orbit scattering can be addressed by treating\nthe terms proportional to $\\lambda$ in Eq.\\ (\\ref{eq:HSA}) as a small\nperturbation. To second order in $\\lambda$ we find,\n\\begin{equation}\n {\\cal G} =\n 4 - {4 \\lambda^2} \\sum_{\\nu \\neq \\mu} \n a_{\\mu\\nu}^{\\rm T} a_{\\mu\\nu}^{\\vphantom{no superscript {\\rm T}}}\n {1 \\over (\\varepsilon_{\\nu} - \\varepsilon_{\\mu})^2},\n \\label{eq:pert}\n\\end{equation}\nwhere $\\Delta$ is the mean level spacing and $a_{\\mu\\nu}$ is an\nantisymmetric $3 \\times 3$ matrix proportional to the matrix elements\nof the perturbation in the eigenbasis $\\{ |\\psi_{\\nu} \\rangle \\}$ of\n$H(0) = S$, $(a_{\\mu\\nu})_{ij} = N^{-1/2} \\langle\n\\psi_{\\mu} | A_k | \\psi_{\\nu} \\rangle \\varepsilon_{kij}$,\nwhere $\\varepsilon_{kij}$ is the antisymmetric tensor.\nWe first consider the change in the principal $g$-factors due to\nthe matrix element $a_{\\mu\\nu}$ coupling the level\n$\\varepsilon_{\\mu}$ to a close neighboring level $\\varepsilon_{\\nu}$\nwhere $\\nu = \\mu + 1$ or $\\mu - 1$. (Level repulsion rules out\nthe possibility that both levels $\\varepsilon_{\\mu \\pm 1}$\nare very close.) In view of the energy denominators in Eq.\\\n(\\ref{eq:pert}), we may expect that this contribution is\ndominant. Taking only the relevant matrix element $a_{\\mu \\nu}$ into\naccount, we find\n\\begin{equation}\n g_3 = 2,\\ \\ g_1 = g_2 = 2 - {\\case{1}{2} \\lambda^2} \n {(\\varepsilon_{\\mu} - \\varepsilon_{\\nu})^{-2}} \n {\\rm tr}\\, a_{\\mu\\nu}^{\\rm T} \n a_{\\mu\\nu}^{\\vphantom{no superscript {\\rm T}}},\n\\end{equation}\nwhere $\\nu = \\mu \\pm 1$. \nSince the spacing distribution $P(|\\varepsilon_{\\mu}\n- \\varepsilon_{\\nu}|) \\approx \\pi \\Delta^{-2}|\\varepsilon_{\\mu} -\n\\varepsilon_{\\nu}|$ for small $\\varepsilon_{\\mu} - \\varepsilon_{\\nu}$\n\\cite{Mehta}, we find that the distribution $P(g)$ of both $g_1$ and\n$g_2$ has tails $P(g) = (3 \\lambda^2 / 2 \\pi) (2 - g)^{-2}$ for $2-g\n\\gg \\lambda^2$. The main effect of contributions from the other\nenergy levels in Eq.\\ (\\ref{eq:pert}) is a reduction of $g_3$\nbelow $2$, and a separation of $g_1$ and $g_2$. This is illustrated\nin Fig.\\ \\ref{fig:3}.\nThe three regimes of weak, intermediate, and strong spin-orbit\nscattering are compared in Fig.\\ \\ref{fig:2}, using a numerical \nevaluation of the distributions of the three principal $g$-values.\n\\begin{figure}\n\\epsfxsize=0.99\\hsize\n%\\hspace{0.1\\hsize}\n\\epsffile{fig3.eps}\\vspace{-0.5cm}\n\n\\caption{\\label{fig:2} Distributions of the principal $g$-factors\n$g_1$, $g_2$, $g_3$ for $\\lambda=0.6$, $2.0$, and $7.7$. \nThe data points are obtained from numerical simulation of Eq.\\ \n(\\protect\\ref{eq:HSA})\nwith $N=100$.} \n\\end{figure}\n\nWe gratefully acknowledge discussions with T. A. Arias, D. Davidovic,\nK. M. Frahm, Y. Oreg, D. C. Ralph, and M. Tinkham. \nUpon completion of this project,\nwe learned of Ref.\\ \\onlinecite{Matveev}, \nwhich contains some overlap with our work.\nThis work was supported in part by the\nNSF through the Harvard MRSEC (grant DMR 98-09363), and by grant DMR\n99-81283.\n\n\\begin{references}\n\n\\bibitem{Ralph} D. C. Ralph, C. T. Black, and M. Tinkham, Phys.\nRev. Lett. {\\bf 74}, 3241 (1995); {\\em ibid} {\\bf 78}, 4087 (1997);\n%C. T. Black, D. C. Ralph, and M. Tinkham, Phys.\n%Rev. Lett. {\\bf 76}, 688 (1996).\n\n\\bibitem{Davidovic} D. Davidovic and M. Tinkham, Phys. Rev. Lett.\n{\\bf 83}, 1644 (1999); cond-mat/9910396.\n\n\\bibitem{Salinas} D. G. Salinas, S. Gu\\'eron, D. C. Ralph, C. T.\nBlack, and M. Tinkham, Phys. Rev. B {\\bf 60}, 6137 (1999).\n\n\\bibitem{Halperin} W. P. Halperin, Rev. Mod. Phys. {\\bf 58}, 533\n(1986).\n\n\\bibitem{Slichter} C. P. Slichter, {\\em Principles of Magnetic\nResonance} (Springer, Berlin, 1980).\n\n\\bibitem{Matveev} K. A. Matveev, L. I. Glazman, and A. I. Larkin,\ncond-mat/0001431.\n\n\\bibitem{Mehta}\nM. L. Mehta, {\\it Random Matrices} (Academic, New York, 1991).\n\n\\bibitem{Pandey}\nA. Pandey and M. L. Mehta, Commun. Math. Phys. {\\bf 87}, 449 (1983).\n\n\\bibitem{French}\nJ. B. French, V. K. B. Kota, A. Pandey, and S. Tomsovic,\nAnn.\\ Phys.\\ (N. Y.) {\\bf 181}, 198 (1988).\n\n\\bibitem{Sommers}\nH.-J. Sommers and S. Iida, Phys.\\ Rev.\\ E {\\bf 49}, 2513 (1994).\n\n\\bibitem{Falko}\nV. I. Fal'ko and K. B. Efetov, Phys.\\ Rev.\\ B {\\bf 50}, 11267 (1994);\nPhys.\\ Rev.\\ Lett.\\ {\\bf 77}, 912 (1996).\n\n\\bibitem{VanLangen}\nS. A. van Langen, P. W. Brouwer, and C. W. J. Beenakker,\nPhys. Rev. E {\\bf 55}, 1 (1997).\n\n\n\\bibitem{quaternion}\nA quaternion is a $2 \\times 2$ matrix $q$ of the form\n$ \n q = q_0 \\openone + i \\sum_{j} q_j \\sigma_j,\n$\nwhere the $q_j$ are real numbers ($j=0,1,2,3$).\n\n\\bibitem{spinres} For example, if the principal axes of ${\\cal G}$ are\nlabeled $\\hat e_1$, $\\hat e_2$, and $\\hat e_3 = \\hat e_1 \\times \\hat e_2$,\nand we apply a static field $B \\hat e_3$, then a\nresonant AC field $\\vec b \\propto \\mbox{Re}\\, [(g_2 \\hat e_1 + i \\eta\n\\hat e_2 g_1) e^{-i \\omega t}]$, with $\\omega = g_3 |\\mu_B| B/\\hbar > 0$,\nwill produce spin flips for $\\eta=1$ but not for $\\eta=-1$.\n\n\\bibitem{Klaus}\nK. M. Frahm, private communication.\n\n\\bibitem{Brezin}\nE. Br\\'ezin, S. Hikami, and A. Zee, Nucl. Phys. B {\\bf 464}, 411 (1996).\n\n\\bibitem{phaserigidity}\nThe same is true for the GOE--GUE crossover, where the relevant \nquantity is the ``phase-rigidity'' $\\rho = |\\langle {\\cal T} \\psi |\n\\psi \\rangle|^2$. The distribution $P(\\rho)$ for large magnetic fields \nequals $P(\\rho)$ in the GUE, with $N$ replaced by $2 \\alpha^2$, where\n$\\alpha$ is a crossover parameter analogous to $\\lambda$, see Ref.\\\n\\onlinecite{VanLangen}.\n\n\n\\end{references}\n\n\\end{document}"
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[
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"name": "cond-mat0002139.extracted_bib",
"string": "\\bibitem{Ralph} D. C. Ralph, C. T. Black, and M. Tinkham, Phys.\nRev. Lett. {\\bf 74}, 3241 (1995); {\\em ibid} {\\bf 78}, 4087 (1997);\n%C. T. Black, D. C. Ralph, and M. Tinkham, Phys.\n%Rev. Lett. {\\bf 76}, 688 (1996).\n\n\n\\bibitem{Davidovic} D. Davidovic and M. Tinkham, Phys. Rev. Lett.\n{\\bf 83}, 1644 (1999); cond-mat/9910396.\n\n\n\\bibitem{Salinas} D. G. Salinas, S. Gu\\'eron, D. C. Ralph, C. T.\nBlack, and M. Tinkham, Phys. Rev. B {\\bf 60}, 6137 (1999).\n\n\n\\bibitem{Halperin} W. P. Halperin, Rev. Mod. Phys. {\\bf 58}, 533\n(1986).\n\n\n\\bibitem{Slichter} C. P. Slichter, {\\em Principles of Magnetic\nResonance} (Springer, Berlin, 1980).\n\n\n\\bibitem{Matveev} K. A. Matveev, L. I. Glazman, and A. I. Larkin,\ncond-mat/0001431.\n\n\n\\bibitem{Mehta}\nM. L. Mehta, {\\it Random Matrices} (Academic, New York, 1991).\n\n\n\\bibitem{Pandey}\nA. Pandey and M. L. Mehta, Commun. Math. Phys. {\\bf 87}, 449 (1983).\n\n\n\\bibitem{French}\nJ. B. French, V. K. B. Kota, A. Pandey, and S. Tomsovic,\nAnn.\\ Phys.\\ (N. Y.) {\\bf 181}, 198 (1988).\n\n\n\\bibitem{Sommers}\nH.-J. Sommers and S. Iida, Phys.\\ Rev.\\ E {\\bf 49}, 2513 (1994).\n\n\n\\bibitem{Falko}\nV. I. Fal'ko and K. B. Efetov, Phys.\\ Rev.\\ B {\\bf 50}, 11267 (1994);\nPhys.\\ Rev.\\ Lett.\\ {\\bf 77}, 912 (1996).\n\n\n\\bibitem{VanLangen}\nS. A. van Langen, P. W. Brouwer, and C. W. J. Beenakker,\nPhys. Rev. E {\\bf 55}, 1 (1997).\n\n\n\n\\bibitem{quaternion}\nA quaternion is a $2 \\times 2$ matrix $q$ of the form\n$ \n q = q_0 \\openone + i \\sum_{j} q_j \\sigma_j,\n$\nwhere the $q_j$ are real numbers ($j=0,1,2,3$).\n\n\n\\bibitem{spinres} For example, if the principal axes of ${\\cal G}$ are\nlabeled $\\hat e_1$, $\\hat e_2$, and $\\hat e_3 = \\hat e_1 \\times \\hat e_2$,\nand we apply a static field $B \\hat e_3$, then a\nresonant AC field $\\vec b \\propto \\mbox{Re}\\, [(g_2 \\hat e_1 + i \\eta\n\\hat e_2 g_1) e^{-i \\omega t}]$, with $\\omega = g_3 |\\mu_B| B/\\hbar > 0$,\nwill produce spin flips for $\\eta=1$ but not for $\\eta=-1$.\n\n\n\\bibitem{Klaus}\nK. M. Frahm, private communication.\n\n\n\\bibitem{Brezin}\nE. Br\\'ezin, S. Hikami, and A. Zee, Nucl. Phys. B {\\bf 464}, 411 (1996).\n\n\n\\bibitem{phaserigidity}\nThe same is true for the GOE--GUE crossover, where the relevant \nquantity is the ``phase-rigidity'' $\\rho = |\\langle {\\cal T} \\psi |\n\\psi \\rangle|^2$. The distribution $P(\\rho)$ for large magnetic fields \nequals $P(\\rho)$ in the GUE, with $N$ replaced by $2 \\alpha^2$, where\n$\\alpha$ is a crossover parameter analogous to $\\lambda$, see Ref.\\\n\\onlinecite{VanLangen}.\n\n\n"
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cond-mat0002140
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Thermodynamics of the Spin-1/2 Antiferromagnetic Uniform Heisenberg Chain
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"author": "A. Kl\\\"umper$^{1,2}$ and D. C. Johnston$^3$"
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\hglue 0.15in We present a new application of the traditional thermodynamic Bethe ansatz to the spin-1/2 antiferromagnetic uniform Heisenberg chain and derive exact nonlinear integral equations for just {\em two} functions describing the elementary excitations. Using this approach the magnetic susceptibility $\chi$ and specific heat $C$ versus temperature $T$ are calculated to high accuracy for $5\times10^{-25}\leq T/J\leq 5$. The $\chi(T)$ data agree very well at low $T$ with the asymptotically exact theoretical low-$T$ prediction of S. Lukyanov, Nucl.\ Phys.\ B {522}, 533 (1998). The unknown coefficients of the second and third lowest-order logarithmic correction terms in Lukyanov's theory for $C(T)$ are estimated from the $C(T)$ data.
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"string": "\\def\\e{{\\rm e}}\n\\def\\be{\\begin{equation}}\n\\def\\ee{\\end{equation}}\n\\def\\bea {\\begin{eqnarray}}\n\\def\\eea {\\end{eqnarray}}\n%\\documentstyle[preprint,prl,aps]{revtex}\n%\\documentstyle[prl,aps]{revtex}\n%\\documentstyle[preprint,prb,aps]{revtex}\n\\documentstyle[prl,aps,twocolumn]{revtex}\n\n\\begin{document}\n\\input{epsf.tex}\n%\\draft\n%\\preprint{}\n\\wideabs{\n\\title{Thermodynamics of the Spin-1/2 Antiferromagnetic\nUniform Heisenberg Chain}\n\\author{ A. Kl\\\"umper$^{1,2}$ and D. C. Johnston$^3$}\n\\address{$^1$Institut f\\\"ur Theoretische Physik, Universit\\\"at zu K\\\"oln,\nZ\\\"ulpicher Strasse 77, 50937 K\\\"oln, Germany}\n\\address{$^2$Fachbereich Physik, Universit\\\"at Dortmund, 44221\nDortmund, Germany}\n\\address{$^3$Ames Laboratory and Department of Physics and Astronomy, Iowa\nState University, Ames, Iowa 50011}\n\\date{Revised Manuscript Submitted to Physical Review Letters on 28\nJanuary 2000}\n\\maketitle\n\\begin{abstract}\\hglue 0.15in\n We present a new application of the traditional thermodynamic Bethe\n ansatz to the spin-1/2 antiferromagnetic uniform Heisenberg chain\n and derive exact nonlinear integral equations for just {\\em two}\n functions describing the elementary excitations. Using this approach\n the magnetic susceptibility $\\chi$ and specific heat $C$ versus\n temperature $T$ are calculated to high accuracy for\n $5\\times10^{-25}\\leq T/J\\leq 5$. The $\\chi(T)$ data agree very well\n at low $T$ with the asymptotically exact theoretical low-$T$\n prediction of S. Lukyanov, Nucl.\\ Phys.\\ B {\\bf 522}, 533 (1998).\n The unknown coefficients of the second and third lowest-order\n logarithmic correction terms in Lukyanov's theory for $C(T)$ are\n estimated from the $C(T)$ data.\n\\end{abstract}\n%\\vspace{1cm}\n\\pacs{PACS numbers: 75.40.Cx, 75.20.Ck, 75.10.Jm, 75.50.Ee}\n}\n\nThe spin $S = 1/2$ antiferromagnetic (AF) uniform Heisenberg chain has a\nlong and distinguished history in condensed matter physics and exhibits\nunusual static and dynamical properties unique to one-dimensional\nspin systems. It has been used as a testing ground for many theoretical\napproaches. The Hamiltonian is ${\\cal H} = J \\sum_{i}\n\\bbox{S}_i\\cdot\\bbox{S}_{i+1}$, where $J>0$ is the AF Heisenberg exchange\ninteraction between nearest-neighbor spins. In this paper we usually set\n$k_{\\rm B} = 1$ and $g \\mu_{\\rm B} = 1$ where $k_{\\rm B}$ is Boltzmann's\nconstant, $g$ is the spectroscopic splitting factor of the spins and\n$\\mu_{\\rm B}$ is the Bohr magneton; also, the reduced temperature\n$t\\equiv T/J$ where $T$ is the absolute temperature.\n\nThe $S = 1/2$ Heisenberg chain is known to be exactly\nsolvable\\cite{Bethe31}, i.e.\\ all eigenvalues can be obtained from the\nso-called Bethe ansatz equations. Despite the amazing property of being\nintegrable, the Heisenberg chain has defied many attempts to calculate\nphysical observables including thermodynamic quantities. A rather direct\nevaluation of the partition function was constructed in\\cite{TakTBA} and\nis known as the ``thermodynamic Bethe ansatz'' (TBA),\nbut this did not allow for\nhigh accuracy calculations especially in the low temperature region. The\nfundamental problem in \\cite{TakTBA} is the necessity to deal with\ninfinitely many coupled nonlinear integral equations for which the\ntruncation procedures are difficult to control.\n\nThe possibility to accurately calculate the physical properties of the\n$S = 1/2$ Heisenberg chain improved following the development of the\npath integral formulation of the transfer matrix treatment of quantum\nsystems\\cite{SuzukiI87}. On the basis of a Bethe ansatz\nsolution\\cite{Tak91} to the quantum transfer matrix, Eggert, Affleck\nand Takahashi in 1994 obtained numerically exact results for the\nmagnetic susceptibility $\\chi(t)$ down to much lower temperatures than\nbefore and compared these with their low-$t$ results from conformal\nfield theory\\cite{Eggert1994}. They found, remarkably, that\n$\\chi(t\\to 0)$ has infinite slope: their conformal field theory\ncalculations showed that the leading order $t$ dependence is\n$\\chi(t\\to 0) = \\chi(0)\\{1 + 1/[2\\ln(t_0/t)]\\}$, where the value of\n$t_0$ is not predicted by the field theory. Such log terms are called\n``logarithmic corrections''. From their comparison of their field\ntheory and Bethe ansatz calculations which extended down to $t =\n0.003$, Eggert, Affleck and Takahashi estimated $t_0 \\approx\n7.7$\\cite{Eggert1994}. Their numerical $\\chi(t)$ values are up to\n$\\sim 10$\\% larger than the former Bonner-Fisher \\cite{Bonner1964}\nextrapolation for $t\\lesssim 0.25$.\n\nLukyanov has recently presented an exact asymptotic field theory for\n$\\chi(t)$ and the specific heat $C(t)$ at low $t$, including the exact\nvalue of $t_0$ \\cite{Lukyanov1997}. These results are claimed to be exact\nin the sense of a renormalization group treatment close to a fixed point\nwhere only few operators are responsible for perturbations. Questions\narising about such calculations are whether these operators have been\ncorrectly identified and whether the effective theory has been properly\nevaluated. A meaningful test of Lukyanov's theory is only possible using\nnumerical data of very high accuracy and at extremely low temperatures,\nsuch as we have attained in our numerical calculations to be presented\nbelow.\n\nIn this Letter we present a new application of the traditional TBA to the\nspin-1/2 Heisenberg chain and derive exact nonlinear integral equations\n[Eqs.~(\\ref{NLIE}--\\ref{kernel}) below] involving just {\\em two}\nfunctions describing the elementary excitations. Our derivation evolved\nfrom earlier work by one of us using the powerful lattice\napproach\\onlinecite{Klumper1993,Klumper1998}. By means of a lattice path\nintegral representation of the finite temperature Heisenberg chain and\nthe formulation of a suitable quantum transfer matrix, a set of\nnumerically well-posed expressions for the free energy was derived. A\nserious disadvantage of this approach lies in the complicated and\nphysically non-intuitive mathematical constructions, which strongly\ninhibits generalizations to other integrable, notably itinerant fermion\nmodels. The present work is a new analytic derivation of the\nfinitely-many integral equations of \\cite{Klumper1993,Klumper1998} by\nmeans of the intuitive TBA approach. Our Eqs.~(\\ref{NLIE}--\\ref{kernel})\nare identical to those obtained in \\cite{Klumper1993} by a rigorous,\nhowever much more involved method. In our new construction, we assume\nthat magnons (on paths $C_\\pm$) are elementary excitations and contain\nall information about the thermodynamics. Bound states are implicitly\ntaken into account by use of the exact scattering phase probed in the\nanalyticity strip. The {\\em a posteriori} success of our reasoning is\nimportant for two reasons. First, our construction is as simple as\nthe standard TBA, however avoiding the problems of dealing with density\nfunctions for (up to) infinitely many bound states. This may be of great\nadvantage in the study of more complicated systems. Second, we have a\nsimple particle approach to the Heisenberg chain which will allow for a\nstudy of transport properties like the Drude weight which has not been\npossible within the path integral approach \\cite{Klumper1993}.\n\nWe also demonstrate here that using our integral equations one\ncan improve the accuracy and extend the temperature range of\nnumerical calculations of $\\chi(t)$ and $C(t)$ for the $S = 1/2$\nHeisenberg chain {\\em on the lattice} far beyond those of previous\ncalculations. We find agreement of our data with the above theory\nof Lukyanov \\cite{Lukyanov1997} for $\\chi(t)$ to high accuracy ($\\lesssim\n1\\times 10^{-6}$) over a temperature range spanning 18 orders of\nmagnitude, $5\\times 10^{-25}\\leq t \\leq 5\\times 10^{-7}$; the agreement\nin the lower part of this temperatures range is much better, ${\\cal\nO}(10^{-7})$. For $C(t)$, the logarithmic correction in Lukyanov's\ntheory is insufficient to describe our numerical data accurately even\nat very low $t$, so we estimate the coefficients of the next two\nlogarithmic correction terms in his theory from our $C(t)$ data.\n\n\\noindent\n{\\em Derivation of integral equations}\n\nWe start with the partially anisotropic Hamiltonian ${\\cal H} = J\n\\sum_{i}({S}^x_i{S}^x_{i+1}+ {S}^y_i{S}^y_{i+1}+\n\\cos(\\gamma){S}^z_i{S}^z_{i+1})-h\\sum_{i}{S}^z_i$ with $0<\\gamma<\\pi/2$\nand magnetic field $h$. The dynamics of\nthe magnons, i.e.\\ the elementary excitations above the ferromagnetic\nstate, constitute the Bethe ansatz. Momentum $p$ and energy $\\epsilon$\nare suitably parametrized in terms of the spectral parameter $x$\n\\be\np(x)=i\\log\\frac{\\sinh(x-i\\gamma/2)}{\\sinh(x+i\\gamma/2)}, \\quad\n\\epsilon(x)=J\\frac{\\sin\\gamma}{2} p'(x)-h,\n\\ee\nwhere real values are obtained for Im$x=0$ and\n$-\\pi/2$, defining magnon bands of type ``{$+$}\" and ``{$-$}\".\n\nAny two magnons with spectral parameters $x$ and $y$ scatter with phase shift\n$\\Theta(x-y)$ where\n\\be\n\\label{phase}\n\\Theta(z)=-i\\log\\frac{\\sinh(z-i\\gamma)}{\\sinh(z+i\\gamma)}.\n\\ee\nNext we apply the standard TBA\\cite{TakTBA}\njust to the\nmagnons and ignore bound states! However,\nthe magnons on \\hbox{band $-$} are considered\nfor spectral parameter $x$ with \\hbox{Im $x$} $=-\\gamma$ hence avoiding the\nbranch cut in the scattering phase.\n\nThe density functions for particles $\\rho_j$\nand holes $\\rho_j^h$ for the bands $j=+$, $-$ give rise to the definition\nof the ratio function $\\eta_j=\\rho_j^h/\\rho_j$. Our analysis shows that\n$\\eta_+$ and $\\eta_-$ are analytic continuations of each other. Quantitatively\nwe find $\\eta_-(x+i\\gamma)=\\eta_+(x)=:\\eta(x)$ subject to\nthe non-linear integral equation\n\\be\n\\log\\eta(x)=\\frac{\\epsilon(x)}{T}+\\int_C\\kappa(x-y)\\log(1+\\eta^{-1}(y))dy\n\\label{NLIE}\n\\ee\nwhere $\\kappa(x)=\\frac{1}{2\\pi}\\Theta'(x)$ and\n$C$ is a contour consisting of the paths $C_+$ and $C_-$\nwith Im $y=0$ and $-\\gamma$\nencircled in clockwise manner.\n%Due to analyticity $C$ may be deformed\n%as long as the singularities of the integrand are avoided.\nSubstituting $\\log(1+\\eta^{-1})=\\log(1+\\eta)-\\log\\eta$ on the contour $C_+$\nand resolving for $\\log\\eta$ we find\n\\bea\n\\log\\eta(x)=-\\frac{\\bar\\epsilon(x)}{T}\n&+&\\int_{C_+}\\bar\\kappa(x-y)\\log(1+\\eta(y))dy\\cr\n&-&\\int_{C_-}\\bar\\kappa(x-y)\\log(1+\\eta^{-1}(y))dy\n\\eea\nwith \n%$e_0(x)=\\frac{\\pi}{\\gamma}/\\cosh\\frac{\\pi}{\\gamma}x$\n%$\\bar p(x)=i\\log\\tanh\\frac{\\pi}{2\\gamma}(x-i\\gamma/2)$\n\\bea\n\\bar\\epsilon(x)&=&J\\frac{\\sin\\gamma}{2}e_0(x)\n+\\frac{\\pi}{2(\\pi-\\gamma)}h,\\ \ne_0(x)=\\frac{\\frac{\\pi}{\\gamma}}{\\cosh\\frac{\\pi}{\\gamma}x},\\\\\n\\bar\\kappa(x)&=&\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty\\frac\n{\\sinh(\\frac{\\pi}{2}-\\gamma)k}\n{2\\cosh\\frac{\\gamma}{2}k\\sinh\\frac{\\pi-\\gamma}{2}k}\\e^{ikx}dk~.\n\\label{kernel}\n\\eea\nFinally, ignoring $T$ and $h$ independent contributions\nwe obtain the free energy as\n\\be\nf=-\\frac{T}{2\\pi}\\int_{-\\infty}^\\infty e_0(x)\n\\log\\left[(1+\\eta(x))(1+\\eta^{-1}(x-i\\gamma))\\right] dx.\n\\label{EqFE}\n\\ee\n\n\\noindent\n{\\em Numerical study of low-$T$ behavior}\n\nLukyanov's low-$t$ asymptotic expansion of $\\chi(t)$ is\\cite{Lukyanov1997}\n%\\label{EqsLukyanov:all}\n\\begin{eqnarray}\n\\chi_{\\rm lt,g}(t)J = {1\\over\\pi^2}\\bigg\\{1 &+& {g\\over 2} + {3 g^3\\over\n32} + {\\cal O}(g^4)\\nonumber\\\\ &+& {\\sqrt{3}\\over \\pi} t^2 [1 + {\\cal\nO}(g)]\\bigg\\}~,\\label{EqsLukyanov:a}\n\\end{eqnarray}\nwhere $g(t/t_0)$ obeys the transcendental equation\n$\n\\sqrt{g}\\exp({1/g}) = {t_0/ t},\n$\nwith a unique value of $t_0$ given by\n$t_0 = \\sqrt{\\pi/ 2}\\exp(\\gamma+1/4)\\approx\n2.866$\nwhere $\\gamma$ is \\mbox{Euler's} constant. His\nexpansion for the free energy per spin at $h = 0$ \\cite{Lukyanov1997}\nyields the specific heat per spin as\n\\begin{eqnarray} C_{\\rm lt,g}(t) = {2 t\\over 3 }\\Big[1 &+& {3\\over 8}\\,g^3\n+ {\\cal O}(g^4)\\Big]\\nonumber\\\\\n\\nonumber\\\\ &+& {(2)3^{5/2}t^3\\over 5\\pi}[1 + {\\cal O}(g)]~,\n\\label{EqCLusnikov}\n\\end{eqnarray}\nwhere the exact prefactor $2t/3$ was found by Affleck in\n1986\\cite{Affleck1986}, and the prefactor 3/8 in the logarithmic\ncorrection term agrees with \\cite{Klumper1998,Affleck1989,Karbach95}.\n\nNumerical data for $\\chi(t)$ and $C(t)$ were obtained using\nour free energy expression~(\\ref{EqFE}). \nThese data are considerably more accurate than those presented previously\nin \\cite{Klumper1998}.\nOur $\\chi(t)J$ data, and the\nexact value $1/\\pi^2$ at $t = 0$ \\cite{Griffiths1964}, are plotted in\nFig.~\\ref{Fig1}. The calculations have an absolute accuracy of $\\approx\n1\\times 10^{-9}$. The data show a maximum at a temperature $t^{\\rm max} =\n0.6\\,408\\,510(4)$ with a value $\\chi^{\\rm max}J = 0.146\\,926\\,279(1)$,\nyielding the $J$-independent product $\\chi^{\\rm max}T^{\\rm max} =\n0.0\\,941\\,579(1)$. These values are consistent within the errors with\nthose found by Eggert, Affleck and Takahashi\\cite{Eggert1994}, but are\nmuch more accurate.\n\nThe differences between our low-$t$ Bethe ansatz $\\chi(t)J$ calculations\nand Lukyanov's theoretical $\\chi_{\\rm lt,g}(t)J$ prediction in\nEq.~(\\ref{EqsLukyanov:a}) are shown in Fig.~\\ref{Fig2}. The\nerror bar on each data point is the estimated uncertainty in $\\chi_{\\rm\nlt,g}J$ arising from the presence of the unknown ${\\cal O}(g^4)$ and\nhigher-order terms in Eq.~(\\ref{EqsLukyanov:a}), which was arbitrarily set\nto $g^4(t)/\\pi^2$; the uncertainty in the $t^2$ contribution,\n$\\sim\\sqrt{3}t^2g(t)/\\pi^3$, is negligible at low $t$ compared to this.\nAt the lower temperatures, the data agree extremely well with the\nprediction of Lukyanov's theory. At the highest temperatures,\nhigher order $t^n$ terms also become important. Irrespective of these\nuncertainties in the theoretical prediction at high temperatures, we can\nsafely conclude directly from Fig.~\\ref{Fig2} that our numerical\n$\\chi(t)$ data are in agreement with the theory of\nLukyanov\\cite{Lukyanov1997} to within an absolute accuracy of $1\\times\n10^{-6}$ (relative accuracy $\\approx 10$ ppm) from $t = 5\\times 10^{-25}$\nto $t = 5\\times 10^{-7}$. The agreement at the lower temperatures,\n${\\cal O}(10^{-7})$, is much better than this.\n\nOur $C(t)$ data for $t\\leq 2$ are shown in the inset of\nFig.~\\ref{Fig3} and have an\nestimated accuracy of $3\\times 10^{-10}C(t)$. The data show a\nmaximum with a value $C^{\\rm max} = 0.3\\,497\\,121\\,235(2)$ at a\ntemperature $t_C^{\\rm max}= 0.48\\,028\\,487(1)$. The electronic specific\nheat coefficient $C(t)/t$ is plotted in Fig.~\\ref{Fig3}. These data\nexhibit a maximum with a value $(C/t)^{\\rm max}= 0.8\\,973\\,651\\,576(5)$ at\n$t_{\\rm C/t}^{\\rm max}= 0.30\\,716\\,996(2)$. The existence of low-$t$ log\ncorrections to $C(t)$ is revealed in the top plot of $\\Delta C(t)/t$ in\nFig.~\\ref{Fig4}, where $\\Delta C(t) = C(t) - 2 t/3$ and $2t/3$ is the\nlow-$t$ limit of $C(t)$. The influence of the $g^3$ log correction term\nin Eq.~(\\ref{EqCLusnikov}) is evaluated by subtracting it in the plot of\n$\\Delta C(t)/t$ as shown by the middle curve in Fig.~\\ref{Fig4}. The $t\n= 0$ singularity is still present but with reduced amplitude; this\ndemonstrates that additional logarithmic correction terms are important\nwithin the accuracy of the data.\n\nWe estimate the unknown coefficients of the next two logarithmic\ncorrection ($g^4,\\ g^5$) terms in Eq.~(\\ref{EqCLusnikov}) from our $C(t)$\ndata as follows. From Eq.~(\\ref{EqCLusnikov}), if we plot the data as\n$[C(t)/t - (2/3)(1+3g^3/8)]/g^4$ vs $g$ and fit the lowest-$t$ data by a\nstraight line, the $y$-intercept gives the coefficient of the\n$g^4$ term and the slope gives the coefficient of the $g^5$ term. We\nfitted a straight line to the data in such a plot for $5\\times10^{-25}\\leq\nt\\leq 5\\times10^{-9}$ as shown by the weighted linear fit in\nFig.~\\ref{Fig5} where the parameters of the fit are given in the figure.\nBy subtracting the influences of these two logarithmic correction terms\nfrom the middle data set as shown in the bottom data set in\nFig.~\\ref{Fig4}, the singular behavior as $t\\to 0$ is largely removed,\nleaving a behavior which is close to a $t^2$ dependence as predicted by\nthe last term in Eq.~(\\ref{EqCLusnikov}). Further discussion of the\npredictions of \\cite{Lukyanov1997}, and high-accuracy fits\n($0\\leq t\\leq 5$) to our $C(t)$ and $\\chi(t)$ data and the respective\nexact $t = 0$ values, will be presented elsewhere\\cite{Johnston1999}.\n\nIn conclusion, we have presented an analytic approach to\nthe thermodynamics of the $S = 1/2$ AF Heisenberg chain on the basis of a\nfinite number of elementary excitations. We envisage that this approach\ncan be generalized to study a variety of other systems such as Hubbard\nand $t$-$J$ models, quantum spin chains with higher symmetries and\nsystems with orbital degrees of freedom. Our free energy expression has\nallowed numerical calculations of $\\chi(t)$ and $C(t)$ for the Heisenberg\nchain to be carried out to much higher accuracy and to much lower\ntemperatures than heretofore attained. Our $\\chi(t)$ data are in\nexcellent agreement with the theory of Lukyanov\\cite{Lukyanov1997} at low\n$t$. The logarithmic correction in Lukyanov's theory for $C(t)$ is found\ninsufficient to describe our $C(t)$ data accurately even at very low $t$.\nHowever, the $t$ dependence of the deviation agrees with the form of his\ntheory, which enabled us to estimate the unknown coefficients of the next\ntwo logarithmic correction terms in his theory for $C(t)$ from our $C(t)$\ndata. Thus we have verified Lukyanov's theory\\cite{Lukyanov1997} of a\ncritical system perturbed by marginal operators and have given evidence\nthat his asymptotic expansion can be systematically extended to higher\norder.\n\nThe authors acknowledge valuable discussions with U.~L\\\"ow and K.\nFabricius. Comparison of our results with their numerical data for\nthe thermodynamics of finite systems proved essential to achieve high\naccuracy in the treatment of the nonlinear integral equations. D.C.J.\nthanks the University of Cologne and the Stuttgart Max-Planck-Institut\nf\\\"ur Festk\\\"orperforschung for their hospitality. A.K.\nacknowledges financial support by the {\\it Deutsche\nForschungsgemeinschaft} under grant No.~Kl~645/3 and by the research\nprogram of the Sonderforschungsbereich 341, K\\\"oln-Aachen-J\\\"ulich. Ames\nLaboratory is operated for the U.S. Department of Energy by Iowa State\nUniversity under Contract No.\\ W-7405-Eng-82. The work at Ames\nwas supported by the Director for Energy Research, Office of\nBasic Energy Sciences.\n\n\\begin{references}\n\n\\bibitem{Bethe31} H. A. Bethe, Z. Phys.\\ {\\bf 71}, 205 (1931).\n\n\\bibitem{TakTBA} M. Takahashi, Prog.\\ Theor.\\ Phys.\\ {\\bf 46}, 401\n(1971); {\\it ibid.}~{\\bf 50}, 1519 (1973).\n\n\\bibitem{SuzukiI87} M. Suzuki and M. Inoue, Prog.\\ Theor.\\ Phys.\\ {\\bf\n78}, 787 (1987).\n\n\\bibitem{Tak91} M. Takahashi, Phys.\\ Rev.\\ B {\\bf 43}, 5788 (1991); {\\it\nibid.}\\ {\\bf 44}, 12\\,382 (1991).\n\n\\bibitem{Eggert1994}S. Eggert, I. Affleck, and M. Takahashi, Phys.\\ Rev.\\\nLett.\\ {\\bf 73}, 332 (1994).\n\n\\bibitem{Bonner1964}J. C. Bonner and M. E. Fisher, Phys.\\ Rev.\\\n{\\bf 135}, A640 (1964).\n\n\\bibitem{Lukyanov1997}S. Lukyanov, Nucl.\\ Phys.\\ B {\\bf 522}, 533 (1998).\n\n\\bibitem{Klumper1993}A. Kl\\\"umper, Z. Phys. B {\\bf 91}, 507 (1993).\n\n\\bibitem{Klumper1998}A. Kl\\\"umper, Eur.\\ Phys.\\ J. B {\\bf 5}, 677 (1998).\n\n\\bibitem{Affleck1986}I. Affleck, Phys.\\ Rev.\\ Lett.\\ {\\bf 56}, 746 (1986).\n\n\\bibitem{Affleck1989}I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman,\nJ.~Phys.\\ A {\\bf 22}, 511 (1989).\n\n\\bibitem{Karbach95}M. Karbach and K.-H. M\\\"utter, J. Phys.\\ A {\\bf 28},\n4469 (1995).\n\n\\bibitem{Griffiths1964}R. B. Griffiths, Phys.\\ Rev.\\ {\\bf 133}, A768\n(1964); C. N. Yang and C. P. Yang, Phys.\\ Rev.\\ {\\bf 150}, 327 (1966).\n\n\\bibitem{Johnston1999}D. C. Johnston {\\it et al.}, unpublished.\n\n\\end{references}\n\n% Figure 1\n\\begin{figure}\n\\epsfxsize=3in\n\\centerline{\\epsfbox{Fig1.eps}}\n\\vglue 0.1in % space between figure and caption\n\\caption{Magnetic susceptibility $\\chi$ at low temperature $T$ for the\n spin $S = 1/2$ antiferromagnetic uniform Heisenberg chain. In the\n inset $\\chi(T)$ is shown on a larger temperature scale.}\n\\label{Fig1}\n\\end{figure}\n\n% Figure 2\n\\begin{figure}\n\\epsfxsize=2.8in\n\\centerline{\\epsfbox{Fig2.eps}}\n\\vglue 0.1in\n\\caption{Semilog plot vs temperature $t$ at low $t$ of the difference\nbetween our Bethe ansatz magnetic susceptibility $\\chi J$ data and the\nprediction $\\chi_{\\rm lt,g}J$ of Lukyanov's\ntheory\\protect\\cite{Lukyanov1997}. The error bars are the estimated\nuncertainties in $\\chi_{\\rm lt,g}(t)J$.}\n\\label{Fig2}\n\\end{figure}\n\n% Figure 3\n\\begin{figure}\n\\epsfxsize=3in\n\\centerline{\\epsfbox{Fig3.eps}}\n\\vglue 0.1in\n\\caption{Electronic specific heat coefficient $C/T$\n versus temperature $T$ for the $S = 1/2$ AF uniform Heisenberg chain.\n In the inset the specific heat $C$ versus $T$ is shown.}\n\\label{Fig3}\n\\end{figure}\n\n% Figure 4\n\\begin{figure}\n\\epsfxsize=3in\n\\centerline{\\epsfbox{Fig4.eps}}\n\\vglue 0.1in\n\\caption{Difference $\\Delta C/t$ between our Bethe ansatz electronic\nspecific heat coefficient data and the exact coefficient 2/3 at $t =\n0$ (top data set), versus temperature $t$. Successive data sets show the\ninfluence of subtracting cumulative logarithmic correction terms.}\n\\label{Fig4}\n\\end{figure}\n\n% Figure 5\n\\begin{figure}\n\\epsfxsize=2.8in\n\\centerline{\\epsfbox{Fig5.eps}}\n\\vglue 0.1in\n\\caption{Plot showing the estimation of the coefficients of the $g^4$ and\n$g^5$ logarithmic correction terms in Eq.~(\\protect\\ref{EqCLusnikov}).\nThe error bars on the data points are smaller than the symbols.}\n\\label{Fig5}\n\\end{figure}\n\n\\end{document}\n\n"
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[
{
"name": "cond-mat0002140.extracted_bib",
"string": "\\bibitem{Bethe31} H. A. Bethe, Z. Phys.\\ {\\bf 71}, 205 (1931).\n\n\n\\bibitem{TakTBA} M. Takahashi, Prog.\\ Theor.\\ Phys.\\ {\\bf 46}, 401\n(1971); {\\it ibid.}~{\\bf 50}, 1519 (1973).\n\n\n\\bibitem{SuzukiI87} M. Suzuki and M. Inoue, Prog.\\ Theor.\\ Phys.\\ {\\bf\n78}, 787 (1987).\n\n\n\\bibitem{Tak91} M. Takahashi, Phys.\\ Rev.\\ B {\\bf 43}, 5788 (1991); {\\it\nibid.}\\ {\\bf 44}, 12\\,382 (1991).\n\n\n\\bibitem{Eggert1994}S. Eggert, I. Affleck, and M. Takahashi, Phys.\\ Rev.\\\nLett.\\ {\\bf 73}, 332 (1994).\n\n\n\\bibitem{Bonner1964}J. C. Bonner and M. E. Fisher, Phys.\\ Rev.\\\n{\\bf 135}, A640 (1964).\n\n\n\\bibitem{Lukyanov1997}S. Lukyanov, Nucl.\\ Phys.\\ B {\\bf 522}, 533 (1998).\n\n\n\\bibitem{Klumper1993}A. Kl\\\"umper, Z. Phys. B {\\bf 91}, 507 (1993).\n\n\n\\bibitem{Klumper1998}A. Kl\\\"umper, Eur.\\ Phys.\\ J. B {\\bf 5}, 677 (1998).\n\n\n\\bibitem{Affleck1986}I. Affleck, Phys.\\ Rev.\\ Lett.\\ {\\bf 56}, 746 (1986).\n\n\n\\bibitem{Affleck1989}I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman,\nJ.~Phys.\\ A {\\bf 22}, 511 (1989).\n\n\n\\bibitem{Karbach95}M. Karbach and K.-H. M\\\"utter, J. Phys.\\ A {\\bf 28},\n4469 (1995).\n\n\n\\bibitem{Griffiths1964}R. B. Griffiths, Phys.\\ Rev.\\ {\\bf 133}, A768\n(1964); C. N. Yang and C. P. Yang, Phys.\\ Rev.\\ {\\bf 150}, 327 (1966).\n\n\n\\bibitem{Johnston1999}D. C. Johnston {\\it et al.}, unpublished.\n\n"
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cond-mat0002142
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Finite-size calculations of spin-lattice relaxation rates in Heisenberg spin-ladders
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Calculations of nuclear spin-lattice relaxation rates are carried out by means of exact diagonalization on small ($2\times 6$) antiferromagnetic Heisenberg ladders, using the simplest forms permitted by symmetry for the hyperfine couplings for the three nuclear sites in Cu$_2$O$_3$ ladders. Several values of the rung/chain exchange ratio $J_{\perp}/J_{\parallel}$ have been considered. Comparisons with experimental results, field theoretic calculations, and the Gaussian approximation highlight some open problems.
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"string": "%% ****** Start of file template.aps ****** %\n%%\n%%\n%% This file is part of the APS files in the REVTeX 4 distribution.\n%% Version 4.0 beta 2 of REVTeX, September 14, 1999.\n%%\n%%\n%% Copyright (c) 1999 The American Physical Society.\n%%\n%% See the REVTeX 4 README file for restrictions and more information.\n%%\n%\n% This is a template for producing files for use with REVTEX 4.0 beta\n% Copy this file to another name and then work on that file.\n% That way, you always have this original template file to use.\n%\n% Group addresses by affiliation. Use superscriptaddress for long\n% author lists or if there are many overlapping affiliations\n% For Phys. Rev. look and feel change preprint to twocolumn\n\n\\documentclass[aps,prb,preprint,groupedaddress,showpacs,amsfonts]{revtex4}\n%\\documentclass[aps,prb,twocolumn,groupedaddress,showpacs,amsfonts]{revtex4}\n%\\documentclass[aps,twocolumn,groupedaddress,showpacs,amsfonts]{revtex4}\n%\\documentclass[aps,preprint,superscriptaddress,showpacs]{revtex4}\n%\\documentclass[aps,twocolumn,groupedaddress]{revtex4}\n\\usepackage{bm}\n\\usepackage{graphicx}\n\n\\begin{document}\n% You should use BibTeX and revtex.bst for references\n\\bibliographystyle{apsrev}\n% marks overfull lines with blackboxes\n%\\draft - no longer supported, use the 'draft' option instead\n\n% Use the \\preprint command to place your local institutional report\n% number on the title page in preprint mode.\n% Multiple \\preprint commands are allowed.\n%\\preprint{}\n\n%Title of paper\n\\title[FINITE-SIZE CALCULATIONS OF SPIN-LATTICE RELAXATION RATES IN $\\ldots$]\n{Finite-size calculations of spin-lattice relaxation rates in Heisenberg\nspin-ladders}\n% Optional argument for running titles on pages\n%\\title[]{}\n\n% repeat the \\author .. \\affiliation etc. as needed\n% \\email, \\thanks, \\homepage, \\altaffiliation all apply to the current\n% author. Explanatory text should go in the []'s, actual e-mail\n% address or url should go in the {}'s for \\email and \\homepage.\n% Please use the appropriate macro for the type of information\n\n% \\affiliation command applies to all authors since the last\n% \\affiliation command. The \\affiliation command should follow the\n% other information\n\n\\author{Martin P. Gelfand}\n\\email[]{gelfand@lamar.colostate.edu}\n%\\homepage[]{Your web page}\n%\\thanks{}\n%\\altaffiliation{}\n\\author{Mohan Mahadevan}\n\\altaffiliation[Permanent address: ]%\n{KLA-Tencor, 3 Technology Drive, Malpitas, California 95035}\n\n\\affiliation{Department of Physics, Colorado State University, Fort Collins, Colorado 80523}\n\n%Collaboration name if desired (requires use of superscriptaddress\n%option in \\documentclass). \\noaffiliation is required (may also be\n%used with the \\author command).\n%\\collaboration{}\n%\\noaffiliation\n\n\\date{\\today}\n\n\\begin{abstract}\nCalculations of nuclear spin-lattice relaxation rates are carried\nout by means of exact diagonalization on small ($2\\times 6$)\nantiferromagnetic Heisenberg ladders, using the simplest forms permitted\nby symmetry\nfor the hyperfine couplings for the three nuclear sites in Cu$_2$O$_3$ ladders.\nSeveral values of the rung/chain exchange ratio\n$J_{\\perp}/J_{\\parallel}$ have been considered.\nComparisons with experimental results, field theoretic calculations,\nand the Gaussian approximation highlight some open problems.\n\\end{abstract}\n% insert suggested PACS numbers in braces on next line\n\\pacs{75.10.Jm, 75.40.Gb, 75.40.Mg, 76.60.-k}\n\n%\\maketitle must follow title, authors, abstract and \\pacs\n\\maketitle\n\n% body of paper here - Use proper section commands (\\section,\\subsection)\n% References should be done using the \\cite, \\ref, and \\label commands\n\\section{Introduction}\n\\label{sec:Introduction}\nSpin-ladder systems, in particular the two-leg, $S=1/2$,\nantiferromagnetic variety, have been the subject of considerable\ntheoretical and experimental investigation.\\cite{dagotto96}\nSpin ladders are appealing because they are one-dimensional\nsystems and thus can be effectively investigated using many \npowerful theoretical tools,\nwhile offering a wider parameter space of ``simple,\" and potentially\nexperimentally realizable, Heisenberg Hamiltonians than spin chains. \nThe simplest Heisenberg ladder Hamiltonian has the form\n\\begin{equation}\n{\\cal H}=\\sum_n J_\\parallel (\\bm{S}_{n,1}\\cdot\\bm{S}_{n+1,1}\n+ \\bm{S}_{n,2}\\cdot\\bm{S}_{n+1,2}) + J_\\perp \\bm{S}_{n,1}\\cdot\\bm{S}_{n,2}\n\\label{eq:slham}\n\\end{equation}\nwhich offers a dimensionless parameter $J_\\perp/J_\\parallel$ which is\nin principle tunable by chemistry or pressure.\nIn addition, compounds containing weakly coupled\nCu$_2$O$_3$ ladders are appealing because of possible connections\nwith cuprate superconductivity.\n\nThe present work was motivated by the nuclear spin-lattice relaxation \nmeasurements in La$_6$Ca$_8$Cu$_{24}$O$_{41}$, an undoped ladder\ncompound, by Imai {\\it et al.}\\cite{imai98} These measurements were\ncarried out for all of the nuclear sites on the ladder, namely the\ncopper, the ``rung'' oxygen, and the ``ladder'' (or ``chain'') oxygen,\nover a wide temperature range, from low temperatures up to nearly\n$900\\,\\rm K$. Because the principal exchange interactions in cuprates\nare so large, on the order of $1000\\,\\rm K$, it is quite challenging to \ndo experimental work at temperatures significantly greater than the \nspin gap ($\\Delta\\approx 500\\,\\rm K$).\n\nThe experimental results (see Figure 1(c) of Imai {\\it et al.}\\cite{imai98})\nhave the following noteworthy features.\nAt temperatures below about $425\\,\\rm K$, the relaxation rates for all three\nsites follow a common (activated) temperature dependence\nup to a scale factor.\nHowever, on increasing $T$ the copper $1/T_1$ (which we will refer\nto as $1/\\hbox{}^{\\rm Cu}T_1$) exhibits a rather sharp\ndeparture from that of the two oxygen sites ($1/\\hbox{}^{\\rm O(1)}T_1$\nand $1/\\hbox{}^{\\rm O(2)}T_1$ for ladder and rung, respectively). \nThere seems to be\na nearly discontinuous decrease in the derivative of $1/\\hbox{}^{\\rm Cu}T_1$;\nmoreover, above $425\\,\\rm K$ the $1/\\hbox{}^{\\rm Cu}T_1$ data appear nearly\nlinear with an almost vanishing intercept.\nThe relaxation rates for the two oxygen sites, in contrast,\nexhibit no particular features in the vicinity of $425\\,\\rm K$.\n\nSeveral aspects of the wavevector dependence of the low-frequency\nspin susceptibility can be gleaned directly from the data.\n\nOne can express the spin-lattice relaxation rate in terms of\nthe dynamic structure factor for the Cu$^{2+}$ spins\n\\begin{equation}\n{1\\over\\hbox{}^n T_1} \\propto \n\\int d\\bm{q} H_n(\\bm{q})S(\\bm{q},\\omega_n)\n\\end{equation}\nwhere $H_n$ is the hyperfine form factor associated with nucleus $n$, \n$\\omega_n$ is the NMR frequency (which we will take to be zero in\neverything that follows), and $S$ is the structure factor. The \nproportionality constants can be neglected for our present purposes.\nThe spin correlations are isotropic, so there is no need to consider \nthe various components, $S^{xx}$ and so forth, individually.\nThe hyperfine interactions are not isotropic, so the orientation\nof the magnetic field in the NMR experiment does affect the results;\nhowever, all of the results of present interest can be obtained \nwith a single field orientation, which then specifies $H_n(\\bm{q})$ uniquely.\nThe largest hyperfine couplings are between a given nuclear site \nand the closest spins; at that level of approximation, and taking\nthe intra- and inter-chain lattice constants to be of unit length, one has\n\\begin{eqnarray}\nH_{\\rm Cu}=A^2,\\ H_{\\rm O(1)}=4C^2\\cos^2(q_x/2),\\nonumber\\\\\nH_{\\rm O(2)}=4F^2\\cos^2(q_y/2)+D^2\n\\end{eqnarray}\nwhere $C$, $F$, and $D$ are the hyperfine couplings identified\nin Fig.~1(a) of Imai {\\it et al.},\\cite{imai98} $A$ is the on-site\nhyperfine interaction for copper, and we have elided the \norientation dependence of the hyperfine interactions\n(so, for example, $A^2$ should really be $A_x^2 + A_y^2$ if the\nstatic field is along the $z$ axis).\n\nThe essential difference between copper and oxygen sites is that\nin the latter the hyperfine interaction in the vicinity of $\\bm{q}=(\\pi,\\pi)$\nis much smaller than in the vicinity of $\\bm{q}=(0,0)$.\nIf, at all temperatures of experimental relevance, $S(\\bm{q},0)$\nhad most of its weight in the vicinity of $\\bm{q}=(0,0)$, then\nthen all three relaxation rates would have tracked one another.\nThe marked decrease of $1/\\hbox{}^{\\rm Cu}T_1$ relative to the other two\nrelaxation rates at $425\\,\\rm K$\nindicates that this cannot be the case, and in fact suggests\nthat at temperatures below $425\\,\\rm K$ the ratio of the spectral weight near\n$(\\pi,\\pi)$ to that near $(0,0)$ is roughly constant and of order unity, while\nabove $425\\,\\rm K$ the ratio falls markedly. (The {\\em decrease} is\ncrucial. If there were an increase in $1/\\hbox{}^{\\rm Cu}T_1$ relative\nto the oxygen rates with increasing $T$, one could ascribe\nthat to a turn-on of $S((\\pi,\\pi),0)$ for $T\\agt\\Delta$ but\n$S((\\pi,\\pi),0)$ might have been negligible compared to \n$S((0,0),0)$ at lower temperatures.)\n\nWhy the emphasis on $\\bm{q}=(0,0)$ and $(\\pi,\\pi)$? In gapped systems\nsuch as spin ladders, the low-energy spin fluctuations are Raman\nprocesses, and at low temperatures one needs to consider only\nthe lowest energy magnons, namely those near $\\bm{q}=(\\pi,\\pi)$.\nSpin fluctuations near $(0,0)$ are associated with \ntwo-magnon processes, and those near $(\\pi,\\pi)$ with\nthree-magnon processes, and on the face of it one would\nbe justified in neglecting the three-magnon processes entirely\nat low temperatures: see Ref.~\\onlinecite{ivanov99} and\nreferences cited therein.\nHowever, as we have just seen, this appears to be inconsistent\nwith the experimental data for La$_6$Ca$_8$Cu$_{24}$O$_{41}$,\nand it is also inconsistent with the quantum Monte Carlo\ncalculations of spin-lattice relaxation in a particular \nHeisenberg ladder ($J_\\perp/J_\\parallel=1$)\nby Sandvik, Dagotto, and Scalapino,\\cite{sandvik96}\nat least at temperatures greater than half the magnon gap.\n\nAn extensive theoretical treatment of spin dynamics in\ngapped one-dimensional Heisenberg models, including\nspin ladders, has been presented by Damle and Sachdev.\\cite{damle98}\nTheir analysis of $S(\\bm{q},\\omega\\approx0)$ was restricted\nto $\\bm{q}$ near $(0,0)$, but they did find the quite interesting\nresult that the activation energy for $1/T_1$ is larger, by a\nfactor of $3/2$, than the activation energy for the uniform\nstatic susceptibility (which is simply the spin gap).\nAn analysis of $S(\\bm{q},\\omega\\approx0)$ for $\\bm{q}$ near\n$(\\pi,\\pi)$, for systems with $J_\\parallel \\gg J_\\perp$ \nhas been presented by Ivanov and Lee.\\cite{ivanov99}\nTheir results are suggestive of a fairly sharp crossover from\nlow- to high-temperature regimes at $T\\approx\\Delta$, and also\nindicate that the $(\\pi,\\pi)$ contribution to $1/T_1$ ``overshoots\"\nits $T=\\infty$ value and thus decreases as $T\\to\\infty$.\n\nIn the present work, we have applied\nexact diagonalization to evaluate spin-lattice\nrelaxation rates, following the method of \nSokol, Gagliano and Bacci.\\cite{sokol93} \nWe have considered three different ladder Hamiltonians, namely\n$J_\\perp/J_\\parallel=0.5$, 1.0, and 2.0, and have obtained \n$1/T_1$ for Cu, O(1), and O(2) sites taking the simplest \nconceivable hyperfine couplings, namely $A=C=F=1$,\nwith all other interactions neglected.\nAll of the calculations were for rather small systems, $2\\times6$,\nsuch that exact diagonalization could be carried out in an\nextremely straightforward manner.\n\nIt was noted above that calculations of spin lattice\nrelaxation rates for spin ladders have already been carried out\nby means of large scale quantum Monte Carlo,\\cite{sandvik96} but \nthose calculations were limited to the Cu sites. \nOur goal is somewhat different than that of Sandvik, Dagotto, and\nScalapino's work.\nWe are not trying to fit the data in detail, rather \nwe want to see what can\nbe learned from modest numerical calculations. One reason \nnot to fit the data is that to get the gap correct to\n10\\% by exact diagonalization for $J_\\perp/J_\\parallel=0.5$\nwould require a system at least $2\\times12$. Another is that\nwe do not treat the spin diffusion\ncontribution to the relaxation rates correctly: our\ncalculations effectively introduce an artificial cut-off so that\nwe obtain a finite spin-lattice relaxation rate.\nFinally, the precise form of the\nspin Hamiltonian for the cuprate ladder compounds is\nstill subject to argument.\nAlthough the Knight-shift results of Imai {\\it et al.}\\cite{imai98}\nappear to be consistent with the simple spin-ladder Hamiltonian\nof Eq.~(\\ref{eq:slham}) for $J_\\perp/J_\\parallel\\approx0.5$,\nit has been suggested by Brehmer {\\it et al.}\\cite{brehmer99} that\ninstead $J_\\perp/J_\\parallel\\approx1$ and in addition \nthere is a modest amount of plaquette ``ring exchange\" in the\nHamiltonian. A quantum-chemical analysis of the exchange\ninteractions in various cuprates\\cite{mizonu98} provides some\nsupport for the latter proposal, since it concludes\nthat $J_\\perp/J_\\parallel\\approx0.9$ for Sr$_{14}$Cu$_{24}$O$_{41}$\n(which is a lightly-self-doped version of the undoped\nLa$_6$Ca$_8$Cu$_{24}$O$_{41}$ compound).\n\nTo be precise, the goals of our calculation are as follows.\nFirst, we want to verify that $S(\\bm{q},0)$ has significant\nweight near $\\bm{q}=(\\pi,\\pi)$ as well as near $(0,0)$\nand see if there are any noticeable trends with varying $J_\\perp/J_\\parallel$.\nSecond, we want to explore the crossover from low to high temperature\nbehavior in $1/T_1$: can we see anything like the experimental results,\nor like the theoretical results of Ivanov and Lee?\nThird, we want to keep our eyes open for any unanticipated patterns\nthat might emerge in the numerical results.\n\n\\section{Method of Calculation and Results}\n\nThe finite-size calculations of spin-lattice relaxation rates\nare carried out following Sokol, Gagliano, and Bacci.\\cite{sokol93}\nRather than repeating their discussion of the method let us make\na few remarks. We take $J_\\parallel$ as the unit of energy.\n\nThe first step in the calculation is a complete diagonalization\nof the Hamiltonian and evaluation of matrix elements for certain\nlocal spin operators (depending on which nuclear site one\nis interested in). For the $2\\times 6$ lattices\nall of the calculations could be done using the simplest possible\nrepresentations of the states in terms of local $S^z$ values; it\nwas not even necessary to use translational invariance to classify\nstates by wave vector.\n\nThe second step is the construction of an auxiliary function which\nSokol {\\it et al.}\\ refer to as $I(\\omega)$. This is implicitly dependent\non $T$ and the hyperfine couplings.\nWe considered temperatures ranging from 0.3 to 50.\nTypically we constructed $I(\\omega)$ at intervals of 0.02 in $\\omega$\nup to at least $\\omega=0.6$.\n\nFinally, one needs to estimate the zero-frequency derivative\nof $I(\\omega)$, because $1/T_1$ is is proportional to \n$T(\\partial I/\\partial\\omega)|_{\\omega=0}$. At high temperatures\n$I(\\omega)$ is quite smooth, but at temperatures comparable to the\ngap significant structure develops (see Fig.~\\ref{fig:Iomega}).\nIn order to avoid introducing spurious temperature dependences\ninto $1/T_1$ it is important to use a consistent procedure for\nextracting the derivative from the data. What we did was to \nfit a zero-intercept line through all the data points up to a\ncutoff $\\omega_{\\rm max}$, weighting all points equally in the fit.\nWe did all of the calculations using both $\\omega_{\\rm max}=0.5$ and\n0.3. While there are noticeable differences in the results using\nthese two cutoffs, as shown in Fig.~\\ref{fig:compcutoffs}, our\nconclusions turn out the same no matter which is chosen. The use of a\nmuch smaller cutoff, which might seem to be preferred on the grounds\nthat one is really looking for a zero-frequency derivative, is not\nbeneficial. The structure that develops in $I(\\omega)$ as $T$ is\nlowered, making it look like a Devil's staircase, is\na finite-size artifact and must be averaged over, using a suitably\nlarge $\\omega_{\\rm max}$, to obtain results that are \nrepresentative of the thermodynamic limit.\n\n%\\begin{figure}\n%\\includegraphics[width=3.4in]{Iomega.eps}\n%\\caption{The auxiliary function $I(\\omega)$ for the Cu site with\n%$J_\\perp=1.0$ and\n%$T=0.5$ (squares), 1.0 (circles), and 5.0 (diamonds).}\n%\\label{fig:Iomega}\n%\\end{figure}\n\n%\\begin{figure}\n%\\includegraphics[width=3.4in]{compcutoffs.eps}\n%\\caption{Estimated values of $1/\\hbox{}^{\\rm Cu}T_1$ for \n%$J_\\perp=1.0$\n%as a function of temperature, taking $\\omega_{\\rm max}=0.3$ ($+$)\n%and 0.5 ($\\times$).}\n%\\label{fig:compcutoffs}\n%\\end{figure}\n\nWe now turn to the results of the calculations for the three\nnuclear sites and three values of $J_\\perp$ considered (0.5, 1.0,\nand 2.0). In every case we take $\\omega_{\\rm max}=0.5$.\nIn Fig.~\\ref{fig:allT} we present results on a linear \ntemperature scale, for $T\\leq2$. The behavior of the spin-lattice\nrelaxation rate at high temperatures is a bit\nsurprising: comparing the plots in Fig.~\\ref{fig:allT}(a) through\n(c) it is apparent that while the Cu and O(1) rates decrease strongly\nas $J_\\perp$ increases, the trend for the O(2) rate is different. \nThis is made more explicit in Fig.~\\ref{fig:highT}, where we show\n$1/T_1$ for all three sites as a function of $J_\\perp$ at $T=50$\n(effectively infinite temperature). In contrast, at low temperatures\n$1/T_1$ decreases with increasing $J_\\perp$ at all sites, as\none would expect since the spin gap is an increasing function of $J_\\perp$.\n\n%\\begin{figure}\n%\\includegraphics[width=3.4in]{plotall_2.eps}\n%\\caption{Spin lattice relaxation rates as a function of temperature (in\n%units of $J_\\parallel$)\n%for copper (circles), ladder oxygen (squares) and rung oxygen \n%(diamonds) sites, for\n%$J_\\perp/J_\\parallel=0.5$, 1.0, and 2.0 in (a), (b), and (c) respectively.\n%The upside-down triangle on each graph indicates the value of\n%the spin gap $\\Delta$ for the corresponding system.}\n%\\label{fig:allT}\n%\\end{figure}\n\n\\section{Discussion and Conclusions}\n\\label{sec:conclusions}\nIt is evident that for $J_\\perp=0.5$ and 1.0, \n$1/T_1$ for all three sites is nearly equal for temperatures\nbelow the spin gap. (Of course we do not claim that this holds\nto arbitrarily low temperatures, just that it seems correct\nfor temperatures as low as we dare to estimate $1/T_1$.) \nBecause of our choice of hyperfine interactions,\nthis suggests that in such cases the weight in $S(\\bm{q},0)$ for\n$\\bm{q}\\approx(\\pi,\\pi)$ is approximately three times that for \n$\\bm{q}\\approx(0,0)$. This is in quantitative agreement with the results\nof Sandvik {\\it et al.}\\cite{sandvik96} at $J_\\perp=1.0$.\nHowever, the story is rather different at $J_\\perp=2.0$, where\nthe spin-lattice relaxation rates for all three sites, including\nthe two oxygen sites, are significantly\ndifferent even at $T=\\Delta/2$. In the strong-coupling limit,\nthen, the simple picture for $S(\\bm{q},0)$ in which its weight\nis concentrated at $(0,0)$ and $(\\pi,\\pi)$ does not work even\nfor temperatures that are a modest fraction of $\\Delta$. \n\nWhat can we say about the low-to-high temperature crossover\nin the spin-lattice relaxation rates? First of all, the sort\nof behavior seen experimentally, in which $1/T_1$ for the oxygen\nsites track each other closely while $1/\\hbox{}^{\\rm Cu}T_1$ splits\noff, appears to be a special feature of $J_\\perp\\approx1$ in\nthe present calculations; it is not at all generic and does\nnot hold for the putative experimental value $J_\\perp\\approx0.5$. \nSecond, in no case does $1/\\hbox{}^{\\rm Cu}T_1$ exhibit any sort of\nsharp ``break'' as seen experimentally; nor does $1/\\hbox{}^{\\rm Cu}T_1$\nexhibit linear-in-$T$ behavior (with zero intercept, or otherwise)\nin the high temperature regime, even over a restricted temperature\nrange (say $\\Delta$ to $2\\Delta$). Finally, in no case does\n$1/\\hbox{}^{\\rm Cu}T_1$ exhibit an ``overshoot'' during the crossover:\nthe spin-lattice relaxation rate associated with all sites\nmonotonically increases with $T$.\n\nOur calculations thus suggest that there are quite a few open\nproblems in this field. \nAlmost none of the prominent experimental facts concerning\n$1/\\hbox{}^{\\rm Cu}T_1$ in La$_6$Ca$_8$Cu$_{24}$O$_{41}$\nare reproduced in our finite-size calculations. \nFurthermore, the work of Ivanov and Lee\\cite{ivanov99} does not\nseem to have much to say about our results, either. Their\ncalculation is controlled only in the $J_\\perp\\ll1$ regime, so\nwe should only look at the $J_\\perp=0.5$ data. Here we have no\nevidence of overshoot in $1/\\hbox{}^{\\rm Cu}T_1$, and no reason\nto believe that one can just examine the spectral weight near\n$(\\pi,\\pi)$ since $1/\\hbox{}^{\\rm O(2)}T_1$ ``peels off'' from\n$1/\\hbox{}^{\\rm O(1)}T_1$ in a manner not very different from\n$1/\\hbox{}^{\\rm Cu}T_1$.\n\nAt this point we face several alternatives. It is possible that\nour results are simply unreliable, because we are\nconsidering systems that are too small (especially for $J_\\perp=0.5$)\nand our procedure for estimating $dI(\\omega)/d\\omega$ is flawed.\nWe cannot rule this out, but we strongly suspect that the trends in the\nresults as a function of $J_\\perp$ are robust.\nIt is possible that the \nspin Hamiltonian for the ladders in\nLa$_6$Ca$_8$Cu$_{24}$O$_{41}$ is more complicated than\nthe model we have considered. Whether the Hamiltonian of\nBrehmer {\\it et al.}\\cite{brehmer99} can reproduce the\nspin-lattice relaxation data requires another calculation.\nAnother possibility that must be considered, given the\nremarkably sharp feature in $1/\\hbox{}^{\\rm Cu}T_1$ found\nin the experimental data, is that La$_6$Ca$_8$Cu$_{24}$O$_{41}$ undergoes,\nby coincidence, a subtle structural transition at $425\\,\\rm K$.\nThis could introduce an anomalously strong $T$-dependence\nto the hyperfine interactions, though why the effect should\nbe so much stronger in $H_{\\rm Cu}(\\bm{q})$ than \n$H_{\\rm O(1)}(\\bm{q})$ and $H_{\\rm O(2)}(\\bm{q})$\nis difficult to envision.\n\nLet us now turn to the results of our calculations for spin-lattice \nrelaxation at very high temperatures, shown in Fig.~\\ref{fig:highT}.\nThe most natural way to think about these results is in terms of\nthe Gaussian approximation.\\cite{anderson53,moriya56a,moriya56b}\nThe basic idea of this approach is to assume that \n$\\int d\\bm{q} H_n({\\bm q}) S({\\bm q},\\omega)$ is a \nGaussian function of $\\omega$,\nand then evaluate the frequency cumulants of this function\nby means of short-time expansions of time-dependent correlation functions.\nAt $T=\\infty$ the calculations are especially simple, because\nthe expectation values of correlators $\\langle \\bm{S}_i \\cdot \\bm{S}_j\\rangle$\nvanish for sites $i\\neq j$. For three sites of interest in\nHeisenberg ladders, the Gaussian approximation yields the following\nexchange dependences of the spin-lattice relaxation rates at $T=\\infty$:\n\\begin{equation}\n1/\\hbox{}^{\\rm Cu}T_1 \\propto 1/\\sqrt{1+\\textstyle{1\\over2}J_\\perp^2}\\ ,\n\\end{equation}\n\\begin{equation}\n1/\\hbox{}^{\\rm O(1)}T_1 \\propto 1/\\sqrt{1+J_\\perp^2}\\ ,\n\\end{equation}\nand $1/\\hbox{}^{\\rm O(2)}T_1$ does not have any $J_\\perp$ dependence\nat all. (Recall that $J_\\parallel \\equiv 1$; in all of these\nresults there is an overall factor of $1/J_\\parallel$.)\nIf this last result seems peculiar, let us note that it\ncan be derived in another way, by considering the strong-$J_\\perp$\nlimit. Then one most naturally thinks about the states in terms of\nsinglets and triplets on the rungs. The relevant energy scale\nfor the dynamics of the total spin on a rung,\nwhich is relevant to $1/\\hbox{}^{\\rm O(2)}T_1$, would seem to be\nproportional to $J_\\parallel$ (that is, the bandwidth in lowest-order\nperturbation theory for a triplet\nexcitation in a single background\\cite{barnes93}), and \nwith the hypothesis of a single energy scale in\n$\\int d\\bm{q} H_{\\rm O(2)}S(\\bm{q},\\omega)$\none reproduces the Gaussian approximation result.\n\nWe see in Fig.~\\ref{fig:highT} that $1/T_1$ for the copper and \nladder oxygen sites decreases with increasing $J_\\perp$, qualitatively\nin agreement with the Gaussian approximation, although the dependence\non $J_\\perp$ is not as strong as that approximation suggests.\nFurthermore, $1/\\hbox{}^{\\rm O(2)}T_1$ exhibits an increase with\n$J_\\perp$. The rather poor performance of the Gaussian approximation\nis somewhat disappointing, considering how well it works for\nestimating spin-lattice relaxation rates in square-lattice\nHeisenberg antiferromagnets.\\cite{gelfand93,sokol93} It is not too surprising,\nperhaps, given that the dynamic correlations in the $S=1/2$ Heisenberg chain\nare far from Gaussian at $T=\\infty$.\\cite{roldan86}\nSo, there is yet another open problem in the area of low-energy spin dynamics\nof Heisenberg ladders.\n\n%\\begin{figure}\n%\\includegraphics[width=3.4in]{highT.eps}\n%\\caption{Spin lattice relaxation rates at $T=50$ as a function of\n%$J_\\perp$.}\n%\\label{fig:highT}\n%\\end{figure}\n\n% If in twocolumn mode, this environment will move to single column\n% format so that long equations can be displayed. Use\n% sparingly. Requires multicol.sty (automatically read in by the\n% twocolumn option).\n%\\begin{widetext}\n% put long equation here\n%\\end{widetext}\n\n% If you have acknowledgements this puts in the proper section\n\\acknowledgements\nThis work was supported by the US National Science Foundation\nthrough grant DMR 94--57928. We thank T. Imai for several stimulating\ndiscussions and also for communicating the results of his group's\nexperiments prior to publication.\n\n% Create the reference section using BibTeX\n\\bibliography{spinladder}\n\n% figures follow here or may be put into the text as floats.\n% Use the graphics or\n% graphicx packages distributed with LaTeX2e. See the LaTeX Graphics\n% Companion by Michel Goosens, Sebastian Rahtz, and Frank Mittelbach\n% for instance.\n%\n% Here is an example of the general form of a figure:\n% Fill in the caption in the braces of the \\caption{} command. Put the label\n% that you will use with \\ref{} command in the braces of the \\label{} command.\n%\n\\begin{figure}\n\\includegraphics{Iomega.eps}\n\\caption{The auxiliary function $I(\\omega)$ for the Cu site with\n$J_\\perp=1.0$ and\n$T=0.5$ (squares), 1.0 (circles), and 5.0 (diamonds).}\n\\label{fig:Iomega}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{compcutoffs.eps}\n\\caption{Estimated values of $1/\\hbox{}^{\\rm Cu}T_1$ for \n$J_\\perp=1.0$\nas a function of temperature, taking $\\omega_{\\rm max}=0.3$ ($+$)\nand 0.5 ($\\times$).}\n\\label{fig:compcutoffs}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{plotall_2.eps}\n\\caption{Spin lattice relaxation rates as a function of temperature (in\nunits of $J_\\parallel$)\nfor copper (circles), ladder oxygen (squares) and rung oxygen \n(diamonds) sites, for\n$J_\\perp/J_\\parallel=0.5$, 1.0, and 2.0 in (a), (b), and (c) respectively.\nThe upside-down triangle on each graph indicates the value of\nthe spin gap $\\Delta$ for the corresponding system.}\n\\label{fig:allT}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{highT.eps}\n\\caption{Spin lattice relaxation rates at $T=50$ as a function of\n$J_\\perp$.}\n\\label{fig:highT}\n\\end{figure}\n\n% tables follow here or maybe be put in the text\n%\n% Here is an example of the general form of a table:\n% Fill in the caption in the braces of the \\caption{} command. Put the label\n% that you will use with \\ref{} command in the braces of the \\label{} command.\n% Insert the column specifiers (l, r, c, d, etc.) in the empty braces of the\n% \\begin{tabular}{} command.\n%\n% \\begin{table}\n% \\caption{}\n% \\label{}\n% \\begin{tabular}{}\n% \\end{tabular}\n% \\end{table}\n\n\\end{document}\n%\n% ****** End of file template.aps ******\n\n"
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"string": "\\begin{thebibliography}{10}\n\\expandafter\\ifx\\csname bibnamefont\\endcsname\\relax\n \\def\\bibnamefont#1{#1}\\fi\n\\expandafter\\ifx\\csname bibfnamefont\\endcsname\\relax\n \\def\\bibfnamefont#1{#1}\\fi\n\\expandafter\\ifx\\csname url\\endcsname\\relax\n \\def\\url#1{\\texttt{#1}}\\fi\n\\expandafter\\ifx\\csname urlprefix\\endcsname\\relax\\def\\urlprefix{URL }\\fi\n\\expandafter\\ifx\\csname bibinfo\\endcsname\\relax \\def\\bibinfo#1#2{#2}\\fi\n\\expandafter\\ifx\\csname eprint\\endcsname\\relax \\def\\eprint#1{#1}\\fi\n\n\\bibitem{dagotto96}\n\\bibinfo{author}{\\bibfnamefont{E.}~\\bibnamefont{Dagotto}} \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{T.~M.} \\bibnamefont{Rice}},\n \\bibinfo{journal}{Science} \\textbf{\\bibinfo{volume}{271}},\n \\bibinfo{pages}{618} (\\bibinfo{year}{1996}).\n\n\\bibitem{imai98}\n\\bibinfo{author}{\\bibfnamefont{T.}~\\bibnamefont{Imai}},\n \\bibinfo{author}{\\bibfnamefont{K.~R.} \\bibnamefont{Thurber}},\n \\bibinfo{author}{\\bibfnamefont{K.~M.} \\bibnamefont{Shen}},\n \\bibinfo{author}{\\bibfnamefont{A.~W.} \\bibnamefont{Hunt}}, \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{F.~C.} \\bibnamefont{Chou}},\n \\bibinfo{journal}{Phys. Rev. Lett.} \\textbf{\\bibinfo{volume}{81}},\n \\bibinfo{pages}{220} (\\bibinfo{year}{1998}).\n\n\\bibitem{ivanov99}\n\\bibinfo{author}{\\bibfnamefont{D.~A.} \\bibnamefont{Ivanov}} \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{P.~A.} \\bibnamefont{Lee}},\n \\bibinfo{journal}{Phys. Rev. B} \\textbf{\\bibinfo{volume}{59}},\n \\bibinfo{pages}{4803} (\\bibinfo{year}{1999}).\n\n\\bibitem{sandvik96}\n\\bibinfo{author}{\\bibfnamefont{A.~W.} \\bibnamefont{Sandvik}},\n \\bibinfo{author}{\\bibfnamefont{E.}~\\bibnamefont{Dagotto}}, \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{D.~J.} \\bibnamefont{Scalapino}},\n \\bibinfo{journal}{Phys. Rev. B} \\textbf{\\bibinfo{volume}{53}},\n \\bibinfo{pages}{R2934} (\\bibinfo{year}{1996}).\n\n\\bibitem{damle98}\n\\bibinfo{author}{\\bibfnamefont{K.}~\\bibnamefont{Damle}} \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{S.}~\\bibnamefont{Sachdev}},\n \\bibinfo{journal}{Phys. Rev. B} \\textbf{\\bibinfo{volume}{57}},\n \\bibinfo{pages}{8307} (\\bibinfo{year}{1998}).\n\n\\bibitem{sokol93}\n\\bibinfo{author}{\\bibfnamefont{A.}~\\bibnamefont{Sokol}},\n \\bibinfo{author}{\\bibfnamefont{E.}~\\bibnamefont{Gagliano}}, \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{S.}~\\bibnamefont{Bacci}},\n \\bibinfo{journal}{Phys. Rev. B} \\textbf{\\bibinfo{volume}{47}},\n \\bibinfo{pages}{14646} (\\bibinfo{year}{1993}).\n\n\\bibitem{brehmer99}\n\\bibinfo{author}{\\bibfnamefont{S.}~\\bibnamefont{Brehmer}},\n \\bibinfo{author}{\\bibfnamefont{H.-J.} \\bibnamefont{Mikeska}},\n \\bibinfo{author}{\\bibfnamefont{M.}~\\bibnamefont{M\\\"uller}},\n \\bibinfo{author}{\\bibfnamefont{N.}~\\bibnamefont{Nagaosa}}, \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{S.}~\\bibnamefont{Uchida}},\n \\bibinfo{journal}{Phys. Rev. B} \\textbf{\\bibinfo{volume}{60}},\n \\bibinfo{pages}{329} (\\bibinfo{year}{1999}).\n\n\\bibitem{mizonu98}\n\\bibinfo{author}{\\bibfnamefont{Y.}~\\bibnamefont{Mizonu}},\n \\bibinfo{author}{\\bibfnamefont{T.}~\\bibnamefont{Tohyama}}, \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{S.}~\\bibnamefont{Maekawa}},\n \\bibinfo{journal}{Phys. Rev. B} \\textbf{\\bibinfo{volume}{58}},\n \\bibinfo{pages}{R14713} (\\bibinfo{year}{1998}).\n\n\\bibitem{anderson53}\n\\bibinfo{author}{\\bibfnamefont{P.~W.} \\bibnamefont{Anderson}} \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{P.~R.} \\bibnamefont{Weiss}},\n \\bibinfo{journal}{Rev. Mod. Phys.} \\textbf{\\bibinfo{volume}{25}},\n \\bibinfo{pages}{269} (\\bibinfo{year}{1953}).\n\n\\bibitem{moriya56a}\n\\bibinfo{author}{\\bibfnamefont{T.}~\\bibnamefont{Moriya}},\n \\bibinfo{journal}{Prog. Theor. Phys.} \\textbf{\\bibinfo{volume}{16}},\n \\bibinfo{pages}{23} (\\bibinfo{year}{1956}).\n\n\\bibitem{moriya56b}\n\\bibinfo{author}{\\bibfnamefont{T.}~\\bibnamefont{Moriya}},\n \\bibinfo{journal}{Prog. Theor. Phys.} \\textbf{\\bibinfo{volume}{16}},\n \\bibinfo{pages}{641} (\\bibinfo{year}{1956}).\n\n\\bibitem{barnes93}\n\\bibinfo{author}{\\bibfnamefont{T.}~\\bibnamefont{Barnes}},\n \\bibinfo{author}{\\bibfnamefont{E.}~\\bibnamefont{Dagotto}},\n \\bibinfo{author}{\\bibfnamefont{J.}~\\bibnamefont{Riera}}, \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{E.~S.} \\bibnamefont{Swanson}},\n \\bibinfo{journal}{Phys. Rev. B} \\textbf{\\bibinfo{volume}{47}},\n \\bibinfo{pages}{3196} (\\bibinfo{year}{1993}).\n\n\\bibitem{gelfand93}\n\\bibinfo{author}{\\bibfnamefont{M.~P.} \\bibnamefont{Gelfand}} \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{R.~R.~P.} \\bibnamefont{Singh}},\n \\bibinfo{journal}{Phys. Rev. B} \\textbf{\\bibinfo{volume}{47}},\n \\bibinfo{pages}{14413} (\\bibinfo{year}{1993}).\n\n\\bibitem{roldan86}\n\\bibinfo{author}{\\bibfnamefont{J.~M.~R.} \\bibnamefont{Roldan}},\n \\bibinfo{author}{\\bibfnamefont{B.~M.} \\bibnamefont{McCoy}}, \\bibnamefont{and}\n \\bibinfo{author}{\\bibfnamefont{J.~H.~H.} \\bibnamefont{Perk}},\n \\bibinfo{journal}{Physica} \\textbf{\\bibinfo{volume}{136A}},\n \\bibinfo{pages}{255} (\\bibinfo{year}{1986}).\n\n\\end{thebibliography}\n"
}
] |
cond-mat0002143
|
Extremal-point Densities of Interface Fluctuations
|
[
{
"author": "Z. Toroczkai"
},
{
"author": "$^{1,2}$ G. Korniss"
},
{
"author": "$^3$ S. Das Sarma$^{1}$"
},
{
"author": "and R.K.P. Zia$^2$"
}
] |
We introduce and investigate the stochastic dynamics of the density of local extrema (minima and maxima) of non-equilibrium surface fluctuations. We give a number of exact, analytic results for interface fluctuations described by linear Langevin equations, and for on-lattice, solid-on-solid surface growth models. We show that in spite of the non-universal character of the quantities studied, their behavior against the variation of the microscopic length scales can present generic features, characteristic to the macroscopic observables of the system. The quantities investigated here present us with tools that give an entirely un-orthodox approach to the dynamics of surface morphologies: a statistical analysis from the short wavelength end of the Fourier decomposition spectrum. In addition to surface growth applications, our results can be used to solve the asymptotic scalability problem of massively parallel algorithms for discrete event simulations, which are extensively used in Monte-Carlo type simulations on parallel architectures.
|
[
{
"name": "xpds.tex",
"string": "\\tolerance=10000 \n\\documentstyle[amstex,pre,aps,multicol,epsf]{revtex}\n\\setlength{\\parskip}{0.0cm}\n\\def\\R{{\\mathbb R}} \n\\def\\N{{\\mathbb N}} \n\\def\\Z{{\\mathbb Z}}\n\\newcommand{\\bb}[1]{\\begin{equation} #1 \\end{equation}}\n\\newcommand{\\e}{\\mbox{ }}\n\\newcommand{\\ba}[1]{\\begin{eqnarray} #1 \\end{eqnarray}}\n\n\\begin{document}\n\n\\title{Extremal-point Densities of Interface Fluctuations}\n\\author{Z. Toroczkai,$^{1,2}$ G. Korniss,$^3$ S. \nDas Sarma$^{1}$, and\nR.K.P. Zia$^2$}\n\\address{$^1$Department of Physics, University of \nMaryland, College Park, \nMD 20742-4111 \\\\\n$^2$Department of Physics, Virginia Polytechnic \nInstitute and State \nUniversity, Blacksburg, VA 24061-0435 \\\\\n$^3$Supercomputer Computations Research Institute, \nFlorida State \nUniversity, Tallahassee, Florida 32306-4130 \\\\}\n\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nWe introduce and investigate the stochastic dynamics of \nthe density of local extrema (minima and maxima) of non-equilibrium\nsurface fluctuations. We give a number of exact, analytic\nresults for interface fluctuations described by linear \nLangevin equations, and for on-lattice,\nsolid-on-solid surface growth models. We show that in spite\nof the non-universal character of the quantities studied, their\nbehavior against the variation of the microscopic length scales \ncan present generic features, characteristic to the macroscopic \nobservables of the system. The quantities investigated here present \nus with tools that give an entirely un-orthodox approach to the \ndynamics of surface morphologies: a statistical analysis from \nthe short wavelength end of the Fourier decomposition spectrum. \nIn addition to surface growth applications, our results can be used \nto solve the asymptotic scalability problem of massively parallel \nalgorithms for discrete event simulations, which are extensively \nused in Monte-Carlo type simulations on parallel architectures. \n\\end{abstract}\n\n\n\n\\section{Introduction and Motivation}\n\nThe aim of statistical mechanics is to relate the macroscopic \nobservables to the microscopic properties of the system. Before \nattempting any such derivation one always has to specify the \nspectrum of length-scales the analysis will comprise: while \n`macroscopic' is usually defined in a unique way by the \nevery-day-life length scale, the `microscopic' is never so \nobvious, and the choice of the best lower-end scale is highly \nnon-universal, it is system dependent, usually left to our \nphysical `intuition', or it is set by the limitations of the \nexperimental instrumentation. It is obvious that in order to \nderive the laws of the gaseous matter we do not need to employ\nthe physics of elementary particles,\nit is enough to start from an effective microscopic \nmodel (or Hamiltonian) on\nthe level of molecular interactions. Then starting from the equations of\nmotion on the microscopic level and using a statistic and probabilistic\napproach, the macroscale physics is derived. In this `long wavelength' \napproach most of the microscopic, or short wavelength \ninformation is usually redundant and it is scaled away. \n\nSometimes however, microscopic quantities are important and \ndirectly contribute to macroscopic observables, e.g., \nthe nearest-neighbor correlations in driven \nsystems determine the current, \nin model B the mobility, in kinetic Ising model the \ndomain-wall velocity, in parallel computation the utilization\n(efficiency) of conservative parallel algorithms, etc. Once a lower length\nscale is set on which we can define an effective microscopic dynamics, it\nbecomes meaningful to ask questions about local properties {\\em at this\nlength scale}, e.g. nearest neighbor correlations, contour\ndistributions, extremal-point densities, etc. These quantities are obviously\nnot universal, however their {\\em behavior} against the variation of the\nlength scales can present qualitative and universal features.\nHere we study the dynamics of macroscopically rough surfaces via \ninvestigating an intriguing miscroscopic quantity: the density of \nextrema (minima) and its finite size effects. We derive a \nnumber of analytical results about these quantities for a \nlarge class of non-equilibrium surface fluctuations described by \nlinear Langevin equations, \nand solid-on-solid (SOS) lattice-growth models. Besides their\nobvious relevance to surface physics our technique can be used\nto show \\cite{KTNR} the asymptotic scalability of conservative\nmassively parallel algorithms for discrete-event simulation, i.e., the fact \nthat the {\\em efficiency} of such computational schemes \ndoes not vanish with \nincreasing the number of processing elements, but it has a \nlower non-trivial \nbound. The solution of this problem is not only of practical \nimportance from \nthe point of view of parallel computing, but it has \nimportant consequences for\nour understanding of systems with {\\em asynchronous} parallel dynamics, \nin general. \nThere are numerous dynamical systems both man-made, and \nfound in the nature,\nthat contain a ``substantial amount'' of parallelism. For example, \n\n1) in wireless cellular communications the call arrivals and departures are \nhappening in continuos time (Poisson arrivals), and the discrete events (call\narrivals) are {\\em not} synchronized by a global clock. Nevertheless, calls \ninitiated in cells substantially far from each other can be processed \nsimultaneously by the parallel simulator {\\em without} \nchanging the poissonian nature of the underlying process. \nThe problem of designing efficient dynamic channel allocation schemes \nfor wireless networks is a very difficult one and \ncurrently it is done by modelling the network as a system of interacting\ncontinuous time stochastic automata on parallel architectures \n \\cite{GLNW}. \n \n2) in magnetic systems the discrete events are the spin flip attempts \n(e.g., Glauber dynamics for Ising systems). While traditional single spin \nflip dynamics may seem inherently serial, systems with short range interactions \ncan be simulated in parallel: spins far from each other with {\\em different} \nlocal simulated times can be updated simultaneously. \nFast and efficient parallel Monte-Carlo \nalgorithms are extremely welcome when studying metastable decay and hysteresis \nof kinetic Ising ferromagnets below their critical temperature, see \\cite{KNR} \nand references therein. \n \n3) financial markets, and especially the stock market \nis an extremely dynamic, high connectivity network of relations, thousands of \ntrades are being made asynchronously every minute. \n \n4) the brain. The human \nbrain, in spite of its low weight of approx. 1kg, and volume of \n1400 $cm^3$, it \ncontains about 100 billion neurons, each neuron being connected \nthrough synapses to approximately 10,000 other neurons. \nThe total number of synapses in a human brain is about 1000 trillion\n($10^{15}$). The neurons of a single human brain placed\nend-to-end would make a ``string'' of an enourmous lenght: \n250,000 miles \\cite{http}. Assuming that each neuron of a single human cortex\ncan be in two states only (resting or acting), the total number of\ndifferent brain configurations would be ${2^{10}}^{11}$. \nAccording\nto Carl Sagan, this number is greater than the total number of protons \nand electrons of the known universe, \\cite{http}.\nThe brain does an incredible amount of parallel computation: it\nsimultaneously manages all of \nour body functions, we can talk and walk at the same time, etc. \n \n5) evolution of networks such as the internet, has parallel dynamics: the \nlocal network connectivity changes concurrently as many sites are attached \nor removed in different locations. As a matter of fact the physics of such \ndynamic networks is a currently heavily investigated and a rapidly emerging \nfield \\cite{alb}. \n \nIn order to present the basic ideas and notions in the simplest way, in the \nfollowing we will restrict ourselves to one dimensional interfaces that have no \noverhangs. The restriction on the overhangs may actually be lifted with \na proper parametrization of the surface, a problem to which we will return \nbriefly in the concluding section. The first visual impression when we look \nat a rough surface $h(x,t)$ is the extent of the fluctuations perpendicular to \nthe substrate, in other words, the {\\em width} of the fluctuations. The width \n(or the rms of the height $h$ of the fluctuations) is probably the most \nextensively studied quantity in interface physics, due to the fact that its \ndefinition is simply quantifiable and therefore measurable: \n\\begin{equation} \nw(L,t) = \\sqrt{\\;\\overline{\\left[h(x,t)\\right]^2}- \n\\left[\\overline{h(x,t)}\\right]^2 }\\;, \\label{width} \n\\end{equation} \nwhere the overline denotes the average over the substrate. \nIt is well-known that this quantity characterizes the long wavelength \nbehavior of the fluctuations, the high frequency components being averaged out \nin (\\ref{width}). The short wavelength end of the spectrum has been ignored \nin the literature mainly because of its non-universal character, and also \nbecause it seemed to lack such a simple quantifiable definition as the width \n$w$. \n \nIn the following we will present a quantity that is almost as simple and \nintuitive as the width $w$ but it characterizes the high frequency \ncomponents of the fluctuations and it is simply quantifiable. \nFor illustrative purposes let us consider the classic Weierstrass function \ndefined as the $M \\to \\infty$ limit of the smooth functions $W_M(a,b;x)$: \n\\bb{ \nW(a,b;x) = \\lim_{M \\to \\infty} W_M(a,b;x)=\\lim_{M \\to \\infty} \n\\sum_{m=0}^{M} a^{-m} \\cos\\left( b^m x\\right) \n\\;,\\;\\;\\;a,b > 1 \\label{weier} \n}\\e \nFigure 1a shows the graph of $W_M$ at $a=2$ and $b=3$ (arbitrary values) \nfor $M=0,1,2,3,4$ in the interval $x\\in [0,4\\pi]$. As one can see, by\nincreasing $M$ we are\nadding more and more detail to the graph of the function on finer and finer \nlength-scales. Thus $M$ plays the role of a regulator for the microscopic \ncut-off length which is $b^{-M}$, and for $M = \\infty$ and $b > a$, the\nfunction becomes nowhere differentiable as it was shown by Hardy \n\\cite{Hardy}. \n \nComparing the graphs of $W_M$ for lower $M$ values with those for \nhigher $M$-s we observe that the width effectively does not\nchange, however \nthe curves look qualitatively very different. This is obvious from\n(\\ref{weier}): adding an extra term will not change the long wavelength\nmodes, but adds a higher frequency component to the \nFourier spectrum of the graph. \nWe need to operationally define a quantifiable parameter which makes a \ndistinction between a much `fuzzier' graph, such as $M=4$ and \na smoother one, such as $M=1$.\nThe natural choice based on Figure 1a is the \n{\\em number of local minima} (or \nextrema) in the graph of function. In Figure 1b we present the number of \nlocal minima $u_M$ vs. \n$M$ for two different values of $b$, $b=2.8$ and $b=1.8$, while \nkeeping $a$ at the same value of $a=2$. \nFor all $b$ values (not only for these two) the leading behaviour is \nexponential: $u_M \\sim \\lambda^M$. The inset in Figure 1b shows the \ndependence of the rate $\\lambda$ as a function of $b$ for fixed $a$. We \nobserve that for $b>a$, $\\lambda=b$, but below \n$b=a$ the dependence crosses \nover to another, seemingly \nlinear function. For $b>a$ the amplitude of the \nextra added term in $W_{M+1}$ is large enough to \nprevent the cancellation of \nthe newly appearing minima by the drop in the \nlocal slope of $W_M$. At $b \n\\leq a$ the number of cancelled extrema starts to \nincrease drastically with an \nexponential trend, leading to the crossover seen in \nthe inset of Figure 2b. \nIt has been shown that the fractal dimension of the Weierstrass\nfunction for $b>a$ is given by $D_0 = \\ln{b} / \\ln{a}$ \\cite{Hunt},\n\\cite{Hughes}. For $b \\leq a$ the Weierstrass curve becomes\nnon-fractal with a dimension of $D_0=1$. By varying $b$ with respect\nto $a$, we are crossing a fractal-smooth transition at $b=a$.\nThe very intriguing observation we just come across is that\neven though we are in the smooth regime ($b<a$) the density of minima\nis still a {\\em diverging} quantity (the $b=1.8$ curve in Figure 1b).\nIt is thus possible to have an infinite number of `wrinkles' in the\nWeierstrass function without having a diverging length, without having a\nfractal in the classical sense. The transition from fractal to smooth, as\n$b$ is lowered appears as a non-analyticity in the divergence rate\nof the curve's wrinkledness.\nA rigorous analytic treatment of this problem seems to \nbe highly non-trivial and \nwe propose it as an open question to the interested reader. \n\n\\begin{figure}\n\\begin{minipage}{3.2 in}\\epsfxsize=3.1 in \n\\epsfbox{xpd1a.ps} \n\\end{minipage} \n\\hspace*{-0.25cm} \n\\begin{minipage}{3.5 in}\\epsfxsize=3.4 \nin \\epsfbox{xpd1b.ps} \n\\end{minipage} \n\\vspace*{0.5cm} \n\\caption{a) the function $W_M(a,b;x)$ at $a=2$, \n$b=3$ and $M=0,1,2,3,4$. b) \nThe scaling $u_M \\sim \\lambda^M$ of the number of local minima for \n$b=2.8$ (plusses) and $b=1.8$ (crosses). The inset shows $\\lambda$ \nvs. $b$ while keeping $a$ constant, $a=2$.} \n\\end{figure} \n\n The simple example shown above suggests that there \nis novel and non-trivial \nphysics lying behind the analysis of extremal \npoint densities, and it \ngives extra information on the morphology of interfaces. \nGiven an interface $h(x,t)$, we \npropose a quantitative form that characterizes the density of minima via a \n`partition-function' like expression, which is hardly more complex than\nEq. (\\ref{width}) and gives an alternate description of the surface\nmorphology: \n\\bb{ \nu_q(L,[h]) = \\frac{1}{L} \\sum_{i} \\left[ K(x_i)\\right]^q\\;,\\;\\;\\;q>0\\;,\\;\\;\\; \n\\mbox{$x_i$ are non-degenerate minima of $h$} \\label{minima} \n}\\e \nwith $K(x_i)$ denoting the {\\em curvature} of $h$ at the local \n(non-degenerate) \nminimum point $x_i$. The variable $q$ can be conceived as `inverse \ntemperature'. Obviously, for $q=0$ we obtain the number of local minima \nper unit interface length. \nThe rigorous mathematical description and definitions lying at the basis of \n(\\ref{minima}) is being presented in Section IV. \nThe quantity in Eq. (\\ref{minima}) \nis reminiscent to the partition function used in the definition of the \nthermodynamical formalism of one dimensional chaotic maps \\cite{Ruelle} \nand \nalso to the definition of the dynamical or R\\'enyi \nentropies of these chaotic \nmaps. In that case, however the curvatures at the minima are replaced by \ncylinder intervals and/or the visiting probabilities of these cylinders. \n \n We present a detailed analysis for the above quantity in case of a \nlarge class of linear Langevin equations of type $\\partial h / \\partial t = \n-\\nu (-\\nabla^2 h)^{z/2}+\\eta(x,t)$, where $\\eta$ is a Gaussian noise term, \nand $z$ a positive real number. These Langevin equations are found to \ndescribe faithfully the fluctuations of monoatomic steps on various substrates, \nsee for a review Ref. \\cite{EDW}. One of the interesting conclusions \nwe came to by studying the extremal-point densities on such equations \nis that depending on the value of $z$ the typical surface morphology \ncan be fractal, or locally smooth, and the two\nregimes are separated by a critical $z$ value, $z_c$. In the fractal case, the\ninterface will have infinitely many minima and cusps just as in the case of\nthe nondifferentiable Weierstrass function (\\ref{weier}), and the extremal\npoint densities become infinite, or if the problem is discretized onto a\nlattice with spacing $a$, a power-law diverging behavior is observed as $a\n\\to 0^+$ for these densities. This sudden change of the `intrinsic roughness' \nmay be conceived as a phase transition even in an experimental situation. \nChanging a parameter, such as the temperature, the {\\em law} describing \nthe fluctuations can change since the mechanism responsible for the \nfluctuations can change character as the temperature varies. \nFor example, it has been recently shown using Scanning Tunneling \nMicroscope (STM) measurements \\cite{Giesen},\nthat the fluctuations of single atom layer steps on $Cu$ (111) below \n$T =300\\mbox{ }^oC$ correspond to the perifery diffusion mechanism \n($z=4$), but above this temperature (such as $T = 500\\mbox{ }^oC$ \nin their measurements) the mechanism is attachement-detachment \nwhere $z=2$, see also Ref. \\cite{PT,Ted}. \n \n \nThe paper is organized as follows: \nIn Sections II and III we define and investigate on several well known\non-lattice models the minimum point density and derive exact results in the\nsteady-state ($t\\to \\infty$) including finite size effects. As a practical\napplication of these on-lattice results, we briefly present in Subsection\nIII.B a lattice surface-growth model which exactly describes the evolution of\nthe simulated time-horizon for conservative massively parallel schemes in\nparallel computation, and solve a long-standing asymptotic scalability\nquestion for these update schemes. In Section IV we lay down a more\nrigorous mathematical treatment for extremal point densities, and stochastic \nextremal point densities on the continuum, with a detailed derivation for \na large class of linear Langevin equations (which are in fact the continuum \ncounterparts of the discrete ones from Section II). The more rigorous treatment \nallows for an exact analytical evaluation not only in the steady-state, but \nfor all times! We identify novel characteristic dimensions that separate \nregimes with divergent extremal point densities from convergent ones and \nwhich give a novel understanding of the short wavelength physics behind \nthese kinetic roughening processes. \n \n \n\\section{Linear surface growth models on the lattice} \n \nIn the present Section we focus on discrete, one \ndimensional models from the \nlinear theory of kinetic roughening \\cite{Barabasi,Krug}. Let us consider \na one dimensional substrate consisting of $L$ lattice sites, with \nperiodic boundary conditions. For simplicity the lattice constant is \ntaken as unity, which clearly, represents \nthe lower cut-off length for the effective equation of motion. \nFor the moment let us study the discretized counterpart of the \ngeneral Langevin equation that describes the linear theory \nof Molecular Beam Epitaxy (MBE) \\cite{MBE,DT}: \n\\begin{equation} \n\\partial_{t} h_{i}(t) = \\nu\\nabla^2 h_{i}(t) - \\kappa\\nabla^4 h_{i}(t) \n+ \\eta_{i}(t) \\;, \\label{h_evolution} \n\\end{equation} \nwhere $\\eta_{i}(t)$ is Gaussian white noise with \n\\begin{equation} \n\\langle\\eta_{i}(t)\\eta_{i}(t')\\rangle = 2D\\;\\delta_{i,j}\\;\\delta(t-t') \\;, \n\\end{equation} \nand $\\nabla^2$ is the discrete Laplacian operator, i.e., \n$\\nabla^2 f_j = f_{j+1}+f_{j-1} -2f_{j}$, \napplied to an arbitrary lattice function $f_j$. \nThis equation arrises in MBE with both surface diffusion \nmechanism (the 4th order, or curvature term) and desorption mechanism \n(the 2nd order, or diffusive term) present and it has been studied \nextensively by several authors \\cite{MBE,Majaniemi}. Stability requires $\\nu \n\\geq 0$ and $\\kappa \\geq 0$ (as a matter of fact, on the lattice is \nenough to have $\\nu > 0$ and $\\kappa \\geq -\\nu /2$). Starting from a\ncompletely flat initial condition, the interface roughens until the\ncorrelation length $\\xi$ reaches the size of the system $\\xi \\simeq L$, when\nthe roughening saturates over into a steady-state regime. The process of\nkinetic roughening is controlled by the intrinsic length scale\n\\cite{Majaniemi}, $\\sqrt{\\kappa/\\nu}$. Below this lengthscale the roughening\nis dominated by the surface diffusion or Mullins \\cite{Mullins} term (the\n4th order operator) but above it is characterized by the evaporation piece\n(the diffusion) or Edwards-Wilkinson \\cite{EW} term. Since eq.\n(\\ref{h_evolution}) is translationaly invariant and linear in $h$ it can be \nsolved via the discrete Fourier-transform: \n\\begin{equation} \\tilde{h}_k=\\sum_{j=0}^{L-1} \ne^{-ikj}h_i \\;,\\;\\;k=\\frac{2\\pi n}{L}\\;,\\;\\; n=0,1,2,\\ldots,L-1 \\;. \n\\end{equation} \nThen eq. (\\ref{h_evolution}) translates into \n\\begin{equation} \n\\partial_{t} \\tilde{h}_{k}(t) = -\\left[2\\nu (1-\\cos(k)) \n + 4\\kappa (1-\\cos(k))^{2}\\right]\\tilde{h}_{k}(t)+ \\tilde{\\eta}_{k}(t) \\;, \n\\label{h_k_evolution} \n\\end{equation} \nwith \n\\bb{ \n\\langle\\tilde{\\eta}_{k}(t)\\tilde{\\eta}_{k'}(t')\\rangle = \n2D L \\;\\delta_{(k+k')\\; \\mbox{mod}\\; 2\\pi, 0}\\;\\delta(t-t') \n}\\e \nFollowing the definition of the equal-time structure factor for $S(k,t)$, \nnamely \n\\bb{ \nS^{h}(k,t) L \\delta_{(k+k')\\; \\mbox{mod}\\; 2\\pi, 0} \\equiv \n\\langle\\tilde{h}_{k}(t)\\tilde{h}_{k'}(t)\\rangle \\;, \n}\\e \none obtains for an initially flat interface: \n\\begin{equation} \nS^{h}(k,t) = S^{h}(k) \n\\left(1 - e^{-(4\\nu (1-\\cos(k)) + 8\\kappa (1-\\cos(k))^{2})t} \n\\right)\\;. \n\\end{equation} \nIn the above equation \n\\begin{equation} \nS^{h}(k)\\equiv \\lim_{t\\rightarrow\\infty} S^{h}(k,t) = \n \\frac{D}{2\\nu (1-\\cos(k)) + 4\\kappa (1-\\cos(k))^{2}} \n\\label{S_h} \n\\end{equation} \nis the steady-state structure factor. \n \nFor $\\nu \\neq 0$, and in the asymptotic scaling limit where \n$L\\gg \\sqrt{\\kappa/\\nu}$, the model belongs to the EW universality \nclass and the roughening exponent is $\\alpha=1/2$ (it is defined through the \nscaling $L^{2\\alpha}$ \nof the interface width $\\langle w_{L}^{2}(t)\\rangle = \n(1/L)\\langle\\sum_{i=1}^{L}[h_{i}(t)-\\overline{h}]^2 \\rangle$ in the steady \nstate). The presence of the curvature term does not change the universal \nscaling properties for the surface width, and one finds the same exponents as \nfor the pure EW ($\\kappa=0$) case in eq. (\\ref{h_evolution}). \nFor $\\nu = 0$ the surface is purely curvature driven ($z=4$) and the model \nbelongs to a different universality class where the steady-state width scales \nwith a roughness exponent of $\\alpha=3/2$. \n \nIn the following we will be mostly interested in some local steady-state \nproperties of the surface ${h_i}$. In particular, we want to find the density \nof local minima for the \nsurface described by (\\ref{h_evolution}). The operator which measures this \nquantity is \\begin{equation} \nu = \\frac{1}{L}\\sum_{i=1}^{L} \\Theta(h_{i-1}-h_{i}) \\Theta(h_{i+1}-h_{i}) \\;. \n\\end{equation} \nThis expression motivates the introduction of the local slopes, \n$\\phi_i=h_{i+1} -h_{i}$. In this representation the operator for the density of \nlocal minima (for the original surface) is \n\\begin{equation} \nu = \\frac{1}{L}\\sum_{i=1}^{L} \\Theta(-\\phi_{i-1}) \\Theta(\\phi_{i}) \\;, \n\\end{equation} \nand its steady-state average is \n$\\langle u\\rangle=\\langle\\Theta(-\\phi_{i-1}) \\Theta(\\phi_{i})\\rangle \n= \\langle\\Theta(-\\phi_{1}) \\Theta(\\phi_{2})\\rangle$, \ndue to translational invariance. The average density of local minima is the \nsame as the probability that at a randomly chosen site of the lattice \nthe surface exhibits a local minimum. It is governed by the \nnearest-neighbor two-slope distribution, which is \nalso Gaussian and fully determined by $\\langle\\phi_{1}^2\\rangle= \n\\langle\\phi_{2}^2\\rangle$ and $\\langle\\phi_{1}\\phi_{2}\\rangle$: \n\\begin{equation} \nP^{nn}(\\phi_1,\\phi_2) \\propto \ne^{-\\frac{1}{2}\\phi_j A^{\\rm nn}_{jk}\\phi_k} \\;,\\;\\;\\; j,k=1,2 \\;, \n\\end{equation} \nwhere \n\\begin{equation} \nA^{\\rm nn} = \\left(\\begin{array}{cc} \n\\langle\\phi_1^2\\rangle & \\langle\\phi_1\\phi_2\\rangle \\\\ \n\\langle\\phi_1\\phi_2\\rangle & \\langle\\phi_1^2\\rangle \n\\end{array} \\right)^{-1} \n\\end{equation} \nAs we derive in Appendix A, the density of local minima only depends \non the ratio $\\langle\\phi_1\\phi_2\\rangle/\\langle\\phi_1^2\\rangle$: \n\\begin{equation} \n\\langle u\\rangle = \\frac{1}{2\\pi}\\arccos\\left( \n \\frac{\\langle\\phi_1\\phi_2\\rangle}{\\langle\\phi_1^2\\rangle}\\right) \\;. \n\\label{min_dens} \n\\end{equation} \nFinite-size effects of $\\langle u\\rangle$ are obviously carried through those\nof the correlations. First we find the steady-state structure factor \nfor the slopes. Since $\\tilde{\\phi}_{k}=(1-e^{-ik})\\tilde{h}_{k}$, we have \n$S^{\\phi}(k)=2(1-\\cos(k))S^{h}(k)$. Then from (\\ref{S_h}) one obtains: \n\\begin{equation} \nS^{\\phi}(k) = \\frac{D}{\\nu + 2\\kappa (1-\\cos(k))}\\;,\\;\\;\\mbox{for}\\;\\; \nk \\neq 0\\;,\\;\\;\\;\\mbox{and}\\;\\;\\;S^{\\phi}(k)=0\\;,\\;\\;\\mbox{for}\\;\\; \nk=0\\;. \\label{S_phi} \n\\end{equation} \nThe former automatically follows \nfrom the $\\sum_{i=1}^{L}\\phi_{i}=0$ relation. Then we obtain the \nslope-slope correlations \n\\begin{equation} \nC^{\\phi}_{L}(l) \\equiv \\langle\\phi_i\\phi_{i+l}\\rangle = \n\\frac{1}{L}\\sum_{n=1}^{L-1} e^{i \\frac{2\\pi n}{L} l} \n S^{\\phi}\\left(\\frac{2\\pi n}{L}\\right) \\;.\\label{C_L} \n\\end{equation} \nWith the help of Poisson summation formulas, in Appendix B we show a \nderivation for the exact spatial correlation function, which yields \n\\begin{equation} \nC^{\\phi}_{L}(l) = \\frac{D}{\\nu + 2\\kappa}\\left\\{ \n\\frac{b^{|l|}}{\\sqrt{1-a^2}} - \\frac{1}{1-a}\\frac{1}{L} + \n\\frac{b^{L}}{1-b^{L}}\\frac{b^{l}+b^{-l}}{\\sqrt{1-a^2}} \n\\right\\} \\;,\\;\\;\\;|l|\\leq L \\;, \\label{C_phi} \n\\end{equation} \nwith \n\\begin{equation} \na \\equiv \\frac{2\\kappa}{\\nu + 2\\kappa}\\;,\\;\\;\\;\\mbox{and}\\;\\;\\; \nb \\equiv \\frac{1-\\sqrt{1-a^2}}{a}\\;. \n\\end{equation} \nWe have $|a| \\leq 1$ and $b \\leq 1$. \nThe second term in the bracket in (\\ref{C_phi}) gives a {\\em uniform} power \nlaw correction, while the third one gives an exponential correction to the \ncorrelation function in the thermodynamic limit. \nFor $\\nu\\neq 0$ and $L \\rightarrow \\infty$ one obtains \n\\begin{equation} \nC^{\\phi}_{\\infty}(l) = \\frac{D}{\\nu + 2\\kappa} \\frac{b^{|l|}}{\\sqrt{1-a^2}} = \n\\frac{D}{\\nu + 2\\kappa} \\frac{e^{-|l|/\\xi^{\\phi}_{\\infty}}}{\\sqrt{1-a^2}} \n\\end{equation} \nwhere we define the correlation length of the slopes for an infinite system \nas: \n\\begin{equation} \n\\xi^{\\phi}_{\\infty} \\equiv -\\frac{1}{\\ln(b)} \\;. \n\\end{equation} \nIn the $\\nu \\rightarrow 0$ limit it becomes the \nintrinsic correlation length which diverges as $\\nu^{-1/2}$: \n$ \n\\xi^{\\phi}_{\\infty} \\stackrel{\\nu\\rightarrow 0}{\\simeq} \n\\sqrt{\\kappa / \\nu} \n$ \nand \n\\begin{equation} \nC^{\\phi}_{\\infty}(l) \\stackrel{\\nu\\rightarrow 0}{\\simeq} \n\\frac{D}{2\\kappa}\\left( \\sqrt{\\frac{\\kappa}{\\nu}} - |l| \\right) \\simeq \n\\frac{D}{2\\kappa}\\left( \\xi^{\\phi}_{\\infty} - |l| \\right) \\;. \n\\label{C_inf_div} \n\\end{equation} \nIn this limit the slopes (separated by any finite distance) become highly \ncorrelated, and one may start to anticipate that the density of local minima \nwill vanish for the original surface $\\{h_i\\}$. \nIn the following two subsections we investigate the density of local minima \nand its finite-size effects for the Edwards-Wilkinson and the Mullins cases. \n \n\\subsection{Density of local minima for Edwards-Wilkinson term dominated \nregime} \n \nTo study the finite size effects for the local minimum density, we neglect \nthe exponentially small correction in (\\ref{C_phi}), so in the \n{\\em asymptotic} limit, where \n$L\\gg\\xi^{\\phi}_{\\infty}$, $C^{\\phi}_{L}(l)$ decays exponentially with \n{\\em uniform} finite-size corrections: \n\\begin{equation} \nC^{\\phi}_{L}(l) \\simeq \\frac{D}{\\nu + 2\\kappa}\\left\\{ \n\\frac{b^{|l|}}{\\sqrt{1-a^2}} - \\frac{1}{1-a}\\frac{1}{L} \n\\right\\} \\; \n\\end{equation} \nThis holds for the special case $\\kappa=0$ as well, (in fact, there the \nexponential correction exatly vanishes) leaving \n\\begin{equation} \nC^{\\phi}_{L}(l) = \\frac{D}{\\nu}\\left(\\delta_{l,0}-\\frac{1}{L}\\right) \\;. \n\\end{equation} \nNow, emplying eq. (\\ref{min_dens}), we can obtain the density of minima as: \n\\begin{equation} \n\\langle u\\rangle_{L}= \n\\frac{1}{2\\pi}\\arccos\\left( \\frac{C^{\\phi}_{L}(1)}{C^{\\phi}_{L}(0)} \\right) \n\\simeq \\frac{1}{2\\pi}\\arccos(b) + \n\\frac{1}{2\\pi}\\sqrt{\\frac{1-b}{1+b}}\\sqrt{\\frac{1+a}{1-a}}\\frac{1}{L} \\;, \n\\label{EW_min_dens} \n\\end{equation} \nAgain, for the $\\kappa=0$ case one has a compact exact expression and the \ncorresponding large $L$ behavior: \n\\begin{equation} \n\\langle u\\rangle_{L} = \\frac{1}{2\\pi}\\arccos\\left( -\\frac{1}{L-1}\\right) \n\\simeq \\frac{1}{4} + \\frac{1}{2\\pi}\\frac{1}{L} \\;, \\label{ezaz} \n\\end{equation} \nwhich can also be obtained by taking the $\\kappa\\rightarrow0$ limit in \n(\\ref{EW_min_dens}). \nTo summarize, as long as $\\nu\\neq 0$, the model belongs to the EW universality \nclass, and in the steady state, the density of local minima behaves as \n\\begin{equation} \n\\langle u\\rangle_{L} \\simeq \\langle u\\rangle_{\\infty} + \\frac{\\rm const.}{L}\\;, \n\\label{EW2_min_dens} \n\\end{equation} \nwhere $\\langle u\\rangle_{\\infty}$ is the value of the density of local \nminima in the thermodynamic limit: \n\\begin{equation} \n\\langle u\\rangle_{\\infty} =\\frac{1}{2\\pi}\\arccos(b) \\;. \n\\end{equation} \nNote that this quantity can be small, but does not vanish if \n$\\nu$ is close but not equal to $0$. Further, the system exhibits the scaling \n(\\ref{EW2_min_dens}) for asymptotically large systems, where \n$L\\gg\\xi^{\\phi}_{\\infty}$. It is important to see in detail how \n$\\langle u\\rangle_{\\infty}$ behaves as $\\nu \\rightarrow 0$: \n\\begin{equation} \n\\langle u\\rangle_{\\infty} \\stackrel{\\nu\\rightarrow 0}{\\simeq} \n\\frac{1}{2\\pi}\\arccos\\left(1-\\sqrt{2}\\sqrt{1-a}\\right) \\simeq \n\\frac{1}{2\\pi}\\arccos\\left(1-\\sqrt{\\frac{\\nu}{\\kappa}}\\right) \\simeq \n\\frac{1}{2\\pi} \\left(2 \\sqrt{\\frac{\\nu}{\\kappa}}\\right)^{1/2} \\simeq \n\\frac{\\sqrt{2}}{2\\pi} \\frac{1}{\\sqrt{\\xi^{\\phi}_{\\infty}}} \\;. \n\\label{u_inf_lim} \n\\end{equation} \nThus, the density of local minima for an {\\em infinite} system vanishes as we \napproach the purely curvature driven ($\\nu \\rightarrow 0$) limit. \nSimply speaking, the local slopes become ``infinitely'' correlated, such that \n$C^{\\phi}_{\\infty}(l)$ diverges [according to eq. (\\ref{C_inf_div})], and the \nratio $C^{\\phi}_{\\infty}(l)/C^{\\phi}_{\\infty}(0)$ for any fixed $l$ tends to \n$1$. This is the physical picture behind the vanishing density of local minima. \n \n \n\\subsection{Density of local minima for the Mullins term dominated regime} \n \nHere we take the $\\nu \\rightarrow 0$ limit {\\em first} and then study the \nfinite size effects in the purely curvature driven model. The slope \ncorrelations are finite for finite $L$ as can be seen from eq. (\\ref{C_L}), \nsince the $n = 0$ term is not included in the sum! Thus, in the exact \nclosed formula (\\ref{C_phi}) a careful limiting procedure has to be \ntaken which indeed yields the internal cancellation of the apparently \ndivergent terms. Then one obtains the exact slope correlations \nfor the $\\nu=0$ case: \n\\begin{equation} \nC^{\\phi}_{L}(l) = \\frac{D}{2\\kappa} \\left\\{ \n\\frac{L}{6}\\left(1-\\frac{1}{L^2}\\right) -|l|\\left(1-\\frac{|l|}{L}\\right) \n\\right\\} \\; \n\\end{equation} \nand for the local minimum density: \n\\begin{equation} \n\\langle u\\rangle_{L} = \\frac{1}{2\\pi}\\arccos\\left(1-\\frac{6}{L+1}\\right) \n\\simeq \\frac{\\sqrt{3}}{\\pi}\\frac{1}{\\sqrt{L}} \\label{u4d}\n\\end{equation} \nIt vanishes in the thermodynamic limit, and hence, one observes that \nthe limits $\\nu \\rightarrow 0$ and $L \\rightarrow \\infty$ are \ninterchangable. For $\\nu=0$, $L$ is directly associated with the correlation \nlength and we can define $\\xi^{\\phi}_L \\equiv L/6$. Then the correlations and \nthe density of local minima takes the same scaling form as eqs. \n(\\ref{C_inf_div}) and (\\ref{u_inf_lim}): \n\\begin{equation} \nC^{\\phi}_{L}(l) \\simeq \n\\frac{D}{2\\kappa}\\left( \\xi^{\\phi}_{L} - |l| \\right) \\;, \n\\end{equation} \nand \n\\begin{equation} \n\\langle u\\rangle_{L} \n\\simeq \\frac{\\sqrt{2}}{2\\pi}\\frac{1}{\\sqrt{\\xi^{\\phi}_{L}}} \\;. \n\\end{equation} \n \n \n\\subsection{Scaling considerations for higher order equations} \n \nLet us now consider another equation but with a generalized \nrelaxational term that includes the Edwards Wilkinson \nand the noisy Mullins equation as particluar cases: \n\\begin{equation} \n\\partial_{t} h_{i}(t) = -\\nu \\left( -\\nabla^2\\right)^{z/2} h \n+ \\eta_{i}(t) \\;. \\label{kinetic} \n\\end{equation} \nwhere $z$ is a positive real number (not necessarily integer). \nOther $z$ values of experimental interest are $z=1$, relaxation \nthrough plastic flow, \\cite{Mullins,Krug}), and $z=3$ terrace-diffusion \nmechanism \\cite{PT,Ted,EDW}. For early times, such that \n$t\\ll L^z$, the interface width $\\langle w_{L}^{2}(t)\\rangle $ increases \nwith time as \n\\begin{equation} \n\\langle w_{L}^{2}(t)\\rangle \\sim t^{2\\beta}\\;, \n\\end{equation} \nwhere $\\beta=(z-1)/2z$ \\cite{Barabasi,Krug}. \nIn the $t \\rightarrow \\infty$ limit, where $t\\gg L^z$, the interface width \nsaturates for a finite system, but diverges with $L$ according to \n$\\langle w_{L}^{2}(\\infty)\\rangle \\sim L^{2\\alpha}$ \nwhere $\\alpha=(z-1)/2$ is the roughness exponent \\cite{Barabasi,Krug}. \n \nFor $z=4$ (curvature driven interface) we saw that the slope fluctuation \nbehaves as $C^{\\phi}_{L}(0)=\\langle\\phi_{i}^2\\rangle \\sim L$. For higher $z$ \nfor the slope-slope correlation function one can deduce \n\\begin{equation} \nC^{\\phi}_{L}(l) = \n\\frac{D}{L}\\sum_{n=1}^{L-1} \\frac{e^{i \\frac{2\\pi n}{L} l}} \n{\\nu \\left[2\\left(1-\\cos\\left(\\frac{2\\pi n}{L}\\right)\\right) \n\\right]^{\\frac{z-2}{2}}} \n\\;. \n\\end{equation} \nIt is divergent in the $L \\rightarrow \\infty$ limit, as a result of \ninfinitely small wave-vectors $\\sim 1/L$, and we can see that \n\\begin{equation} \nC^{\\phi}_{L}(0)\\sim L^{z-3} \\;. \n\\label{C_scaling} \n\\end{equation} \nIt is also useful to define the slope difference correlation function \n\\begin{equation} \nG^{\\phi}_{L}(l) \\equiv \\langle(\\phi_{i+l} - \\phi_{i})^2 \\rangle \n\\end{equation} \nfor which one can write \n\\begin{equation} \nG^{\\phi}_{L}(l) = \n\\frac{D}{L}\\sum_{n=1}^{L-1} \\frac{2\\left(1-\\cos\\left(\\frac{2\\pi n}{L}l\\right)\\right)} \n{\\nu \\left[2\\left(1-\\cos\\left(\\frac{2\\pi n}{L}\\right)\\right) \n\\right]^{\\frac{z-2}{2}}} \n\\;. \n\\end{equation} \nFor the small wave-vector behavior we can again deduce that for $z>5$ \n\\begin{equation} \nG^{\\phi}_{L}(l) \\sim L^{z-5} \\, l^2 \\;. \n\\label{anomalous} \n\\end{equation} \nOne may refer to this form as ``anomalous'' scaling \\cite{Krug} for the slope \ndifference correlation function in the following sense. \nFor $z<5$ the scaling form for $G^{\\phi}_{L}(l)$ follows that of \n$C^{\\phi}_{L}(0)$ [eq. (\\ref{C_scaling})], i.e., \n$G^{\\phi}_{L}(l) \\sim l^{z-3}$. For $z>5$ [eq. (\\ref{anomalous})] it obviously \nfeatures a different $l$ dependence and an additional power of $L$, and it \ndiverges in the $L \\rightarrow \\infty$ limit. \n \nHaving these scaling functions for large $L$, we can easily obtain the \nscaling behavior for the average density of local minima. Exploiting the \nidentity \n\\begin{equation} \nC^{\\phi}_{L}(l) = C^{\\phi}_{L}(0) - \\frac{1}{2}G^{\\phi}_{L}(l) \n\\end{equation} \nwe use the general form for the local minimum density: \n\\begin{equation} \n\\langle u\\rangle = \\frac{1}{2\\pi}\\arccos\\left( \n\\frac{C^{\\phi}_{L}(1)}{C^{\\phi}_{L}(0)} \\right) = \n\\frac{1}{2\\pi}\\arccos\\left( 1-\\frac{1}{2} \n\\frac{G^{\\phi}_{L}(1)}{C^{\\phi}_{L}(0)} \\right) \\simeq \n\\frac{1}{2\\pi}\\arccos\\left( 1-\\frac{\\rm const.}{L^2} \\right) \\sim \n\\frac{1}{L} \n\\end{equation} \nNote that this is the scaling behavior for {\\em all} $z>5$. \nIt simply shows the trivial lower bound for $\\langle u\\rangle$: since there is \nalways at least one minima (and one maxima) among the $L$ sites, it can \nnever be smaller than $1/L$. \n \n\\subsection{The average curvature at local minima} \n\nThe next natural question to ask is how the average curvature, $K$ at the \nminimum points scales with the system size for the general system \ndescribed by eq. (\\ref{kinetic}). This can be evaluated as the \nconditional average of the local curvature at the local minima: \n\\begin{equation} \n\\langle K\\rangle_{\\min} = \\langle (\\phi_{i}-\\phi_{i-1}) \\rangle_{\\min} = \n\\frac{\\langle (\\phi_{i}-\\phi_{i-1})\\Theta(-\\phi_{i-1}) \n\\Theta(\\phi_{i})\\rangle }{\\langle\\Theta(-\\phi_{i-1}) \n\\Theta(\\phi_{i})\\rangle} = \n\\frac{\\langle (\\phi_{2}-\\phi_{1})\\Theta(-\\phi_{1}) \\Theta(\\phi_{2})\\rangle \n} \n{\\langle u \\rangle} \n\\label{curvature} \n\\end{equation} \nwhere translational invariance is exploited again. \nThe numerator in (\\ref{curvature}) can be obtained after performing the \nsame basis transformation (Appendix A) that was essential to find \n$\\langle u \\rangle$. Then after elementary integrations we find \n\\begin{equation} \n\\langle K\\rangle_{\\min} = \\frac{1}{\\langle u \\rangle} \n\\frac{1}{\\sqrt{2\\pi}} \n\\frac{C^{\\phi}_{L}(0)-C^{\\phi}_{L}(1)}{\\sqrt{C^{\\phi}_{L}(0)}} = \n\\frac{\\sqrt{2\\pi}}{\\sqrt{C^{\\phi}_{L}(0)}}\\; \n\\frac{C^{\\phi}_{L}(0)-C^{\\phi}_{L}(1)}{ \n\\arccos(C^{\\phi}_{L}(1)/C^{\\phi}_{L}(0))} \n\\end{equation} \nUsing the explicit results for the slope correlation function for $z=2$ \nand \n$z=4$, and the scaling forms for it for higher $z$ given in the previous \nsubsections, one can easily deduce the following. \nFor $z<5$ the average curvature at the local minimum points on a lattice \ntends \nto a {\\em constant} in the thermodynamic limit. For $z=2$ \n\\begin{equation} \n\\langle K\\rangle_{\\min} \\simeq \n\\frac{2\\sqrt{2}}{\\sqrt{\\pi}}\\sqrt{ \\frac{D}{\\nu}} \n+ {\\cal O}\\left(\\frac{1}{L}\\right)\\;, \n\\end{equation} \nand for $z=4$ \n\\begin{equation} \n\\langle K\\rangle_{\\min} \\simeq \\sqrt{\\pi}\\sqrt{ \\frac{D}{2\\nu}} \n+ {\\cal O}\\left(\\frac{1}{L}\\right)\\;. \\label{ke4}\n\\end{equation} \nThe behavior of this quantity drastically changes for $z>5$, where it \n{\\em diverges} with the system size as: \n\\begin{equation} \n\\langle K\\rangle_{\\min} \\sim L^{\\frac{z-5}{2}} \n\\end{equation}. \n \n\\section{Other lattice models and an application to parallel computing} \n\n\\subsection{The single-step model} \n \nIn the single-step model the height differences (i.e., the local slopes) are \nrestricted to $\\pm 1$, and the evolution consists of particles of height $2$ \nbeing deposited at the local minima. While the full dynamic behavior of the \nmodel belongs to the KPZ universality class, in one dimension the steady state \nis governed by the EW Hamiltonian \\cite{single_step}. Thus, the roughness \nexponent is $\\alpha=1/2$, and we expect the finite-size effects for \n$\\langle u\\rangle$ to follow eq. (\\ref{EW2_min_dens}). \nThe advantage of this model is that \nit can be mapped onto a hard-core lattice gas for which the {\\em steady-state} \nprobability distribution of the configurations is known exactly \n\\cite{single_step,Spitzer}. This enables us to \nfind arbitrary moments of the local minimum density operator. \nSince $\\phi_i=\\pm 1$, it can be simly written as \n\\begin{equation} \nu = \\frac{1}{L} \\sum_{i=1}^{L}\\frac{1-\\phi_{i-1}}{2}\\; \\frac{1+\\phi_i}{2} = \n \\frac{1}{L} \\sum_{i=1}^{L} (1-n_{i-1}) n_i\\;, \n\\label{ss_min_dens_op} \n\\end{equation} \nwhere $n_i=(1+\\phi_i)/2$, corresponds to the hard core lattice gas occupation \nnumber. The constraint $\\sum_{i=1}^{L}\\phi_i=0$ translates to \n$\\sum_{i=1}^{L}n_i=L/2$. \nNote that here $\\langle u\\rangle=\\langle(1-n_{i-1})n_i\\rangle$ is proportional \nto the average current. Knowing the exact steady-state probability \ndistribution \\cite{single_step,Spitzer}, one can easily find that \n\\begin{equation} \n\\langle n_i \\rangle= \\frac{1}{2} \n\\;,\\;\\;\\; \n\\langle n_i n_j \\rangle_{i\\neq j} \n = \\frac{1}{4}\\;\\frac{L-2}{L-1} \n\\end{equation} \nThus the exact finite-size effects for the local minimum density: \n\\begin{equation} \n\\langle u\\rangle_L = \\frac{1}{4}\\;\\frac{L}{L-1} \n= \\frac{1}{4} + \\frac{1}{4L} + {\\cal O}(L^{-2})\\;, \n\\end{equation} \nin qualitative agreement with (\\ref{ezaz}). \n \n \n\\subsection{The Massively Parallel Exponential Update model} \n \n \nOne of the most challenging areas in parallel computing \\cite{parallel} \nis the efficient implementation of dynamic Monte-Carlo algorithms \nfor discrete-event simulations on massively parallel architectures. \nAs already mentioned in the Introduction, it has numerous practical \napplications ranging from magnetic systems (the discrete events \nare spin flips) to queueing networks ( the discrete events are job arrivals). \n A parallel architecture by definition contains (usually) a large number of \nprocessors, or processing elements (PE-s). During the simulation \neach processor has to tackle only a fraction of the full computing \ntask (e.g., a specific block of spins), and the algorithm has to ensure \nthrough synchronization that the underlying dynamics is not altered. \nIn a wide range of models the discrete events are Poisson arrivals. Since this \nstochastic process is reproducible (the sum of two Poisson processes is a \nPoisson process again with a new arrival frequency), the Poisson streams \ncan be simulated simultaneously on each subsystem carried by each PE. \nAs a consequence, the simulated time is {\\em local} and {\\em random}, \nincremented by exponentially distributed random variables on each PE. \nHowever, the algorithm has to ensure that causality accross the boundaries of \nthe neighboring blocks is not violated. This \nrequires a comparison between the neighboring simulated times, and waiting, if \nnecessary (conservative approach). In the simplest scenario (one site/PE), \nthis means that only those PEs will be allowed to attempt the update the state \nof the underlying site and increment their local time, where the local \nsimulated time is a {\\em local minimum} regarding the full simulated time \nhorizon of the system, $\\{\\tau_i\\}$, $i=1,..,L$ (for simplicity we consider a \nchain-like connectivity among the PE-s but connectivities of higher degree can \nbe treated as well). One can in fact think of the time horizon as a \nfluctuating surface with height variable $\\tau_i$. \nOther examples where the update attempts are independent Poisson arrivals \ninclude arriving calls \nin the wireless cellular network of a large metropolitan area \\cite{GLNW}, or \nthe spin flip attempts in an Ising ferromagnet. This extremely robust parallel \nscheme was introduced by Lubachevsky, \\cite{Luba} and it is applicable to a \nwide range of stochastic cellular automata with local dynamics where the \ndiscrete events are Poisson arrivals. The local random time increments is, \nin the language of the associated surface, equivalent to depositing random \namounts of `material' (with an exponential distribution) at the local minima \nof the surface, see Figure 2. This in fact defines a simple surface growth \nmodel which we shall refer to as `the massively parallel exponential update \nmodel' (MPEU). \nThe main concern about a parallel implementation is its efficiency. Since \nin the next time step only a fraction of PE-s will get updated, i.e., those \nthat are in the local minima of the time horizon, while the rest are in idle, \nthe efficiency is nothing but the average number of non-idling PE-s divided by \nthe total number of PE-s ($L$), i.e., {\\em the average number of minima per \nunit length}, or the minimum-point density, $u$. \nThe fundamental question of the so called {\\em scalability} arises: will the \nefficiency of the algorithm go to zero as the number of PE-s is increased \n($L\\to \\infty$) indefinitely, or not? If the efficiency has a non-zero lower \nbound for $L\\to \\infty$ the algorithm is called {\\em scalable}, and \ncertainly this is the preferred type of scheme. Can one design in principle \nsuch efficient algorithms? \n\n\\begin{figure}\n\\hspace*{3.5cm}\\epsfxsize=4 in \n\\epsfbox{mpeu.ps} \n\\vspace*{0.5cm} \n\\caption{The MPEU model. The arrows show the local minima where \nrandom amounts of material will be deposited in the next time step.} \n\\end{figure} \n\nAs mentioned in the Introduction, we know of one\nexample that nature provides with an efficient algorithm for a very large\nnumber of processing elements: the human brain with its $10^{11}$ PE-s is the\nlargest parallel computer ever built. Although the intuition suggests that\nindeed there are scalable parallel schemes, it has only been proved recently,\nsee for details Ref. \\cite{KTNR}, by using the aforementioned analogy with\nthe simple MPEU surface growth model. \nWhile the MPEU model {\\em exactly} mimics the evolution of the simulated \ntime-horizon, it can also be considered as a primitive model for ion \nsputtering of surfaces (etching dynamics): to see this, define a new height \nvariable via $h_i \\equiv -\\tau_i$, i.e. flip Figure 2 upside down. This means \nthat instead of depositing material we have to take, `etch', and this has to \nbe done at the local {\\em maxima} of the $\\{h_i\\}$ surface. In sputtering \nof surfaces by ion bombardement an incoming ion-projectile will most likely \n`break off' a piece from the top of a mound instead from a valley, very \nsimilar to our `reversed' MPEU model. It was shown that the sputtering process \nis described by the KPZ equation, \\cite{bru,Barabasi}. \nThis qualitative argument is in complete agrrement with the extensive MC \nsimulations and a coarse-grained approximation of Ref. \\cite{KTNR} that MPEU, \nsimilar to the single-step model, it also belongs to the KPZ dynamic \nuniversality class; in one dimension the macroscopic landscape is governed by \nthe EW Hamiltonian. \n \nThe slope varaibles $\\phi_i$ for MPEU are not independent in the \n$L\\rightarrow\\infty$ limit, but short-ranged. This already guarantees that \nthe steady-state behavior is governed by the EW Hamiltonian, and \nthe density of local minima does not vanish in the thermodynamic limit. \nOur results confirm that the finite-size effects for $\\langle u\\rangle$ \nfollow eq. (\\ref{EW2_min_dens}): \n\\begin{equation} \n\\langle u\\rangle_L \\simeq \\langle u\\rangle_{\\infty} + \\frac{\\rm const.}{L} \n\\end{equation} \nwith $\\langle u\\rangle_{\\infty}=0.24641(7)$, see Fig. 3. \n\n\\begin{figure}\n\\hspace*{3.5cm}\\epsfxsize=4 in \n\\epsfbox{util.ps} \n\\vspace*{0.5cm} \n\\caption{Density of minima vs. $1/L$ for the MPEU model.} \n \\end{figure} \n\n\nWe conclude that the basic algorithm \n(one site per PE) is scalable for one-dimensional arrays. The same \ncorrespondence can be applied to model the performance of the algorithm for \nhigher-dimensional logical PE topologies. While this will involve the typical \ndifficulties of surface-growth modeling, such as an absence of exact results \nand very long simulation times, it establishes potentially fruitful \nconnections between two traditionally separate research areas. \n \n\\subsection{The larger curvature model} \n\nIn this subsection we briefly present a curvature driven SOS surface\ndeposition model known in the literature as the larger curvature model,\nand show a numerical analysis of the density of minima on this model.\n This model was originally introduced by Kim and Das Sarma \\cite{KSDS}\nand Krug \\cite{Krug2} independently, as an atomistic deposition\nmodel which fully conforms to the behaviour of the\ncontinuum fourth order linear Mullins equation ($\\nu=0$, $\\kappa > 0$ in\nEq. \\ref{h_evolution}). Note that the discrete analysis we presented in\nSection II is based on the discretization of the continuum equation\nusing the simplest forward Euler differencing scheme. The larger curvature\nmodel, however, is a {\\em growth} model where the freshly deposited particles\ndiffuse on the surface according to the rules of the model until they are \nembedded. Since in all the quantities studied so far, the correspondence\n(on the level of scaling) between the larger curvature model and the Langevin\nequation is very good, we would expect that the dynamic scaling properties of\nthe density of minima for both the model and the equation to be identical. \n\n The large curvature model has rather simple rules: a freshly deposited atom\n(let us say at site $i$) will be incorporated at the nearest neighbor site\nwhich has largest curvature (i.e., $K_i=h_{i+1}+h_{i-1}-2h_i$ is maximum).\nIf there are more neighbors with the same maximum curvature, then \none is chosen randomly. If the original site ($i$) \nis among those with maximum curvature, then the atom is incorporated at $i$.\n\nFigure 4 shows the scaling of the density of minima $<u>_L$ in the steady\nstate, vs. $1/\\sqrt{L}$. According to Eq. (\\ref{u4d}), for the fourth order\nequation on the lattice, the behavior of the density of minima in the steady\nstate scales with system size as $1/\\sqrt{L}$. And indeed, Figure 4 shows the\nsame behaviour for the larger curvature model, as expected. Note that this\nbehavior sets in at rather small system sizes already, at about $L=100$,\nmeaning that the finite system size effects are rather small for the larger\ncurvature model. This is a very fortunate property since increasing the system\nsize means decreasing the density of minima, therefore relative statistical\nerrors will increase. \n\n\\begin{figure}\n\\hspace*{3.5cm}\\epsfxsize=4 in \n\\epsfbox{evL.ps} \n\\vspace*{0.5cm} \n\\caption{Density of minima in the steady state for the larger curvature\nmodel.} \\end{figure} \n\nThis can only be improved by better statistics, i.e.,\nwith averages over larger number of runs. This becomes however quickly\na daunting task, since the cross-over time toward the steady state scales\nwith system size as $L^4$. As we shall see in Section V.A, a matematically\nrigorous approach to the continuum equation yields the same $1/\\sqrt{L}$\nbehaviour. Since the density of minima does decay to zero, an algorithm\ncorresponding to the larger curvature model (or the Mullins equation)\nwould {\\em not} be asymptotically scalable.\n\nFinally, we would like to make a brief note \nabout the observed morphologies in the steady state for the Mullins equation\n, or the related models.\nIt has been shown previously \\cite{ALF} that in the steady state the\nmorphology tipically shows a single large mound (or macroscopic groove).\nAt first sight this may appear as a surprise, since we have shown that the\nnumber of minima (or maxima) diverges as $\\sqrt{L}$ (the density vanishes as\n$1/\\sqrt{L}$. There is however no contradiction, because that refers to a\na mound that expands throughout the system, i.e. it is a long wavelenght\nstructure, whereas the number of minima measures {\\em all} the minima, and\nthus it is a short wavelenght characteristic. In the steady state we indeed\nhave a single large, macroscopic groove, however, there are numerous small\ndips and humps generated by the constant coupling to the noise.\n\n\\section{Extremal-point densities on the continuum} \n \n Let us consider a continuous and at least two times differentiable function \n$f:[0,L] \\to \\R$. We are interested in counting the total number of extrema \nof $f$ in the $[0,L] $ interval. The \ntopology of continuous curves in one dimension allows for three possibilities \non the nature of a point $x_i$ for which $f'|_{x_i} = 0 $. \nNamely, $x_i$ is a local minimum if $f''|_{x_i}>0 $,\na local maximum if $f''|_{x_i}<0$ and it is degenerate if\n$f''|_{x_i}=0$. We call the point\n$x_i$ a degenerate flat of order $k$, if $f^{(j)}|_{x_i} = 0$ for\n$j=1,2,..,k$ and $f^{(k+1)}|_{x_i} \\neq 0$, $k\\geq 2$, assuming that the\nhigher order derivatives $f^{(j)}$ implied exist. The counter-like quantity \n\\begin{equation} \nc(L,[f]) \\equiv \\frac{1}{L} \\int\\limits_0^L dx \\; |f''|\\; \\delta(f')\\label{counter} \n\\end{equation} \nwhere $\\delta$ is the Dirac-delta, \ngives the number of extremum points per unit \nlength in the interval $[0,L]$, which in the limit of $L \\to 0$ is the \nextremum point density of $f$ in the origin. \nFor our purposes $L$ will always be a finite number, however, for the sake\nof briefness we shall refer to $c$ simply as the density of extrema.\nNote that counting the extrema\nof a function $f$ is equivalent to counting the zeros of its derivate $f'$.\nThe divergence of $c$ for finite $L$ implies either the existence of\ncompletely flat regions (infinitely degenerate), \nor an ``infinitely wrinkled'' region, such as for the truncated\nWeierstrass function shown in Fig. 2 ( in this latter case the divergence is\nunderstood by taking the limit $M \\to \\infty$). As already explained in the\nIntroduction this infinitely wrinkled region does not necessarily imply\nthat the curve is fractal, but if the curve is fractal, then regions of\ninfinite wrinkledness must exist. The divergence or non-divergence of\n$c$ can be used as an indicator of the existence of such regions (completely\nflat or infinitely wrinkled).\n\nOne can make the following precise statement related to the counter $c$: \nif $x_i$ is an extremum point of $f$ of at most finite degeneracy $k$, and if\nthere exist a small enough $\\epsilon$, such that $f$ is \nanalytic in the neighborhood $[x_i-\\epsilon, x_i+\\epsilon]$, and there \nare no other extrema in this neighborhood, then \n\\begin{equation} \nI(x_i) \\equiv \\int\\limits_{x_i-\\epsilon}^{x_i+\\epsilon}dx\\; |f''|\\; \n\\delta(f') = 1\\;,\\;\\;\\;0 < \\epsilon \\ll 1 \\label{stat1} \n\\end{equation} \nIn the following we give a proof to this statement. \n \nUsing Taylor-series expansions around $x_i$, one writes: \n\\begin{eqnarray} \n&&f'(x)=\\frac{a_k}{k!} (x-x_i)^k+ \n\\frac{a_{k+1}}{(k+1)!} (x-x_i)^{k+1}+... \\label{exp1}\\\\ \n&& f''(x)=\\frac{a_k}{(k-1)!} (x-x_i)^{k-1}+ \n\\frac{a_{k+1}}{k!} (x-x_i)^{k}+...\\label{exp2} \n\\end{eqnarray} \nwhere we introduced the shorthand notation $a_j \\equiv f^{(j+1)}|_{x_i}$. \nFor the non-degenerate case of $k=1$, (\\ref{stat1}) \nfollows from a classical property of the \ndelta function, namely: \n\\begin{equation} \n\\delta(g(x)) = \n\\sum_i |g'(x_i)|^{-1} \\; \\delta(x-x_i)\\; \n, \\;\\;\\;\\mbox{ $x_i$ are {\\em simple} \nzeros of $g$}. \\label{dprop} \n\\end{equation} \nLet us now assume that $x_i$ is degenerate of order $k$ ($k \\geq 2$). \nUsing the expansions (\\ref{exp1}), (\\ref{exp2}), the variable change \n$u=x-x_i$, and the \nwell-known property $\\delta(ax)=|a|^{-1} \\delta(x)$, we obtain: \n\\begin{equation} \nI(x_i) = k \\int\\limits_{-\\epsilon}^{\\epsilon} du\\; |u|^{k-1}\\; \n\\left|1+\\sum_{j=1}^{\\infty} \\frac{(k-1)!}{(k-1+j)!}\\; \n\\frac{a_{k+j}}{a_k}\\;u^j \\right|\\;\\delta\\left( |u|^{k} \n\\left[ 1+\\sum_{j=1}^{\\infty} \\frac{k!}{(k+j)!}\\; \n\\frac{a_{k+j}}{a_k}\\;u^j\\right]\\right) \\label{31} \n\\end{equation} \nNext we split the integral (\\ref{31}) in two: \n$\\int_{-\\epsilon}^{\\epsilon} ... = \\int_{-\\epsilon}^{0} ... \n+\\int_{0}^{\\epsilon}$, make the variable change $u \\to -u$ \nin the first one, and then $z=u^k$ in both integrals. The final \nexpression can then be written in the form: \n\\begin{equation} \nI(x_i) = \\int\\limits_{-\\epsilon^k}^{\\epsilon^k} dz\\; \n|A(z)|\\;\\delta(zB(z)) \\label{33}\\;, \n \\end{equation} \nwhere \n\\begin{equation} \nA(z)=1+\\sum_{j=1}^{\\infty} \\frac{(k-1)!}{(k-1+j)!}\\; \n\\frac{a_{k+j}}{a_k}\\;z^j |z|^{\\frac{j}{k}-j}\\;,\\;\\;\\;\\mbox{and}\\;\\;\\; \nB(z)=1+\\sum_{j=1}^{\\infty} \\frac{k!}{(k+j)!}\\; \n\\frac{a_{k+j}}{a_k}\\;z^j |z|^{\\frac{j}{k}-j} \\label{34} \n\\end{equation} \nWe have $A(0)=B(0)=1$, and \n\\begin{equation} \n[zB(z)]' = 1 +\\sum_{j=1}^{\\infty} \\frac{k!}{(k+j)!}\\; \n\\frac{a_{k+j}}{a_k}\\;\\left(\\frac{j}{k}+1\\right) z^j |z|^{\\frac{j}{k}-j} \n\\;,\\;\\;\\;\\rightarrow\\;\\;\\;\\left.[zB(z)]'\\right|_{z=0} = 1\\;. \n\\end{equation} \n(Take the derivatives separately to the right and to the left of $z=0$). \nThus, since $z=0$ is a {\\em simple} zero of $zB(z)$, property \n(\\ref{dprop}) can be applied for sufficiently small $\\epsilon$: \n\\begin{equation} \nI(x_i) = |A(0)|=1 \n\\end{equation} \nproving our assertion. \nNote that because of \n(\\ref{stat1}), $c$ counts {\\em all} the non-degenerate and the finitely \ndegenerate points as well, giving the equal weight of unity to each. Can we \ncount the non-degenerate extrema separately? The answer is affirmative, if \none considers instead of (\\ref{counter}) the following quantity: \n\\begin{equation} \nc_q(L,[f]) \\equiv \\frac{1}{L} \\int\\limits_0^L dx \\; |f''|^{q+1}\\;\\delta (f') \n\\;,\\;\\;\\;q > 0\\label{counterq} \n\\end{equation} \nPerforming the same steps as above we obtain for a degenerate point: \n\\begin{equation} \nI_q(x_i) \\equiv \\int\\limits_{x_i-\\epsilon}^{x_i+\\epsilon}dx\\; |f''|^{q+1}\\; \n\\delta (f') = \\left[ \\frac{|a_k|}{(k-1)!}\\right]^q \n\\int\\limits_{-\\epsilon^k}^{\\epsilon^k} dz\\; |z|^{q\\left(1-\\frac{1}{k}\\right)} \n\\;|A(z)|^{q+1}\\;\\delta(zB(z)) \\label{37}\\;.\\label{stat2} \n\\end{equation} \nSince $k \\geq 2$, $q\\left(1-\\frac{1}{k}\\right) \\geq \\frac{1}{2}q > 0$, \ni.e., \n\\begin{equation} \nI_q(x_i) =0\\;,\\;\\;\\;\\mbox{for $x_i$ degenerate}. \n\\end{equation} \nThis means, that $q > 0$ eliminates the degenerate points from the count. \nTo non-degenerate points ($k=1$) (\\ref{counterq}) gives the weight of \n\\begin{equation} \nI_q(x_i) =|a_1|^q = \\Big|f''|_{x_i}\\Big|^q\\;,\\;\\;\\; \n\\mbox{for $x_i$ \nnon-degenerate}. \n\\end{equation} \nIn other words, \n\\begin{equation} \nc_q(L,[f]) \n=\\frac{1}{L} \\sum_i \\left| K(x_i) \\right|^q\\;,\\;\\;\\; \n\\mbox{$q>0$, $x_i$ non-degenerate \nextrema of $f$} \\label{40} \n\\end{equation} \nwhere $K(x) = f''$ is the {\\em curvature} of $f$ at $x$. \nThe limit $q\\to 0^+$ in (\\ref{40}) gives the \nextremum point density $\\overline{c}(L,[f])$ \nof $f$ of non-degenerate extrema: \n\\begin{equation} \n\\overline{c}(L,[f]) = \\lim_{q \\to 0^+} c_q(L,[f]) = \\lim_{q \\to 0^+} \n\\frac{1}{L} \\int\\limits_0^L dx \\; |f''|^{q+1}\\;\\delta (f') \\label{42} \n\\end{equation} \nIt is important \nto note, that taking the $q \\to 0^+$ limit in (\\ref{40}) {\\em is not \nequivalent} to taking $q=0$ in (\\ref{counterq}), i.e., the \nlimit and the integral on the rhs of (\\ref{42}) are not \ninterchangeable! The difference is \nthe set of degenerate points! \n \nUntil now, we did not make any distiction between maxima and minima. \nIn a natural way, we expect that the quantity: \n\\begin{equation} \nu(L,[f]) \\equiv \\frac{1}{L} \\int\\limits_0^L dx \\; f''\\; \n\\delta (f')\\; \\theta(f'') \\label{43} \n\\end{equation} \nwhere $\\theta(x)$ is the Heaviside step-function, will give the \ndensity of minima (due to the step function, \nhere we can drop the absolute values). \nHowever, performing a similar derivation as above, one concludes \nthat (\\ref{43}) is a little bit ill-defined, in the sense that the weight \ngiven to degenerate points depends on the definition of the step-function in \nthe origin (however, $u(L,[f])$ is bounded). Introducing a $q-regulator$ as above, \nthe weight of degenerate points is pulled down to zero: \n\\begin{equation} \nu_q(L,[f]) \\equiv \\frac{1}{L} \\int\\limits_0^L dx \\; [f'']^{q+1}\\; \n\\delta (f')\\; \\theta(f'') \\;,\\;\\;\\;q>0.\\label{44} \n\\end{equation} \nand \n\\begin{equation} \nu_q(L,[f]) =\\frac{1}{L} \\sum_i \\left[ K(x_i) \\right]^q\\;,\\;\\;\\; \n\\mbox{$q>0$, $x_i$ non-degenerate \nminima of $f$} \\label{45} \n\\end{equation} \nNote that in the equation above the absolute values are not needed, since \nwe are summing over the curvatures of all local {\\em minima}. \nThe density $\\overline{u}(L,[f]) $ \nof non-degenerate minima of $f$ is obtained \nafter taking the limit $q\\to 0^+$: \n\\begin{equation} \n\\overline{u}(L,[f]) = \\lim_{q\\to 0^+} u_q(L,[f]) \\label{47} \n\\end{equation} \nand the limit is not interchangeable with the integral in (\\ref{44}). \nTo obtain densities for maxima, one only has to replace \nthe argument $f''$ of the Heaviside function with $-f''$. \n \n\\subsection{Stochastic extremal-point densities} \n \nWe are interested to explore the previously introduced quantities \nfor a stochastic function, subject to time evolution, $h(x,t)$. \nThis function may be for example the solution to a Langevin equation. \nWe define the two basic quantities in the same way as before, \nexcept that now one performs a stochastic average over the noise, \nas well: \n\\begin{eqnarray} \nC_q(L,t)= \\left\\langle \n\\frac{1}{L} \\int\\limits_0^L dx \\; \n\\left|\\frac{\\partial^2 h}{\\partial x^2}\\right|^{q+1} \n\\delta \\left( \\frac{\\partial h}{\\partial x}\\right)\\right\\rangle \n\\;,\\;\\;\\;\\mbox{and}\\;\\;\\; \nU_q(L,t)= \\left\\langle \n\\frac{1}{L} \\int\\limits_0^L dx \\; \n\\left[\\frac{\\partial^2 h}{\\partial x^2}\\right]^{q+1} \n\\delta \\left( \\frac{\\partial h}{\\partial x}\\right)\\; \n\\theta \\left( \\frac{\\partial^2 h}{\\partial x^2}\\right) \n\\right\\rangle \n\\end{eqnarray} \nFor systems preserving translational invariance, the stochastic \naverage of the integrand becomes $x$-independent, and the integrals \ncan be dropped: \n\\begin{eqnarray} \n& &C_q(L,t)= \\left\\langle \n\\left|\\frac{\\partial^2 h}{\\partial x^2}\\right|^{q+1} \n\\delta \\left( \\frac{\\partial h}{\\partial x}\\right)\\right\\rangle \\label{Cq}\\\\ \n& &U_q(L,t)= \\left\\langle \n\\left[\\frac{\\partial^2 h}{\\partial x^2}\\right]^{q+1} \n\\delta \\left( \\frac{\\partial h}{\\partial x}\\right)\\; \n\\theta \\left( \\frac{\\partial^2 h}{\\partial x^2}\\right) \n\\right\\rangle \\label{Uq} \n\\end{eqnarray} \nAccording to (\\ref{40}) and (\\ref{45}), $C_q(L,t)$ and $U_q(L,t)$ can be thought of \nas time dependent ``partition functions'' for the non-degenerate \nextremal-point densities of the underlying stochastic process, with $q$ \nplaying the role of ``inverse temperature'': \n\\begin{eqnarray} \n& &C_q(L,t)= \\left\\langle \\frac{1}{L} \\sum_i \\left| K(x_i) \\right|^q \n\\right\\rangle\\;,\\;\\;\\; \\mbox{$q>0$, $x_i$ non-degenerate \nextrema} \\\\ \n& &U_q(L,t)=\\left\\langle \\frac{1}{L} \\sum_i \\left[ K(x_i) \\right]^q \n\\right\\rangle\\;,\\;\\;\\; \\mbox{$q>0$, $x_i$ non-degenerate \nminima} \n\\end{eqnarray} \nIt is important to mention that in the above equations the average $<..>$ \nand the summation are {\\em not} interchangeable: particular \nrealizations of $h$ have particular sets of minima. \n\nTwo values for $q$ are of special interest: when $q\\to 0^+$ \nand $q=1$. In the first case we obtain the stochastic average of the \ndensity of non-degenerate extrema and minima: \n\\begin{equation} \n\\overline{C}(L,t) = \\lim_{q\\to 0^+} C_q(L,t)\\;,\\;\\;\\; \n\\mbox{and} \\;\\;\\;\\overline{U}(L,t) = \\lim_{q\\to 0^+} U_q(L,t)\\;, \\label{bar} \n\\end{equation} \nand in the second case we obtain the stochastic average of the mean \ncurvature at extrema and minima: \n\\begin{equation} \n\\overline{K}_{ext}(L,t) = \\frac{C_1(L,t)}{\\overline{C}(L,t) }\\;,\\;\\;\\; \n\\mbox{and} \\;\\;\\;\\overline{K}_{min}(L,t) = \n\\frac{U_1(L,t)}{\\overline{U}(L,t) }\\;, \\label{curv} \n\\end{equation} \n(we need to normalize with the number of extrema/minima per unit \nlength to get the curvature per extremum/minimum). \n \nIn the following we explore the quantities (\\ref{Cq})-(\\ref{curv}) \nfor a large class of linear Langevin equations. \nTo simplify the calculations, we will assume that \n$q$ is a positive integer. Then we will attempt analytic \ncontinuation on the final result as a function of $q$. In the \ncalculations we will make extensive use of the standard integral \nrepresentations of the delta and step functions: \n\\begin{eqnarray} \n&&\\delta(y) = \\int\\limits_{-\\infty}^{\\infty} \\frac{dz}{2\\pi}\\; \ne^{izy} = \\sum_{n=0}^{\\infty} \n \\int\\limits_{-\\infty}^{\\infty} \n \\frac{dz}{2\\pi}\\; \n\\frac{(iz)^n}{n!}\\;y^n\\;, \\label{51a} \\\\ \n&&\\theta(y)=\\lim_{\\epsilon \\to 0^+} \n\\int\\limits_{-\\infty}^{\\infty} \\frac{dz}{2\\pi}\\; \n\\frac{e^{izy} }{\\epsilon+iz} = \\lim_{\\epsilon \\to 0^+} \n\\sum_{n=0}^{\\infty} \\int\\limits_{-\\infty}^{\\infty} \\frac{dz}{2\\pi}\\; \n\\frac{1}{\\epsilon+iz}\\;\\frac{(iz)^n}{n!}\\;y^n \\label{51b} \n\\end{eqnarray} \nIf $q$ is a positive integer, we may drop the absolute value signs \nin (\\ref{Uq}). In (\\ref{Cq}) we can only do that for odd $q$. \nThe absolute values make the calculation of stochastic averages \nvery difficult. We can get around this problem by employing the \nfollowing identity: \n\\begin{equation} \n|y|^n = y^n \\left\\{ (-1)^n + \\theta(y) \\left[ \n1-(-1)^n \n\\right] \\right\\} \\label{52} \n\\end{equation} \nThis brings (\\ref{Cq}) to \n\\begin{equation} \nC_q(L,t)=\\left[ 1-(-1)^q \\right] U_q(L,t)+(-1)^{q+1}B_q(L,t) \\label{relation} \n\\end{equation} \nwhere \n\\begin{equation} \nB_q(L,t)=\\left\\langle \n\\frac{1}{L} \\int\\limits_0^L dx \\; \n\\left(\\frac{\\partial^2 h}{\\partial x^2}\\right)^{q+1} \n\\delta \\left( \\frac{\\partial h}{\\partial x}\\right)\\right\\rangle \n=\\left\\langle \n\\left(\\frac{\\partial^2 h}{\\partial x^2}\\right)^{q+1} \n\\delta \\left( \\frac{\\partial h}{\\partial x}\\right)\\right\\rangle \n\\end{equation} \nObviously for $q$ odd integer, $B_q=C_q$. For $q$ even, \n$B_q$ is an interesting quantity by itself. In this case the \nweight of an extremum $x_i$ is $sgn (K(x_i)) |K(x_i)|^q$. \nIf the analytic continuation can be performed, then the \n$q \\to 0^+$ limit will tell us if there are more \nnon-degenerate maxima than minima (or otherwise) in average. \nUsing the integral representations (\\ref{51a}) and (\\ref{51b}): \n\\begin{eqnarray} \n&&B_q(L,t)=\\sum_{n=0}^{\\infty} \n \\int\\limits_{-\\infty}^{\\infty} \n \\frac{dz}{2\\pi}\\;\\frac{(iz)^n}{n!}\\; \n\\left\\langle \n\\left(\\frac{\\partial^2 h}{\\partial x^2}\\right)^{q+1} \n\\left( \\frac{\\partial h}{\\partial x}\\right)^n \n\\right\\rangle \\label{55a}\\\\ \n&&U_q(L,t)=\\lim_{\\epsilon \\to 0^+} \n\\sum_{n_1=0}^{\\infty}\\sum_{n_2=0}^{\\infty} \n \\int\\limits_{-\\infty}^{\\infty} \n \\frac{dz_1}{2\\pi}\\;\\frac{(iz_1)^{n_1}}{n_1!} \n\\int\\limits_{-\\infty}^{\\infty} \n \\frac{dz_2}{2\\pi}\\;\\frac{(iz_2)^{n_2}}{n_2!}\\; \n\\frac{1}{\\epsilon+iz_2}\\;\n\\left\\langle \n\\left(\\frac{\\partial^2 h}{\\partial x^2}\\right)^{n_2+q+1} \n\\left( \\frac{\\partial h}{\\partial x}\\right)^{n_1} \n\\right\\rangle \\label{55b} \n\\end{eqnarray} \n \n\\section{Extremal-point densities of linear stochastic evolution \nequations} \n \n Next we calculate the densities (\\ref{55a}), (\\ref{55b}) for the \nfollowing type of linear stochastic equations: \n\\begin{equation} \n\\frac{\\partial h}{\\partial t} = - \\nu \\left( - \\nabla^2 \\right)^{z/2} h + \n\\eta(x,t)\\;,\\;\\;\\;\\;\\nu, D, z > 0\\;,\\;\\;\\;x\\in[0,L] \\label{egyenlet} \n\\end{equation} \nwith initial condition \n$h(x,0)=0$, for all $x \\in [0,L]$. \n$\\eta$ is a white noise term drawn from a Gaussian distribution with \nzero mean $\\langle \\eta(x,t) \\rangle = 0$, and covariance : \n\\begin{equation} \n\\langle \\eta(x,t)\\eta(x',t') \\rangle = 2D \\delta(x-x')\\;\\delta(t-t')\\;, \n\\end{equation} \nWe also performed our calculations with other noise types, such as volume \nconserving and long-range correlated, however the details are \nto lengthy to be included in the present paper, it will be the subject\nof a future publication. As\nboundary condition we choose periodic boundaries: \\begin{eqnarray} \nh(x+nL,t) = h(x,t) \\;,\\;\\;\\; \n \\eta(x+nL,t) = \\eta(x,t)\\;,\\;\\;\\;\\mbox{for all $n\\in \\Z$} \n\\end{eqnarray} \n \nThe general solution to (\\ref{egyenlet}) is obtained simply with the \nhelp of Fourier series \\cite{Krug}. The Fourier series and its \ncoefficients for a function $f$ defined on $[0,L]$ is \n\\begin{eqnarray} \nf(x)= \\sum_{k} \\tilde{f}(k)\\;e^{ikx}\\;,\\;\\;\\; \n\\tilde{f}(k)=\\frac{1}{L} \\int\\limits_{-L}^{L} dx\\; f(x)\\;e^{-ikx} \n\\label{Fseries} \n\\end{eqnarray} \nwhere $k=\\frac{2\\pi}{L}n$, $n= ..,-2,-1,0,1,2,..$. \nThe Fourier coefficients of the general solution to (\\ref{egyenlet}) \nare: \n\\begin{equation} \n\\tilde{h}(k,t) = \n\\int\\limits_{0}^{t} dt'\\;e^{-\\nu |k|^z (t-t')}\\tilde{\\eta}(k,t') \\label{9} \n\\end{equation} \nThe correlations of the noise in momentum space are: \n\\begin{equation} \n\\left\\langle \\tilde{\\eta}(k,t)\\tilde{\\eta}(k',t')\\right\\rangle \n=\\frac{2D}{L}\\;\\delta_{k,-k'}\\;\\delta(t-t')\\;. \n\\end{equation} \n \n \nDue to the Gaussian character of the noise, the \ntwo-point correlation of the solution (\\ref{9}) is also delta-correlated and \nit completely characterizes the statistical \nproperties of the stochastic dynamics (\\ref{egyenlet}). \nIt is given by: \n\\begin{equation} \n\\left\\langle \\tilde{h}(k,t)\\tilde{h}(k',t')\\right\\rangle = \nS(k,t) \\delta_{k,-k'} \\label{hcor} \n\\end{equation} \nwhere $S(k,t)$ is the structure factor given by:\\\\ \n\\begin{equation} \nS(k,t)= \n\\frac{D}{\\nu L |k|^z}\\left[1- e^{-2\\nu |k|^zt}\\right]\\;. \\label{sn} \n\\end{equation} \nEquation (\\ref{egyenlet}) has been analyzed in great detail by a number \nof authors, see Ref. \\cite{Krug} for a review. It was shown \nthat there exist un upper critical dimension $d_c = z$ for the noisy \ncase of Eq. (\\ref{egyenlet}) which separates the rough regime with \n$d< z$ from the non-roughening regime $d>z$. In one dimension, \nthe rough regime corresponds to the condition $z > 1$, which we \nshall assume from now on, since this is where the interesting physics lies. \n \n Next, we evaluate the quantities (\\ref{Cq})-(\\ref{curv}) via directly \ncalculating the expressions in (\\ref{55a}) and (\\ref{55b}). This \namounts to computing averages of type: \n\\begin{equation} \nQ_{N,M} = \\left\\langle \\left( \\frac{\\partial^2 h}{\\partial x^2} \n\\right)^N \\left( \\frac{\\partial h}{\\partial x} \n\\right)^M \\right\\rangle \\label{avg} \n\\end{equation} \nExpressing $h$ with its Fourier series according to (\\ref{Fseries}), \nwe write: \n\\begin{eqnarray} \n&&\\left( \\frac{\\partial h}{\\partial x} \n\\right)^M = i^M \\sum_{k_1}...\\sum_{k_M} \nk_1...k_M\\; \n\\tilde{h}(k_1,t)...\\tilde{h}(k_M,t)\\;e^{i(k_1+...+k_M)x} \\\\ \n&&\\left( \\frac{\\partial^2 h}{\\partial x^2} \n\\right)^N = (-1)^N \\sum_{k_1'}...\\sum_{k_N'} \n{k_1'}^2...{k_N'}^2\\; \n\\tilde{h}(k_1',t)...\\tilde{h}(k_N',t)\\;e^{i(k_1'+...+k_N')x} \n\\end{eqnarray} \nwhich then is inserted in (\\ref{avg}). Thus in Fourier \nspace one needs to \ncalculate averages of type $\\langle \\tilde{h}(k_1,t)...\\tilde{h}(k_M,t) \n\\tilde{h}(k_1',t)...\\tilde{h}(k_N',t) \\rangle$. According to \n(\\ref{hcor}) $\\tilde{h}$ is anti-delta-correlated, therefore \nthese averages can be performed in the standard way \n\\cite{itzi} which is by taking all the possible pairings of indices and \nemploying (\\ref{hcor}). In our case there are three types \nof pairings: $\\{ k_j,k_l\\}$, $\\{ k_j,k_l'\\}$, and $\\{ k_j',k_l'\\}$. \nLet us pick a `mixed' pair $\\{ k_j,k_l'\\}$ containing a primed and \na non-primed index. The corresponding contribution in the \n$Q_{N,M}$ will be: \n\\begin{equation} \n\\sum_{k_j}\\sum_{k_l'} k_j {k_l'}^2 S(k_l',t) \ne^{i(k_j+k_l')x} \\delta_{k_j,-k_l'} \\label{cont} \n\\end{equation} \nSince the structure factor $S(k,t)$ is an even function in \n$k$, (\\ref{cont}) becomes \n$\\sum_{k_j} {k_j}^3 S(k_j,t) = 0$, because the summand is an odd \nfunction of $k_j$ and the summation is symmetric around zero. Thus, \nit is enough to consider non-mixed index-pairs, only. This means, that \n$Q_{N,M}$ decouples into: \n\\begin{equation} \nQ_{N,M} = \\left\\langle \\left( \n\\frac{\\partial^2 h}{\\partial x^2} \n\\right)^N \\right\\rangle \n\\left\\langle \n\\left( \\frac{\\partial h}{\\partial x} \n\\right)^M \\right\\rangle \\label{avg1} \n\\end{equation} \nThe averages are calculated easily, and we find: \n\\begin{eqnarray} \n\\left\\langle \\left( \\frac{\\partial h}{\\partial x} \n\\right)^M \\right\\rangle = \\left\\{ \n\\begin{array}{l} \n(M-1)!! \\;\\left[ \\phi_2(L,t) \\right]^{M/2}\\;,\\;\\;\\;\\mbox{for $M$ even }\\;,\\\\ \n\\\\ \n0\\;,\\;\\;\\;\\mbox{for $M$ odd } \n\\end{array} \n\\right. \\label{phi2} \n\\end{eqnarray} \nand \n\\begin{eqnarray} \n\\left\\langle \\left( \\frac{\\partial^2 h}{\\partial x^2} \n\\right)^N \\right\\rangle = \\left\\{ \n\\begin{array}{l} \n(N-1)!! \\;\\left[ \\phi_4(L,t) \\right]^{N/2}\\;,\\;\\;\\;\\mbox{for $N$ even }\\;,\\\\ \n\\\\ \n0\\;,\\;\\;\\;\\mbox{for $M$ odd } \n\\end{array} \n\\right. \\label{phi4} \n\\end{eqnarray} \nwhere \n\\begin{equation} \n\\phi_m(L,t) \\equiv \\sum_{k} |k|^m\\;S(k,t) \\label{phi} \n\\end{equation} \nEmploying (\\ref{phi2}), and (\\ref{phi4}) in (\\ref{55a}), it follows that \nif $q$ is an even integer, $q=2s$, $s=1,2,..$: \n\\begin{equation} \nB_{2s}(t) = 0\\;,\\;\\;\\;s=1,2,... \\label{77a} \n\\end{equation} \nwhereas for $q$ odd integer, $q=2s-1$, $s=1,2,..$: \n\\begin{equation} \nB_{2s-1}(t) = C_{2s-1}(t) = \n\\frac{2^{s-\\frac{1}{2}}}{\\pi}\\;\\Gamma \n\\left(s+\\frac{1}{2}\\right)\\; \n\\frac{\\left[ \\phi_4(L,t) \\right]^{s}}{ \n\\sqrt{\\phi_2(L,t)}} \n\\;,\\;\\;\\;s=1,2,... \\label{77b} \n\\end{equation} \nwhere we used the identity $2^p(2p-1)!!/(2p)!=1/p!$, and performed \nthe Gaussian integral. \n \nThe calculation of $U_q$ is a bit trickier. The sum over $n_1$ in (\\ref{55b}) \nis easy and leads to the Gaussian $e^{-\\phi_2(L,t) z_1^2 /2}$. However, \nthe sum over $n_2$ is more involved. Let us make the temporary \nnotation for the sum over $n_2$: \n\\begin{equation} \nR_q = \\sum_{n_2=0}^{\\infty} \\frac{(iz_2)^{n_2}}{n_2!}\\;(2r-1)!!\\; \n\\left[ \\phi_4(L,t) \\right]^{r}\\;,\\;\\;\\;n_2+q+1=2r \n\\end{equation} \nWe have to distinguish two cases according to the parity of $q$:\\\\ \n1) $q$ is odd, $q=2s-1$, $s=1,2,..$. In this case $R_q$ becomes \n\\begin{equation} \nR_{2s-1} = \\sum_{r=s}^{\\infty} \\frac{(iz_2)^{2(r-s)}}{[2(r-s)]!} \n\\;\\frac{(2r)!}{r!}\\; \n\\left[ \\frac{1}{2}\\phi_4(L,t) \\right]^{r}=(z_2)^{-2s}(-1)^s \n\\left\\{ \\frac{\\partial^{2s}}{\\partial x^{2s}} \n\\left[ e^{-\\phi_4(L,t)z_2^2x^2/2}\\right] \\right\\}_{x=1} \n\\end{equation} \nThe Hermite polynomials are defined via the Rodrigues formula as: \n\\begin{equation} \nH_n(x) = (-1)^n e^{x^2} \\frac{d^n}{dx^n}\\left(e^{-x^2}\\right) \n\\label{Rodrigues} \n\\end{equation} \nUsing this, we can express $R_{2s-1}$ with the help of Hermite \npolynomials: \n\\begin{equation} \nR_{2s-1} = (-1)^s \\left[ \\frac{1}{2} \\phi_4(L,t) \\right]^s \\; \nH_{2s}\\left( \\sqrt{\\frac{1}{2} \\phi_4(L,t) } z_2 \\right)\\; \ne^{-\\phi_4(L,t)z_2^2x^2/2} \\label{rodd} \n\\end{equation} \n\\noindent 2) $q$ is even, $q=2s$, $s=1,2,..$. The calculations are analogous \nto the odd case: \n\\begin{equation} \nR_{2s} = \\sum_{r=s+1}^{\\infty} \\frac{(iz_2)^{2(r-s)-1}}{[2(r-s)-1]!} \n\\;\\frac{(2r)!}{r!}\\; \n\\left[ \\frac{1}{2}\\phi_4(L,t) \\right]^{r}=(iz_2)^{-2s-1} \n\\left\\{ \\frac{\\partial^{2s+1}}{\\partial x^{2s+1}} \n\\left[ e^{-\\phi_4(L,t)z_2^2x^2/2}\\right] \\right\\}_{x=1} \n\\end{equation} \nor via Hermite polynomials: \n\\begin{equation} \nR_{2s} = i (-1)^s \\left[ \\frac{1}{2} \\phi_4(L,t) \\right]^{s+\\frac{1}{2}} \\; \nH_{2s+1}\\left( \\sqrt{\\frac{1}{2} \\phi_4(L,t) } z_2 \\right)\\; \ne^{-\\phi_4(L,t)z_2^2x^2/2} \\label{reven} \n\\end{equation} \nIn order to obtain $U_q$ we have to do the integral over $z_2$ in \n(\\ref{55b}). This can be obtained after using the formula: \n\\begin{equation} \n\\int\\limits_{-\\infty}^{\\infty} dx\\;(x\\pm ic)^{\\nu}\\;H_n(x)\\;e^{-x^2} = \n2^{n-1-\\nu}\\sqrt{\\pi}\\;\\frac{\\Gamma \\left( \n\\frac{n-\\nu}{2} \\right)}{\\Gamma(-\\nu)}\\;e^{\\pm \\frac{i\\pi}{2}(\\nu+n)}\\;, \n\\;\\;\\;c \\to 0^+. \n\\end{equation} \nFinally, the densities for the minima read as: \n\\begin{eqnarray} \n&&U_{2s}(t)=\\frac{2^{s-1}}{\\pi}\\;\\Gamma(s+1)\\; \n\\frac{\\left[ \\phi_4(L,t) \\right]^{s+\\frac{1}{2}}}{ \n\\sqrt{\\phi_2(L,t)}} \\label{77c}\\\\ \n&&U_{2s-1}(t)=\\frac{2^{s-\\frac{3}{2}}}{\\pi}\\;\\Gamma\\left(s+\\frac{1}{2}\\right)\\; \n\\frac{\\left[ \\phi_4(L,t) \\right]^s}{ \n\\sqrt{\\phi_2(L,t)}} \\label{77d} \n\\end{eqnarray} \nFormulas (\\ref{77a}), (\\ref{77b}), (\\ref{77c}), and (\\ref{77d}) combined with \n(\\ref{relation}) \ncan be condensed very simply, and we obtain the general result as: \n\\begin{eqnarray} \n&&U_q(L,t) = \\frac{2^{\\frac{q}{2}-1}}{\\pi}\\; \n\\Gamma\\left(\\frac{q}{2}+1\\right)\\; \n\\frac{\\left[ \\phi_4(L,t) \\right]^{\\frac{q+1}{2}}}{ \n\\sqrt{\\phi_2(L,t)}} \\label{78a}\\\\ \n&&C_q(L,t)=2 U_q(L,t) \\label{78b} \n\\end{eqnarray} \nEquations (\\ref{78a}), (\\ref{78b}) with together with (\\ref{77d}) \nfully solve the problem for the density of non-degenerate \nextrema. Eq. (\\ref{78b}) is an expected result in \none dimension, because Eq. (\\ref{egyenlet}) preserves the up-down symmetry. \nThe density of non-degenerate minima is: \n\\begin{equation} \n\\overline{U}(L,t)=\\lim_{q\\to 0^+} U_q(L,t)=\\frac{1}{2 \\pi} \\; \n\\sqrt{\\frac{\\phi_4(L,t)}{\\phi_2(L,t)}} \\label{genU} \n\\end{equation} \nand the stochastic average of the mean curvature at a minimum point is: \n\\begin{equation} \n\\overline{K}(L,t)=\\frac{U_1(L,t)}{\\overline{U}(L,t)}=\\sqrt{\\frac{\\pi}{2}}\\; \n\\sqrt{\\phi_4(L,t)} \\label{genK} \n\\end{equation} \ni.e., the average curvature at a minimum is proportional to the \nsquare root of the fourth moment of the structure factor. \nIn the following section we exploit the physical information behind \nthe above expressions for the stochastic process (\\ref{egyenlet}). \nAt some parameter values\na few, or all the quantities above may diverge. In this case \nwe introduce a microscopic lattice cut-off $0 < a \\ll 1$, and analyze the \nlimit $a \\to 0^+$ in the final formulas. This in fact corresponds to placing \nthe whole problem on a lattice with lattice constant $a$. \n It has been shown in Ref. \\cite{Krug} that for the class \nof equations (\\ref{egyenlet}) there are three important length-scales \nthat govern the statistical behavior of the interface $h$: the lattice \nconstant $a$, the system size $L$, and the {\\em dynamical correlation \nlength} $\\xi$ defined by: \n\\begin{equation} \n\\xi(t) \\equiv (2\\nu t)^{1/z} \\label{xi} \n\\end{equation} \n According to (\\ref{phi}) and (\\ref{sn}) the function $\\phi_m(L,t)$ becomes:\n\\bb{ \n\\phi_m(L,t)=\\frac{2D}{\\nu L}\\sum_{n=0}^{\\infty} \n\\left(\\frac{2\\pi n}{L}\\right)^{m-z}\\left[1- e^{\\textstyle -\\left(\\xi \n \\frac{2\\pi n}{L}\\right)^z}\\right] \n\\;,\\;\\;\\;m=2,4 \\label{phin} \n}\\e \nThe $n=0$ term can be dropped from the sum above, because it is zero \neven for $m < z$ (expand the exponential and then take $n=0$). However, \nthe whole sum may diverge depending on $m$ and $z$. In order to handle \nall the cases, including the divergent ones we introduce a microscopic \nlattice cut-off $a$, $0<a \\ll 1$, and then analyze the limit $a \\to 0^+$ in the \nfinal expressions. This is in fact equivalent to putting the whole problem on \na lattice of lattice spacing $a$. Appropiately, (\\ref{phin}) becomes: \n\\bb{ \n\\phi_m(L,t)=\\frac{2D}{\\nu L}\\sum_{n=1}^{\\frac{L}{2a}} \n\\left(\\frac{2\\pi n}{L}\\right)^{m-z}\\left[1- e^{\\textstyle -\\left(\\xi \n \\frac{2\\pi n}{L}\\right)^z}\\right] \n\\;,\\;\\;\\;m=2,4 \\label{phina} \n}\\e \n \n\\subsection{Steady-state regime.} \n \nPutting $\\xi=\\infty$ in (\\ref{phina}) $\\phi_m$ \ntakes a simpler form: \n\\bb{ \n\\phi_m(L,\\infty)=\\frac{2D}{\\nu L} \n\\left(\\frac{2\\pi}{L}\\right)^{m-z} \n\\sum_{n=1}^{\\frac{L}{2a}} \nn^{m-z} \n\\;,\\;\\;\\;m=2,4 \\label{phinb} \n}\\e \nAs $a \\to 0^+$, $\\phi_m$ becomes proportional to $\\zeta(z-m)$. For \n$z-m > 1$ $\\phi_m$ is convergent, otherwise it is divergent. In the divergent \ncase we quote the following results: \n\\bb{ \n\\sum_{n=1}^{N} n^s = \\ln{N} + {\\cal C} + {\\cal \nO}(1/N)\\;,\\;\\;\\;\\mbox{if}\\;\\;s=-1 \\label{euler} \n}\\e \nand \n\\bb{ \n\\sum_{n=1}^{N} n^s =\\frac{N^{s+1}}{s+1}\\Big[1 + \n{\\cal O}(1/N)\\Big] \\;,\\;\\;\\;\\mbox{if}\\;\\;s > -1 \\label{cesaro} \n}\\e \nwhich we will use to derive the leading behaviour of the extremal point \ndensities when $L/a \\to \\infty$. From equations (\\ref{phinb}), \n(\\ref{78a}), (\\ref{genU}) and (\\ref{genK}) follows: \n\\bb{ \nU_q(L,\\infty) = \\Gamma\\left(\\frac{q}{2} +1 \\right) \n\\left(\\frac{2D}{\\pi \\nu} \\right)^{ \\frac{q}{2}} \\left( \n2\\pi \\right)^{\\frac{q}{2}(5-z)} \nL^{-1-\\frac{q}{2}(5-z)} \n\\left[ \\sum_{n=1}^{L/2a} n^{4-z}\\right]^{\\frac{q+1}{2}} \n\\left[ \\sum_{n=1}^{L/2a} n^{2-z}\\right]^{-\\frac{1}{2}}\\;, \n\\label{uqst} \n}\\e \n\\bb{ \n\\overline{U}(L,\\infty) = \\frac{1}{L} \\sqrt{ \n \\sum_{n=1}^{L/2a} n^{4-z}\\left(\n\\sum_{n=1}^{L/2a} n^{2-z}\\right)^{-1} }\\;, \n\\label{u0st} \n}\\e \nand \n\\bb{ \n\\overline{K}(L,\\infty) = \\sqrt{\\frac{D}{2} \\left( \n2\\pi \\right)^{5-z} L^{z-5} \n\\sum_{n=1}^{L/2a} n^{4-z}}\\;.\\label{kst} \n}\\e \nThe convergency (divergency) properties of the sums in \nEqs. (\\ref{uqst}-\\ref{kst}) for $a \\to 0^+$ generate two critical \nvalues for $z$, namely $z=3$ and $z=5$. In the three regions \nseparated by these values we obtain {\\em qualitatively} different \nbehaviors for the extremal-point densities. \n \n{\\em i)} $z > 5$. All quantities are convergent as $a \\to 0^+$. We \nhave: \\bb{ \nU_q(L,\\infty) = \\Gamma\\left(\\frac{q}{2}+1 \\right) \n\\left(\\frac{2D}{\\pi \\nu} \\right)^{\\frac{q}{2}} (2\\pi)^{\\frac{q}{2}(5-z)} \n\\frac{[\\zeta(z-4)]^{\\frac{q+1}{2}}}{[\\zeta(z-2)]^{\\frac{1}{2}}} \n\\;L^{-1+\\frac{q}{2}(z-5)}\\;,\\;\\;\\;\\;z>5 \\label{uqzg5} \n}\\e \n\\bb{ \n\\overline{U}(L,\\infty) = \\frac{1}{L} \n\\sqrt{\\frac{\\zeta(z-4)}{\\zeta(z-2)}}\\;, \\;\\;\\;\\; \\label{u0zg5} \n}\\e \n\\bb{ \n\\overline{K}(L,\\infty) = (2 \\pi)^{\\frac{5-z}{2}} \n\\sqrt{\\frac{D}{2 \\nu} \n\\zeta(z-4)}\\;L^{\\frac{z-5}{2}}\\;,\\;\\;\\;\\; \\label{kzg5} \n}\\e \nEq. (\\ref{u0zg5}) shows that there are a finite number of minima \n($\\sqrt{\\zeta(z-4)/\\zeta(z-2)}$) in the steady state, independently of the \nsystem size $L$. ($\\overline{U}(L,\\infty)$ is the number of minima per unit \nlength, and $L\\overline{U}(L,\\infty)$ is the number of minima on the substrate \nof size $L$). The mean curvature $\\overline{K}(L,\\infty)$ diverges with \nsystem size as $L^{(z-5)/2}$. This is consistent with the fact that \nthe system size grows as $L$, \nthe width grows as $L^{(z-1)/2}$, i.e., faster than $L$, and thus the peaks \nand minima should become sleeker and sharper as $L \\to \\infty$, expecting \ndiverging curvatures in minima and maxima. However, this is not always \ntrue, since the sleekness of the humps and mounds does not \nnecessarily imply large curvatures in minima and maxima if the {\\em shape} of \nthe humps also changes as $L$ changes, i.e., there is lack of {\\em \nself-affinity}. The existence of $z=5$ as a critical value is a non-trivial results \ncoming from the presented analysis. \n \n{\\em ii)} $z = 5$. According to \n(\\ref{euler}), $\\phi_4(L,\\infty)$ diverges logarithmically as $a \\to 0^+$. One \nobtains: \n\\bb{ \nU_q(\\infty) \\simeq \\Gamma\\left(\\frac{q}{2}+1 \\right) \n\\left(\\frac{2D}{\\nu \\pi} \\right)^{\\frac{q}{2}} \\frac{1}{\\sqrt{\\zeta(3)}} \n\\frac{1}{L} \n\\left( \\ln{\\frac{L}{2a}} + {\\cal C}\\right)^{\\frac{q+1}{2}}\\;, \n \\label{uqze5} \n}\\e \n\\bb{ \n\\overline{U}(L,\\infty) = \\frac{1}{\\sqrt{\\zeta(3)}}\\; \n\\frac{1}{L} \\sqrt{ \\ln{\\frac{L}{2a}} + {\\cal C}}\\; \n\\;, \\label{u0ze5} \n}\\e \n\\bb{ \n\\overline{K}(L,\\infty) = \n\\sqrt{\\frac{D}{2 \\nu} \n\\left( \n\\ln{\\frac{L}{2a}} + {\\cal C}\\right)}\\;. \\label{kze5} \n}\\e \n Eq. (\\ref{u0ze5}) shows that the although the density of minima \nvanishes, the number of minima is no longer a constant but {\\em diverges} \nlogarithmically with system size $L$. The mean curvature still diverges, but \nlogarithmically, when compared to the power law divergence of (\\ref{kzg5}). \n \nFor the mean curvature $\\overline{K}(\\infty)$ in (\\ref{kst}) $z=5$ is the only \ncritical value, since it only depends on $\\phi_4$. For $z < 5$, using Eq. \n(\\ref{cesaro}) we arrive to the result that the mean curvature in a minimum \npoint approaches to an $L$-independent constant for $L/a \\to \\infty$ with \ncorrections on the order of $a/L$: \n\\bb{ \n\\overline{K}(L,\\infty) \\simeq \n\\left( \\frac{\\pi}{a}\\right)^{\\frac{5-z}{2}} \n\\sqrt{\\frac{D}{2\\nu (5-z)}} \\;, \\;\\;\\;z<5 \\label{kzl5} \n}\\e \nWe arrived to the same conclusion in Section II.D when we studied the\nsteady state of the discretized version of the continuum equation.\nCoincidentally, for $z=4$ the two constant values from (\\ref{kzl5})\nand (\\ref{ke4}) are identical ($a=1$ by definition in (\\ref{ke4}).\n\n{\\em iii)} $3 < z < 5$. In this case $\\phi_4(L,\\infty) \\to \\infty$ \nand $\\phi_2(L,\\infty) < \\infty$ as $a \\to 0^+$, and: \n\\bb{ \nU_q(L,\\infty) \\simeq \\Gamma\\left(\\frac{q}{2}+1 \\right) \n\\left(\\frac{2D}{\\nu \\pi} \\right)^{\\frac{q}{2}} \n\\left(\\frac{\\pi}{a}\\right)^{\\frac{q}{2}(5-z)} \n\\left(\\frac{1}{2a}\\right)^{\\frac{5-z}{2}} \n\\frac{L^{-\\frac{z-3}{2}}} \n{(5-z)^{\\frac{q+1}{2}}\\sqrt{\\zeta(z-2)}}\\;,\\label{uq3lzl5} \n}\\e \nand \n\\bb{ \n\\overline{U}(L,\\infty) \\simeq \\left(\\frac{1}{2a}\\right)^{\\frac{5-z}{2}}\\; \n\\frac{L^{-\\frac{z-3}{2}}}{ \\sqrt{ (5-z) \\zeta(z-2)}}\\; \n\\;, \\label{u03lzl5} \n}\\e \nand the mean curvature is just given by (\\ref{kzl5}). \n \nComparing Eqs. (\\ref{uqzg5}), (\\ref{uqze5}), and (\\ref{uq3lzl5}) we can make \nan interesting observation: while for $z \\geq 5$ the dependence on the system \nsize $L$ is coupled to the `inverse temperature' $q$, for $3 < z < 5$ the \ndependence on $L$ {\\em decouples} from $q$, i.e., it becomes independent \nof the inverse temperature! Eq. (\\ref{u03lzl5}) shows that the density \nof minima vanishes with system size as a power law with an \nexponent $(z-3)/2$ \nbut the number of minima of the substrate diverges as a power law with an \nexponent of $(5-z)/2$. \n \n{\\em iv)} $z=3$. In this case $\\phi_4(L,\\infty) \\to \\infty$ \nand $\\phi_2(L,\\infty) \\to \\infty$ logarithmically as $a \\to 0^+$. \nOne obtains: \n\\bb{ \nU_q(L,\\infty) \\simeq \n\\frac{1}{2\\sqrt{2} a} \n\\Gamma\\left(\\frac{q}{2}+1 \\right) \n\\left(\\frac{\\pi D}{\\nu a^2} \\right)^{\\frac{q}{2}} \n\\frac{1}{\\sqrt{\\ln{\\frac{L}{2a}}+{\\cal C}}} \n\\;,\\label{uqze3} \n}\\e \nand \n\\bb{ \n\\overline{U}(L,\\infty)\\simeq \n\\frac{1}{2\\sqrt{2} a} \n\\frac{1}{\\sqrt{\\ln{\\frac{L}{2a}}+{\\cal C}}} \n\\;,\\label{u0ze3} \n}\\e \nwith a logarithmically vanishing density of minima, and \nthe dependence on the \nsystem size in (\\ref{uqze3}) is not coupled to $q$. \n \n \n{\\em v)} $1 < z < 3$. Now both $\\phi_4$ and $\\phi_2$ \ndiverge as $a \\to 0^+$. \nEmploying (\\ref{cesaro}), yields: \n\\bb{ \nU_q(L,\\infty) \\simeq \n\\frac{1}{2 a} \n\\Gamma\\left(\\frac{q}{2}+1 \\right) \n\\left(\\frac{2 D}{\\pi \\nu} \\right)^{\\frac{q}{2}} \n\\left(\\frac{\\pi}{a}\\right)^{\\frac{q}{2}(5-z)} \n\\frac{\\sqrt{3-z}}{(5-z)^{\\frac{q+1}{2}}} \n\\;,\\label{uqzl3} \n}\\e \nand \n\\bb{ \n\\overline{U}(L,\\infty)\\simeq \n\\frac{1}{2 a} \\sqrt{ \\frac{3-z}{5-z}} \n\\;.\\label{u0zl3} \n}\\e \nNote, that in leading order, both $U_q(L,\\infty)$ and the density of minima \n$\\overline{U}(L,\\infty)$ become system size independent! The system size \ndependence comes in as {\\em corrections} on the order of $a/L$ and higher. \nThe fact that the efficiency of the massively parallel algorithm presented in \nSection III.B is not vanishing is due precisely to the above phenomenon: \nthe fluctuations of the time horizon in the steady state belong to the $z=2$ \nclass (Edwards-Wilkinson universality), and according to the results under \n{\\em iv)}, the density of minima (or the efficiency of the parallel \nalgorithm) converges to a non-zero constant, as $L \\to \\infty$, ensuring the \nscalability of the algorithm. An algorithm that would map into a $z \\geq 3$ \nclass would have a vanishing efficiency with increasing the number of \nprocessing elements. In particular, for $z=2$, one obtains from (\\ref{u0zl3}) \n$\\overline{U}(L,\\infty) \\simeq (a 2\\sqrt{3})^{-1} = 0.2886.../a$. Note that \nthe utilization we obtained is somewhat different from the discrete case \nwhich was $0.25$. This is due to the fact that this number is non-universal \nand it may show differences depending on the discretization scheme used, \nhowever it cannot be zero. \n \nAnother important conclusion can be drawn from the final results enlisted \nabove: at and below $z=5$, all the quantities {\\em diverge} when \n$a \\to 0^+$, and keep $L$ fixed. This means that the higher the resolution \nthe more details we find in the morphology, just as for an\ninfinitely wrinkled, or a fractal-like surface. We call this transition\naccross $z=5$ a `wrinkle' transition. As shown in the Introduction,\nwrinkledness can assume two phases depending on whether \nthe curve is a fractal or not and the transition between these two pases\nmay be conceived\nas a phase transition. \nHowever, one may be able to scale the system size $L$\nwith $a$ such that the quantities calculated will not diverge in this limit.\nThis is possible only in the regime $3< z <5$, when we impose: \n\\begin{equation} \n L^{z-3}a^{(q+1)(5-z)} = const. \n\\end{equation} \nThis shows that the rescaling cannot be done for all inverse temperatures\n$q$ at the same time. \nIn particular, for the density of minima and $z=4$, $La=const$. \n \n\\subsection{Scaling regime} \n \nIn order to obtain the temporal behavior of the extremal-point densities \nwe will use the Poisson summation formula (\\ref{ZP}) from Appendix B \non (\\ref{phina}). After simple changes of variables in the integrals \nthis leads to: \n\\ba{ \n\\phi_m(L,t) = \\frac{D}{\\nu L} \\left( \\frac{\\pi}{a}\\right)^{m-z} \n\\left[ 1-e^{-\\left( \\xi \\frac{\\pi}{a}\\right)^{z}}\\right] \n+\\frac{D}{\\pi\\nu} \\xi^{-(m-z+1)} \\int\\limits_{0}^{\\pi \\xi / a} \ndx\\; x^{m-z} \\left(1-e^{-x^z} \\right)+ && \\nonumber \\\\ \n\\frac{2D}{\\pi\\nu} \\xi^{-(m-z+1)} \\sum_{n=1}^{\\infty} \n\\int\\limits_{0}^{\\pi\\xi / a} dx\\; x^{m-z} \n\\cos{\\left( \\frac{L}{\\xi}nx\\right)} \n \\left(1-e^{-x^z} \\right) && \\label{phic} \n}\\e \nThis expression shows, that the scaling properties of the dynamics \nare determined by the dimensionless {\\em ratios} $L/\\xi$ and $\\xi/a$. \nThe scaling regime is defined by $a \\ll \\xi \\ll L$. \n \nAs we have seen in the previous section, $\\phi_m$ is convergent \nfor $z > m+1$ but diverges when $z \\leq m+1$, as $a \\to 0^+$. \nIn the convergent case, the lattice spacing $a$ can be taken as zero, and thus \nthe first term on the rhs. of (\\ref{phic}) vanishes and the time dependence \nof the infinite system-size piece of $\\phi_m$ (the first integral term \nin (\\ref{phic})) assumes \nthe {\\em clean} power-law behaviour of $t^{(z-m-1)/z}$ (with a {\\em positive} \nexponent). In the divergent case, however, the non-integral term of \n(\\ref{phic}) does not vanish, and the time-dependence will not be a clean \npower-law. \nEven the integral terms will present corrections to the \npower-law $t^{-(m+1-z)/z}$ (which has now a {\\em negative} exponent), since \nthe limits for integration contain $\\xi$. The first intergal on the rhs of \n(\\ref{phic}) for $z \\neq m+1$ can be calculated exactly: \n\\bb{ \n\\int\\limits_{0}^{\\pi \\xi / a}\\!\\! \ndx\\; x^{m-z} \\left(1-e^{-x^z} \\right) = \n\\frac{1}{m-z+1} \\left( \\frac{\\pi \\xi}{a}\\right)^{m-z+1}\\!\\! - \n\\frac{1}{z}\\Gamma\\left( \\frac{m-z+1}{z}\\right) \n+\\frac{1}{z}\\Gamma\\left( \\frac{m-z+1}{z},\\left( \\frac{\\pi \n\\xi}{a}\\right)^z\\right),\\;\\;z\\neq m+1 \\label{intt} \n}\\e \nwhere $\\Gamma(\\alpha,x)$ is the incomplete Gamma function. In our case \n$(\\pi \\xi / a)^z$ is a large number, and therefore we can use the asymptotic \nrepresentation of $\\Gamma(\\alpha,x)$ for large $x$, see \\cite{RG}, pp. \n951, equation 8.357. According to this, for large $x$, $\\Gamma(\\alpha,x) \\sim \nx^{\\alpha-1}e^{-x}$, i.e., it can become arbitrarily small, with an exponential \ndecay. This term can therefore be neglected from (\\ref{intt}), compared \nto the other two terms, even in the divergent case. \nInserting (\\ref{intt}) into (\\ref{phic}), we will see that also the \nnon-integral piece of (\\ref{phic}) can be neglected compared to the term \ngenerated by the first on the rhs of (\\ref{intt}), since in the \nscaling regime $a \\ll \\xi \\ll L$, and thus the ratio $a/L$ can be neglected \ncompared to $(m-z+1)^{-1}$. (This is needed only in the divergent \nregime, $z < m+1$.) Thus, one obtains: \n\\bb{ \n\\phi_m(L,t) \\simeq \\frac{D}{\\pi\\nu (m-z+1)} \\left( \\frac{\\pi}{a}\\right)^{m-z+1} \n\\!\\!-\\frac{D}{\\pi\\nu z} \n\\Gamma\\left( \\frac{m-z+1}{z} \\right) \n\\xi^{z-m-1} \\left[ 1- E_m\\left( \\frac{L}{\\xi}, \n\\frac{\\xi}{a}\\right) \\right],\\;\\;z\\neq m+1\\label{phic1} \n}\\e \nwhere \n\\bb{ \nE_m(\\lambda,\\rho) = \\frac{2z}{\\Gamma\\left( \\frac{m-z+1}{z} \\right)} \n\\sum_{n=1}^{\\infty} \n\\int\\limits_{0}^{\\pi\\rho} dx\\; x^{m-z} \n\\cos{( \\lambda n x)} \n \\left(1-e^{-x^z} \\right),\\;\\;z\\neq m+1 \n}\\e \nThe oscillating terms \ncondensed in $E_m$ will give the finite-size corrections, as long as \n$L/\\xi \\gg 1$. \n \nThe $z=m+1$ case (divergent) can also be calculated, however, instead \nof (\\ref{intt}) now we have: \n\\bb{ \n\\int\\limits_{0}^{\\pi \\xi / a}\\!\\! \n\\frac{dx}{x}\\;\\left(1-e^{-x^z} \\right) = \n\\ln{\\left( \\frac{\\pi \\xi}{a}\\right)} +\\frac{{\\cal C}}{z} -\\frac{1}{z} \n\\mbox{Ei}\\left( - \\left( \\frac{\\pi \\xi}{a}\\right)^z\\right) \n,\\;\\;z = m+1\\;. \\label{intl} \n}\\e \nwhere $\\mbox{Ei}(x)$ is the exponential integral function. According to \nthe large-$x$ expansion of the exponential integral function, see \\cite{RG}, \npp. 935, equation 8.215, $\\mbox{Ei}(-x) \\sim -x^{-1}e^{-x}$, it is vanishing \nexponentially fast, thus it can be neglected in the expression of $\\phi_m$ \nin the scaling limit: \n\\bb{ \n\\phi_m(L,t) \\simeq \n\\frac{D}{\\pi \\nu} \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)} + \n\\frac{D{\\cal C}}{\\pi\\nu z} + \n\\frac{D}{\\pi \\nu} F_m\\left( \\frac{L}{\\xi}, \n\\frac{\\xi}{a}\\right)\\;,\\;\\; z = m+1 \\label{phic2} \n}\\e \nwhere \n\\bb{ \nF_m(\\lambda,\\rho) = 2 \\sum_{n=1}^{\\infty} \n\\int\\limits_{0}^{\\pi\\rho} \\frac{dx}{x}\\; \n\\cos{( \\lambda n x)} \n \\left(1-e^{-x^z} \\right),\\;\\;z = m+1 \n}\\e \n \nThus in the scaling limit, the temporal behavior of $\\phi_m$ becomes \na logarithmic time dependence plus a constant, as long as $L/ \\xi \\gg 1$. \n \nObserve that for $z < m+1$ the first term \non the rhs. of (\\ref{phic1}), reproduces exactly the diverging term \n(as $a \\to 0^+$) of the steady-state expression \n(\\ref{phinb}) which can be seen after employing (\\ref{euler}) in (\\ref{phinb}). \nThis means that for $\\xi \\to \\infty$, $E_m(L/\\xi,\\xi/a)$ diverges slower than \n$\\xi^{m+1-z}$ (this is how the saturation occurs). \nSimilarly, for $z = m+1$ \nthe first term on the rhs of (\\ref{phic2}) \n(after replacing $\\xi$ with $L$ ) \nreproduces the diverging term (as $a \\to 0^+$) of the steady-state expression \n(\\ref{phinb}) which can be seen after employing \n(\\ref{cesaro}) in (\\ref{phinb}). This means, that in the saturation (or \nsteady-state) regime the remaining terms from (\\ref{phic2}) must behave as \n$const.+{\\cal O}(a/L)+\\ln{(L/\\xi)}$, as $\\xi \\to \\infty$ while keeping $L$ and \n$a$ fixed. \n \nJust as in the case of steady-state one has to distinguish 5 cases \ndepending on the values of $z$, with respect to the critical \nvalues 3 and 5. For the sake of simplicity of writing, we \nwill omit the arguments of $E_m(\\lambda,\\rho)$ and $F_m(\\lambda,\\rho)$. \n \n{\\em i)} $z>5$. We have: \n\\bb{ \n\\phi_m(L,t) = \\frac{D}{\\pi \\nu (z-m-1)} \\Gamma\\left(\\frac{m+1}{z}\\right) \n\\xi^{z-m-1} (1- E_m)\\;,\\;\\;\\;m=2,4 \\label{phic3} \n}\\e \nFrom Eqs. (\\ref{78a}), (\\ref{genU}) and (\\ref{genK}), it follows: \n\\bb{ \nU_q(L,t) = \\frac{1}{2\\pi} \\Gamma\\left( \\frac{q}{2}+1\\right) \n\\left( \\frac{2D}{\\pi \\nu}\\right)^{\\frac{q}{2}} \n\\left[\\frac{\\Gamma\\left(\\frac{5}{z}\\right)}{z-5} \\right]^{\\frac{q+1}{2}} \n\\left[ \\frac{z-3}{\\Gamma\\left(\\frac{3}{z}\\right)}\\right]^{\\frac{1}{2}} \n[\\xi(t)]^{-1-\\frac{q}{2}(z-5)} \n\\frac{( 1-E_4)^{\\frac{q+1}{2}}} \n{(1- E_2)^{\\frac{1}{2}}}\\;, \\label{tuqzg5} \n}\\e \n\\begin{equation} \n\\overline{U}(L,t) = \\frac{1}{2\\pi} \\sqrt{ \n\\frac{(z-3)\\Gamma\\left(\\frac{5}{z} \\right)} \n{(z-5)\\Gamma\\left(\\frac{3}{z} \\right)}}\\; \n\\left[ \\xi(t)\\right]^{-1} \n\\sqrt{\\frac{1-E_4}{1- E_2}}\\;, \\label{tu0zg5} \n\\end{equation} \nand \n\\begin{equation} \n \\overline{K}(L,t)=\\sqrt{\\frac{D\\Gamma\\left(\\frac{5}{z} \\right)}{2\\nu(z-5)}} \n\\left[ \\xi(t)\\right]^{\\frac{z-5}{2}} \\sqrt{1-E_4}\\; \n \\label{tkzg5} \n \\end{equation} \nand therefore the time-behaviour is a clean power-law: \n$U_q(L,t)$ {\\em decays} as $\\sim t^{-[2+q(z-5)]/2z}$, $\\overline{U}(L,t) \\sim \nt^{-1/z}$, and $\\overline{K}(L,t)$ diverges as $\\sim t^{(z-5)/2z}$, for $L/\\xi \n\\gg 1$. \n \n{\\em ii)} $z=5$. In this case $\\phi_4$ takes the form (\\ref{phic2}) but \n$\\phi_2$ is still given by (\\ref{phic1}). The quantities of interest \nbecome: \n\\bb{ \nU_q(L,t) \\simeq \\frac{\\Gamma\\left( \\frac{q}{2}+1\\right)}{2\\pi} \n\\left( \\frac{2D}{\\pi \\nu}\\right)^{\\frac{q}{2}} \n\\sqrt{\\frac{2}{\\Gamma\\left( \\frac{3}{5}\\right)}}\\;\\xi^{-1} \n\\left[ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}\\right]^{\\frac{q+1}{2}} \n\\frac{ \\left\\{ 1+ \\left[ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}\\right]^{-1} \n\\left(\\frac{{\\cal C}}{5}+F_4 \\right) \\right\\}^{\\frac{q+1}{2}} \n}{\\sqrt{1-E_2}}\\;, \\label{tuqze5} \n}\\e \n\\bb{ \n\\overline{U}(L,t) \\simeq \n\\frac{1}{2\\pi} \\sqrt{\\frac{2}{\\Gamma\\left( \\frac{3}{5}\\right)}}\\; \n\\xi^{-1} \\sqrt{ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}}\\; \n\\sqrt{\\frac{1+ \\left[ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}\\right]^{-1} \n\\left(\\frac{{\\cal C}}{5}+F_4 \\right)} \n{\\sqrt{1-E_2}}}\\;,\\label{tu0ze5} \n}\\e \nand \n\\begin{equation} \n \\overline{K}(L,t)=\\sqrt{\\frac{D}{2\\nu}} \n\\sqrt{ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}}\\; \n\\sqrt{1+ \\left[ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}\\right]^{-1} \n\\left(\\frac{{\\cal C}}{5}+F_4 \\right)} \n\\; \\label{tkze5} \n \\end{equation} \nOne can observe that the leading temporal behaviour has logarithmic \ncomponent due to the borderline situation: $U_q(L,t)$ {\\em decays} as \n$\\sim t^{-1/5}(\\ln{t})^{(q+1)/2}$, $\\overline{U}(L,t) \\sim \nt^{-1/5}(\\ln{t})^{1/2}$, and $\\overline{K}(L,t)$ {\\em diverges} as $\\sim \n(\\ln{t})^{1/2}$. \n \n{\\em iii)} $3<z<5$. \n\\bb{ \nU_q(L,t) \\simeq \\frac{\\Gamma\\left( \\frac{q}{2}+1\\right)}{2\\pi} \n\\left( \\frac{2D}{\\pi \\nu}\\right)^{\\frac{q}{2}} \n\\sqrt{\\frac{z-3}{\\Gamma\\left( \\frac{3}{z}\\right)}} \n(5-z)^{-\\frac{q+1}{2}} \\left( \n\\frac{\\pi}{a}\\right)^{\\frac{q+1}{2}(5-z)} \n\\xi^{-\\frac{z-3}{2}} \\; \n\\frac{ \\left[ 1-\\Gamma\\left( \\frac{5}{z}\\right) \n\\left( \\frac{a}{\\pi \\xi}\\right)^{5-z} (1-E_4) \\right]^ \n{\\frac{q+1}{2}}}{\\sqrt{1-E_2}}\\;, \\label{tuq3lzl5} \n}\\e \n\\bb{ \n\\overline{U}(L,t) \\simeq \n\\frac{1}{2\\pi} \\sqrt{\\frac{z-3}{(5-z) \n\\Gamma\\left( \\frac{3}{z}\\right)}}\\; \n\\left( \\frac{\\pi }{a}\\right)^{\\frac{5-z}{2}} \n\\xi^{-\\frac{z-3}{2}} \\; \n\\sqrt{\\frac{1-\\Gamma\\left( \\frac{5}{z}\\right) \n\\left( \\frac{a}{\\pi \\xi}\\right)^{5-z} (1-E_4) } \n{1-E_2}} \n\\;,\\label{tu03lzl5} \n}\\e \n\\bb{ \n\\overline{K}(L,t) \\simeq \n\\sqrt{\\frac{D}{2\\nu (5-z)}} \\left( \n\\frac{\\pi }{a} \n\\right)^{\\frac{5-z}{2}} \n\\sqrt{1-\\Gamma\\left( \\frac{5}{z}\\right) \n\\left( \\frac{a}{\\pi \\xi}\\right)^{5-z} (1-E_4) } \\label{tk3lzl5} \n}\\e \n\n\\begin{figure}\n\\hspace*{3.5cm}\\epsfxsize=4 in \n\\epsfbox{uvt.ps} \n\\vspace*{0.5cm} \n\\caption{Density of minima for the larger curvature model as a function \nof time (the nr of deposited layers), for two system sizes, $L=100$ \n(diamonds) and $L=120$ (crosses). The\nstraight line corresponds to the behavior $t^{-1/8}$.} \n\\end{figure} \n\nAn important conclusion that can be drawn from these \nexpressions is that below $z=5$, the leading time-dependence \nof the partition function $U_q(L,t)$ becomes {\\em independent} \nof the inverse temperature $q$ and it presents a clean power-law \ndecay $\\sim t^{-(z-3)/2z}$ which is the same also for \n$\\overline{U}(L,t)$. In particular, for $z=4$ this means a $t^{-1/8}$\ndecay which is very well verified by the larger curvature model from\nSection III.C, see Figure 5. \nAlso notice from Eq. (\\ref{tu03lzl5}) that the leading term is system\nsize independent. And indeed, this property is also in a very good agreement\nwith the numerics on the larger curvature model from Figure 5,\nwhere the two data sets for $L=100$ and $L=120$ practically coincide.\n\nSince the mean curvature depends on $\\phi_4$, only, for all \ncases below $z=5$ the dependence is given by the same \nformula (\\ref{tk3lzl5}) (just need to replace the corresponding \nvalue for $z$). \n\n \n{\\em iv)} $z=3$. This is another borderline situation, \nthe corresponding expressions are found easily: \n\\bb{ \nU_q(L,t) \\simeq \\frac{\\Gamma\\left( \\frac{q}{2}+1\\right)} \n{2\\sqrt{2}\\pi} \\left( \\frac{2D}{\\pi \\nu}\\right)^{\\frac{q}{2}} \n\\left( \\frac{\\pi}{a}\\right)^{q+1} \n\\left[ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}\\right]^{-\\frac{1}{2}} \n\\frac{ \\left[ \n1-\\Gamma\\left( \\frac{5}{3}\\right) \n\\left( \\frac{a}{\\pi \\xi}\\right)^2 (1-E_4)\\right]^{\\frac{q+1}{2}}} \n{ \\sqrt{ 1+ \\left[ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}\\right]^{-1} \n\\left(\\frac{{\\cal C}}{3}+F_2 \\right) }}\\;, \\label{tuqze3} \n}\\e \n\\bb{ \n\\overline{U}(L,t) \\simeq \\frac{1}{2\\sqrt{2}a} \n\\left[ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}\\right]^{-\\frac{1}{2}} \n\\sqrt{\\frac{1-\\Gamma\\left( \\frac{5}{3}\\right) \n\\left( \\frac{a}{\\pi \\xi}\\right)^2 (1-E_4)} \n{1+ \\left[ \\ln{\\left( \\frac{\\pi \\xi}{a}\\right)}\\right]^{-1} \n\\left(\\frac{{\\cal C}}{3}+F_2 \\right)}}\\; \\label{tu0ze3} \n}\\e \nand the leading time dependences are: $U_q(L,t) \\sim \n(\\ln{t})^{-1/2}$, $\\overline{U}(L,t) \n\\sim (\\ln{t})^{-1/2}$. \n \n{\\em v)} $1<z<3$. \n\\bb{ \nU_q(L,t) \\simeq \\frac{\\Gamma\\left( \\frac{q}{2}+1\\right)} \n{2\\sqrt{2}\\pi} \\left( \\frac{2D}{\\pi \\nu}\\right)^{\\frac{q}{2}} \n\\sqrt{\\frac{3-z}{(5-z)^{q+1}}} \\left( \n\\frac{\\pi}{a}\\right)^{1+\\frac{q}{2}(5-z)} \n\\frac{ \\left[ 1-\\Gamma\\left( \\frac{5}{z}\\right) \n\\left( \\frac{a}{\\pi \\xi}\\right)^{5-z} (1-E_4) \\right]^{\\frac{q+1}{2}} } \n{\\sqrt{1-\\Gamma\\left( \\frac{3}{z}\\right) \n\\left(\\frac{a}{\\pi \\xi}\\right)^{3-z} (1-E_2)}}\\;,\\label{tuqzl3} \n}\\e \n\\bb{ \n\\overline{U}(L,t) \\simeq \n\\frac{1}{2a} \\sqrt{\\frac{3-z}{(5-z)}}\\; \n\\sqrt{\\frac{1-\\Gamma\\left( \\frac{5}{z}\\right) \n\\left( \\frac{a}{\\pi \\xi}\\right)^{5-z} (1-E_4) } \n{1-\\Gamma\\left( \\frac{3}{z}\\right) \n\\left(\\frac{a}{\\pi \\xi}\\right)^{3-z} (1-E_2)}} \n\\;,\\label{tu0zl3} \n}\\e \nIn this case the partition function and the density of minima \nall converge to a constant which in leading order is independent \nof the system size. The density of minima was shown in Section \nII to have this property in the steady-state. Here we see not only that \nbut also the fact that {\\em all} $q$-moments show the same behavior, and \neven more, the time behavior before reaching the steady-state \nconstant is not a clean power-law, but rather a decaying correction in the \napproach to this constant. The leading term in the temporal correction \nis of $\\sim t^{-(3-z)/z}$ and the next-to-leading has \n$\\sim t^{-(5-z)/z}$. \n\n\n\n\\section{Conclusions and outlook}\n\nIn summary, based on the analytical results presented, a \nshort wavelength based analysis of interface fluctuations can provide \nus with novel type of information and give an alternative description \nof surface morphologies. This analysis gives a more detailed characterization \nand can be used to distinguish interfaces that are `fuzzy' from those \nthat locally appear to be smooth, and the central quantities, the \nextremal-point densities are numerically and analytically accessible. \n The partition function-like formalism enables us to access a wide range \nof $q$-momenta of the local curvatures distribution. In the case \nof the stochastic evolution equations studied we could exactly relate these \n$q$-momenta to the structure function of the process via the \nsimple quantities $\\phi_2$ and $\\phi_4$. The wide spectrum of results \naccessed through this technique shows the richness of short wavelength \nphysics. This physics is there, and the long wavelength approach just \nsimply cannot reproduce it, but instead may suggest an oversimplified \npicture of the reality. For example, the MPEU model has been shown to belong \nin the steady state to the EW universality class, however, \n{\\em it cannot be described exactly} by the EW equation in {\\em all} \nrespects, not even in the steady-state! For example, the utilization (or \ndensity of minima) of the MPEU model is 0.24641 which for the \nEW model on a lattice is 0.25. Also, if one just simply looks at the \nsteady-state configuration, one observes high {\\em skewness} \nfor the MPEU model \\cite{KTNR},\nwhereas the EW is completely up-down symmetric. \nThis can also be\nshown by comparing the calculated two-slope correlators. \nFor a number of models that belong to the KPZ equation \nuniversality class, this broken-symmetry property vs. the EW case has been\nextensively investigated by Neergard and den Nijs \\cite{NN}. \nThe difference on the short wavelength \nscale between two models that otherwise belong to the same universality \nclass lies in the existence of irrelevant operators (in the RG sense). \nAlthough these operators do not change universal properties, \nthe quantities associated with them may be of very practical interest. \nThe parallel computing example shows that the fundamental \nquestion of algorithmic scalability is answered based on the \nfact that the simulated time horizon in the steady state belongs to the EW \nuniversality class, thus it has a {\\em finite} density of local minima. \nThe actual value of the density of local minima in the thermodynamic limit, \nhowever, strongly depends on the details of the microscopics, which in \nprinciple can be described in terms of irrelevant operators \\cite{NN}. \n \nThe extremal-point densities introduced in the present paper may actually \nhave a broader application than stochastic surface fluctuations. \nThe main geometrical characterization of fractal curves is based on the \nconstruction of their Haussdorff-Besikovich dimension, or the \n`box-counting' dimension: one covers the set with small boxes of linear size \n$\\epsilon$ and then track the divergence of the number of \nboxes needed to cover \nin a minimal way the whole set as $\\epsilon$ is lowered to zero. For example, \na smooth line in the plane has a dimension of unity, but the Weierstrass curve \nof (\\ref{weier}) has a dimension of $\\ln{b}/\\ln{a}$ (for $b>a$). The actual\nlength of a fractal curve whose dimension is larger than unity will diverge\nwhen $\\epsilon \\to 0^+$. The total length at a given resolution $\\epsilon$ is\na {\\em global} property of the fractal, it does not tell us about the way `it\ncurves'. The novel measure we propose in (\\ref{minima}) is meant to\ncharacterize the distribution of a local property of the curve, its\n{\\em bending} which in turn is a measure of the curve's wrinkledness. For\nsimplicity we formulated it for functions, i.e., for curves which are\nsingle-valued in a certain direction. This can be remedied and generalized by\nintroducing a parametrization $\\gamma \\in [0,1]$ of the curve, and then \nplotting the local {\\em curvature} vs. this parameter $K(\\gamma)$. The plot\nwill be a single valued function on which now (\\ref{minima}) is easily\ndefined. \n\nOther desirable extensions of the present technique are: 1) to include a \nstatistical description of the degeneracies of higher order, and 2) to repeat \nthe analysis for higher (such as $d=2$) substrate dimensions. The latter \nis promising an even richer spectrum of novelties, since in higher dimensions \nthere is a plethora of singular points ($\\nabla f ={\\bf 0} $) which are \nclassified by the eigenvalues of the Hessian matrix of the function in the \nsingular point. Deciphering the statistical behaviour of these various \nsingularities for randomly evolving surfaces is an interesting challenge. \nThe studies performed by Kondev and Henley \\cite{Kondev} on the distribution of\ncontours on random Gaussian surfaces should come to a good aid in\nachieving this goal.\nIn particular we may find the method developed here useful in studying the\nspin-glass ground state, and the spin-glass transition problem. \nAnd at last but not the least, we invite the reader to consider instead of the\nLangevin equations studied here, noisy wave equations, with a second\nderivative of the time component, or other stochastic evolution equations. \n \n \n \n \n\\section*{Acknowledgements} \n \nWe thank S. Benczik, M.A. Novotny, P.A. Rikvold, \nZ. R\\'acz, B. Schmittmann, T. T\\'el, E.D. \nWilliams, and I. \\v{Z}uti\\'{c} for stimulating discussions. This work was \nsupported by NSF-MRSEC at University of Maryland, \nby DOE through SCRI-FSU, and \nby NSF-DMR-9871455. \n \n \n\\appendix %A \n\\section{$\\langle\\Theta(-x_1)\\Theta(x_2)\\rangle$ for general coupled Gaussian \nvariables} \nThe expression we derive in this appendix, despite its simplicity, is probably \nthe most important one concerning the extremal-point densities of \none-dimensional Gaussian interfaces on a lattice. If the correlation matrix \nfor two possibly coupled Gaussian variables is given by \n\\begin{eqnarray} \n\\langle x_{1}^{2}\\rangle = \\langle x_{1}^{2}\\rangle & = & d >0 \\\\ \n\\langle x_{1}x_{2}\\rangle & = & c \\nonumber \n\\end{eqnarray} \nthen the distribution follows as \n\\begin{equation} \nP(x_{1},x_{2}) = \\frac{1}{2\\pi\\sqrt{\\cal D}} \n\\exp\\left\\{ -\\frac{1}{2{\\cal D}} \\left(d x_{1}^{2}+d x_{2}^{2} \n- 2cx_{1}x_{2}\\right) \\right\\} = \\frac{1}{2\\pi\\sqrt{\\cal D}} \n\\exp\\left\\{ -\\frac{d}{2{\\cal D}} \\left(x_{1}^{2}+x_{2}^{2} \n- 2\\frac{c}{d}x_{1}x_{2}\\right) \\right\\} \\;, \n\\end{equation} \nwhere ${\\cal D}\\equiv d^2 - c^2>0$. We aim to find the average of the \nstochastic variable $u=\\Theta(-x_1)\\Theta(x_2)$: \n\\begin{equation} \n\\langle u\\rangle = \\langle\\Theta(-x_1)\\Theta(x_2)\\rangle = \n\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}d\\!x_{1} d\\!x_{2} \\, \n\\Theta(-x_1)\\Theta(x_2) P(x_{1},x_{2}) \n\\end{equation} \nwhich is simply the total weight of the density $P(x_{1},x_{2})$ in the \n$x_1 < 0$, $x_2 > 0$ quadrant. If $c = 0$, the density is \nisotropic, and $\\langle u\\rangle=1/4$. In the general case it is \nconvenient to find a new set of basis vectors, where the probability density \nis isotropic (of course the shape of the original quadrant will \ntransform accordingly). Introducing the following linear transformation \n\\begin{eqnarray} \nx_{1} & = & \\sqrt{\\frac{{\\cal D}}{2}}\\left( \n\\frac{y_1}{\\sqrt{d+c}} + \\frac{y_2}{\\sqrt{d-c}} \\right) \\\\ \nx_{2} & = & \\sqrt{\\frac{{\\cal D}}{2}}\\left( \n-\\frac{y_1}{\\sqrt{d+c}} + \\frac{y_2}{\\sqrt{d-c}} \\right) \\;, \\nonumber \n\\end{eqnarray} \nand exploiting that $\\Theta(\\lambda x)=\\Theta(x)$ for $\\lambda > 0$ \nwe have \n\\begin{equation} \n\\langle u\\rangle = \n\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}d\\!y_{1} d\\!y_{2}\\, \n\\Theta\\left( -\\frac{y_1}{\\sqrt{d+c}} - \\frac{y_2}{\\sqrt{d-c}} \\right) \n\\Theta\\left( -\\frac{y_1}{\\sqrt{d+c}} + \\frac{y_2}{\\sqrt{d-c}} \\right) \n\\frac{1}{2\\pi}\\exp\\left\\{-\\frac{1}{2}(y_1^2 + y_2^2)\\right\\} \n\\end{equation} \nNow the probability density for the new variables, $y_1,y_2$, is isotropic, \nand $\\langle u\\rangle=\\theta/(2\\pi)$, \nwhere $\\theta$ is the angle enclosed by the following two unit vectors: \n\\begin{equation} \n{\\bf v}_1 = \\frac{1}{\\sqrt{2d}} \\left(\\begin{array}{r} \n-\\sqrt{d+c} \\\\ \\sqrt{d-c} \\end{array}\\right) \\;,\\;\\;\\; \n{\\bf v}_2 = \\frac{1}{\\sqrt{2d}} \\left(\\begin{array}{r} \n-\\sqrt{d+c} \\\\ -\\sqrt{d-c} \\end{array}\\right) \\;. \n\\end{equation} \nFrom their dot product one obtains \n\\begin{equation} \n\\cos(\\theta) = \\frac{{\\bf v}_1\\cdot{\\bf v}_2}{|{\\bf v}_1||{\\bf v}_2|} = \n\\frac{c}{d}. \n\\end{equation} \nand, thus, for $\\langle u\\rangle$: \n\\begin{equation} \n\\langle u\\rangle = \\frac{1}{2\\pi}\\arccos\\left(\\frac{c}{d}\\right) \\;. \n\\end{equation} \n \n \n \n\\section{Poisson summation formulas} %B \n \n In this Appendix we recall the well-known Poisson summation formula \nand adapt it for functions with finite support in $\\R$. In the theory \nof generalized functions \\cite{Jones} the following identity is proven: \n\\begin{equation} \n\\sum_{m=-\\infty}^{\\infty} \\delta(x-m) = \\sum_{m=-\\infty}^{\\infty} \ne^{2\\pi i m x} \\label{identity} \n\\end{equation} \nLet $f: [\\alpha,\\beta] \\to \\R$ be a continuous function with continuous \nderivative on the interval $[\\alpha,\\beta]$. Multiply Eq. (\\ref{identity}) on \nboth sides with $f(x)$, then integrate both sides from $\\alpha$ to $\\beta$. In \nthe evaluation of the left hand side we have to pay attention to whether any, \nor both the numbers $\\alpha$ and $\\beta$ are integers, or non-integers. In the \ninteger case the contribution of the end-point is calculated via the identity: \n\\begin{equation} \n\\int\\limits_{n}^{n+r} \ndx\\;\\delta(x-n)\\;f(x)=\\frac{1}{2}\\;f(n)\\;,\\;\\;\\;\\forall\\;r>0 \\label{ident2} \n\\end{equation} \nAssuming that $f$ is absolutely integrable if $\\beta=\\infty$, and choosing \n$\\alpha=0$, the classical Poisson summation formula is obtained: \n\\begin{equation} \n\\sum_{n=0}^{\\infty} f(n)=\\frac{1}{2}f(0)+\\int\\limits_{0}^{\\infty} dx\\;f(x)+ \n2\\sum_{m=1}^{\\infty} \n\\int\\limits_{0}^{\\infty} dx\\;f(x)\\;\\cos(2\\pi m x) \\label{CP} \n\\end{equation} \nLet us write also explicitely out the case when both $\\alpha$ and $\\beta$ are \nintegers: \\begin{equation} \n\\sum_{n=\\alpha}^{\\beta}f(n) = \\frac{1}{2}[f(\\alpha) + f(\\beta)]+ \n\\int\\limits_{\\alpha}^{\\beta}dx\\;f(x) +2\\sum_{m=1}^{\\infty} \n\\int\\limits_{\\alpha}^{\\beta} dx\\;f(x)\\;\\cos(2\\pi m x)\\;, \n\\;\\;\\;\\mbox{when}\\;\\;\\;\\alpha,\\beta \\in \\Z \\label{ZP} \n\\end{equation} \nNext we apply these equations to give an exact closed expression for \nthe slope correlation function for {\\em finite} $L$ [eq. (\\ref{C_L})]: \n\\begin{equation} \nC^{\\phi}_L(l)= \n\\frac{D}{L}\\sum_{n=1}^{L-1} \\frac{e^{i\\left( \\frac{2\\pi n}{L}\\right) \nl}}{\\nu+2\\kappa \\left[ 1-\\cos\\left( \\frac{2\\pi n}{L}\\right) \\right]} \n\\end{equation} \nwhere $l \\in \\{0,1,2,..,L-1 \\}$, $\\nu,\\kappa \\in \\R^+$. Let us denote \n\\begin{equation} \na=\\frac{2 \\kappa}{\\nu+2\\kappa}\\;. \n\\end{equation} \nWe have $|a| <1$, and \n\\begin{equation} \nC^{\\phi}_L(l)=\\frac{Da}{2\\kappa L}\\sum_{n=1}^{L-1} \\frac{e^{i\\left( \\frac{2\\pi \nn}{L}\\right) l}}{1-a\\cos\\left( \\frac{2\\pi n}{L}\\right)} \n\\end{equation} \nIn order to apply the Poisson summation formula (\\ref{ZP}), we introduce \nthe function: \n\\begin{equation} \nf(x)=\\frac{a}{2\\kappa L}\\sum_{n=1}^{L-1} \\frac{e^{i\\left( \\frac{2\\pi \nx}{L}\\right) l}}{1-a\\cos\\left( \\frac{2\\pi x}{L}\\right)} \\;, \n\\;\\;\\;1\\leq x \\leq L-1 \n\\end{equation} \nand identify in (\\ref{ZP}) $\\alpha \\equiv 1$ and $\\beta \\equiv L-1$. \nThe non-integral terms of (\\ref{ZP}) give: \n\\begin{equation} \n\\frac{1}{2}[f(1) + f(L-1)] = \\frac{a}{2 \\kappa L}\\;\\frac{\\cos\\left( \n\\frac{2\\pi}{L}l\\right)}{1- a\\cos\\left( \n\\frac{2\\pi}{L}l\\right)} \\label{first} \n\\end{equation} \nThe next term becomes: \n\\begin{equation} \n\\int\\limits_{1}^{L-1}dx\\;f(x)=\\frac{a}{2\\kappa \\sqrt{1-a^2}}\\; \n\\left( \\frac{1-\\sqrt{1-a^2}}{a}\\right)^l-\\frac{a}{2 \\pi \\kappa} \n\\int\\limits_{0}^{2 \\pi / L}dx\\; \\frac{\\cos{xl}}{1-a\\cos{x}} \n\\label{second} \n\\end{equation} \nwhere during the evaluation of the integral we made a simple \nchange of variables and used a well-known integral from \nrandom walk theory \\cite{Hughes}, \\cite{RG}: \n\\begin{equation} \n\\int\\limits_{-\\pi}^{\\pi}dx\\;\\frac{e^{ixl}}{1-a\\cos{x}}= \n\\frac{2 \\pi }{ \\sqrt{1-a^2}}\\; \n\\left(\\frac{1-\\sqrt{1-a^2}}{a}\\right)^l\\;,\\;\\;\\;l\\geq 0 \\label{rwalk} \n\\end{equation} \nThe sum over the integrals in (\\ref{ZP}) can also be \nevaluated, and one obtains: \n\\begin{equation} \n2\\sum_{n=1}^{\\infty} \n\\int\\limits_{1}^{L-1} dx\\;f(x)\\cos(2\\pi n x) = \n\\frac{a \\left(b^l + b^{-l} \\right)}{2 \\kappa \\sqrt{1-a^2}}\\;\\frac{b^L}{1-b^L} \n-\\frac{a}{2 \\pi \\kappa} \\sum_{n=1}^{\\infty} \\int\\limits_{-2 \\pi /L}^{2 \\pi /L} \ndx\\;\\cos(nLx)\\;\\frac{e^{ilx}}{1-a\\cos{x}} \\label{akarmi} \n\\end{equation} \nwhere \n\\begin{equation} \nb=\\frac{1-\\sqrt{1-a^2}}{a}\\;,\\;\\;\\;\\mbox{and}\\;\\;\\;|b|<1 \n\\end{equation} \nTo compute the sum on the rhs of (\\ref{akarmi}) we recall another \nidentity from the theory of generalized functions (see Ref. \\cite{Jones}, \npage 155): \n\\begin{equation} \n\\sum_{n=1}^{\\infty} e^{i n x} = \\pi \\sum_{m=-\\infty}^{\\infty} \n\\delta(x-2m\\pi) + \n\\frac{i}{2} ctg\\left( \\frac{x}{2}\\right)-\\frac{1}{2} \\label{identity'} \n\\end{equation} \nCombining (\\ref{identity'}) and identity (\\ref{identity}), one obtains: \n\\begin{equation} \n\\sum_{n=1}^{\\infty} \\cos(nx) = \\pi \\sum_{m=-\\infty}^{\\infty} \n\\delta(x-2m\\pi) + \\frac{1}{2} \\label{identity''} \n\\end{equation} \nPeforming the sum over $n$ directly in the rhs of (\\ref{akarmi}) via \n(\\ref{identity''}), yields: \n\\begin{equation} \n-\\frac{a}{2 \\pi \\kappa} \\sum_{n=1}^{\\infty} \\int\\limits_{-2 \\pi /L}^{2 \\pi /L} \ndx\\;\\frac{\\cos(nLx)e^{ilx}}{1-a\\cos{x}} = \n-\\frac{a}{2 \\kappa L}\\sum_{m=-\\infty}^{\\infty} \\int\\limits_{-2\\pi}^{2\\pi} \ndy\\;\\frac{e^{ily}\\delta (y-2m\\pi) }{1-a\\cos{y}} + \\frac{a}{2\\pi \\kappa} \n\\int\\limits_{0}^{2\\pi /L} dx\\;\\frac{\\cos{lx}}{1-a\\cos{x}} \\label{tt} \n\\end{equation} \nOnly $m=\\pm 1, 0$ contribute in (\\ref{tt}). With the help of \n(\\ref{ident2}): \n\\begin{equation} \n-\\frac{a}{2 \\pi \\kappa} \\sum_{n=1}^{\\infty} \\int\\limits_{-2 \\pi /L}^{2 \\pi /L} \ndx\\;\\frac{\\cos(nLx)e^{ilx}}{1-a\\cos{x}} = \n-\\frac{a}{2 \\kappa L}\\left\\{\\frac{1}{1-a}+ \n\\frac{\\cos\\left(\\frac{2\\pi}{L}l\\right)}{1-a\\cos\\left(\\frac{2\\pi}{L}\\right)} \n\\right\\} + \\frac{a}{2\\pi \\kappa} \n\\int\\limits_{0}^{2\\pi /L} dx\\;\\frac{\\cos{lx}}{1-a\\cos{x}}\\label{ttt} \n\\end{equation} \nUsing (\\ref{ttt}) in (\\ref{akarmi}), we can add the result to the \nrest of the contributions (\\ref{first}) and (\\ref{second}) to obtain \nthe final expression [eq. (\\ref{C_phi})] after the cancellations. \n \n \n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n \n\\begin{references} \n \n\\bibitem{KTNR} G.\\ Korniss, Z. Toroczkai, M.\\ A.\\ Novotny, and P.\\ A.\\ Rikvold, \nPhys. Rev. Lett., {\\bf 84}, 1351 (2000). \n \n\\bibitem{GLNW} A.G. Greenberg, B.D. Lubachevsky, D.M. \nNicol, and P.E. 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ACM {\\bf 33}, 30 (1990). \n \n\\bibitem{Luba} B.\\ D.\\ Lubachevsky, Complex Systems {\\bf 1}, 1099 (1987); \nJ. Comput. Phys. {\\bf 75}, 103 (1988). \n \n\\bibitem{single_step} P.\\ Meakin, P.\\ Ramanlal, L.\\ M.\\ Sander, and \nR.\\ C.\\ Ball, Phys. Rev. A {\\bf 34}, 5091 (1986); \nM.\\ Plischke, Z.\\ R\\'acz, and D.\\ Liu, Phys. Rev. B {\\bf 35}, 3485 (1987);\nL.M. Sander and H. Yan, Phys. Rev. A {\\bf 44}, 4885 (1991).\n \n\\bibitem{KSDS} J. M. Kim and S. Das Sarma, Phys. Rev. Lett. {\\bf 72}, 2903\n(1994)\n\n\\bibitem{Krug2} J. Krug, Phys. Rev. Lett. {\\bf 72}, 2907 (1994)\n\n\\bibitem{ALF} J.G. Amar, P-M Lam, and F. Family, Phys. Rev. E {\\bf 47}, 3242\n(1993); M. Siegert and M. Plischke, Phys. Rev. Lett. {\\bf 68}, 2035 (1992).\n\n\\bibitem{Spitzer} F.\\ Spitzer, Adv. Math. {\\bf 5}, 246 (1970). \n \n\\bibitem{itzi} C. Itzykson and J-M. Drouffe, {\\em Statistical Field Theory}, \nCambridge University Press, 1989, vol. 1. \n \n\\bibitem{Jones} D.S. Jones, {\\em The theory of generalized functions}, page \n153 \n \n\\bibitem{RG} I.S. Gradshteyn and I.M. Ryzhik, {Table of Integrals, Series, and \nProducts}, Ed. Alan Jeffrey, Academic Press, 1994. \n \n\\bibitem{NN} J. Neergard and M. den Nijs , J.Phys.A {\\bf 30}, 1935 (1997). \n\n\\bibitem{Kondev} J. Kondev and C.L. Henley, Phys. Rev. Lett. 74, 4580 (1995).\n \n\\end{references} \n \n\n\\end{document} \n \n \n"
}
] |
[
{
"name": "cond-mat0002143.extracted_bib",
"string": "\\bibitem{KTNR} G.\\ Korniss, Z. Toroczkai, M.\\ A.\\ Novotny, and P.\\ A.\\ Rikvold, \nPhys. Rev. Lett., {\\bf 84}, 1351 (2000). \n \n\n\\bibitem{GLNW} A.G. Greenberg, B.D. Lubachevsky, D.M. \nNicol, and P.E. Wright, {\\em Proceedings, 8th Workshop on \nParallel and Distributed Simulation (PADS '94)}, Edinburgh, \nUK, (1994) p. 187. \n \n\n\\bibitem{KNR} \nG.\\ Korniss, M.\\ A.\\ Novotny, and P.\\ A.\\ Rikvold, \nJ. Comput. Phys., {\\bf 153}, 488 (1999). \n\n\n\\bibitem{http} See for example \nhttp://www.csuchico.edu/psy/BioPsych/neurotransmission.html\n \n\n\\bibitem{alb} R. Albert, H. Jeong, and A-L. Barab\\'asi , \nNature {\\bf 401} 130-131 (1999); A-L. Barab\\'asi, R. Albert, and H. \nJeong, Physica A {\\bf 272} 173-187 (1999); \nA-L. Barab\\'asi and R. Albert, Science {\\bf 286} 509-512 (1999). \n \n\n\\bibitem{Hardy} G.H. Hardy, Trans.Am.Math.Soc. {\\bf 17} 301 (1916). \n\n\n\\bibitem{Hunt} B.R. Hunt, Proc. Amer. Math. Soc. {\\bf 126} 791 (1998).\n\n\n\\bibitem{Hughes} B. D. Hughes, {Random Walks and Random Environments, \nVolume 1: Random Walks}, Clarendon Press, Oxford, 1995. \n \n\n\\bibitem{Ruelle} D. Ruelle, {\\em Thermodynamic Formalism} (Addison-Wesley, \nReading, 1978); T. Bohr, and D. Rand, Physica D {\\bf 25} 387 (1987). \n \n\n\\bibitem{Giesen} M. Giesen, and G.S. Icking-Konert, Surf.Sci. {\\bf 412/413} \n645 (1998). \n \n\n\\bibitem{PT} Z. Toroczkai, and E.D. Williams, Phys.Today {\\bf 52}(12) 24 \n(1999). \n \n\n\\bibitem{Ted} S.V. Khare and T.L. Einstein, Phys.Rev.B {\\bf 57} \n 4782 (1998); \n \n\n\\bibitem{EDW} H-C Jeong, and E.D. Williams, Surf.Sci.Rep. {\\bf 34} 171 (1999). \n \n\n\\bibitem{Barabasi} A-L.\\ Barab\\'asi and H.\\ E.\\ Stanley, \n{\\it Fractal Concepts in Surface Growth} \n(Cambridge University Press, Cambridge, 1995). \n \n\n\\bibitem{MBE} J. Villain, J.Phys. I {\\bf 1} 19 (1991); \nZ.W. Lai and S. Das Sarma, Phys. Rev.Lett. {\\bf 66} 2348 (1991);\n D.D. Vvedensky, A. Zangwill, C.N. Luse, and M.R. Wilby, \nPhys.Rev.E {\\bf 48} 852 (1993). \n \n\n\\bibitem{DT} S. Das Sarma and P. Tamborenea, Phys.Rev.Lett. {\\bf 66}, \n325 (1991); D.E. Wolf and J. Villain, Europhys.Lett. {\\bf 13}, 389 (1990). \n \n\n\\bibitem{Majaniemi} S. Majaniemi, T. Ala-Nissila, and J. Krug, Phys.Rev. B \n{\\bf 53}, 8071 (1996) \n \n\n\\bibitem{Krug} J. Krug, Adv. Phys., {\\bf 46}, 137 (1997); S. Das Sarma, C.J.\nLanczycki, R. Kotlyar, and S. V. Ghaisas, Phys. Rev. E {\\bf 53}, 359 (1996).\n \n\n\\bibitem{Mullins} W.W. Mullins, J.Appl.Phys. {\\bf 28}, 333 (1957); \nW.W. Mullins, {\\em ibid} {\\bf 30}, 77 (1957); W.W. Mullins in \n{\\em Metal Surfaces}, American Society of Metals, Metals Park, OH, \nEds. W.D.Robertson, N.A. Gjostein, 1962, pp. 17. \n \n\n\\bibitem{EW} S.\\ F.\\ Edwards and D.\\ R.\\ Wilkinson, Proc. R. Soc. London, \nSer A {\\bf 381}, 17 (1982). \n \n\n\\bibitem{KPZ} M.\\ Kardar, G.\\ Parisi, and Y.-C.\\ Zhang, Phys. Rev. Lett. \n{\\bf 56}, 889 (1986). \n \n\n\\bibitem{bru} B. Bruisma, in {\\em Surface Disordering: Growth, Roughening \nand Phase Transitions}, Eds. R. Jullien, J. Kert\\'esz, P. Meakin, and D.E. Wolf \n(Nova Science, New York, 1992). \n \n\n\\bibitem{parallel} R.M. Fujimoto, Commun. ACM {\\bf 33}, 30 (1990). \n \n\n\\bibitem{Luba} B.\\ D.\\ Lubachevsky, Complex Systems {\\bf 1}, 1099 (1987); \nJ. Comput. Phys. {\\bf 75}, 103 (1988). \n \n\n\\bibitem{single_step} P.\\ Meakin, P.\\ Ramanlal, L.\\ M.\\ Sander, and \nR.\\ C.\\ Ball, Phys. Rev. A {\\bf 34}, 5091 (1986); \nM.\\ Plischke, Z.\\ R\\'acz, and D.\\ Liu, Phys. Rev. B {\\bf 35}, 3485 (1987);\nL.M. Sander and H. Yan, Phys. Rev. A {\\bf 44}, 4885 (1991).\n \n\n\\bibitem{KSDS} J. M. Kim and S. Das Sarma, Phys. Rev. Lett. {\\bf 72}, 2903\n(1994)\n\n\n\\bibitem{Krug2} J. Krug, Phys. Rev. Lett. {\\bf 72}, 2907 (1994)\n\n\n\\bibitem{ALF} J.G. Amar, P-M Lam, and F. Family, Phys. Rev. E {\\bf 47}, 3242\n(1993); M. Siegert and M. Plischke, Phys. Rev. Lett. {\\bf 68}, 2035 (1992).\n\n\n\\bibitem{Spitzer} F.\\ Spitzer, Adv. Math. {\\bf 5}, 246 (1970). \n \n\n\\bibitem{itzi} C. Itzykson and J-M. Drouffe, {\\em Statistical Field Theory}, \nCambridge University Press, 1989, vol. 1. \n \n\n\\bibitem{Jones} D.S. Jones, {\\em The theory of generalized functions}, page \n153 \n \n\n\\bibitem{RG} I.S. Gradshteyn and I.M. Ryzhik, {Table of Integrals, Series, and \nProducts}, Ed. Alan Jeffrey, Academic Press, 1994. \n \n\n\\bibitem{NN} J. Neergard and M. den Nijs , J.Phys.A {\\bf 30}, 1935 (1997). \n\n\n\\bibitem{Kondev} J. Kondev and C.L. Henley, Phys. Rev. Lett. 74, 4580 (1995).\n \n"
}
] |
cond-mat0002144
|
Hamiltonian dynamics and geometry of phase transitions in classical XY models
|
[
{
"author": "Monica Cerruti-Sola$^{1,3,}$\\cite{moni}"
},
{
"author": "Cecilia Clementi$^{2,}$\\cite{cecilia}"
},
{
"author": "and Marco Pettini$^{1,3,}$\\cite{marco}"
}
] |
The Hamiltonian dynamics associated to classical, planar, Heisenberg XY models is investigated for two- and three-dimensional lattices. Besides the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively new information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests to consider other observables of geometric meaning tightly related with the largest Lyapunov exponent. The numerical computation of these observables - unusual in the study of phase transitions - sheds a new light on the microscopic dynamical counterpart of thermodynamics also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces $\Sigma_E$ of phase space can be naturally established. In this framework, an approximate formula is worked out, determining a highly non-trivial relationship between temperature and topology of the $\Sigma_E$. Whence it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of the $\Sigma_E$. This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.
|
[
{
"name": "xy.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% REVTEX FILE\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentstyle[aps,pre,amsfonts,graphicx]{revtex}\n\n % \\setlength{\\textwidth}{6.5in}\n % \\setlength{\\textheight}{9.0in}\n % \\setlength{\\topmargin}{0.0in}\n % \\setlength{\\oddsidemargin}{0.0in}\n\n\\newcommand{\\beq}{\\begin{equation}}\n\\newcommand{\\eeq}{\\end{equation}}\n\\def\\erre{\\hbox{\\rm\\rlap{I}\\kern.1em R}}\n\n\\newcommand{\\be}{\\begin{equation}}\n\n\\begin{document}\n\n\\title{Hamiltonian dynamics and geometry of phase transitions in classical\nXY models}\n\n\\author{Monica Cerruti-Sola$^{1,3,}$\\cite{moni}, \nCecilia Clementi$^{2,}$\\cite{cecilia},\n and Marco Pettini$^{1,3,}$\\cite{marco}}\n\\address{\n$^1$Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze,\n Italy \\\\\n$^2$Department of Physics, University of California at San Diego, \nLa Jolla, CA 92093-0319, USA \\\\\n$^3$Istituto Nazionale per la Fisica della Materia, Unit\\`a di Ricerca di\nFirenze, Italy }\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\nThe Hamiltonian dynamics associated to classical, planar, Heisenberg XY \nmodels is investigated for two- and three-dimensional lattices.\nBesides the conventional signatures of phase transitions, here obtained\nthrough time averages of thermodynamical observables in place of ensemble \naverages, qualitatively new information is derived from the \ntemperature dependence of Lyapunov exponents. A Riemannian geometrization\nof newtonian dynamics suggests to consider other observables of geometric \nmeaning tightly related with the largest Lyapunov exponent. The numerical\ncomputation of these observables - unusual in the study of phase transitions -\nsheds a new light on the microscopic dynamical counterpart of \nthermodynamics also pointing to the existence of some major change in the \ngeometry of the mechanical manifolds at the thermodynamical transition. \nThrough the microcanonical definition of the entropy, a relationship between\nthermodynamics and the extrinsic geometry of the constant energy surfaces \n$\\Sigma_E$ of phase space can be naturally established. In this framework, \nan approximate formula is worked out, determining a highly non-trivial \nrelationship between temperature and topology of the $\\Sigma_E$. Whence it can\nbe understood that the appearance of a phase transition must be tightly\nrelated to a suitable major topology change of the $\\Sigma_E$. This contributes\nto the understanding of the origin of phase transitions in the\nmicrocanonical ensemble.\n\\end{abstract}\n\\pacs{PACS: 05.45.+b; 05.20.-y}\n\\newpage\n\n\\section{Introduction}\n\\label{intro}\nThe present paper deals with the study of the microscopic Hamiltonian \ndynamical phenomenology associated to thermodynamical phase transitions.\nThis general subject is addressed in the special case of planar, classical\nHeisenberg XY models in two and three spatial dimensions.\nA preliminary presentation of some of the results and ideas contained in \nthis paper has been already given in \\cite{CCCP}.\n\nThere are several reasons to tackle the Hamiltonian dynamical counterpart of\nphase transitions. On the one hand, we might wonder whether our knowledge\nof the already wide variety of dynamical properties of Hamiltonian systems\ncan be furtherly enriched by considering the dynamical signatures, if any,\nof phase transitions. On the other hand, it is {\\it a-priori} conceivable\nthat also the theoretical investigation of the phase transition phenomena\ncould benefit of a direct investigation of the natural microscopic dynamics.\nIn fact, from a very general point of view, we can argue that in those times\nwhere microscopic dynamics was completely unaccessible to any kind of \ninvestigation, statistical mechanics has been invented just to replace \ndynamics. During the last decades, the advent of powerful computers \nhas made\npossible, to some extent, a direct access to microscopic dynamics through the\nso called molecular dynamical simulations of the statistical properties of\n\"macroscopic\" systems.\n\nMolecular dynamics can be either considered as a mere alternative to Monte\nCarlo methods in practical computations, or it can be also seen as a possible\nlink to concepts and methods (those of nonlinear Hamiltonian dynamics) \nthat could deepen our insight about phase transitions. \nIn fact, by construction,\nthe ergodic invariant measure of the Monte Carlo stochastic dynamics, commonly\nused in numerical statistical mechanics, is\nthe canonical Gibbs distribution, whereas there is no general result that \nguarantees the ergodicity and mixing of natural (Hamiltonian) dynamics. \nThus, the general interest for any contribution that helps in clarifying \nunder what conditions equilibrium statistical mechanics correctly describes\nthe average properties of a large collection of particles, safely replacing\ntheir microscopic dynamical description. \n\nActually, as it has been already shown and confirmed by the \nresults reported below, there are some intrinsically dynamical observables\nthat clearly signal the existence of a phase transition.\nNotably, Lyapunov exponents appear as sensitive measurements for phase\ntransitions.\nThey are also probes of a hidden geometry of the dynamics, because Lyapunov \nexponents depend on the geometry of certain ``mechanical manifolds''\nwhose geodesic flows coincide with the natural motions.\nTherefore, a peculiar energy -- or temperature -- dependence of the largest \nLyapunov exponent at a phase transition point also reflects some important\nchange in the geometry of the mechanical manifolds.\n\nAs we shall discuss throughout the present paper, also the topology of these\nmanifolds has been discovered to play a relevant role in the phase transition\nphenomena (PTP).\n\nAnother strong reason of interest for the Hamiltonian dynamical\ncounterpart of PTP is related to the equivalence problem of statistical \nensembles. Hamiltonian dynamics has its most natural and tight relationship\nwith microcanonical ensemble. Now, the well known equivalence among\nall the statistical ensembles in the thermodynamic limit is valid in general \nin the absence of thermodynamic singularities, i.e. in the absence of phase \ntransitions. This is not a difficulty for statistical mechanics \nas it might seem at first sight\n\\cite{Gallavotti}, rather, this is a very interesting and intriguing point.\n\nThe inequivalence of canonical and microcanonical ensembles in presence of \na phase transition has been analytically shown for a particular model by \nHertel and Thirring \\cite{Thirring}, it is mainly revealed by the \nappearance\nof negative values of the specific heat and has been discussed by several\nauthors \\cite{Lyndenbell,Gross}.\n\nThe microcanonical description of phase transitions seems also to offer many\nadvantages in tackling first order phase transitions \\cite{Gross2}, and \nseems considerably\nless affected by finite-size scaling effects with respect to the canonical\nensemble description \\cite{Gross1}.\nThis non-equivalence problem, together with certain advantages of the\nmicrocanonical ensemble, strenghtens the interest for the Hamiltonian dynamical\ncounterpart of PTP. Let us briefly mention the existing contributions in the\nfield. \n\nButera and Caravati \\cite{Butera},\nconsidering an XY model in two dimensions, found that \nthe temperature dependence of the largest Lyapunov exponent changes just\nnear the critical temperature $T_c$ of the\nKosterlitz-Thouless phase transition. Other interesting aspects of the\nHamiltonian dynamics of the XY model in two dimensions have been extensively\nconsidered in \\cite{Leoncini}, where a very rich phenomenology is reported.\nRecently, the behaviour of Lyapunov exponents\nhas been studied in Hamiltonian dynamical systems: {\\it i)} with long-range\ninteractions \\cite{Rapisarda,Ruffo,Antoni}, {\\it ii)} describing either \nclusters of particles or magnetic or gravitational models exhibiting phase \ntransitions, {\\it iii)} in classical\nlattice field theories with $O(1)$, $O(2)$ and $O(4)$ global symmetries\nin two and three space dimensions \\cite{CCP1,CCCPPG}, {\\it iv)} in the XY \nmodel in two and three space dimensions\n\\cite{CCCP}, {\\it v)} in the \"$\\Theta$ - transition\" of homopolymeric chains \n\\cite{polimeri}.\nThe pattern of $\\lambda(T)$ close to the critical temperature $T_c$ is\nmodel-dependent. The behaviour of Lyapunov exponents near the transition point\nhas been considered also in the case of first- order phase transitions\n\\cite{Dellago,Mehra}. It is also worth mentioning the very intriguing result\nof Ref.\\cite{Berry}, where a glassy transition is accompanied by a sharp\njump of $\\lambda(T)$. \n\n$\\lambda(T)$ always detects a phase transition and, even if\nits pattern close to the critical temperature $T_c$ is\nmodel-dependent, it can be used as an order parameter -- of dynamical origin --\nalso in the absence of a standard order parameter (as in the case of the\nmentioned \"$\\Theta$-transition\" of homopolymers and of the glassy transition\nin amorphous materials).\nThis appears of great prospective interest also in the light of recently\ndeveloped analytical methods to compute Lyapunov exponents (see Section IV). \n\nAmong Hamiltonian models with long-range\ninteractions exhibiting phase\ntransitions, the most extensively studied is the\nmean-field XY model \\cite{Ruffo,Firpo,Ruffo_prl,Ruffo_talk}, whose\nequilibrium statistical mechanics\nis exactly described, in the thermodynamic limit, by mean-field\ntheory \\cite{Ruffo}. In this system, \nthe theoretically predicted temperature dependence of the largest Lyapunov \nexponent $\\lambda$ displays a non-analytic behavior at the phase transition \npoint. \n\nThe aims of the present paper are\n\\begin{itemize}\n\\item \nto investigate the dynamical phenomenology of Kosterlitz-Thouless and of \nsecond order phase transitions in the $2d$ and $3d$ classical\nHeisenberg XY models \nrespectively;\n\\item\nto highlight the microscopic dynamical counterpart of phase transitions\nthrough the temperature dependence of the Lyapunov exponents, also providing\nsome physical interpretation of abstract quantities involved in\nthe geometric theory of chaos (in particular among vorticity, Lyapunov \nexponents and sectional curvatures of configuration space);\n\\item\nto discuss the hypothesis that phase transition phenomena could be originated\nby suitable changes in the topology of the constant energy hypersurfaces of \nphase space, therefore hinting to a mathematical characterization of phase \ntransitions in the microcanonical ensemble.\n\\end{itemize}\n\nThe paper is organized as follows: Sections $II$ and $III$ are devoted to \nthe dynamical investigation of the $2d$ and $3d$ XY models respectively.\nIn Section $IV$ the geometric description of chaos is considered, with \nthe analytic derivation of the temperature dependence of the largest Lyapunov \nexponent, the geometric signatures of a second-order phase transition and \nthe topological hypothesis. \nSection $V$ contains a presentation of the relationship between the extrinsic\ngeometry and topology of the energy hypersurfaces of phase space and \nthermodynamics; the results of some numeric computations are also reported.\nFinally, Section $VI$ is devoted to summarize the achievements reported in \nthe present paper and to discuss their meaning.\n\n\\medskip\n\\section{$2d$ XY model}\n\\medskip\n\nWe considered a system of planar, classical ``spins'' (in fact rotators) \non a square lattice of $N=n\\times n$ sites, and interacting through\nthe ferromagnetic interaction $V=-\\sum_{\\langle i,j\\rangle}J{\\bf S}_{i}\n\\cdot {\\bf S}_{j}$ (where $\\vert{\\bf S}_i\\vert =1)$. \nThe addition of standard, i.e. quadratic, kinetic energy\nterm leads to the following choice of the Hamiltonian\n\\beq\nH= \\sum_{i,j=1}^{n} \\left \\{ \\frac{p_{i,j}^2}{2}+J[2- \\cos(q_{i+1,j}-q_{i,j})\n-\\cos(q_{i,j+1}-q_{i,j})]\\right\\}~~,\n\\label{xy2d}\n\\eeq\nwhere $q_{i,j}$ are the angles with respect to a fixed direction on\nthe reference plane of the system.\nIn the usual definition of the XY model both the kinetic term and the\nconstant term $2JN$ are lacking; however, their contribution does not modify\nthe thermodynamic averages (because they usually depend only on the \nconfigurational partition function, \n$Z_C=\\int\\prod_{i=1}^N dq_i\\exp[-\\beta V(q)]$, \nthe momenta being trivially integrable when the kinetic energy is quadratic).\nThus, as we tackle classical systems, the choice of a quadratic\n kinetic energy term\nis natural because it corresponds to $\\frac{1}{2}\\sum_{i=1}^N \\vert\n{\\bf\\dot S}_i\\vert^2$, written in terms of the momenta $p_{i,j}$ canonically\nconjugated to the lagrangian coordinates $q_{i,j}$. The constant term $2JN$ is \nintroduced to make the low energy expansion of Eq. (\\ref{xy2d}) coincident\nwith the\nHamiltonian of a system of weakly coupled harmonic oscillators.\n\nThe theory predicts for this model a Kosterlitz-Thouless phase transition\noccurring at a critical temperature\n estimated around $T_c\\sim J$. Many Monte Carlo\nsimulations of this model have been done in order to check the predictions\nof the theory. Among them, we quote those of\nTobochnik and Chester \\cite{TobChes} and of Gupta and Baillie \\cite{Gupta}\nwhich, on the basis of accurate numerical analysis, confirmed the predictions \nof the theory and fixed the critical temperature at $T_c=0.89$ ($J=1$).\n\nThe analysis of the present work is based on the numerical integration\nof the equations of motion derived from Hamiltonian (\\ref{xy2d}).\nThe numerical integration is performed by means of\na bilateral, third order, symplectic algorithm \\cite{Lapo}, and it is \nrepeated at several values of the energy density\n$\\epsilon = E/N$ ($E$ is the total energy of the system which depends upon \nthe choice of the initial conditions). \nWhile the Monte Carlo simulations perform statistical averages in the canonical\nensemble, Hamiltonian dynamics has its statistical counterpart in the \nmicrocanonical ensemble. Statistical averages are here replaced by time \naverages of relevant observables. In this perspective, from the microcanonical\ndefinition of temperature $1/T=\\partial S/\n\\partial E$, where $S$ is the entropy, \ntwo definitions of temperature are available: $T=\\frac{2}{N}\\langle K\\rangle$\n(where $K$ is the kinetic energy per degree of freedom), if \n$S=\\log \\int\\prod_{i=1}^N dq_i dp_i \\Theta (H(p,q) - E)$, where $\\Theta\n(\\cdot)$ is the Heaviside step function, and\n$\\tilde T=\\left[\\left(\\frac{N}{2}-1\\right)\\langle K^{-1}\\rangle\\right]^{-1}$,\nif $S=\\log \\int\\prod_{i=1}^N dq_i dp_i \\delta (H(p,q) - E)$ \\cite{Pearson}.\n$T$ (or $\\tilde T$) are numerically\ndetermined by measuring the time average of the kinetic energy $K$ \nper degree of freedom (or its inverse), i.e. \n$T=\\lim_{t\\rightarrow\\infty}\\frac{2}{N}\n\\frac{1}{t}\\int_0^td\\tau K(\\tau)$ (and similarly for $\\tilde T$).\nThere is no appreciable difference in the outcomes of the computations of \ntemperature according to these two definitions.\n\n\\medskip\n\\subsection{Dynamical analysis of thermodynamical observables}\n\\medskip\n\n\\subsubsection{Order parameter}\n\nThe order parameter for a system of planar ``spins'' whose Hamiltonian \nis invariant under the action of the group\n$O(2)$, is the bidimensional vector\n\\beq\n{\\bf M}= (\\sum_{i,j=1}^{n}{\\bf S}^x_{i,j},\n\\sum_{i,j=1}^{n}{\\bf S}^y_{i,j})\\equiv (\\sum_{i,j=1}^{n}\\cos q_{i,j},\n\\sum_{i,j=1}^{n} \\sin q_{i,j}),\n\\label{order_par}\n\\eeq\nwhich describes the mean spin orientation field.\nAfter the \nMermin-Wagner theorem, we know that no symmetry-breaking transition can\noccur in one and two dimensional systems with a continuous symmetry and\nnearest-neighbour interactions. This means that, at any non-vanishing \ntemperature,\nthe statistical average of the total magnetization vector is necessarily\nzero in the thermodynamic limit.\nHowever, a vanishing magnetization is not necessarily expected\nwhen computed by means of Hamiltonian dynamics at\nfinite $N$.\nIn fact, statistical averages are equivalent to averages computed through\nsuitable markovian Monte Carlo dynamics that {\\it a-priori} can reach\nany region of phase space, \nwhereas in principle a true ergodicity breaking is possible in\nthe case of differentiable dynamics. Also an \"effective\" ergodicity breaking\nof differentiable dynamics is possible, when the relaxation times -- of time\nto ensemble averages -- are very fastly increasing with $N$ \\cite{Palmer}.\n\nThis model has two integrable limits: \ncoupled harmonic oscillators and free rotators, \nat low and high temperatures respectively. Hereafter, $T$ is meant in units\nof the coupling constant $J$.\n\nFor a lattice of $N = 10 \\times10$ sites,\nFigure \\ref{figura.spin2d.10e10} shows that \nat low temperatures ($T<0.5$)-- being the system almost harmonic --\nwe can observe a persistent\nmemory of the total magnetization associated with the initial condition,\nwhich, on the typical time scales of our numeric simulations ($10^6$ units \nof proper time), looks almost frozen.\n\nBy raising the temperature above a first threshold $T_0\\simeq 0.6$,\nthe total magnetization vector -- observed on the same time scale -- \nstarts rotating on the plane where it is confined.\nA further increase of the temperature induces a faster rotation of the\nmagnetization vector together with a slight reduction of its average modulus.\n\nAt temperatures slightly greater than $1$, we observe that already at \n$N=10 \\times10$ a random variation of the direction and of the modulus of\nthe vector ${\\bf M}(t)$ sets in. \n\nAt $T>1.2$, we observe a fast relaxation and, at high temperatures \n($T\\simeq 10$), a sort\nof saturation of chaos.\n\nAt a first glance, the results reported in Fig. \n\\ref{figura.spin2d.10e10} could suggest \nthe presence of a phase transition associated with the breaking of the\n$O(2)$ symmetry.\nIn fact, having in mind the Landau theory, the ring-shaped distribution of\nthe instantaneous magnetization shown\nby Fig. \\ref{figura.spin2d.10e10} is the typical signature of an\n$O(2)$-broken symmetry phase and the spot-like patterns around zero are\nproper to the unbroken symmetry phase.\n\nThe apparent contradiction of these results with the Mermin-Wagner theorem\nis resolved by checking whether the observed phenomenology is stable with\n$N$. Thus, some simulations have been performed at larger values of $N$. \nAt any temperature, we found that the average modulus \n$\\langle\\vert{\\bf M}(t)\\vert\\rangle_t$ of the vector ${\\bf M}(t)$, \ncomputed along the trajectory, systematically\ndecreases by increasing $N$. However, \nfor temperatures lower than $T_0$, the $N$-dependence \nof the order parameter is very weak, whereas,\nfor temperatures greater than $T_0$, the $N$-dependence \nof the order parameter is rather strong.\nIn Fig. \\ref{fig.spin2d.t0.74} two extreme cases\n($N=10 \\times10$ and $N=200 \\times200$) are shown for $T = 0.74$.\nThe systematic trend of $\\langle\\vert{\\bf M}(t)\\vert\\rangle$ \ntoward smaller values at increasing $N$ is \nconsistent with its expected vanishing in the limit $N\\rightarrow\\infty$.\n\nAt $T = 1$, Fig. \\ref{fig.spin2d.t1} shows that,\nwhen the lattice dimension is greater than $50\\times 50$,\n${\\bf M}(t)$ displays random variations both in direction (in the interval\n[0,$2\\pi$]) and in \nmodulus (between zero and a value which is smaller at larger \n$N$).\n\n\\medskip\n\\subsubsection{Specific heat}\n\\medskip\nBy means of the recasting of a standard formula which relates the average \nfluctuations of a generic observable computed in canonical and \nmicrocanonical ensembles \\cite{LPV}, and by specializing it to the\nkinetic energy fluctuations, one obtains a microcanonical estimate \nof the canonical specific heat\n\\begin{equation}\nc_{V}(T)=\\frac{C_V}{N}\\rightarrow \n\\cases{\n%\\left\\{\\begin{array}{1}\nc_{V}(\\epsilon)=\\displaystyle{ \\frac{k_{B}d}{2}\\left[1-\\frac{N\nd}{2}\\frac{\\langle K^{2}\\rangle -\\langle K\\rangle\n^{2}}{\\langle K\\rangle ^{2}}\\right]^{-1}} ~~,\\cr\nT=T(\\epsilon)\\cr }\n%\\end{array} \\right.\n\\label{specheat}\n\\end{equation}\nwhere $d$ is the number of degrees of freedom for each particle.\nTime averages of the kinetic energy fluctuations computed at any\ngiven value of the energy density $\\epsilon$ yield $C_V(T)$, according\nto its parametric definition in Eq.(\\ref{specheat}).\n\n>From the microcanonical definition $1/C_V=\\partial T(E)/\\partial E$ of the\nconstant volume specific heat, a formula can be worked out \\cite{Pearson},\nwhich is exact at \n{\\it any} value of $N$ (at variance with the expression (\\ref{specheat})).\n It reads\n\\begin{equation}\nc_V=\\frac{C_V}{N}=[N - (N - 2)\\langle K\\rangle\\langle K^{-1}\\rangle]^{-1}\n\\label{cvmicro}\n\\end{equation}\nand it is the natural expression to be used in Hamiltonian\ndynamical simulations of\nfinite systems.\n\nThe numerical simulations of the Hamiltonian dynamics of the $2d$\nXY model -- computed with both Eqs.(\\ref{specheat}) and (\\ref{cvmicro}) -- \nyield a cuspy pattern for $c_V(T)$ peaked at $T\\simeq 1$\n(Fig. \\ref{calspec_2d}).\nThis is in good agreement with the outcomes of canonical Monte Carlo \nsimulations reported in Ref. \\cite{TobChes,Gupta},\nwhere a pronounced peak of $c_{V}(T)$ was detected at $T \\simeq 1.02$.\n\nBy varying the lattice dimensions, the peak height remains\nconstant, in agreement with the absence of a symmetry-breaking phase\ntransition. \n\\medskip\n\\subsubsection{Vorticity}\n\\medskip\nAnother thermodynamic observable which can be studied \nis the vorticity of the system. Let us briefly recall that if\nthe angular differences of nearby ``spins'' are small, we can suppose the\nexistence of a continuum limit function $\\theta({\\bf r})$ that conveniently\nfits a given spatial configuration of the system.\nSpin waves correspond to regular patterns of $\\theta({\\bf r})$, whereas the\nappearance of a singularity in $\\theta({\\bf r})$ corresponds to a topological \ndefect, or a vortex, in the ``spin'' configuration. When such a defect is \npresent, along any closed path ${\\cal C}$ that contains the centre of the \ndefect, one has\n\\begin{equation}\n\\oint_{\\cal C}\\nabla\\theta({\\bf r})\\cdot d{\\bf r}= 2\\pi q~,~~~~q=0,\\pm 1,\\pm 2,\n\\dots\n\\end{equation}\nindicating the presence of a vortex whose intensity is $q$. For \na lattice model with periodic boundary conditions, there is an equal number \nof vortices and antivortices (i.e. vortices rotating in opposite directions).\nThus, the vorticity of our model can be defined as the mean total number of \nequal sign vortices per unit volume.\nIn order to compute the vorticity ${\\cal V}$ as a function of temperature,\nwe have averaged the number of positive vortices along the numerical phase \nspace trajectories. On the lattice, ${\\bf r}$ is replaced by the multi-index \n${\\bf i}$ and $\\nabla_\\mu\\theta_i= q_{{\\bf i}+\\mu} - q_{\\bf i}$, then the\nnumber of elementary vortices is counted: the discretized version of\n$\\oint_\\square\\nabla\\theta\\cdot d{\\bf r}=1$ amounts to one elementary vortex \non a plaquette. Thus ${\\cal V}$ is obtained by summing over all the plaquettes.\n\nOur results are in agreement with the values obtained \nby Tobochnik and Chester \\cite{TobChes}\nby means of Monte Carlo simulations with $N=60\\times 60$. \n\nAs shown in Fig. \\ref{fig.vort2d},\non the $10\\times 10$ lattice, \nthe first vortex shows up at $T\\sim0.6$\nand on the $40\\times 40$ lattice \nat $T\\sim0.5$, when the\nsystem changes its dynamical behavior,\nincreasing its chaoticity (see next Subsection).\n At lower temperatures, vortices are less probable,\ndue to the fact\nthat the formation of vortex has a minimum energy cost.\nBelow $T\\sim1$, the vortex density steeply grows \nwith a power law ${\\cal V}(T)\\sim T^{10}$.\n The growth of ${\\cal V}$ then slows down, until the saturation is reached at \n$T\\sim10$.\n\n\\medskip\n\\subsection{Lyapunov exponents and chaoticity}\n\\medskip\n\nThe values of the largest Lyapunov exponent $\\lambda_{1}$\nhave been computed using the standard tangent dynamics equations [see Eqs.\n(\\ref{eqdintang}) and (\\ref{bgs})], and are reported \nin Fig. \\ref{xy2d.lyap.num.fig}.\n\nBelow $T\\simeq 0.6$, \nthe dynamical behavior is nearly the same as that of harmonic\noscillators and the excitations of the system are only ``spin-waves''.\n\nIn the interval $[0., 0.6]$, the observed temperature dependence\n$\\lambda_1(T) \\sim T^2$ is equivalent to the \n$\\lambda_1(\\epsilon) \\sim\\epsilon^2$ dependence (since at low temperature \n$T(\\epsilon) \\propto \\epsilon$), already found -- analytically and\n numerically --\nin the quasi-harmonic regime of other systems and characteristic of weakly\nchaotic dynamics \\cite{CCP}.\n \nAbove $T \\simeq0.6$, vortices begin to form and correspondingly the largest\nLyapunov exponent signals a \"qualitative\" change of the dynamics through a\nsteeper increase vs. $T$.\n\nAt $T \\simeq0.9$, where the theory predicts a Kosterlitz - Thouless phase \ntransition, $\\lambda_1(T)$ displays an inflection point.\n\nFinally, at high temperatures, the power law $\\lambda_{1}(T)\\sim T^{-1/6}$\nis found. \n\n\n\\medskip\n\\section{$3d$ XY model}\n\\medskip\n\nIn order to extend the dynamical investigation \nto the case of second-order phase transitions, we have\nstudied a system described by an Hamiltonian having at the same time\nthe main characteristics\nof the $2d$ model and the differences necessary to the appearance\nof a spontaneous symmetry-breaking below a certain critical temperature.\n The model\nwe have chosen is such that the spin rotation is constrained on a plane and\nonly the lattice dimension has been increased, in order to elude the\n``no go'' conditions of the Mermin-Wagner theorem.\nThis is simply achieved by tackling a system\ndefined on a cubic lattice of $N=n\\times n\\times n$ sites\nand described by the Hamiltonian\n\\begin{eqnarray}\nH&=& \\sum_{i,j,k=1}^{n}\n\\{ \\frac{p_{i,j,k}^{2}}{2}+J[3-\\cos(q_{i+1,j,k}-q_{i,j,k})-\\nonumber\\\\\n&-&\\cos(q_{i,j+1,k}-q_{i,j,k})-\\cos(q_{i,j,k+1}-q_{i,j,k})] \\}~~.\n\\end{eqnarray}\n\n\\medskip\n\\subsection{Dynamical analysis of thermodynamical observables}\n\\medskip\nThe basic thermodynamical phenomenology of a second-order phase transition is \ncharacterized \nby the existence of equilibrium\nconfigurations that make the order parameter bifurcating away from zero\nat some critical temperature $T_{c}$ and by a divergence of \nthe specific heat $c_{V}(T)$ at\nthe same $T_{c}$. \nTherefore, this is the obvious starting point for the Hamiltonian dynamical\napproach.\n\n\\medskip\n\\subsubsection{Order parameter}\n\\medskip\nBelow a critical value of the temperature,\nthe symmetry-breaking in a system invariant under the action of the\n$O(2)$ group, appears\nas the selection\n -- by the average magnetization vector of Eq. (\\ref{order_par})--\n of a preferred direction \n among all the possible, energetically equivalent choices. \nBy increasing the lattice dimension, the symmetry breaking is therefore\ncharacterized by a sort of simultaneous \"freezing\"\n of the direction of the order parameter ${\\bf M}$ and of the\nconvergence of its modulus to a non-zero value.\n\nFigure \\ref{mag3d.e2} shows that in the $3d$ lattice, \nat $T < 2$, i.e. in the broken-symmetry phase (as we shall see in the \nfollowing), \nthe dynamical simulations yield a thinner spread of the longitudinal\nfluctuations by increasing $N$ -- that is, $| {\\bf M} |$\noscillates by exhibiting a trend\nto converge to a\nnon-zero value -- and that the transverse \nfluctuations damp, ``fixing'' the direction of the oscillations.\nThis direction depends on the initial conditions.\n\nMoreover, the dynamical analysis provides us with a better detail than a \nsimple distinction between regular and chaotic dynamics. \nIn fact, it is possible to \ndistinguish between three different dynamical regimes \n(Fig. \\ref{fig.1.spin3d.9}).\n\nAt low temperatures, up to $T \\simeq 0.8$, one observes the\npersistency of the initial direction and of an equilibrium value of the \nmodulus $| {\\bf M} |$ close to one.\n\nAt $0.8 < T < 2.2$, one observes transverse\noscillations, whose amplitude increases with temperature.\n\nAt $T > 2.2$, \nthe order parameter exhibits the features typical of an unbroken symmetry\nphase.\nIn fact, it displays fluctuations peaked at zero, whose\ndispersion decreases by increasing the temperature (bottom of\nFig. \\ref{fig.1.spin3d.9}) and,\nat a given temperature, by increasing the lattice volume (Fig.\n\\ref{3d.fig.spin.altat}a,b). \n\nWe can give an estimate of the order parameter by evaluating the average \nof the modulus $\\langle | {\\bf M}(t)|\\rangle\n= \\rho(T)$. At $T < 2.2$, the $N$-dependence is given mainly by \nthe rotation of the vector, while the longitudinal oscillations are moderate,\nas shown in Fig. \\ref{parord.3d.fig}. At temperatures above $T \\simeq2.2$, \nwe observe the squeezing of $\\rho(T)$ to a small value.\n\nThe existence of a second order phase transition can be recognized\nby comparing the temperature behavior and the $N$-dependence of \nthe thermodynamic observables computed for\nthe $2d$ and the $3d$ models.\nBoth systems exhibit\nthe rotation of the magnetization\nvector and small fluctuations of its modulus when they are considered on small\nlattices.\nIn the $2d$ model the average modulus of the order parameter is theoretically \nexpected to vanish logarithmically with $N$, what seems qualitatively \ncompatible with the weak $N$ dependence shown in\nFig. \\ref{fig.spin2d.t0.74}, whereas in the $3d$ model we observe a stability\nwith $N$ of $\\langle| {\\bf M} |\\rangle$, suggesting the convergence\nto a non-zero value of the order parameter also in the limit\n$N\\rightarrow\\infty$, as shown in Fig. \\ref{mag3d.e2}.\n\n$T \\simeq2.2$ is an approximate value of the critical temperature $T_c$ of the\nsecond-order phase transition. This value will be refined in the following\nSubsection. No finite-size scaling analysis has been performed for two\ndifferent reasons: {\\it i)} our main concern is a qualitative phenomenological\nanalysis of the Hamiltonian dynamics of phase transitions rather than a \nvery accurate quantitative analysis, {\\it ii)} finite-size effects are much \nweaker in the microcanonical ensemble than in the canonical ensemble \n\\cite{Gross1}.\n\n\\medskip\n\\subsubsection{Specific heat}\n\\medskip\nAs in the $2d$ model, numerical simulations of the Hamiltonian dynamics\nhave been performed with both Eqs.(\\ref{specheat}) and (\\ref{cvmicro}).\nThe outcomes\nshow a cusplike pattern of the specific heat, whose peak\n makes possible a better determination of \nthe critical temperature. By increasing the lattice\ndimension up to $N=15\\times 15\\times 15$,\nthe cusp becomes more pronounced, \nat variance with the case of the $2d$ model.\nFig. \\ref{calspec.3d.fig} shows that this occurs\nat the temperature $T_{c}\\simeq2.17$.\n\n\\medskip\n\\subsubsection{Vorticity}\n\\medskip\nThe definition of the vorticity in the $3d$ case is not a simple extension\nof the $2d$ case.\nVortices are always defined on a plane and if all the ``spins'' could freely\nmove\nin the three-dimensional space, the concept of vortices would be meaningless.\nFor the $3d$ planar (anisotropic) model considered here, vortices can be\ndefined and studied on two-dimensional\nsubspaces of the lattice. The variables $q_{i,j,k}$ do not contain any\ninformation about the position of the plane where the reference direction\nto measure the angles $q_{i,j,k}$ is assigned.\nDynamics is completely independent of this choice, which has no effect on the\nHamiltonian. Moreover, as the Hamiltonian is symmetric with respect to the \nlattice axes, the three coordinate-planes are equivalent. This \nequivalence implies that vortices can contemporarily exist on three \northogonal planes. Though the usual pictorial representation of a vortex\ncan hardly be maintained, its mathematical definition is the same as in the \n$2d$ lattice case. Hence three vorticity functions exist and their \naverage values - at a given temperature - should not differ, what is actually\nconfirmed by numerical simulations.\n\nThe vorticity function vs. temperature is plotted in \nFig. \\ref{vort.fig3d}.\nOn a lattice of $10\\times 10\\times 10$ spins, the first vortex is \nobserved at $T \\simeq0.8$.\nThe growth of the average density of vortices is very\nfast up to the critical temperature, above which the saturation is reached.\n\n\\medskip\n\\subsection{Lyapunov exponents and symmetry-breaking phase transition}\n\\medskip\nA quantitative analysis of the dynamical chaoticity is provided by the \ntemperature dependence \nof the largest Lyapunov exponent. \n\nFigure \\ref{lyap.3d.fig} shows the results of this computation. \nAt low temperatures, in the limit of quasi-harmonic oscillators, the scaling\nlaw is again found to be\n $\\lambda_{1}(T)\\sim T^{2}$ and, at high\ntemperatures, the scaling law is again $\\lambda_{1}(T)\\sim T^{-1/6}$, \nas in the $2d$ case.\nIn the temperature range intermediate between $T \\simeq0.8$ and \n$T_c \\simeq2.17$, there is a linear growth of $\\lambda_1(T)$.\nAt the critical temperature, the Lyapunov exponent exhibits an angular \npoint. This makes a remarkable difference between this\nsystem undergoing a second order phase transition and \nits $2d$ version, undergoing a Kosterlitz-\nThouless transition. In fact, the analysis of the $2d$ model has\nshown a mild transition between the different regimes of $\\lambda_{1}(T)$\n(inset of Fig. \\ref{vort.fig3d}),\nwhereas in the $3d$ model this transition \nis sharper (inset of Fig. \\ref{lyap.3d.fig}).\n\nWe have also computed the temperature dependence of the largest Lyapunov \nexponent of Markovian random processes which replace the true dynamics on the \nenergy surfaces $\\Sigma_E$ (see Appendix).\nThe results are shown in Fig. \\ref{randyn.3d.fig}. \nThe dynamics is considered\nstrongly chaotic in the temperature range where the patterns \n$\\lambda_1(T)$ are the same for both random and \ndifferentiable dynamics, i.e. when differentiable dynamics mimics,\nto some extent,\na random process.\nThe dynamics is considered weakly chaotic when the value \n$\\lambda_1$ resulting from\nrandom dynamics is larger than the value $\\lambda_1$\nresulting from differentiable\ndynamics.\n The transition from weak to strong chaos is quite abrupt. \nFigure \\ref{randyn.3d.fig} shows that \nthe pattern of the largest Lyapunov exponent \ncomputed by means of the random dynamics \nreproduces that of the true Lyapunov exponent\nat temperatures $T \\geq T_c$. This means that the setting in\nof strong thermodynamical disorder corresponds to the setting in of strong\ndynamical chaos. \n The ``window'' of strong chaoticity starts at\n$T_c$ and ends at $T \\sim 10$.\nThe existence of a second transition from strong to weak chaos is\ndue to the existence, for $T\\rightarrow\\infty$,\nof the second integrable limit (of free \nrotators), whence chaos cannot remain strong at any $T>T_c$.\n\n\\medskip\n\\section{Geometry of dynamics and phase transitions}\n\\medskip\nLet us briefly recall that the geometrization of the dynamics \nof $N$-degrees-of-freedom systems defined by a Lagrangian\n${\\cal L} = K - V$, in which the kinetic energy is quadratic in the velocities:\n$K=\\frac{1}{2}a_{ij} \\dot{q}^i\\dot{q}^j~$, stems from the fact that\nthe natural motions are the extrema\nof the Hamiltonian action functional ${\\cal S}_H = \n\\int {\\cal L} \\, dt$, \nor of the Maupertuis' action\n${\\cal S}_M = 2 \\int K\\, dt$.\nIn fact, also the geodesics of Riemannian and pseudo-Riemannian \nmanifolds are the extrema of a functional, the arc-length \n$\\ell=\\int ds$, with $ds^2={g_{ij}dq^i dq^j}$. \nHence, a suitable choice of the metric tensor allows for the \nidentification of the arc-length with either ${\\cal S}_H$ or \n${\\cal S}_M$, and of the geodesics with the natural motions of the\ndynamical system. Starting from ${\\cal S}_M$, the ``mechanical manifold''\nis the accessible configuration space endowed with\nthe Jacobi metric \\cite{Pettini} \n\\beq\n(g_J)_{ij} = [E - V(q)]\\,a_{ij}~~,\n\\label{jacobi_metric}\n\\eeq \nwhere $V(q)$ is the potential energy and $E$ is the total energy.\nA description of the extrema of Hamilton's \naction ${\\cal S}_H$ as geodesics of a ``mechanical manifold'' \ncan be obtained using Eisenhart's metric \n\\cite{Eisenhart} on an enlarged configuration spacetime \n($\\{q^0\\equiv t,q^1,\\ldots,q^N\\}$ \nplus one real coordinate $q^{N+1}$), whose arc-length is\n\\begin{equation}\nds^2 = -2V(\\{ q \\}) (dq^0)^2 + a_{ij} dq^i dq^j + 2 dq^0 \ndq^{N+1}~~.\n\\label{ds2E}\n\\end{equation}\nThe manifold has a Lorentzian structure and the dynamical \ntrajectories are those geodesics satisfying the condition\n$ds^2 = C dt^2$, where $C$ is a positive constant. \nIn the geometrical framework, the (in)stability \nof the trajectories is the (in)stability \nof the geodesics, and it is completely determined by the \ncurvature properties of the underlying manifold according to\nthe Jacobi equation \\cite{Pettini,doCarmo}\n\\begin{equation}\n\\frac{\\nabla^2 \\xi^i}{ds^2} + R^i_{~jkm}\\frac{dq^j}{ds} \\xi^k \n\\frac{dq^m}{ds} = 0~~,\n\\label{eqJ}\n\\end{equation}\nwhose solution $\\xi$, usually called Jacobi or geodesic variation field, \nlocally measures the distance between nearby geodesics; \n$\\nabla/ds$ stands for the covariant derivative\nalong a geodesic and $R^i_{~jkm}$ are the components of \nthe Riemann curvature tensor. \nUsing the Eisenhart metric (\\ref{ds2E}),\nthe relevant part of the Jacobi equation \n(\\ref{eqJ}) is \\cite{CCP}\n\\begin{equation}\n\\frac{d^2 \\xi^i}{dt^2} + R^i_{~0k0}\\xi^k = 0~~,~~~~i=1,\\dots ,N\n\\label{eqdintang}\n\\end{equation}\nwhere the only non-vanishing components of the curvature tensor are\n$R_{0i0j}=\\partial^2 V/\\partial q_i \\partial q_j $. Equation \n(\\ref{eqdintang}) is the tangent dynamics equation, which is commonly used to\nmeasure Lyapunov exponents in standard Hamiltonian systems. \nHaving recognized its geometric origin, \nit has been \ndevised in Ref.\\cite{CCP} a geometric reasoning\n to derive from Eq.(\\ref{eqdintang})\nan {\\it effective} scalar stability equation that, {\\it independently} of the\nknowledge of dynamical trajectories, provides an average measure of their\ndegree of instability. \nAn intermediate step in this derivation yields\n\\beq\n\\frac{d^2 \\xi^j}{dt^2} + k_R(t) \\xi^j + \\delta K^{(2)}{(t)} \\xi^j = 0~~,\n\\label{sectional}\n\\eeq\nwhere $k_R=K_R/N$ is the Ricci curvature along a geodesic defined as\n$K_R = \\frac{1}{v^2} R_{ij} {\\dot{q}^i}{\\dot{q}^j}$,\nwith $v^2 = {\\dot{q}^i}{\\dot{q}_i}$ and \n$R_{ij} = R^{k}_{~ikj}$, and $\\delta K^{(2)}$ is the local\n deviation of sectional\ncurvature from its average value \\cite{CCP}.\nThe sectional curvature is defined as $K^{(2)} = R_{~ijkl} \\xi^i \\dot{q}^j\n\\xi^k \\dot{q}^l/ \\parallel\\xi\\parallel^2 \\parallel\\dot{q}\\parallel^2$.\n\nTwo simplifying assumptions are made:\n$(i)$ the ambient manifold is {\\em almost isotropic}, i.e. \nthe components of the curvature tensor --- that for an isotropic manifold\n(i.e. of constant curvature) \nare $R_{ijkm}=k_0(g_{ik} g_{jm} - g_{im} g_{jk})$, $k_0=const$ \n-- can be approximated by \n$R_{ijkm} \\approx k(t)\n(g_{ik} g_{jm} - g_{im} g_{jk})$\nalong a generic geodesic $\\gamma(t)$; $(ii)$\nin the large $N$ limit, the ``effective curvature''\n$k(t)$ can be modeled by a gaussian and $\\delta$-correlated stochastic \nprocess.\nHence, one derives\n an effective\nstability equation, independent of the dynamics and in the\nform of a stochastic oscillator equation\n\\cite{CCP},\n\\begin{equation}\n\\frac{d^2\\psi}{dt^2} + [k_0 + \\sigma_k \\eta(t)] \\, \\psi = 0~~,\n\\label{eqpsi}\n\\end{equation}\nwhere $\\psi^2 \\propto |\\xi|^2$. \n The mean $k_0$ and variance $\\sigma_k$ of $k(t)$ \nare given by \n$k_0 = {\\langle K_R \\rangle}/{N}$\nand $\\sigma^2_k = {\\langle (K_R - \\langle K_R \\rangle)^2 \\rangle}/{N}$,\nrespectively, and the averages $\\langle\\cdot\\rangle$ are geometric averages,\ni.e. integrals computed on the mechanical manifold. These averages are \ndirectly related with microcanonical averages, as it will be seen at the end\nof Section V.\n$\\eta(t)$ is a gaussian $\\delta$-correlated random process of\nzero mean and unit variance.\n\nThe main source of instability of the solutions of Eq.(\\ref{eqpsi}),\nand therefore the main source of Hamiltonian chaos, is \nparametric resonance, which is\n activated by the variations of the Ricci curvature\nalong the geodesics and which takes place also on positively curved manifolds\n\\cite{CerrutiPettini}. The dynamical instability can be enhanced \nif the geodesics encounter regions \nof negative sectional curvatures, such that $k_R + \\delta K^{(2)}\n< 0$, as it is evident from Eq. (\\ref{sectional}).\n\nIn the case of Eisenhart metric, \nit is \n$K_R\\equiv \\Delta V = \\sum_{i=1}^N ({\\partial^2 V}/{\\partial q_i^2})$\nand $K^{(2)} = R_{~0i0j} \\xi^i \\xi^j/ \\parallel \\xi \\parallel^2 \\equiv\n(\\partial^2V/\\partial q^i\\partial q^j)\\xi^i\\xi^j/\\Vert\\xi\\Vert^2$.\nThe exponential growth rate $\\lambda$ of the \nquantity $\\psi^2+\\dot\\psi^2$ of the solutions of Eq. (\\ref{eqpsi}),\nis therefore an estimate of the largest Lyapunov exponent that can be \nanalytically computed. The final result reads \\cite{CCP} \n\\begin{equation}\n\\lambda = \\frac{\\Lambda}{2} - \\frac{2 k_0}{3 \\Lambda}\\,,~~\n\\Lambda = \\left(2\\sigma_k^2 \\tau +\n\\sqrt{\\frac{64 k_0^3}{27} + 4\\sigma_k^4 \\tau^2}~\\right)^\\frac{1}{3}~,\n\\label{lambda}\n\\end{equation}\nwhere\n$\\tau = \\pi\\sqrt{k_0}/(2\\sqrt{k_0(k_0 + \\sigma_k)}\n+\\pi\\sigma_k )$;\nin the limit $\\sigma_k/k_0 \\ll 1$ one finds $\\lambda \\propto \\sigma_k^2~$.\n\n\\subsection{Signatures of phase transitions from geometrization of \ndynamics}\n\nIn the geometric picture, chaos is mainly originated by the parametric \ninstability activated by the fluctuating curvature felt by geodesics,\ni.e. the fluctuations of the (effective)\ncurvature are the source of the instability of the dynamics.\nOn the other hand, as it is witnessed by the derivation of Eq. (\\ref{eqpsi})\nand by the equation itself, a statistical-mechanical-like treatment of the\naverage degree of chaoticity is made possible by the geometrization of the\ndynamics. The relevant curvature properties of the mechanical manifolds\nare computed, at the formal level, as statistical averages, like other\nthermodynamic observables. Thus, we can expect that some precise relationship\nmay exist between geometric, dynamic and thermodynamic quantities.\nMoreover, this implies that phase transitions should correspond to peculiar\neffects in the geometric observables.\n\nIn the particular case of the $2d$ XY model, \nthe microcanonical average kinetic energy $\\langle K \\rangle$\nand the average Ricci curvature \n$\\langle K_R \\rangle$ computed \nwith the Eisenhart metric are linked by the equation \n\\beq\nK_{R}=\n\\left \\langle \\sum _{i,j=1}^{N}\\frac{\\partial^{2} V}{\\partial^2 q_{i,j}}\n\\right \\rangle = 2J\n\\sum_{i,j=1}^{N}\n\\left\\langle\\cos(q_{i+1,j}-q_{i,j})\n+\\cos(q_{i,j+1}-q_{i,j}) \\right\\rangle =2(J-\\langle V \\rangle )~~,\n\\label{kinRicci}\n\\eeq\nso that\n\\beq\nH=N\\epsilon=\\langle K \\rangle + \\langle V \\rangle \\mapsto\\frac{ \\langle K \n\\rangle }{N}=\\epsilon-2J+\\frac{1}{2}\\frac{\\langle K_{R} \\rangle}{N}~~.\n\\label{cRicci.Uinterna}\n\\eeq\nBeing the temperature defined as $T=2\\langle K \\rangle /N$\n(with $k_{B}=1$) and being $d=1$ (because each spin has only one\nrotational degree of freedom), from Eq.(\\ref{specheat}) it follows that\n\n\\beq\nc_{V}=\\frac{1}{2}\\left(1-\\frac{1}{2}\\frac{\\sigma^{2}_k/N}{T^2}\\right)^{-1}.\n\\label{fluttKr.calspec}\n\\eeq\n\nIn the special case of these XY systems,\nit is possible to link the specific heat and the Ricci curvature\nby inserting Eq.(\\ref{cRicci.Uinterna}) into the usual expression for the \nspecific heat at constant volume. Thus, one obtains the equation\n\\beq\nc_{V}=-\\frac{1}{2N}\\frac{\\partial \\langle K_{R} \\rangle(T)}{\\partial T}~.\n\\label{calspec,Ricci}\n\\eeq\nThe appearance of a peak in the specific heat function at the critical \ntemperature\nhas to correspond to a suitable temperature dependence of the Ricci curvature.\n\nIn the $3d$ model, the\npotential energy and the Ricci curvature \nare proportional, according to:\n$\\frac{1}{N} \\langle V \\rangle = 3 - \\frac{1}{2 N} \\langle K_{R} \\rangle~$.\n\nAnother interesting point is the relation between a geometric observable\nand the vorticity function in both models.\nAs already seen in previous sections, the\nvorticity function is a useful signature of the dynamical chaoticity \nof the system. From the geometrical point of view,\nthe enhancement of the instability of the dynamics\nwith respect to the parametric instability due to curvature fluctuations,\nis linked\nto the probability of obtaining negative sectional curvatures \nalong the geodesics (as discussed for $1d$ XY model in Ref.\\cite{CCP}).\nIn fact, when vortices are present in the system, \nthere will surely be two neighbouring spins with an orientation difference \ngreater than $\\pi/2$, such that, if $i,j$ and $i+1,j$ are their coordinates\non the lattice, it follows that\n\\beq\nq_{i+1,j}-q_{i,j} > \\frac{\\pi}{2} \\rightarrow \\cos(q_{i+1,j}-q_{i,j})<0~.\n\\label{coord}\n\\eeq\nThe sectional curvature relative to the plane defined \nby the velocity ${\\bf v}$ along\na geodesic and a generic vector ${\\bf \\xi}\\perp {\\bf v}$ is \n\\beq\nK^{(2)}= \\sum_{i,j,k,l=1}^{N} \\frac{\\partial^2V}\n{\\partial q_{i,j}\\partial q_{k,l}}\n\\frac{\\xi^{i,j}\\xi^{k,l}}{\\|{\\bf \\xi}\\|^{2}}~~.\n\\label{kappa2}\n\\eeq\nFor the $2d$ XY model, it is \n\\beq\nK^{(2)}= \\frac{J}{\\|{\\bf \\xi}\\|^{2}}\\sum_{i,j=1}^{N}\\{\\cos(q_{i+1,j}-q_{i,j})\n[\\xi^{i+1,j}-\\xi^{i,j}]^{2} +\\cos(q_{i,j+1}-q_{i,j})[\\xi^{i,j+1}-\\xi^{i,j}]^{2}\n\\}~.\n\\label{curvsezXY}\n\\eeq\nThus, a large probability of\nhaving a negative value of the cosine of the difference among the directions\nof two close spins corresponds to a larger probability of obtaining\nnegative values of the sectional curvatures along the geodesics; here for $\\xi$\nthe geodesic separation vector of Eq.(\\ref{eqdintang}) is chosen.\n\nIn the $3d$ model, the sectional curvature relative to\nthe plane defined by the velocity ${\\bf v}$ and \na generic vector ${\\bf \\xi}\\perp {\\bf v}$ is \n\\begin{eqnarray}\nK^{(2)}&=& \\frac{J}{\\|\\xi\\|^{2}}\\sum_{i,j,k=1}^{N} \\{\n\\cos(q_{i+1,j,k}-q_{i,j,k})[\\xi^{i+1,j,k}-\\xi^{i,j,k}]^{2}+\n\\nonumber\\\\\n&+&\\cos(q_{i,j+1,k}-q_{i,j,k})[\\xi^{i,j+1,k}-\\xi^{i,j,k}]^{2}+\n\\cos(q_{i,j,k+1}-q_{i,j,k})[\\xi^{i,j,k+1}-\\xi^{i,j,k}]^{2}\\}\n\\label{curvsez3d}\n\\end{eqnarray}\nand again the probability of finding negative values of $K^{(2)}$\nalong a trajectory is limited to the probability of finding vortices.\n\nThe mean values of the geometric quantities entering Eq.(\\ref{eqpsi})\ncan be numerically computed by means of Monte Carlo simulations\nor by means of time averages along the\ndynamical trajectories. In fact, due to the lack of an explicit expression\nfor the canonical partition function of the system,\nthese averages are not analytically computable.\nFor sufficiently high\ntemperatures, the potential energy becomes \nnegligible with respect to the kinetic energy, and each spin is free to move\nindependently from the others. Thus, in the limit of high temperatures,\none can estimate the configurational partition function\n$Z_C = \\int_{-\\pi}^{\\pi} \\prod_{\\bf i} dq_{\\bf i} e^{-\\beta V(q)}$ \nby means of the expression \n\\begin{eqnarray}\nZ_C& =& e^{-2 \\beta JN}\n\\int_{-\\pi}^{\\pi} \\prod_{i,j=1}^{N}\ndq_{i,j}\\exp\\{\\beta J \\sum_{i,j=1}^{N}[\n\\cos(q_{i+1,j}-q_{i,j})+\\cos(q_{i,j+1}-q_{i,j})]\\}\\nonumber\\\\\n&\\sim& e^{-2 \\beta JN}\\int_{-\\pi}^{\\pi} \\prod_{i,j=1}^{N}du_{i,j}dv_{i,j}\n\\exp\\{\\beta J \\sum_{i,j=1}^{N}[\n\\cos(u_{i,j})+\\cos(v_{i,j})]\\}\n\\label{partition}\n\\end{eqnarray}\nafter the introduction of $u_{i,j}= q_{i+1,j}-q_{i,j}$ and \n$v_{i,j}= q_{i,j+1}-q_{i,j}$ as independent variables.\nIn this way, some analytical estimates of the average Ricci\ncurvature $k_{0}(T)$ and of its r.m.s. fluctuations \n$\\sigma^2_k(T)$ have been obtained for the $2d$ model\n (Fig. \\ref {fig.Ricci2d}).\nFor temperatures above the temperature of the Kosterlitz-Thouless\ntransition, these estimates\nare in agreement\nwith the numerical computations on a $N=10\\times 10$ lattice.\nIt is confirmed that Hamiltonian dynamical simulations,\nalready on rather small lattices, \nare useful to predict, with a good approximation, the thermodynamic limit\nbehavior of relevant observables.\nMoreover, the good quality of the high temperature estimate gives\na further information: at the transition temperature,\nthe correlations among the different degrees of freedom are destroyed,\nconfirming the\nstrong chaoticity of the dynamics.\n\nThe same high temperature estimates of $k_0(T)$ and $\\sigma^2_{k}(T)$\nhave been performed for the $3d$ system.\nIn Fig. \\ref{keflutt.3dfig},\nthe numerical determination of $\\sigma^2_{k}(T)$\nshows the appearance of a very pronounced peak at the phase transition point\nwhich is not predicted by the analytic estimate, \nwhereas the average Ricci curvature $k_0(T)$ is in agreement\nwith the analytic values of the high temperature estimate, computed by spin \ndecoupling, above the critical temperature, as in the $2d$ model.\n\n\\medskip\n\\subsection{Geometric observables and Lyapunov exponents}\n\\medskip\nWe have seen that the largest\nLyapunov exponent is sensitive to the phase transition and at the same time\nwe know that\nit is also related to the average curvature properties of the ``mechanical\nmanifolds''.\nThus, the geometric observables $k_0(T)$ and $\\sigma^2_k(T)$\nabove considered can be used to \nestimate the Lyapunov exponents, as well as to detect the phase transition.\n\nIn principle, by means of Eq.(\\ref{lambda}), one can evaluate\nthe largest Lyapunov exponent without any need of dynamics,\nbut simply using \nglobal geometric quantities of the manifold \nassociated to the physical system.\nFor $2d$ and $3d$ XY models, fully analytic computations are possible only\nin the limiting cases of high and low temperatures. Microcanonical averages\nof $k_0$ and $\\sigma^2_k$ at arbitrary $T$ have been numerically computed\nthrough time averages. We can call this hybrid method semi-analytic.\n\nIn Fig. \\ref{prev.Lyap.2d}, the results of the semi-analytic prediction of the \nLyapunov exponents for the $2d$ model are plotted vs. temperature and \ncompared with the numerical outcomes of the tangent dynamics.\nAs one can see, the prediction formulated on the basis of Eq.(\\ref{lambda}) \nunderestimates the numerical values given by the tangent \ndynamics.\nThe semi-analytic prediction can be improved\nby observing that the replacement of the sectional curvature fluctuation\n$\\delta K^{(2)}$ in Eq.(\\ref{sectional}) with a fraction \nof the Ricci curvature [which underlies the derivation of Eq.(\\ref{eqpsi})] \nunderestimates the frequency of occurrence of negative sectional curvatures,\nwhich was already the case of the $1d$ XY model \\cite{CCP}.\nThe correction procedure can be implemented\nby evaluating the probability $P(T)$ of obtaining a negative value of the \nsectional curvature along a generic trajectory and then by operating the \nsubstitution\n\\beq\nK_{R}(T)\\rightarrow \\frac{K_{R}(T)}{1+P(T) \\alpha}~.\n\\label{kappafrac}\n\\eeq\nThe parameter $\\alpha$ is a free parameter to be empirically estimated.\nIts value ranges from $100$ to $200$, without appreciable differences in the\nfinal result.\nIt resumes \nthe non trivial information about the more pronounced tendency\nof the trajectories towards negative\nsectional curvatures with respect to the predictions of the geometric model\ndescribing the chaoticity of the dynamics.\n\nThe probability $P(T)$ is estimated through the occurrence along \na trajectory of negative values \nof the sum of the coefficients that appear in the definition\nof $K^{(2)}$ [Eqs.(\\ref{curvsezXY}) and (\\ref{curvsez3d})]\n\\beq\nP(T)\\sim \\frac{\\int_{-\\pi}^{\\pi}\n\\Theta (-\\cos(q_{k+1,l}-q_{k,l})-\\cos(q_{k,l+1}-q_{k,l}))\n\\exp[-\\beta V({\\bf q})] \\prod_{k,l=1}^{N}dq_{k,l}}\n{\\int_{-\\pi}^{\\pi}\\exp[-\\beta V({\\bf\nq})] \\prod_{k,l=1}^{N}dq_{k,l}}~~,\n\\label{2d.pro.cos.neg}\n\\eeq\naveraged over all the sites\n$\\forall k,l\\in (1,\\ldots,N)$; $\\Theta$ is the step function.\n\nAlternatively, owing to the already remarked relation between vorticity\nand sectional curvature $K^{(2)}$,\n$P(T)$ can be replaced by the average density of\nvortices\n\\beq\nK_{R}(T)\\rightarrow \\frac{K_{R}(T)}{1+\\overline{\\alpha} {\\cal V}(T)}~,\n\\label{correction}\n\\eeq\nwhere $\\overline{\\alpha}$ a free parameter. \nActually,\nin the $2d$ model, the two corrections, one given by\nEq.(\\ref{kappafrac}) with $P(T)$ of Eq. (\\ref{2d.pro.cos.neg}), the other \ngiven by Eq.(\\ref{correction}) with the vorticity function in place of $P(T)$,\nconvey the same information.\nThe semi-analytic predictions of $\\lambda_1(T)$ with correction\nare reported in Fig. \\ref{prev.Lyap.2d}.\n\nIn the limits of high and low temperatures, \n$\\lambda_1(T)$ can be given the analytic forms \n$\\lambda_{1}(T)\\sim T^{-1/6}$ at high \ntemperature, and \n$\\lambda_{1}(T)\\sim T^{2}$ at low temperature.\nIn the former case, the high temperature approximation (\\ref{partition})\nis used, and in the latter case the quasi-harmonic oscillators approximation\nis done. The deviation of $\\lambda_1(T)$ from the quasi-harmonic scaling,\nstarting at $T \\simeq0.6$ and already observed to correspond to the \nappearance of vortices, finds here a simple explanation through the geometry\nof dynamics: vortices are associated with negative sectional curvatures,\nenhancing chaos.\n\nBy increasing the spatial dimension of the system, it becomes more and more\ndifficult to accurately estimate the probability of obtaining negative\nsectional curvatures. \nThe assumption that the occurrence of negative values\nof the cosine of the difference between the directions of two nearby spins\nis nearly equal to $P(T)$, is less effective in the $3d$ \nmodel than in the $2d$ one. \nAgain, the vorticity function can be assumed as an estimate of $P(T)$\n[Eq. (\\ref{correction})].\nThe quality of the results has a weak dependence upon \nthe parameter $\\alpha$.\nThe correction remains good, with $\\alpha$ belonging to a broad interval\nof values ($100 \\div 200$). \nIn the limits of high and low temperatures, the model predicts correctly\nthe same scaling laws of the $2d$ system.\n\nIn Fig. \\ref{prev.Lyap.3d} the semi-analytic\n predictions for the Lyapunov exponents, \nwith and without correction, are \nplotted vs. temperature together with the numerical results\nof the tangent dynamics.\nIt is noticeable that the prediction of Eq. (\\ref{lambda}) is able to give\nthe correct asymptotic behavior of the Lyapunov exponents\nalso at low temperatures, the most difficult part \nto obtain by means of dynamical simulations. \n\n\\medskip\n\\subsection{A topological hypothesis}\n\\medskip\n\nWe have seen in Fig. \\ref{keflutt.3dfig} that a sharp peak of the \nRicci-curvature fluctuations\n$\\sigma_\\kappa^2(T)$ is found for the $3d$ model in correspondence of the \nsecond order phase transition, whereas, for the $2d$ model, \n$\\sigma_\\kappa^2(T)$ appears regular and in agreement with the theoretically \npredicted smooth pattern.\nOn the basis of heuristic arguments, in Refs.\\cite{CCCP,CCCPPG} we suggested\nthat the peak of $\\sigma_\\kappa^2$ observed for the $3d$ XY model, as well as\nfor $2d$ and $3d$ scalar and vector lattice $\\varphi^4$ models, might originate\nin some change of the {\\it topology} of the mechanical manifolds. In fact,\nin abstract mathematical models, consisting of families of surfaces undergoing\na topology change -- i.e. a loss of diffeomorphicity among them -- \nat some critical value of\na parameter labelling the members of the family, we have actually observed the\nappearance of cusps of $\\sigma_K^2$ at the \ntransition point between two subfamilies of surfaces of different topology,\n$K$ being the Gauss curvature.\n\nActually, for the mean-field XY model, where both $\\sigma_\\kappa^2(T)$ and \n$\\lambda_1(T)$ have theoretically been shown to loose analyticity at the phase \ntransition point, a direct evidence of a ``special'' change of the topology\nof equipotential hypersurfaces of configuration space has been given\n\\cite{cegdcp}. Other indirect and direct evidences of the actual involvement\nof topology in the deep origin of phase transitions have been recently\ngiven \n\\cite{fps1,fps2} for the lattice $\\varphi^4$ model.\n\nIn the following Section we consider the extension \nof this\ntopological point of view about phase transitions\nfrom equipotential hypersurfaces of configuration space\nto constant energy hypersurfaces of phase space.\n\n\\medskip\n\\section{Phase space geometry and thermodynamics.}\n\\medskip\n\nIn the preceding Section we have used some elements of intrinsic differential\ngeometry of submanifolds of configuration space to describe the average\ndegree of dynamical instability (measured by the largest Lyapunov exponent).\nIn the present Section we are interested in the relationship\nbetween the extrinsic geometry of the constant energy\nhypersurfaces $\\Sigma_E$ and thermodynamics.\n\nHereafter, phase space is considered as an even-dimensional subset $\\Gamma$\nof ${\\Bbb R}^{2N}$\nand the hypersurfaces $\\Sigma_E=\\{(p_1,\\dots ,p_N,q_1,\\dots ,q_N)\\in{\\Bbb R}\n\\vert H(p_1,\\dots ,p_N,q_1,\\dots ,q_N)=E\\}$ are manifolds that can be equipped\nwith the standard Riemannian metric induced from ${\\Bbb R}^{2N}$. If, for \nexample, a surface is parametrically defined through the equations\n$x^i=x^i(z^1,\\dots ,z^k)$, $i=1,\\dots ,2N$, then the metric $g_{ij}$\n{\\it induced} on the surface is given by $g_{ij}(z^1,\\dots ,z^k)=\n\\sum_{n=1}^{2N}\n\\frac{\\partial x^n}{\\partial z^i} \\frac{\\partial x^n}{\\partial z^j}$. \nThe geodesic flow associated with the metric induced on $\\Sigma_E$ from\n${\\Bbb R}^{2N}$ has nothing to do with the \nHamiltonian flow that belongs to $\\Sigma_E$. Nevertheless, it exists an\nintrinsic \nRiemannian metric $g_S$ of phase space $\\Gamma$ such that the geodesic \nflow of $g_S$, restricted to $\\Sigma_E$, coincides with the Hamiltonian flow\n($g_S$ is the so called Sasaki lift to the tangent bundle\nof configuration space of the Jacobi\nmetric $g_J$ that we mentioned in a preceding Section).\n\nThe link between extrinsic geometry of the $\\Sigma_E$ and thermodynamics\nis estabilished through the microcanonical definition of entropy\n\\beq\nS = k_B \\log \\int_{\\Sigma_E} \\frac{d\\sigma}{\\|{\\nabla H}\\|} ~,\n\\label{entropy}\n\\eeq\nwhere $d\\sigma = \\sqrt{det(g)} dx_1...dx_{2N-1}$ is the invariant volume\nelement of $\\Sigma_E \\subset{\\Bbb R}^{2N}$, ${g}$ is the metric induced from\n${\\Bbb R}^{2N}$ and $x_1...x_{2N-1}$ are the coordinates on $\\Sigma_E$.\n\nLet us briefly recall some necessary definitions and concepts that are needed\nin the study of hypersurfaces of euclidean spaces.\n\nA standard way to investigate the geometry of an hypersurface $\\Sigma^m$ is \nto study the way in which it curves around in ${\\Bbb R}^{m+1}$: this is \nmeasured by the way the normal direction changes as we move from point to\npoint on the surface. The rate of change\nof the normal direction ${\\bf N}$ at a point $x\\in\\Sigma$ is described by\nthe {\\it shape operator} $L_x({\\bf v}) = - \\nabla_{\\bf v}{\\bf N}\n = - (\\nabla N_1\\cdot{\\bf v},\\dots ,\\nabla N_{m+1}\n\\cdot{\\bf v})$, where\n${\\bf v}$ is a tangent vector at $x$ and $\\nabla_{\\bf v}$ is the directional\nderivative of the unit normal ${\\bf N}$.\nAs $L_x$ is an operator of the tangent space at $x$ into\nitself, there are $m$ independent eigenvalues \\cite{thorpe} $\\kappa_1(x),\n\\dots,\\kappa_m(x)$, which are called the principal curvatures of $\\Sigma$ at\n$x$. Their product is the {\\it Gauss-Kronecker curvature}: \n$K_G(x)=\\prod_{i=1}^m\\kappa_i(x)={\\rm det}(L_x)$, and\ntheir sum is the so-called {\\it mean curvature}:\n$M_1(x)=\\frac{1}{m}\\sum_{i=1}^m\\kappa_i(x)$.\nThe quadratic form $L_x({\\bf v})\\cdot{\\bf v}$, associated with the shape \noperator at a point $x$, is called the second fundamental form of $\\Sigma$ at \n$x$.\n\nIt can be \n shown \\cite{doCarmo} that the mean curvature of the energy hypersurfaces\n is given by\n\\begin{equation}\nM_1(x)= -\\frac{1}{2N-1}\\nabla\\cdot\\left(\\frac{\\nabla H(x)}{\\Vert\\nabla H(x)\n\\Vert}\\right)~~,\n\\label{emme1}\n\\end{equation} \nwhere $\\nabla H(x)/\\Vert\\nabla H(x)\\Vert$ is the unit normal to\n$\\Sigma_E$ at a given point $x=(p_1,\\dots ,p_N,q_1,\\dots ,q_N)$, and \n$\\nabla =(\\partial/\\partial p_1,\\dots,\\partial/\\partial q_N)$, \nwhence the explicit expression\n\\begin{eqnarray}\n(2N-1)\\, M_1 = &-&\\frac{1}{\\Vert\\nabla H\\Vert}\\left[ N + \\sum_{\\bf i}\n\\left(\\frac{\\partial^2V}{\\partial q_{\\bf i}^2}\\right)\\right]\\nonumber\\\\\n& +& \\frac{1}{\\Vert\\nabla H\\Vert^3}\n\\left[\\sum_{\\bf i}p_{\\bf i}^2 + \\sum_{\\bf i,j}\\left(\n\\frac{\\partial^2V}{\\partial q_{\\bf i}\\partial q_{\\bf j}}\\right)\\left(\n\\frac{\\partial V}{\\partial q_{\\bf i}}\\right)\\left(\\frac{\\partial V}\n{\\partial q_{\\bf j}}\\right)\\right]~,\n\\label{M1}\n\\end{eqnarray}\nwhere ${\\bf i,j}$ are multi-indices according to the number of spatial \ndimensions.\n \nThe link between geometry and physics stems from the microcanonical definition\nof the temperature\n\\begin{equation}\n\\frac{1}{T}=\\frac{\\partial S}{\\partial E}=\\frac{1}{\\Omega_{\\nu}(E)}\\frac\n{d\\Omega_{\\nu}(E)}{dE}~,\n\\label{temperature}\n\\end{equation}\nwhere we used Eq.(\\ref{entropy}) with $k_B=1$, $\\nu =2N-1$, and \n$\\Omega_{\\nu}(E)=\\int_{\\Sigma_E}d\\sigma /\\Vert\\nabla H\\Vert$. \n>From the formula \\cite{laurence} \n\\begin{equation}\n{d^k \\over dE^k} \\left( \\int_{\\Sigma_E} \\alpha~d\\sigma \\right) (E')\n = \\int_{\\Sigma_{E'}} A^k(\\alpha)\\, d\\sigma~~,\n\\end{equation}\nwhere $\\alpha$ is an integrable function and $A$ is the operator \n$ A(\\alpha)= {\\nabla \\over \\Vert \\nabla H\\Vert}\n \\cdot \\left( \\alpha \\cdot\n {\\nabla H \\over \\Vert \\nabla H\\Vert} \\right),\n$\nit is possible to work out the result\n\\begin{equation}\n\\frac{1}{T}=\n\\frac{1}{\\Omega_{\\nu}(E)}\\frac{d\\Omega_{\\nu}(E)}{dE} = \\frac{1}{\\Omega_{\\nu}}\n\\int_{\\Sigma_E} \\frac{d\\sigma}{\\Vert\\nabla\nH\\Vert} \\left[ 2 \\frac{M_1^\\star}{\\Vert\\nabla H\\Vert} - \\frac{\\triangle H}\n{\\Vert\\nabla H\\Vert^2} \\right]\\simeq \\frac{1}{\\Omega_{\\nu}} \n\\int_{\\Sigma_E} \\frac{d\\sigma}\n{\\Vert\\nabla H\\Vert}\\, \\frac{M_1^\\star}{\\Vert\\nabla H\\Vert} ~~ ,\n\\label{ballerotte}\n\\end{equation}\nwhere $M_1^\\star = \\nabla (\\nabla H/ \\Vert\\nabla H\\Vert)$ is directly\nproportional to the mean curvature (\\ref{emme1}). In the last term of\nEq.(\\ref{ballerotte}) we have neglected a contribution which vanishes\nas ${\\cal O}(1/N)$. Eq. (\\ref{ballerotte}) provides the\nfundamental link between extrinsic geometry and thermodynamics \\cite{nota1}. \nIn fact, the microcanonical average of ${M_1^\\star}/{\\Vert\\nabla H\\Vert}$, \nwhich is a quantity \ntightly related with the mean curvature of $\\Sigma_E$, gives the inverse\nof the temperature, whence other important thermodynamic observables can be \nderived. For example, the constant volume specific heat\n\\begin{equation}\n\\frac{1}{C_V}=\\frac{\\partial T(E)}{\\partial E}~,\n\\label{cv}\n\\end{equation}\nusing Eq.(\\ref{temperature}), yields\n\\begin{equation}\nC_V=-\\left(\\frac{\\partial S}{\\partial E}\\right)^2\\,\\left(\\frac{\\partial^2S}\n{\\partial E^2}\\right)^{-1}~~,\n\\label{cv1}\n\\end{equation}\nbecoming at large $N$ \n\\begin{equation}\nC_V=-\\left<\\frac{{M_1^\\star}}{{\\Vert\\nabla H\\Vert}}\\right>_{mc}^2 \\left[\n \\frac{1}{\\Omega_{\\nu}} \\frac{d}{dE}\n\\int_{\\Sigma_E} \\frac{d\\sigma}\n{\\Vert\\nabla H\\Vert}\\,\\left( \\frac{M_1^\\star}{\\Vert\\nabla H\\Vert}+R(E)\\right)\n- \\left<\\frac{{M_1^\\star}}{{\\Vert\\nabla H\\Vert}}\\right>_{mc}^2\\right]^{-1}~,\n\\label{cv2}\n\\end{equation} \nwhere the subscript $mc$ stands for microcanonical average, and\n$R(E)$ stands for the quantities of order ${\\cal O}(1/N)$ \nneglected in the last term of Eq.(\\ref{temperature})\n(a-priori, its derivative can be non negligible and has to be taken into \naccount).\nEq. (\\ref{cv2}) highlights a more elaborated link between geometry and\nthermodynamics: the specific heat depends upon the microcanonical average of\n${M_1^\\star}/{\\Vert\\nabla H\\Vert}$ and upon the energy variation rate of the \nsurface integral of this quantity. \n\nRemarkably, the relationship between curvature\nproperties of the constant energy surfaces $\\Sigma_E$ and thermodynamic\nobservables given by Eqs.(\\ref{temperature}) and (\\ref{cv2}) can be extended\nto embrace also a deeper and very interesting relationship between \nthermodynamics and {\\it topology} of the constant\nenergy surfaces. Such a relationship can be discovered through a \nreasoning which, though approximate, is highly non-trivial, for it makes use\nof a deep theorem due to Chern and Lashof \\cite{ChernLashof}. \nAs $\\Vert\\nabla H\\Vert =\\{\\sum_i p_i^2 + [\\nabla_iV(q)]^2\\}^{1/2}$ is a\npositive quantity increasing with the energy, we can write\n\\begin{equation}\n\\frac{1}{T}=\n\\frac{1}{\\Omega_{\\nu}}\\frac{d\\Omega_{\\nu}}{dE} \\simeq\\frac{1}{\\Omega_{\\nu}}\n\\int_{\\Sigma_E} \\frac{d\\sigma}\n{\\Vert\\nabla H\\Vert}\\, \\frac{M_1^\\star}{\\Vert\\nabla H\\Vert}\n= D(E)\\frac{1}{\\Omega_{\\nu}} \\int_{\\Sigma_E} d\\sigma \\, M_1~,\n\\label{deformaz}\n\\end{equation} \nwhere we have introduced the factor function $D(E)$ in order to extract the\ntotal mean curvature $\\int_{\\Sigma_E} d\\sigma \\, M_1$; $D(E)$ has been \nnumerically found to be smooth and very close to $\\langle {1}/{\\Vert\n\\nabla H\\Vert^2} \\rangle_{mc}$ (see Section \\ref{5a} and Fig. \n\\ref{D(E)}). Then, \nrecalling the expression of a multinomial expansion \n\\begin{equation}\n (x_1 + \\cdots + x_{\\nu})^{\\nu} = \\sum_{_{\\{n_i\\},\\sum n_k =\\nu}} \n\\frac{\\nu !}{n_1!\\cdots n_{\\nu}!} \\cdot x_1^{n_1} \\cdots x_{\\nu}^{n_{\\nu}}~~ ,\n\\label{somma1}\n\\end{equation}\nand identifying the $x_i$ with the principal curvatures $k_i$, one obtains\n\\begin{equation}\n M_1^{\\nu} = {\\nu}! \\prod_{i=1}^{\\nu} k_i + R = {\\nu}! K + R~~~,\n\\end{equation}\nwhere $K=\\prod_i k_i$ is the Gauss-Kronecker curvature,\nand $R$ is the sum (\\ref{somma1}) without the term with the\nlargest coefficient ($n_k=1, \\ \\forall k$).\nUsing $\\nu ! \\simeq \\nu^{\\nu} e^{-\\nu} \\sqrt{2\\pi\\nu}$, \n\\begin{equation}\n M_1^{\\nu} \\simeq \\nu^{\\nu} e^{-\\nu} \\sqrt{4\\pi N} K + R\n\\label{pallino}\n\\end{equation}\nis obtained. \nThe above mentioned theorem of Chern and Lashof \nstates that\n\\begin{equation}\n\\int_{\\Sigma_E}\n\\vert K\\vert\\,d\\sigma\\geq Vol[{\\Bbb S}_1^{\\nu}]\\sum_{i=0}^{\\nu} \nb_i(\\Sigma_E )~,\n\\label{chern1}\n\\end{equation}\ni.e. the total absolute Gauss-Kronecker curvature of a hypersurface is\nrelated with the sum of all its Betti numbers $b_i(\\Sigma_E)$. The Betti \nnumbers are {\\it diffeomorphism invariants} of fundamental topological \nmeaning \\cite{Betti}, therefore their sum is also a topologic invariant.\n${\\Bbb S}_1^\\nu$ is a hypersphere of unit radius. Combining Eqs.\n(\\ref{pallino}) and (\\ref{chern1}) and integrating on $\\Sigma_E$, we obtain\n\\begin{equation}\n \\int_{\\Sigma_E}\\vert M_1^{\\nu}\\vert\\, d \\sigma \\simeq \n\\nu^{\\nu} e^{-\\nu} \\sqrt{2\\pi\\nu} \n\\int_{\\Sigma_E}\\vert K\\vert d \\sigma + \\int_{\\Sigma_E}\\vert R\\vert d \\sigma\n\\ge {\\cal A}\\,\\sum_{i=0}^{\\nu} b_i(\\Sigma_E ) + {\\cal R}(E)~~,\n\\label{tria}\n\\end{equation}\nwith the shorthands \n${\\cal A}=\\nu^{\\nu} e^{-\\nu} Vol(S_1^{\\nu})$ and ${\\cal R}=\\int_{\\Sigma_E}\n\\vert R\\vert d \\sigma$. \n \nNow, with the aid of the inequality $\\int\\Vert f\\Vert^{1/n}d\\mu\\geq\\Vert\\int f\nd\\mu\\Vert^{1/n}$, we can write\n\\begin{equation}\n\\int_{\\Sigma_E}\\vert M_1\\vert\\, d \\sigma = \\int_{\\Sigma_E}\\vert M_1^{\\nu}\\vert\n^{1/\\nu}\\, d \\sigma \\geq \n\\left\\vert\\int_{\\Sigma_E}M_1^{\\nu}\\,d\\sigma\\right\\vert^{1/\\nu}~~.\n\\label{chern2}\n\\end{equation}\nIf $M_1\\geq 0$ everywhere on $\\Sigma_E$, then \n$\\left\\vert\\int_{\\Sigma_E}M_1^{\\nu}\\,d\\sigma\\right\\vert^{1/\\nu} =\n\\left(\\int_{\\Sigma_E}\\vert M_1^{\\nu}\\vert\\,d\\sigma\\right)^{1/\\nu}$, whence, in \nthe hypothesis that $M_1\\geq 0$ is largely prevailing \\cite{bassi-indici}, \n$\\left\\vert\\int_{\\Sigma_E}M_1^{\\nu}\\,d\\sigma\\right\\vert^{1/\\nu} \\sim\n\\left(\\int_{\\Sigma_E}\\vert M_1^{\\nu}\\vert\\,d\\sigma\\right)^{1/\\nu}$. Under the \nsame assumption, $\\int_{\\Sigma_E}M_1d\\sigma\\sim\\int_{\\Sigma_E}\\vert M_1\\vert\nd\\sigma$ and therefore \n\\begin{equation}\n\\int_{\\Sigma_E} M_1\\, d \\sigma \\sim \\int_{\\Sigma_E}\\vert M_1^{\\nu}\\vert\n^{1/\\nu}\\, d \\sigma \\geq \n\\left\\vert\\int_{\\Sigma_E}M_1^{\\nu}\\,d\\sigma\\right\\vert^{1/\\nu} \\sim \n\\left(\\int_{\\Sigma_E}\\vert M_1^{\\nu}\\vert\\,d\\sigma\\right)^{1/\\nu} \n\\geq \\left[ {\\cal A}\\sum_{i=0}^{\\nu} b_i(\\Sigma_E ) + {\\cal R}(E)\n\\right]^{1/\\nu}~~.\n\\label{chern3}\n\\end{equation}\nFinally,\n\\begin{eqnarray}\n\\frac{1}{T(E)} = \n\\frac{1}{\\Omega_{\\nu}}\\frac{d\\Omega_{\\nu}}{dE}& \\simeq &\n\\left<\\frac{{M_1^\\star}}{{\\Vert\\nabla H\\Vert}}\\right>_{mc} =\\, \n\\frac{1}{\\Omega_{\\nu}} \\int_{\\Sigma_E} \\frac{d\\sigma}\n{\\Vert\\nabla H\\Vert}\\, \\frac{M_1^\\star}{\\Vert\\nabla H\\Vert}\n= D(E)\\frac{1}{\\Omega_{\\nu}} \\int_{\\Sigma_E} d\\sigma \\, M_1\\nonumber \\\\\n&\\geq & \n\\frac{D(E)}{\\Omega_{\\nu}}\\left[ {\\cal A}\n\\sum_{i=0}^{\\nu} b_i(\\Sigma_E ) + {\\cal R}(E) \\right]^{1/\\nu}~~.\n\\label{thermtop}\n\\end{eqnarray} \nEquation (\\ref{thermtop}) has the remarkable property of relating \nthe microcanonical definition of\ntemperature of Eq.(\\ref{deformaz}) with a {\\it topologic invariant} of\n$\\Sigma_E$.\nThe Betti numbers can be exponentially large with $N$ [for example, \nin the case of $N$-tori ${\\Bbb T}^N$, they are $b_k={N\\choose k}$], so that \nthe quantity $(\\sum b_k)^{1/N}$ can converge, at arbitrarily large $N$, to a \nnon-trivial limit (i.e. different from one). \nThus, even though the energy dependence of ${\\cal R}$ is\nunknown, the energy variation of $\\sum b_i(\\Sigma_E)$ must be mirrored -- at \nany arbitrary $N$ -- by the energy variation of the temperature. \nBy considering Eq.(\\ref{cv2}) in the light of Eq.(\\ref{thermtop}), we can\nexpect that \nsome suitably abrupt and major change in the topology of the $\\Sigma_E$ can \nreflect into the appearance of a peak of the specific heat,\nas a consequence of the associated \nenergy dependence of $\\sum b_k(\\Sigma_E)$ and of its derivative with respect \nto $E$. In other words, we see that a link must exist between thermodynamical\nphase transitions and suitable topology changes of the constant energy \nsubmanifolds of the phase space of microscopic variables. \nThe arguments given above, though in a still rough formulation, provide a \nfirst attempt to\nmake a connection between the topological aspects of the {\\it microcanonical} \ndescription of phase transitions and the already proposed {\\it topological \nhypothesis} about topology changes in configuration space and phase transitions\n\\cite{CCCP,CCCPPG,cegdcp,fps1,fps2}. \n\nDirect support to the {\\it topological \nhypothesis} has been given by the analytic study of a mean-field XY model\n\\cite{cegdcp} and by the numerical computation of the Euler characteristic \n$\\chi$ of the equipotential hypersurfaces $\\Sigma_v$ of the configuration \nspace in a $2d$ lattice $\\varphi^4$ model \\cite{fps2}.\nThe Euler characteristic is the alternate sum of all the Betti numbers of\na manifold, so it is another topological invariant, but it identically\nvanishes for odd dimensional manifolds, like the $\\Sigma_E$. \nIn Ref.\\cite{fps2},\n$\\chi (\\Sigma_v)$ neatly reveals the symmetry-breaking phase transition\nthrough a sudden change of\nits variation rate with the potential energy density $v$. A sudden \n``second order'' variation of the topology of the $\\Sigma_v$ appears in both\nRefs.\\cite{cegdcp,fps2} as the requisite for the appearance of a phase \ntransition. These results strenghten the arguments given in the present \nSection about the role of the topology of the constant energy hypersurfaces.\nIn fact, the larger is $N$, the smaller are the relative fluctuations\n$\\langle\\delta^2V\\rangle^{1/2}/\\langle V\\rangle$ and $\\langle\\delta^2K\\rangle\n^{1/2}/\\langle K\\rangle$ of the potential and kinetic energies respectively.\nAt very large $N$, the product manifold\n$\\Sigma^{N-1}_v\\times{\\Bbb S}_t^{N-1}$, with $v\\equiv\\langle V\\rangle$ and\n$t\\equiv\\langle K\\rangle$, $v+t=E$, is a good model manifold to represent \nthe part of $\\Sigma_E$ that is overwhelmingly sampled by the dynamics and that\ntherefore constitutes the effective support of the microcanonical measure\non $\\Sigma_E$. The kinetic energy submanifolds ${\\Bbb S}_t^{N-1}=\\{(p_1,\\dots \n,p_N)\\in{\\Bbb R}^N\\vert\\sum_{i=1}^N\\frac{1}{2}p_i^2=t\\}$ are hyperspheres.\n\nIn other words, at very large $N$ the microcanonical measure \nmathematically extends over a whole energy surface but, as far as physics\nis concerned, a non-negligible contribution to the microcanonical measure is\nin practice given only by a small subset of an energy surface. This subset can\nbe reasonably modeled by the product manifold $\\Sigma^{N-1}_v\\times{\\Bbb S}\n_t^{N-1}$, because the total kinetic and total potential energies - having\narbitrarily small fluctuations, provided that $N$ is large enough - can be\nconsidered almost constant. Thus, since ${\\Bbb S}_t^{N-1}$ at any $t$ is always\nan hypersphere, a change in the topology of $\\Sigma^{N-1}_v$ directly entails \na change of the topology of $\\Sigma^{N-1}_v\\times {\\Bbb S}_t^{N-1}$, that is\nof the effective model-manifold for the subset of $\\Sigma_E$ where the dynamics\nmainly ``lives'' at a given energy $E$.\n\nAt small $N$, the model with a single product manifold is no longer good and \nshould be replaced by the non-countable union $\\bigcup_{v\\in{\\cal I}\\subset\n{\\Bbb R}}\\Sigma^{N-1}_v\\times{\\Bbb S}_{E-v}^{N-1}$, with $v$ assuming all the \npossible values in a real interval ${\\cal I}$. From this fact the smoothing \nof the energy dependence of thermodynamic variables follows. \nNevertheless, the geometric and \ntopologic signals of the phase transition can remain much sharper than the\nthermodynamic signals also at small $N$ $(< 100)$, as it is witnessed by the\n$2d$ lattice $\\varphi^4$ model \\cite{fps1,fps2}.\n\nFinally, let us comment about the relationship between intrinsic geometry, \nin terms of\nwhich we discussed the geometrization of the dynamics, and extrinsic\ngeometry, dealt with in the present Section.\n\nThe most direct and intriguing link is estabilished by the \nexpression for microcanonical averages of generic observables of the kind\n$A(q)$, with $q=(q_1,\\dots,q_N)$,\n\\begin{equation}\n\\langle A\\,\\rangle_{mc} = \\frac{1}{\\Omega_{2N}(E)} \\int_{H(p,q)\\leq E} \nd^Np\\, d^Nq\\, A(q)\n=\\frac{1}{Vol(M_E)}\\int_{V(q)\\leq E} d^Nq\\, [E - V(q)]^{N/2}\\, A(q)~,\n\\label{medie1}\n\\end{equation}\nwhere $M_E=\\{q\\in{\\Bbb R}^N\\vert V(q)\\leq E\\}$. Eq. (\\ref{medie1}) is \nobtained by means of a Laplace-transform method \\cite{Pearson}; it is \nremarkable that $[E-V(q)]^N\\equiv det(g_J)$, where $g_J$ is the Jacobi metric\nwhose geodesic flow coincides with newtonian dynamics (see Section $V$),\ntherefore $d^Nq\\, [E - V(q)]^{N/2}\\equiv d^N q\\, \\sqrt{det (g_J)}$\nis the invariant Riemannian volume element of $(M_E,g_J)$. Thus, \n\\begin{equation}\n\\frac{1}{Vol(M_E)}\\int_{V(q)\\leq E} d^Nq\\, [E - V(q)]^{N/2}\\, A(q) \\equiv\\;\n\\frac{1}{Vol(M_E)}\\int_{M_E} d^Nq\\, \\sqrt{det (g_J)}\\, A(q)~,\n\\label{medie2}\n\\end{equation} \nwhich means that the microcanonical averages $\\langle A(q)\\,\\rangle_{mc}$ \ncan be expressed as Riemannian integrals on the mechanical manifold \n$(M_E,g_J)$. \n\nIn particular, this also applies to the microcanonical definition of entropy \n\\begin{equation}\nS = k_B \\log \\int_{H(p,q)\\leq E} d^N p\\, d^N q\\, =\n k_B \\log \\int_0^E dE^\\prime \\int_{\\Sigma_{E^\\prime}} \\frac{d\\sigma}\n{\\Vert{\\nabla H}\\Vert}~, \n\\label{entropy1}\n\\end{equation}\nwhich is alternative to that given in Eq.(\\ref{entropy}), though equivalent \nto it in the large $N$ limit. We have \n\\begin{eqnarray}\nS &=& k_B \\log \\left[ \\frac{1}{C \\Gamma (N/2+1)} \\int_{V(q)\\leq E} \nd^Nq \\;[E - V(q)]^{N/2}\\right] \\nonumber \\\\\n&\\equiv & k_B \\log \\int_{M_E} d^N q\\, \\sqrt{det (g_J)}+\\, const\\,.~~,\n\\label{entropy2}\n\\end{eqnarray}\nwhere the last term gives the entropy as the logarithm of the \nRiemannian volume of the manifold. \n\nThe topology changes of the surfaces $\\Sigma_v^{N-1}$, that are to be \nassociated with phase transitions, will deeply affect also the geometry of\nthe mechanical manifolds $(M_E, g_J)$ and $(M\\times{\\Bbb R}^2, g_E)$ \nand, consequently, they will affect the average instability properties of\ntheir geodesic flows. In fact, Eq.(\\ref{lambda}) links some curvature\naverages of these manifolds with the numeric value of the largest Lyapunov \nexponent. Loosely speaking, major topology changes of $\\Sigma_v^{N-1}$ will\naffect microcanonical averages of geometric quantities \ncomputed through Eq.(\\ref{medie1}), likewise entropy, computed\nthrough Eq.(\\ref{entropy2}).\n\nThus, the peculiar temperature patterns displayed by the largest Lyapunov \nexponent at a second-order phase transition point -- in the present paper\nreported for the $3d$ $XY$ model, in Ref.\\cite{CCCPPG} reported for\nlattice $\\varphi^4$ models -- appear as reasonable consequences of the deep \nvariations of the topology of the equipotential hypersurfaces of configuration \nspace.\n\nWe notice that topology seems to provide a common ground to the roots of \nmicroscopic dynamics and of thermodynamics and, notably, it can account for\n major qualitative changes simultaneously occurring in both dynamics and\nthermodynamics when a phase transition is present.\n\n\\medskip\n\\subsection{Some preliminary numerical computations}\n\\label{5a}\n\\medskip\nLet us briefly report on some preliminary numerical computations concerning \nthe extrinsic geometry of the hypersurfaces $\\Sigma_E$ in the case of the $3d$ \nXY model. \n\nThe first point about extrinsic geometry that we numerically addressed was to\ncheck whether the inverse of the temperature, that appears in \nEq.(\\ref{deformaz}), can be reasonably factorized into the product of a smooth \n``deformation factor'' $D(E)$ and of the total mean curvature \n$\\int_{\\Sigma_E}M_1d\\sigma$.\nTo this purpose, the two independently computed quantities \n$\\langle 1/\\Vert\\nabla H\\Vert^2\\rangle_{mc}$ and $D(E)=[\\int_{\\Sigma_E}\n(d\\sigma /\\Vert\\nabla H\\Vert )(M_1^\\star/\\Vert\\nabla H\\Vert)]\\,/\\,\n[\\int_{\\Sigma_E}d\\sigma M_1]$ are compared in Fig. \\ref{D(E)},\nshowing that actually $\\int_{\\Sigma_E}(d\\sigma /\\Vert\\nabla H\\Vert )\n(M_1^\\star/\\Vert\\nabla H\\Vert) \\simeq\n\\langle 1/\\Vert\\nabla H\\Vert^2\\rangle_{mc}\\int_{\\Sigma_E}d\\sigma M_1$.\nIn other words, $D(E)\\simeq\\langle 1/\\Vert\\nabla H\\Vert^2\\rangle_{mc}$ \nand no ``singular'' feature in its energy pattern\nseems to exist, what suggests that $\\int_{\\Sigma_E} d\\sigma \\ M_1$ has to \nconvey all the information \nrelevant to the detection of the phase transition.\nThere is no reason to think that the validity of the \nfactorization given in Eq.(\\ref{deformaz}) is limited to the special case \nof the XY model.\n\nThe other point that we tackled concerns an indirect quantification of how\na phase space trajectory curves around and knots on the $\\Sigma_E$ to which \nit belongs. We can expect that the way in which an hypersurface \n$\\Sigma_E$ is ``filled'' by a phase space trajectory living on it will be \naffected by the geometry and the topology of the $\\Sigma_E$.\nIn particular, we computed the normalized autocorrelation function of the time\nseries $M_1[x(t)]$ of the mean curvature at the points of $\\Sigma_E$ visited\nby the phase space trajectory, \nthat is, the quantity\n\\begin{equation}\n\\Gamma (\\tau )=\\langle \\delta M_1(t+\\tau )\\delta M_1(t)\\rangle_t~~,\n\\label{autocor}\n\\end{equation}\nwhere $\\delta M_1(t)=M_1(t)-\\langle M_1(t^\\prime )\\rangle_{t^\\prime}$ is the\nfluctuation with respect to the average (the ``process'' $M_1(t)$ is supposed\nstationary). Our aim was to highlight the extrinsic\ngeometric-dynamical counterpart of a symmetry-breaking phase transition.\n\nThe practical computation of $\\Gamma (\\tau )$ proceeds by working out the \nFourier power spectrum $\\vert \\tilde M_1(\\omega )\\vert^2$ of $M_1[x(t)]$,\nobtained by averaging $15$ spectra computed by an FFT algorithm with a mesh\nof $2^{15}$ points and a sampling time $\\Delta t=0.1$.\nSome typical results for $\\Gamma (\\tau )$, obtained at different temperatures,\nare reported in Fig.\\ref{Gamma}. The patterns $\\Gamma (\\tau )$ display a\nfirst regime of very fast decay, which is not surprising because of the \nchaoticity of the trajectories at \nany energy, followed by a longer tail of slower \ndecay. An autocorrelation time $\\tau_{corr}$ can be defined\nthrough the first intercept of $\\Gamma (\\tau )$ with an almost-zero level\n($\\Gamma =0.01$).\nIn Fig.\\ref{tau} we report the values of $\\tau_{corr}$ so defined vs.\ntemperature.\nIn correspondence of the phase transition (whose critical temperature is \nmarked by a vertical dotted line), $\\tau_{corr}$ changes its temperature \ndependence: by lowering the temperature,\nbelow the transition $\\tau_{corr}(T)$ rapidly increases, \nwhereas it mildly \ndecreases above the transition. Below $T\\simeq 0.9$, where the vortices\ndisappear, the autocorrelation functions of $M_1$ look quite different and \nit seems no longer possible to coherently define a correlation time. \nThis result has an intuitive meaning and confirms that the phase transition\ncorresponds to a change in the microscopic dynamics, as already signaled by\nthe largest Lyapunov exponent; however, notice that the correlation times \n$\\tau_{corr}(T)$ are much longer than the inverse values of the corresponding \n$\\lambda_1(T)$. Qualitatively, $\\lambda_1(T)$ and $\\tau_{corr}^{-1}(T)$\nlook similar, however the two functions are not simply related.\n\\medskip\n\\section{Discussion and perspectives}\n\\medskip\nThe microscopic Hamiltonian dynamics of the classical Heisenberg XY model in\ntwo and three spatial dimensions has been numerically investigated.\nThis has been possible after the addition to the Heisenberg potentials of\na standard (quadratic) kinetic energy term.\nSpecial emphasis has been given to the study of the dynamical counterpart\nof phase transitions, detected through the time averages of conventional \nthermodynamic observables, and to the new mathematical concepts that are\nbrought about by Hamiltonian dynamics.\n\nThe motivations of the present study are given in the Introduction.\nLet us now summarize what are the outcomes of our investigations and comment \nabout their meaning. There are three main topics, tightly related one to \nthe other:\n\\begin{itemize} \n\\item{} the phenomenological description of phase transitions through the \nnatural, microscopic dynamics in place of the usual Monte Carlo stochastic \ndynamics; \n\\smallskip\n\\item{} the investigation, in presence of phase transitions, of certain \naspects of the (intrinsic) geometry of the mechanical manifolds where the\nnatural dynamics is represented as a geodesic flow;\n\\smallskip\n\\item{} the discussion of the relationship between the (extrinsic) \ngeometry of constant energy hypersurfaces of phase space and thermodynamics.\n\\smallskip\n\\end{itemize}\nAbout the first point, we have found that microscopic Hamiltonian dynamics very\nclearly evidences the presence of a second order phase transition through the\ntime averages of conventional thermodynamic observables. Moreover, the \nfamiliar \nsharpening effects, at increasing $N$, of the specific heat peak and \nof the order parameter bifurcation are observed. The evolution of the \norder parameter with respect to the physical time (instead of the\nfictitious Monte Carlo time) is also accessible, showing the appearance of \nGoldstone modes and that, in presence of a second order phase transition,\nthere is a clear tendency to the freezing of transverse fluctuations\nof the order parameter when $N$ is increased. \nThe \"freezing\" is observed together with a reduction of the\nlongitudinal fluctuations, i.e. the rotation of the magnetization \nvector slows down, preparing the breaking of the $O(2)$ symmetry at \n$N\\rightarrow\\infty$. At variance, when a Kosterlitz-Thouless transition is \npresent, at increasing $N$ the magnetization vector has a faster rotation\nand a smaller norm, preparing the absence of symmetry-breaking in the\n$N\\rightarrow\\infty$ limit as expected. \n\nRemarkably, to detect phase transitions, microscopic Hamiltonian dynamics \nprovides us with additional observables of purely dynamical nature, i.e.\nwithout statistical counterpart: Lyapunov exponents. Similarly to what we and \nother authors already reported for other models (see Introduction), also\nin the case of the $3d$ XY model a peculiar \ntemperature pattern of the largest Lyapunov exponent shows up in presence of\nthe second order phase transition, signaled by a ``cuspy'' point. \nBy comparing the patterns $\\lambda_1(T)$ given by Hamiltonian dynamics and by \na suitably defined random dynamics respectively, we suggest that the \ntransition between thermodynamically ordered and disordered phases has its \nmicroscopic dynamical counterpart in a transition between weak and strong \nchaos. Though $a-posteriori$ physically reasonable, this result is far from\nobvious, because the largest Lyapunov exponent measures the average \n{\\it local instability} of the dynamics, which $a-priori$ has little to do\nwith a {\\it collective}, and therefore global, phenomenon such as a phase \ntransition.\nThe effort to understand the reason of such a sensitivity of $\\lambda_1$ to\na second order phase transition and to other kinds of transitions, as\nmentioned in the Introduction, is far reaching. \n\nHere we arrive to the second\npoint listed above. In the framework of a Riemannian geometrization of\nHamiltonian dynamics, the largest Lyapunov exponent is related to the \ncurvature properties of suitable submanifolds of configuration space whose\ngeodesics coincide with the natural motions. In the mathematical light of this\ngeometrization of the dynamics, and after the numerical evidence of a sharp\npeak of curvature fluctuations at the phase transition point, the peculiar \npattern of $\\lambda_1(T)$ is due to some major change occurring to the geometry\nof mechanical manifolds at the phase transition. Elsewhere, we have conjectured\nthat indeed some major change in the {\\it topology} of configuration space\nsubmanifolds should be the very source of the mentioned major change of \ngeometry. \n\nThus, we have made a first attempt to provide an analytic argument\nsupporting this topological hypothesis (third point of the above list).\nThis is based on the appearance of a non trivial relationship between the\ngeometry of constant energy hypersurfaces of phase space with their topology\nand with the microcanonical definition of thermodynamics. Even still in a\npreliminary formulation, our reasoning already seems to indicate the \ntopology of energy hypersurfaces as the best candidate to explain the deep\norigin of the dynamical signature of phase transitions detected through\n$\\lambda_1(T)$.\n\nThe circumstance, mentioned in the preceding Section, of the persistence \nat small $N$ of geometric and topologic signals of the phase transition \nthat are much sharper than the thermodynamic signals is of\nprospective interest for the study of phase transition phenomena in finite,\nsmall systems, a topic of growing interest thanks to the modern developments\n- mainly experimental - about the physics of nuclear, atomic and molecular \nclusters, of conformational phase transitions in homopolymers and proteins,\nof mesoscopic systems, of soft-matter systems of biological interest.\nIn fact, some unambiguous information for small systems - even about the \nexistence itself of a phase transition - could be better obtained by means \nof concepts and mathematical tools outlined here and in the quoted papers.\nHere we also join the very interesting line of thought of Gross and \ncollaborators \\cite{Gross,Gross3} about the microcanonical description of phase\ntransitions in finite systems. \n\nLet us conclude with a speculative comment about another possible direction of\ninvestigation related with this signature of phase transitions through Lyapunov\nexponents. \nIn a field-theoretic framework, based on a path-integral\nformulation of classical mechanics \\cite{reuter,gozzi1,gozzi2}, Lyapunov \nexponents are defined through the expectation values of suitable operators. \nIn the field-theoretic framework, ergodicity breaking appears related to a\nsupersymmetry breaking \\cite{reuter}, and Lyapunov exponents are related to \nmathematical objects that have many analogies with topological concepts \n\\cite{gozzi2}.\n\nThe new mathematical concepts and methods, that the Hamiltonian \ndynamical approach brings about, could hopefully be useful also in the study \nof more ``exotic'' transition phenomena than those tackled in the present \nwork. Besides the above mentioned soft-matter systems, this could be the case \nof transition phenomena occurring in amorphous and disordered materials.\n\n\\medskip\n\\section{acknowledgments}\n\\medskip\nWe warmly thank L. Casetti, E.G.D. Cohen,R. Franzosi and L. Spinelli for\nmany helpful discussions.\nDuring the last year C.C. has been supported by the\nNSF (Grant \\# 96-03839) and by the La Jolla Interfaces in Science\nprogram (sponsored by the Burroughs Wellcome Fund).\nThis work has been partially supported by I.N.F.M., under the PAIS\n{\\it Equilibrium and non-equilibrium dynamics in condensed matter systems},\nwhich is hereby gratefully acknowledged.\n\n\\medskip\n\\section{Appendix A}\n\\medskip\n\nLet us briefly explain how a random markovian dynamics is constructed on a\ngiven constant energy hypersurface of phase space. The goal is to\ncompare the energy dependence of the largest Lyapunov exponent computed\nfor the Hamiltonian flow and for a suitable random walk respectively.\nOne has to devise an algorithm to generate a random walk on a given energy\nhypersurface such that, once the time interval $\\Delta t$ separating two\nsuccessive steps is assigned, the average increments of the coordinates\nare equal to the average increments of the same coordinates for the\ndifferentiable dynamics integrated with a time step $\\Delta t$.\nIn other words, the random walk has to roughly mimick the differentiable \ndynamics with the exception of its possible time-correlations.\n\nOne starts with a random initial configuration of the coordinates\n$q_{i},~ i=1,2,\\ldots,N$, uniformly distributed in the interval\n$[0,2\\pi]$, and with a random gaussian-distributed choice of the \ncoordinates $p_{i}$.\nThe random pseudo-trajectory is generated according to the simple scheme\n\\begin{eqnarray}\n(q_{i})_{(k+1)\\Delta t} &\\mapsto& (q_{i})_{k \\Delta t} +\n\\alpha _{q} G_{i,k} \\Delta t \\nonumber\\\\\n(p_{i})_{(k+1)\\Delta t} &\\mapsto& (p_{i})_{k \\Delta t} +\n\\alpha _{p} G_{i,k} \\Delta t~~,\n\\label{step.micro}\n\\end{eqnarray}\nwhere $\\Delta t$ is the time interval associated to one step $k \\mapsto k+1$ \nin the markovian chain, $G_{i,k}$ are gaussian distributed random numbers with\nzero expectation value and unit variance; the parameters\n$\\alpha_{q}$ and $\\alpha_{p}$ are the variances of the processes\n$(q_i)_k$ and $(p_i)_k$.\nThese variances are functions of the energy per degree of freedom\n$\\varepsilon$. They have to be set equal to the \nnumerically computed average increments of the coordinates obtained \nalong the differentiable trajectories integrated with the same time step \n$\\Delta t$, that is\n\\begin{eqnarray}\n\\alpha_{q}(\\epsilon)&=&\\left\\langle\\left[{\\frac{1}{N}\\sum_{i=1}^{N}\n\\frac{(q_{i}(t+\\Delta t)\n-q_{i}(t))^{2}}{\\Delta t}}\\right]^{1/2}\\right\\rangle_t \\sim\n\\left\\langle\\left[ {\\frac{1}{N}\\sum_{i=1}^{N}p_{i}^{2}}\\right]^{1/2}\n\\right\\rangle_t\\sim \\sqrt{T}\\nonumber\\\\\n\\alpha_{p}(\\epsilon)&=&\\left\\langle\\left[{\\frac{1}{N}\\sum_{i=1}^{N}\n\\frac{(p_{i}(t+\\Delta t)-p_{i}(t))^{2}}{\\Delta t}}\\right]^{1/2}\\right\n\\rangle_t\\sim\n\\left\\langle\\left[{\\frac{1}{N}\\sum_{i=1}^{N} \\dot{p}_{i}^{2}}\\right]^{1/2}\n\\right\\rangle_t~~,\n\\end{eqnarray}\nwhere $T$ is the temperature.\nThen, in order to make minimum the energy fluctuations around any given value\nof the total energy, a criterium to accept or reject a new step along the\nmarkovian chain has to be assigned.\nA similar problem has been considered by Creutz, who developed\na Monte Carlo microcanonical algorithm \\cite{Creutz}, where a \n\"Maxwellian demon\" gives a part of its energy to the system to let it move to \na new configuration, or gains energy from the system,\nif the new proposed configuration produces an energy lowering. If the demon\ndoes not have enough energy to allow an energy increasing update of the\ncoordinates, no coordinate change is performed. In this way, the total \nenergy remains almost constant with only small fluctuations.\nAs usual in Monte Carlo simulations, it is appropriate to fix the parameters \nso as the acceptance rate of the proposed updates of the configurations is \nin the range $30\\%$ -- $60\\%$.\n\nA reliability check of the so defined random walk, and of the \nadequacy of the phase space sampling through the number of steps \nadopted in each run, is obtained by computing the averages of typical \nthermodynamic observables of known temperature dependences.\n\nAn improvement to the above described ``demon'' algorithm has been \nobtained through a simple reprojection on $\\Sigma_E$ of the\nupdated configurations \\cite{Pettini}; \nthe coordinates generated by means of (\\ref {step.micro})\nare corrected with the formulae\n\\beq\nq_{i}(k \\Delta t) \\mapsto q_{i}(k \\Delta t) +\n\\left[\\frac{(\\frac{\\partial H}{\\partial q_{i}}) \\Delta E}\n{\\sum_{i=1}^N (p_{j}^2+\n(\\frac{\\partial H}{\\partial q_{j}})^2 )}\\right]_{x_{R}({k \\Delta t})} \\\n\\eeq\n\\[\np_{i}({k \\Delta t}) \\mapsto p_{i}({k \\Delta t}) -\n\\left[\\frac{p_{i} \\Delta E}{\\sum_{j=1}^N (p_{j}^2+\n(\\frac{\\partial H}{\\partial q_{j}})^2 )}\\right]_{x_{R}({k \\Delta t})}~~, \\\n\\]\nwhere $\\Delta E$ is the difference between the energy of the new configuration\nand the reference energy, and $x_{R}({k \\Delta t})$ denotes the random\nphase space trajectory.\nAt each assigned energy, the computation of the largest Lyapunov exponent\n$\\lambda_{1}^R$ of this random trajectory is obtained by means of the \nstandard definition\n\\begin{equation}\n\\lambda_{1}^R = \\lim_{n \\rightarrow \\infty} \\frac{1}{n \\Delta t}\\sum_{k=1}^n\n\\log \\frac{\\| \\zeta ((k+1) \\Delta t)\\|}{\\| \\zeta (k \\Delta t)\\|}~~,\n\\label{bgs}\n\\end{equation}\nwhere $\\zeta (t)\\equiv (\\xi (t),\\dot\\xi (t))$ is given by the discretized \nversion of the tangent dynamics\n\\begin{equation}\n\\frac{\\xi_i ((k+1)\\Delta t)-2 \\xi_i(k\\Delta t)+\\xi_i ((k-1)\\Delta t)}\n{\\Delta {t}^2}+\n\\left(\\frac{\\partial^2 V}{\\partial q_{i} \\partial q_{j}}\\right)_\n{x_{R}(k \\Delta t)}\n\\xi_j (k \\Delta t) = 0~~.\n\\label{tandynR}\n\\end{equation}\n\nFor wide variations of the parameters ($\\Delta t$ and\nacceptance rate), \nthe resulting values of $\\lambda_{1}^R$ are in very good agreement.\nMoreover, the algorithm is sufficiently stable\n and the final value of $\\lambda_{1}^R$ is independent of\nthe choice of the initial condition.\n\nA more refined algorithm could be implemented by constructing a random \nmarkovian process $q(t_k)\\equiv [q_1(t_k),\\dots ,q_N(t_k)]$ performing an \nimportance sampling of the measure $d\\mu =[E-V(q)]^{N/2-1}\\,dq$ in \nconfiguration space. 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Halicioglu, and W. A. Tiller, Phys. Rev. \nA{\\bf 32}, 3030 (1985).\n\n\\bibitem{Palmer} A thorough discussion about ergodicity breaking, non\ninterchangeability of $t\\rightarrow\\infty$ and $N\\rightarrow\\infty$ limits \ncan be found in: R.G. Palmer, Adv. Phys. {\\bf 31}, 669 (1982).\n\n\\bibitem{LPV} J. Lebowitz, J. Percus, and L. Verlet, Phys. Rev.\n{\\bf 153}, 250 (1967).\n\n\\bibitem{CCP} L. Casetti, C. Clementi and M. Pettini, Phys. Rev. E\n {\\bf 54}, 5969 (1996).\n\n\\bibitem{Pettini} M.Pettini, Phys. Rev. E {\\bf 47}, 828 (1993).\n\n\\bibitem{Eisenhart} L. P. Eisenhart, Ann. of Math. {\\bf 30}, 591 (1939).\n\n\\bibitem{doCarmo} M. P. Do Carmo, {\\it Riemannian Geometry} (Birkh\\\"{a}user,\n Boston, 1992).\n\n\\bibitem{CerrutiPettini} M.Cerruti-Sola and M. Pettini, Phys. Rev.\n E{\\bf 53}, 179 (1996); M.Cerruti-Sola, R. Franzosi and\n M. Pettini, Phys. Rev. E{\\bf 56}, 4872 (1997).\n\n\\bibitem{cegdcp} L. Casetti, E.G.D. Cohen, and M. Pettini, Phys. Rev. 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J. {\\bf 5},\n 5 (1958).\n\n\\bibitem{Betti} M. Nakahara, {\\it Geometry, Topology and Physics}, \n (Adam Hilger, Bristol, 1989). \n\n\\bibitem{bassi-indici} The mathematical details to justify such an assumption\n are discussed in: R. Franzosi, {\\it Geometrical and Topological Aspects\n in the study of Phase Transitions}, PhD thesis, Department \n of Physics, University of Florence, (1999).\n\n\\bibitem{Gross3} D.H.E. Gross and E. Votyakov, {\\it Phase transitions in\n finite systems = topological peculiarities of the microcanonical entropy \n surface}, archived in cond-mat/9904073;\n D.H.E. Gross and E. Votyakov, {\\it Phase transitions in ``small'' systems},\n archived in cond-mat/9911257; D.H.E. Gross, {\\it Phase \n Transitions without thermodynamic limit}, archived in cond-mat/9805391.\n\n\\bibitem{reuter} E. Gozzi and M. Reuter, Phys. Lett. B {\\bf 233}, 383 (1989).\n\n\\bibitem{gozzi1} E. Gozzi, M. Reuter and W. D. Thacker, Chaos, Solitons \\&\n Fractals {\\bf 2}, 441 (1992).\n\n\\bibitem{gozzi2} E. Gozzi and M. Reuter, Chaos, Solitons \\&\n Fractals {\\bf 4}, 1117 (1994).\n\n\\bibitem{Creutz} M. Creutz, Phys. Rev. Lett. {\\bf 50}, 1411 (1983).\n\n\n\\end{thebibliography}\n%---------------------------------------------------------------------------\n\\newpage\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig01.ps}}\n\\caption{ The magnetization vector ${\\bf M}(t)$ computed along a trajectory\nfor the $2d$ XY model at different temperatures on a lattice \nof $N= 10 \\times 10$. Each point represents a vector ${\\bf M}(t)$ at \na certain time $t$. }\n\\label{figura.spin2d.10e10}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig02.ps}}\n\\caption{ The magnetization vector ${\\bf M}(t)$ at the temperature $T=0.74$,\ncorresponding to the specific energy $\\epsilon = 0.8$ and computed in \na time interval $\\Delta t = 10^5$, with a random initial configuration,\non lattices of $N = 10 \\times 10$ (external points)\n and of $N = 200 \\times 200$ \n(internal points). }\n\\label{fig.spin2d.t0.74}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig03.ps}}\n\\caption{ The magnetization vector ${\\bf M}(t)$ at the temperature $T=1$,\ncorresponding to the energy $\\epsilon = 1.2$, computed in\na time interval $\\Delta t = 10^5$, with a random initial configuration on \nlattices of {\\it a)} $N = 10 \\times 10$, {\\it b)} $N = 50 \\times 50$,\n{\\it c)} $N = 100 \\times 100$ and {\\it d)} $N = 200 \\times 200$ sites, \nrespectively. }\n\\label{fig.spin2d.t1}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig04.ps}}\n\\caption{ Specific heat at constant volume \ncomputed by means of Eq. (\\ref{cvmicro})\non a lattice of $N= 10 \\times 10$ (open circles) and of $N= 15\\times 15$\n(full triangles). Starlike squares refer to specific heat values\ncomputed by means of Eq. (\\ref{specheat}) on a lattice of $N= 10 \\times 10$\n. }\n\\label{calspec_2d}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig05.ps}}\n\\caption{ Vorticity function (plotted in {\\it a)} linear scale\nand {\\it b)} logarithmic scale) computed at different temperatures\nfor lattices of $N=10\\times 10$ (open circles) and $N=40\\times 40$\n(full circles). The results of the Monte Carlo\nsimulations for a lattice of $N=60\\times 60$ (crosses)\n are from \\protect\\cite{TobChes}. \nThe dashed line represents the power\nlaw ${\\cal V}(t) \\sim T^{10}$. }\n\\label{fig.vort2d}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig06.ps}}\n\\caption{ The largest Lyapunov exponents computed on different lattice \nsizes:\n$N = 10 \\times 10$ (starred squares), $N= 20 \\times 20$ (open triangles),\n$N= 40 \\times 40$ (open stars), $N= 50 \\times 50$ (open squares) and \n$N = 100 \\times 100$ (open circles). In the inset, symbols have the same\nmeaning. } \n\\label{xy2d.lyap.num.fig}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig07.ps}}\n\\caption{ The magnetization vector ${\\bf M}(t)$, computed at the temperature\n$T = 1.7$, on lattices of different sizes. By increasing the lattice \ndimensions,\nthe longitudinal fluctuations decrease. The time interval $\\Delta t = 3.5\n\\times 10^4 - 8 \\times 10^4$ is the same for the four simulations. }\n\\label{mag3d.e2}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig08.ps}}\n\\caption{ The magnetization vector ${\\bf M}(t)$ computed at different \ntemperatures on a lattice of $N = 10 \\times 10 \\times 10$ spins. }\n\\label{fig.1.spin3d.9}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.50\\linewidth]{Fig09a.ps}}\n\\medskip\n\\centerline{\\includegraphics[width=0.50\\linewidth]{Fig09b.ps}}\n\\caption{ The magnetization vector ${\\bf M}(t)$ computed at \nthe temperature $T = 2.22$ (slightly higher than the critical value)\non lattices of {\\it a)} $N = 10 \\times 10 \\times 10$ \nand {\\it b)}$N = 15 \\times 15 \\times 15$, respectively.\nThe time interval $\\Delta t = 0.5 \\times 10^4 - 1.5 \\times 10^4$ is the same\nfor both simulations. } \n\\label{3d.fig.spin.altat}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig10.ps}}\n\\caption{ The dynamical order parameter,\ndefined as the average of the modulus $| {\\bf M}(t) |$ along a trajectory,\ncomputed on lattices of\n$N = 10 \\times 10 \\times 10$ (full circles)\nand $N = 15 \\times 15 \\times 15$ (open circles). }\n\\label{parord.3d.fig}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig11.ps}}\n\\caption{ Specific heat at constant volume for the $3d$ model, computed\nby means of Eq. (\\ref{cvmicro}) on\nlattices of $N= 8 \\times 8 \\times 8$ (open triangles),\n$N = 10 \\times 10 \\times 10$ (open circles), $N = 12 \\times 12 \\times 12$\n(open stars) and $N= 15 \\times 15 \\times 15$ (open squares) .\nFull circles refer to specific heat values computed by means of Eq. (\\ref\n{specheat}) on a lattice of $N= 10 \\times 10 \\times 10$. \nThe dashed line points out the critical temperature $T_c \\simeq 2.17$ at which\nthe phase transition occurs. }\n\\label{calspec.3d.fig}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig12.ps}}\n\\caption{ Vorticity function at different temperatures along a dynamical\ntrajectory on a lattice of $N = 10 \\times 10 \\times 10$ sites. }\n\\label{vort.fig3d}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig13.ps}}\n\\caption{ The largest Lyapunov exponents computed at different temperatures\nfor the $3d$ model.\nNumerical results are for lattices of $N= 10 \\times 10 \\times 10$ (open \ncircles) and $N= 15 \\times 15 \\times 15$ (open stars). \nIn the inset, symbols have the same meaning. The dashed line points out the\ntemperature $T_c \\simeq 2.17$ of the phase transition. The solid line puts in\nevidence the departure of $\\lambda_1(T)$ from the linear growth. }\n\\label{lyap.3d.fig}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig14.ps}}\n\\caption{ The largest Lyapunov exponents computed by means of the random \ndynamics algorithm (full circles) are plotted \n in comparison with those computed by means of the standard dynamics\n(open stars) for a lattice of $N= 10 \\times 10 \\times 10$. }\n\\label{randyn.3d.fig}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth,angle=90]{Fig15.ps}}\n\\caption{ Time average of Ricci curvature (open circles) and its r.m.s.\n fluctuations\n(full circles) at different temperatures computed for a lattice\nof $N = 40 \\times 40$ sites. Solid lines are the analytic estimates obtained \nfrom a high temperature expansion. } \n\\label{fig.Ricci2d}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth,angle=90]{Fig16.ps}}\n\\caption{ Time average of Ricci curvature (open triangles) and its r.m.s.\n fluctuations\n(full triangles) computed at different\ntemperatures for a lattice of $N = 10 \\times 10 \\times 10$.\nOpen circles and full diamonds refer to a lattice size \nof $N= 15 \\times 15 \\times 15$. Solid lines are the analytic estimates\nin the limit of high temperatures. The dashed line points out the temperature\n$T_c \\simeq 2.17$ of the phase transition. }\n\\label{keflutt.3dfig}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig17.ps}}\n\\caption{ Analytic Lyapunov exponents computed for the $2d$ model by means of \nEq.(\\protect\\ref{lambda}) without correction (dots) and incorporating \nthe correction that accounts for\nthe probability of obtaining negative sectional curvatures\n(full squares) for a lattice size of\n$N = 40 \\times 40$\nare plotted in comparison with the numerical values \nof Fig. \\ref{xy2d.lyap.num.fig}.\n The dashed lines are the asymptotic behaviors at high and low\ntemperatures in the thermodynamic limit. }\n\\label{prev.Lyap.2d}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig18.ps}}\n\\caption{ Analytic Lyapunov exponents computed for the $3d$ model by means of \nEq.(\\protect\\ref{lambda}) without correction (dots) and\nincorporating\nthe correction that accounts for\nthe probability of obtaining negative sectional curvatures (full circles)\nare plotted in comparison with the numerical values of\nFig. \\ref{lyap.3d.fig}.\nThe dashed lines are the\nasymptotic behaviors at high and low temperatures in the thermodynamic \nlimit. }\n\\label{prev.Lyap.3d}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig19.ps}}\n\\caption{ The deformation factor $D(E)=[\\int_{\\Sigma_E}(d\\sigma /\\Vert\n\\nabla H\\Vert )(M_1^\\star/\\Vert\\nabla H\\Vert)]\\,/\\,\n[\\int_{\\Sigma_E}d\\sigma M_1]$ of Eq. (\\ref{deformaz}) (open circles)\nis plotted vs. energy density $E/N$ and compared to the quantity\n$\\langle 1/\\Vert\\nabla H\\Vert^2\\rangle$ \n(open triangles). $N=10 \\times 10\\times 10$. }\n\\label{D(E)}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig20.ps}}\n\\caption{ The normalized autocorrelation functions $\\Gamma(\\tau)$ are plotted\nvs. time $\\tau$ for a lattice of $N=10 \\times 10\\times 10$ and for \nfour different \nvalues of the temperature (from top to bottom: $T=0.49, 1.28, 1.75, 2.16$). }\n\\label{Gamma}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.80\\linewidth]{Fig21.ps}}\n\\caption{ Autocorrelation times $\\tau_{corr}$ are plotted vs. temperature $T$.\nThe vertical dashed line points out the temperature $T_c \\simeq2.17$ at\nwhich the phase transition occurs. }\n\\label{tau}\n\\end{figure}\n\\clearpage\n%-------------------------------------------------------\n\\end{document}\n\n"
}
] |
[
{
"name": "cond-mat0002144.extracted_bib",
"string": "\\begin{thebibliography}{999}\n\n\\bibitem[a]{moni} Electronic address: mcs@arcetri.astro.it\n\n\\bibitem[b]{cecilia} Electronic address: cclementi@ucsd.edu\n\n\\bibitem[c]{marco} Also at INFN, Sezione di Firenze, Italy.\nElectronic address: pettini@arcetri.astro.it\n\n\\bibitem{CCCP} L. Caiani, L. Casetti, C. Clementi and M. Pettini,\n Phys. Rev. Lett. {\\bf 79}, 4361 (1997).\n\n\\bibitem{Gallavotti} G. Gallavotti, {\\it Meccanica Statistica}, \n (Quaderni C.N.R., Roma, 1995).\n\n\\bibitem{Thirring} P. Hertel, and W. Thirring, Ann. Phys. (NY) {\\bf 63},\n 520 (1971).\n\n\\bibitem{Lyndenbell} R.M. Lynden-Bell, in {\\it Gravitational Dynamics},\n O. Lahav, E. Terlevich and R.J. Terlevich, eds.,\n Cambridge Contemporary Astrophysics, (Cambridge\n Univ. Press, 1996)\n\n\\bibitem{Gross} D.H.E. Gross, Phys. Rep. {\\bf 279}, 119 (1997); D.H.E. Gross\n and M.E. Madjet, {\\it Microcanonical vs. canonical thermodynamics},\n archived in cond-mat/9611192\n see also the references quoted in these papers. \n\n\\bibitem{Gross2} D.H.E. Gross, A. Ecker and X.Z. Zhang, Ann. Physik {\\bf 5},\n 446 (1996).\n\n\\bibitem{Gross1} A. H\\\"uller, Z. Physik B{\\bf 95}, 63 (1994); M. Promberger, \nM. Kostner, and A. H\\\"uller, {\\it Magnetic properties of finite systems: \nmicrocanonical finite-size scaling}, archived in cond-mat/9904265.\n\n\\bibitem{Butera} P. Butera and G. Caravati, Phys. Rev. A {\\bf 36}, 962 (1987).\n\n\\bibitem{Leoncini} X. Leoncini, A. Verga, and S. Ruffo, Phys. Rev. E {\\bf 57},\n6377 (1998).\n\n\\bibitem{Rapisarda} A. Bonasera, V. Latora, A. Rapisarda, Phys. Rev. Lett.\n{\\bf 75}, 3434 (1995).\n\n\\bibitem{Ruffo} M. Antoni and S. Ruffo, Phys. Rev E {\\bf 52}, 2361 (1995).\n\n\\bibitem{Antoni} M. Antoni and A. Torcini, Phys. Rev. E {\\bf 57}, R6233 (1998).\n\n\\bibitem{CCP1} L. Caiani, L. Casetti and M. Pettini, J. Phys. A:\n Math. Gen., {\\bf 31}, 3357 (1998).\n\n\\bibitem{CCCPPG} L. Caiani, L. Casetti, C. Clementi, G. Pettini,\n M. Pettini and R. Gatto, Phys. Rev E {\\bf 57}, 3886 (1998).\n\n\\bibitem{polimeri} C. Clementi, {\\it Dynamics of homopolymeric chain models},\n Master Thesis, SISSA/ISAS, Trieste, (1996). \n\n\\bibitem{Dellago} Ch. Dellago, H. A. Posch, W. G. Hoover, Phys. Rev. E \n{\\bf 53}, 1485 (1996); Ch. Dellago, H. A. Posch, Physica A {\\bf 230}, \n364 (1996); Ch.\\ Dellago and H.\\ A.\\ Posch, Physica A {\\bf 237}, 95 (1997); \nCh.\\ Dellago and H.\\ A.\\ Posch, Physica A {\\bf 240}, 68 (1997) .\n\n\\bibitem{Mehra} V. Mehra, R. Ramaswamy, preprint\n{\\tt chao-dyn/9706011}.\n\n\\bibitem{Berry} S. K. Nayak, P. Iena, K. D. Ball and R. S. Berry,\n J. Chem. Phys. {\\bf 108}, 234 (1998).\n\n\\bibitem{Firpo} M.-C. Firpo, Phys. Rev. E {\\bf 57}, 6599 (1998).\n\n\\bibitem{Ruffo_prl} V. Latora, A. Rapisarda, and S. Ruffo, Phys. Rev. Lett.\n{\\bf 80}, 692 (1998).\n\n\\bibitem{Ruffo_talk} V. Latora, A. Rapisarda, and S. Ruffo, Physica D (1998),\nin press.\n\n\\bibitem{TobChes} J. Tobochnik, and G. V. Chester, Phys. Rev. B{\\bf 20}, \n3761 (1979).\n\n\\bibitem{Gupta} R. Gupta, and C.F. Baillie, Phys. Rev. B{\\bf 45}, 2883 (1992).\n\n\\bibitem{Lapo} L. Casetti, Phys. Scr. {\\bf 51}, 29 (1995).\n\n\\bibitem{Pearson} E. M. Pearson, T. Halicioglu, and W. A. Tiller, Phys. Rev. \nA{\\bf 32}, 3030 (1985).\n\n\\bibitem{Palmer} A thorough discussion about ergodicity breaking, non\ninterchangeability of $t\\rightarrow\\infty$ and $N\\rightarrow\\infty$ limits \ncan be found in: R.G. Palmer, Adv. Phys. {\\bf 31}, 669 (1982).\n\n\\bibitem{LPV} J. Lebowitz, J. Percus, and L. Verlet, Phys. Rev.\n{\\bf 153}, 250 (1967).\n\n\\bibitem{CCP} L. Casetti, C. Clementi and M. Pettini, Phys. Rev. E\n {\\bf 54}, 5969 (1996).\n\n\\bibitem{Pettini} M.Pettini, Phys. Rev. E {\\bf 47}, 828 (1993).\n\n\\bibitem{Eisenhart} L. P. Eisenhart, Ann. of Math. {\\bf 30}, 591 (1939).\n\n\\bibitem{doCarmo} M. P. Do Carmo, {\\it Riemannian Geometry} (Birkh\\\"{a}user,\n Boston, 1992).\n\n\\bibitem{CerrutiPettini} M.Cerruti-Sola and M. Pettini, Phys. Rev.\n E{\\bf 53}, 179 (1996); M.Cerruti-Sola, R. Franzosi and\n M. Pettini, Phys. Rev. E{\\bf 56}, 4872 (1997).\n\n\\bibitem{cegdcp} L. Casetti, E.G.D. Cohen, and M. Pettini, Phys. Rev. Lett.\n{\\bf 82}, 4160 (1999).\n\n\\bibitem{fps1} R. Franzosi, L. Casetti, L. Spinelli, and M. Pettini,\n Phys. Rev. E{\\bf 60}, 5009 (1999).\n\n\\bibitem{fps2} R. Franzosi, M. Pettini, and L. Spinelli, {\\it Topology and\nPhase transitions: a paradigmatic evidence}, Phys. Rev. Lett.,\n(1999) submitted; archived in cond-mat/9911235.\n\n\\bibitem{thorpe} J.A. Thorpe, {\\it Elementary Topics in Differential Geometry},\n(Springer, New York, 1979).\n\n\\bibitem{laurence} P. Laurence, Zeit. Angew. Math. Phys.{\\bf 40}, 258 (1989).\n\n\\bibitem{nota1} A similar formula is obtained in: H.H. Rugh, J.Phys.A: Math. \nGen. {\\bf 31}, 7761 (1998), though without using the result of \nRef. \\protect\\cite{laurence}, and in: C. Giardin\\`a, R. Livi, J. Stat. Phys. \n{\\bf 98}, 1027 (1998); however in these papers the relation between temperature\nand mean curvature was not established.\n\n\\bibitem{ChernLashof} S. Chern, and R.K. Lashof, Michigan Math. J. {\\bf 5},\n 5 (1958).\n\n\\bibitem{Betti} M. Nakahara, {\\it Geometry, Topology and Physics}, \n (Adam Hilger, Bristol, 1989). \n\n\\bibitem{bassi-indici} The mathematical details to justify such an assumption\n are discussed in: R. Franzosi, {\\it Geometrical and Topological Aspects\n in the study of Phase Transitions}, PhD thesis, Department \n of Physics, University of Florence, (1999).\n\n\\bibitem{Gross3} D.H.E. Gross and E. Votyakov, {\\it Phase transitions in\n finite systems = topological peculiarities of the microcanonical entropy \n surface}, archived in cond-mat/9904073;\n D.H.E. Gross and E. Votyakov, {\\it Phase transitions in ``small'' systems},\n archived in cond-mat/9911257; D.H.E. Gross, {\\it Phase \n Transitions without thermodynamic limit}, archived in cond-mat/9805391.\n\n\\bibitem{reuter} E. Gozzi and M. Reuter, Phys. Lett. B {\\bf 233}, 383 (1989).\n\n\\bibitem{gozzi1} E. Gozzi, M. Reuter and W. D. Thacker, Chaos, Solitons \\&\n Fractals {\\bf 2}, 441 (1992).\n\n\\bibitem{gozzi2} E. Gozzi and M. Reuter, Chaos, Solitons \\&\n Fractals {\\bf 4}, 1117 (1994).\n\n\\bibitem{Creutz} M. Creutz, Phys. Rev. Lett. {\\bf 50}, 1411 (1983).\n\n\n\\end{thebibliography}"
}
] |
cond-mat0002145
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Sum Rule Approach to Collective Oscillations of Boson-Fermion Mixed Condensate of Alkali Atoms
|
[
{
"author": "T. Miyakawa"
},
{
"author": "T. Suzuki and H. Yabu"
}
] |
The behavior of collective oscillations of a trapped boson-fermion mixed condensate is studied in the sum rule approach. Mixing angle of bosonic and fermionic multipole operators is introduced so that the mixing characters of the low-lying collective modes are studied as functions of the boson-fermion interaction strength. For an attractive boson-fermion interaction, the low-lying monopole mode becomes a coherent oscillation of bosons and fermions and shows a rapid decrease in the excitation energy towards the instability point of the ground state. In contrast, the low-lying quadrupole mode keeps a bosonic character over a wide range of the interaction strengths. For the dipole mode the boson-fermion in-phase oscillation remains to be the eigenmode under the external oscillator potential. For weak repulsive values of the boson-fermion interaction strengths we found that an average energy of the out-of-phase dipole mode stays lower than the in-phase oscillation. Physical origin of the behavior of the multipole modes against boson-fermion interaction strength is discussed in some detail.
|
[
{
"name": "subfin.tex",
"string": "\\documentstyle[prl,aps]{revtex}\n%\\documentstyle[pra,aps,preprint]{revtex}\n\\oddsidemargin=-5mm\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\def\\rvec{\\vec{r}}\n\\def\\xvec{\\vec{x}}\n\\def\\half{{1\\over 2}}\n\\def\\Ekin{E_{\\rm kin}}\n\\def\\Eho{E_{\\rm ho}}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{document}\n%\n\\title{Sum Rule Approach to Collective Oscillations\nof Boson-Fermion Mixed Condensate of Alkali Atoms}\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\\begin{abstract}\nThe behavior of collective oscillations of a trapped boson-fermion mixed \ncondensate is studied in the sum rule approach. Mixing angle of \nbosonic and fermionic multipole operators is introduced so \nthat the mixing characters of the low-lying collective modes are \nstudied as functions of the boson-fermion interaction strength. \nFor an attractive boson-fermion interaction, the low-lying monopole \nmode becomes a coherent oscillation of bosons and fermions and shows \na rapid decrease in the excitation energy towards the instability point of \nthe ground state. In contrast, the \nlow-lying quadrupole mode keeps a bosonic character \nover a wide range of the interaction strengths. For the dipole mode \nthe boson-fermion in-phase oscillation remains to be the eigenmode \nunder the external oscillator potential. For weak repulsive values of \nthe boson-fermion interaction strengths we found that \nan average energy of the out-of-phase dipole mode stays lower \nthan the in-phase oscillation. \n Physical origin of the behavior of the multipole modes against \nboson-fermion interaction strength is discussed in some detail. \n\\end{abstract}\n%\n\\pacs{PACS number: 03.75.Fi, 05.30.Fk,67.60.-g}\n%\n%%%%%%%%%%%\n% MAIN TEXT\n%%%%%%%%%%%\n%\\section{Introduction}\n\n\nCollective oscillation is one of the most prominent phenomena \ncommon to a variety of many-body systems. The realization of \nthe Bose-Einstein condensates (BEC) for trapped Alkali \natoms\\cite{review} offers a possibility to study such phenomena \nof quantum systems under ideal conditions. Up to now the \nexperimental\\cite{Coll} as well as theoretical\\cite{Stringari,Kimura,TBEC} \nstudies of collective motions of BEC have been intensively performed. \nQuite recently the degenerate Fermi gas of trapped $^{40}$K atoms \nhas been realized\\cite{DeMarco}, which motivates the study of collective \nmotion also in Fermi gases\\cite{TFex}. These developments further \nencourage the study of possible boson-fermion mixed condensates \nof trapped atoms\\cite{symbf,hydro}. Now the static \nproperties\\cite{Molmer,Nygaard,Miyakawa}, stability \nconditions\\cite{Minniti,Instbf}, and some \ndynamical properties\\cite{Amoruso,tsurumi} of trapped boson-fermion \ncondensates have been investigated. In the present paper, we study \nthe behavior of collective oscillations \nof a boson-fermion mixed condensate at $T=0$ for both repulsive and \nattractive boson-fermion interactions. We adopt the sum rule \napproach\\cite{Stringari,Kimura} that has proved to be successful \nin the studies of collective excitations of Bose condensates. \nFor the mixed condensate, we introduce a mixing angle \nof the boson/fermion excitation operators so as to allow the \nin- and out-of-phase oscillations of the bose and fermi condensates.\n\n%\\section{sum rule approach}\n\nIn the sum rule approach we first calculate the energy weighted \nmoments $m_p=\\sum_{j}(E_{j}-E_{0})^{p}|\\langle j|F| 0 \\rangle|^{2}$ \nof the relevant multipole operator $F$, where $|j\\rangle$'s represent \nthe complete set of eigenstates of the Hamiltonian with energies $E_{j}$, \nand $|0\\rangle$ denotes the ground state. The excitation energy \nis expressed as $\\hbar\\omega=(m_3/m_1)^{1/2}$ which provides a useful \nexpression for the average energy of the collective \noscillation\\cite{Stringari,TFex}. \nThe moments are calculated from formulae \n$m_{1}=\\half \\langle 0| \\left[F^{\\dagger},[H,F]\\right] |0\\rangle$ and \n$m_{3}=\\half \\langle 0| \\left[[F^{\\dagger},H],\\left[H,[H,F]\\right]\n\\right] |0\\rangle$. We consider three types of multipole operators which \nare defined by \n\\begin{equation}\n F_\\alpha^\\pm \\equiv \\sum_{i=1}^{N_b}f_\\alpha(\\rvec_{bi})\\pm \n \\sum_{i=1}^{N_f} f_\\alpha(\\rvec_{fi}),\\quad (\\alpha=M,D,Q)\n\\end{equation}\nwhere the functions $f_\\alpha$ are defined by $f_M(\\rvec)=r^2$ for \nmonopole, $f_D(\\rvec)=z$ for dipole, and $f_Q(\\rvec)=3z^2-r^2$ for \nquadrupole excitations. The \nindices $b,f$ denote boson/fermion, $N_b, N_f$ the numbers of \nbose/fermi particles, and $\\pm$ correspond to \nthe in-phase and out-of-phase oscillation of the two types of particles. \nWe actually take a linear combination of the form \n\\begin{equation}\n F_{\\alpha}({\\bf r};\\theta) =F^+_{\\alpha} \\cos{\\theta}\n +F^-_{\\alpha}\\sin{\\theta} \\qquad (-\\frac{\\pi}{2} < \\theta\\le \\frac{\\pi}{2})\n\\end{equation}\nparametrized by the mixing angle $\\theta$. We study the value of $\\theta$ \nthat minimizes the calculated frequency $\\omega$ for each $\\alpha$. \n\nWe consider the polarized boson-fermion mixed condensate \nin spherically symmetric harmonic oscillator potential. \n%*** Extension to the deformed potential is straightforward. %***\nThe system is described by the Hamiltonian\n\\begin{equation}\n\\label{bfH}\n H = \\sum_{i=1}^{N_{b}} \\left\\{\\frac{{\\vec{p}}\\,^{2}_{bi}}{2m}\n + \\half m\\omega_0^{2}\\rvec\\,^{2}_{bi}\n + \\half g \\sum_{j=1}^{N_{b}}\\delta(\\rvec_{bi}-\\rvec_{bj}) \\right\\}\n + \\sum_{i=1}^{N_{f}} \\left\\{ \\frac{{\\vec{p}}\\,^{2}_{fi}}{2m}\n +\\half m\\omega_0^{2}\\rvec\\,^{2}_{fi}\\right\\}\n + h \\sum_{i=1}^{N_{b}}\\sum_{j=1}^{N_{f}}\\delta(\\rvec_{bi}-\\rvec_{fj})\n \\end{equation}\nwhere we assume the same oscillator frequencies and masses for bosons and \nfermions for simplicity. \nThe coupling constants $g,h$ are the boson-boson/boson-fermion \ninteraction strengths represented by the $s$-wave scattering lengths \n$a_{bb}$ and $a_{bf}$ as $g=4\\pi\\hbar^{2}a_{bb}/m$\nand $h=4\\pi\\hbar^{2}a_{bf}/m$. The fermion-fermion interaction has been neglected \nas the polarized system is considered. Following the standard calculation \nprocedure the excitation frequencies are obtained as:\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n(i) Monopole\n\\begin{equation}\n\\label{egm}\n\\frac{\\omega_{M}(\\theta)}{\\omega_0}=\\sqrt{2}\\sqrt{1+\\frac{\\Ekin^{+}\n+\\frac{9}{4}E_{bb}+\\frac{9}{4}E_{bf}+2(\\Ekin^{-}+\\frac{9}{4}E_{bb}\n+\\frac{3}{4}\\Delta')\\cos{\\theta}\\sin{\\theta}\n-\\Delta\\sin^2{\\theta}}\n{\\Eho^{+}+2\\Eho^{-}\\cos{\\theta}\\sin{\\theta}}}\n\\end{equation}\n\n(ii) Quadrupole\n\\begin{equation}\n\\label{egq}\n\\frac{\\omega_{Q}(\\theta)}{\\omega_0}=\\sqrt{2}\\sqrt{1+\\frac{\\Ekin^{+}\n+2 \\Ekin^{-}\\cos{\\theta}\\sin{\\theta}-\\frac{2}{5}\\Delta \\sin^2{\\theta}}\n{\\Eho^{+}+2\\Eho^{-}\\cos{\\theta}\\sin{\\theta}}}\n\\end{equation}\n\n(iii) Dipole\n\\begin{equation}\n\\label{egd}\n\\frac{\\omega_{D}(\\theta)}{\\omega_0}=\\sqrt{1-\\frac{4}{3\\hbar\\omega_0}\n\\frac{\\Omega \\sin^2{\\theta}}\n{N^{+}+2 N^{-}\\cos{\\theta}\\sin{\\theta}}}\n\\end{equation}\n\nHere we defined $\\Ekin^{\\pm}\\equiv \\Ekin^{b}\\pm \\Ekin^{f},\n\\Eho^{\\pm}\\equiv \\Eho^{b}\\pm \\Eho^{f}$ and $N^{\\pm}\\equiv \nN_{b}\\pm N_{f}$, where $\\Ekin^{\\{b,f\\}}$ and $\\Eho^{\\{b,f\\}}$ \nare respectively the expectation values of the kinetic and oscillator \npotential energies for boson/fermion in the ground state. Boson-boson \nand boson-fermion interaction energies have been denoted \n by $E_{bb}$ and $E_{bf}$. \nThe quantities $\\Delta,\\Delta'$ and $\\Omega$ are given in terms of the \nboson/fermion densities $n_b(r), n_f(r)$ in the ground state by \n\\begin{equation}\n\\label{DO}\n \\Delta\\equiv h\\int\\!\\! d^3r \\, r^2 \n \\frac{dn_{f}(r)}{dr}\\frac{dn_{b}(r)}{dr},\\quad \n \\Delta'\\equiv h\\int\\!\\! d^3r \\, r \\left[ n_{f}(r)\\frac{dn_{b}(r)}{dr}\n -\\frac{dn_{f}(r)}{dr}n_{b}(r)\\right] , \\quad\n \\Omega \\equiv h\\,\\xi^2\\int\\!\\! d^3r\n \\frac{dn_{f}(r)}{dr}\\frac{dn_{b}(r)}{dr}, \n\\end{equation}\nwhere $\\xi=\\sqrt{\\hbar/m\\omega_0}$. \nOne may use the stationary condition of the ground state, \n\\begin{equation}\n 2\\Ekin^+-2\\Eho+3E_{bb}+3E_{bf}=0, \\quad \n 2\\Ekin^--2\\Eho^-+3E_{bb}+\\Delta'=0\n\\end{equation}\n in order to eliminate in eq.(\\ref{egm}) the dependences on \n$E_{bb},E_{bf}$ and $\\Delta'$. The monopole frequency is then rewritten as \n\\begin{equation}\n\\label{egmm}\n\\frac{\\omega_{M}(\\theta)}{\\omega_0}=\\sqrt{5-\\frac{\\Ekin^{+}\n+2\\Ekin^{-}\\cos{\\theta}\\sin{\\theta}+2\\Delta\\sin^2{\\theta}}\n{\\Eho^{+}+2\\Eho^{-}\\cos{\\theta}\\sin{\\theta}}}.\n\\end{equation} \nWe have checked that the Thomas-Fermi calculation of the ground \nstate adopted below gives rise to a negligible difference if \none evaluates either the expression (\\ref{egmm}) or (\\ref{egm}). \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe calculate the ground state energies and densities of the \nboson-fermion mixed system in the Thomas-Fermi approximation which \nis valid for $gN_b\\gg 1$ and $N_f\\gg 1$ except around the \nsurface region\\cite{Molmer,Nygaard,Miyakawa,Amoruso}. \nWe take harmonic oscillator length $\\xi$ \nand energy $\\hbar\\omega_0$ as units, and define scaled dimensionless \nvariables: the radial distance $x=r/\\xi$, boson/fermion densities \n$\\rho_{b,f}(x)=n_{b,f}(r)\\xi^3/N_{b,f}$, and chemical potentials \n$\\tilde{\\mu}_{b,f}=2\\mu_{b,f}/\\hbar\\omega_0$. We solve the coupled \nThomas-Fermi equations, \n\\begin{equation}\n\\label{tfeq}\n \\tilde{g}N_{b}\\rho_{b}(x)+x^2+\\tilde{h}\n N_{f}\\rho_{f}(x)=\\tilde{\\mu_{b}}, \\quad\n [6\\pi^2N_{f}\\rho_{f}(x)]^{2/3}+x^2+\\tilde{h}N_{b}\\rho_{b}(x)\n =\\tilde{\\mu_{f}}, \n\\end{equation}\nwhere $\\tilde{g}=2g/\\hbar\\omega_0\\xi^3$ and \n$\\tilde{h}=2h/\\hbar\\omega_0\\xi^3$.\n\n\nOne of the most promising candidates for the realization of the \nmixed condensate is \nthe potassium isotope system. Precise values of the scattering \nlengths are not well known at present and different values\nhave been reported \\cite{Cote,Kscat1,Kscat2}.\nWe take for the boson-boson interaction the parameters of the \n$^{41}$K-$^{41}$K system in \\cite{Cote} and a trapping \nfrequency of $450$Hz which gives $\\tilde{g}=0.2$. For the \nboson-fermion interaction we take several values in the range \n$h/g=\\tilde{h}/\\tilde{g}=-2.37 \\sim 3.2$. It should be noted that\nthe interaction strength can be controlled using \nFeshbach resonances\\cite{Fesh}. \n\nWe have performed a numerical calculation for $N_{b}=N_{f}=10^6$. \nIn the ground state the fermions \nhave a much broader distribution than bosons because of the Pauli \nprinciple. Fermions are further squeezed out of the center for a \nrepulsive boson-fermion interaction ($h>0$) . They will eventually \nform a shell-like distribution around the surface of bosons \nfor $h/g\\ge 1$ and will be completely pushed \naway from the center ($n_{f}(0) = 0$) at around $h/g \\sim 3$. \nFor an attractive boson-fermion interaction, on the other hand, the \ncentral densities of the bosons and fermions increase together. \nThe system becomes unstable against collapse at around \n$h/g= -2.37$ due to the strong attractive boson-fermion \ninteraction\\cite{Instbf}. \n\n Figure 1 shows the kinetic energy, the oscillator potential energy, \nand the interaction energy contributions to the ground state energy \n against the parameter $h/g$. The figure shows also \nthe quantities $\\Delta$ and $\\Omega$ which represent the contributions \nof the boson-fermion interaction to the multipole frequencies \n(\\ref{egm})-(\\ref{egd}) and (\\ref{egmm}). One may notice that the \nfermionic kinetic- and potential-energy contributions are \na few times larger than the bosonic contribution \nin the present system. It is noted that $\\Delta$ takes large negative \nvalues at both large negative and positive regions of $h/g$. In the \nformer region the bose and fermi density distributions become coherent \ndue to the attractive interaction, \nand the radial integral in eq.(\\ref{DO}) takes a large positive value. \nIn the opposite case ($h/g\\gg 1$), the same integral changes sign \nbecause the fermions are pushed away from the center by the repulsive \nboson-fermion interaction, thus giving a large negative contribution \nin the bosonic surface region. In the region $0< h/g < 1$, on the \nother hand, the integral is slightly positive and $\\Delta$ takes a \nsmall positive value. The quantity $\\Omega$ follows the same trend as \n$\\Delta$, but the absolute value \nis much smaller than $\\Delta$, as the most important contribution to \nthe integral comes from the surface region where $r\\gg \\xi$. \n\nOnce the ground state parameters are \nobtained the frequencies (\\ref{egm})-(\\ref{egd}) are minimized against \n$\\theta$. \nUsually, the sum rule approach predicts a strength-weighted average \nenergy of eigenstates for a given multipolarity. The calculated energy \n coincides with the true excitation energy if the \nrelevant strength is concentrated in a single state. By adopting the \nminimization procedure we simultaneously determine the character of \nthe low-lying collective mode and the corresponding average energy. The \ncharacter of the mode is given by the value of $\\theta$, for instance, \n$\\theta\\simeq\\pi/4$ for the bosonic- and $-\\pi/4$ for the fermionic-modes, \n$\\theta\\simeq 0$ for the in-phase oscillation and $\\pi/2$ for the \nout-of-phase oscillation of the two types of particles. \n%%%% \nAs there are two kinds of particles involved in eq.(2), one would expect \nan emergence of two types of collective oscillations for each multipole. \nAnother collective mode would have a character orthogonal to the low-lying \nmode as far as the phase relation of the two operators in (2) is concerned. \nIn the present approach we calculate the frequency of the latter mode, \nthe high-lying one, from eqs.(\\ref{egm})-(\\ref{egd}), by using the \noperator $F^+_\\alpha \\sin\\theta_L-F^-_\\alpha\\cos\\theta_L$ for each $\\alpha$, \nwhere the mixing angle $\\theta_L$ is the one determined for the low-lying \nmode. \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nFigure 2 shows frequencies of the lower (solid lines) \nand the higher (dashed lines) modes for (a) monopole, (b) \nquadrupole and (c) dipole cases as functions of $h/g$. \nThe corresponding mixing angles $\\theta$ determined by the minimization \nprocedure for the lower mode are plotted in Fig.3 \nfor each multipolarity as a function of $h/g$. \nBelow we discuss the behavior of the frequencies $\\omega_\\alpha$ \nby defining three regions of $h/g$: \n(I) $h/g<0$, (II) $0<h/g\\lesssim 1$, (III) $1\\lesssim h/g$.\n%\n\n\\vspace{2mm}\\noindent\n{\\it a)monopole}: \\\\\nFor a non-interacting boson-fermion system the low-lying monopole \nmode is the fermionic oscillation with frequency \n$\\omega_M^L=2\\omega_0$, while the higher mode is the bosonic one \nwith $\\omega_M^H=\\sqrt{5}\\omega_0$ in the Thomas-Fermi approximation. \nAround $h\\simeq 0$ one may obtain \n$\\omega^{L}_{M}\\simeq 2 \\omega_0 \\left(1+A \\tilde{h}N^{\\frac{1}{6}}\\right)$,\n$\\omega^{H}_{M}\\simeq \\sqrt{5} \\omega_0 \\left(1-A^{\\prime}\n\\tilde{h}\\tilde{g}^{-\\frac{1}{5}}N^{\\frac{3}{10}}\\right)$, \nwhere $A=3\\cdot 6^{\\frac{1}{6}}/4 \\sqrt{2}\\pi^{2}$ and $A^{\\prime} = 7 \\sqrt{3}\n/20\\pi^{2} \\cdot (8\\pi/15)^{\\frac{1}{5}}$. The mixing angle for the lower mode \nis given as $\\theta_{M} = -\\pi/4 - \\delta_{M}$ \nwith $\\delta_{M} = (5\\cdot 6^{\\frac{1}{6}}/\\sqrt{2}\\pi^{2})\n\\tilde{h}N^{\\frac{1}{6}}$. \nThis behavior is seen in region (II) where the boson-fermion interaction \nis weakly repulsive and the lower monopole mode is of a fermionic \ncharacter, see Fig.3 (solid line). \nBosonic oscillation in this region is more rigid than the \nfermionic one because of the repulsive interaction among bosons. \nIn region (I), the situation is quite different: The low-lying mode becomes \na coherent boson-fermion oscillation as represented by the large negative \nvalue of $\\Delta$, and the excitation energy shows a \nsharp decrease towards the instability point $h/g\\sim -2.37$ of the \nground state\\cite{Instbf}, although $\\omega_M$ does not become exactly zero \nwithin our approximation. \nIn this region the attractive boson-fermion interaction is much more \neffective in the excited state than in \nthe ground state and cancels out the increase in the kinetic energy. \nIn region (III), too, we find that the low-lying mode becomes an in-phase \noscillation. Here, the boson and the fermion densities in the ground \nstate are somewhat separated, and the in-phase motion which keeps this \nseparation is energetically more favorable than the out-of-phase motion \nas seen in the value of $\\Delta$. \n\n\\vspace{2mm}\\noindent\n{\\it b)quadrupole}: \\\\\nFor the quadrupole excitation (Fig.2(b)), the lower \n(higher) energy mode is almost the pure bosonic (fermionic) oscillation \nover the broad range of the $h/g$ values studied, Fig.2 (dashed line). \nTo the first \norder in $\\tilde{h}$ the frequencies of the lower and the higher \nquadrupole modes are given by \n$\\omega^{L}_{Q}\\simeq \\sqrt{2} \\omega_0 \\left(1-B \n\\tilde{h}N^{\\frac{1}{6}}\\right)$,\n$\\omega^{H}_{Q}\\simeq 2 \\omega_0 (1-B^{\\prime}\n\\tilde{h}\\tilde{g}^{\\frac{2}{5}}N^{\\frac{7}{30}})$, \nwhere $B=6^{\\frac{1}{6}}/4 \\sqrt{2} \\pi^{2}$ and $B^{\\prime}=(15/8\\pi)^{\\frac{2}{5}}\n/(14\\cdot 6^{\\frac{1}{6}}\\sqrt{2}\\pi^{2})$.\nThe corresponding mixing angle for the lower mode is given by \n$\\theta_{Q} = \\pi/4 + \\delta_{Q}$ with \n$\\delta_{Q} = 2^{3}B^{\\prime} \\tilde{h} \\tilde{g}^{\\frac{2}{5}}N^{\\frac{7}{30}}$. \nFor the quadrupole mode similar mechanisms as for the monopole mode \nare at work concerning \nthe dependence on $|h/g|$. The role of the boson-fermion \ninteraction is however much reduced compared with the monopole case \nas seen by the factor 2/5 of eq.(\\ref{egq}), which reflects that the \nquadrupole oscillation has five different components. Thus \nthe quadrupole mode \nobtains an in-phase character only at large values of $|h/g|$. In the \nother region of $|h/g|$ the low-lying mode becomes a simple bosonic oscillation. \nThis is because the fermionic mode costs a larger kinetic energy and \nfavors $\\theta=\\pi/4$ as \nseen in the term $(\\Ekin^b-\\Ekin^f)\\sin\\theta\\cos\\theta$ in \neq.(\\ref{egq}).\n\n\n\\vspace{2mm}\\noindent\n{\\it c)dipole}: \\\\\nGeneral arguments\\cite{review} show that for a harmonic oscillator \nexternal potential a uniform shift of the ground state density \ngenerates an eigenstate of the system, corresponding to the boson-fermion \nin-phase dipole oscillation with frequency $\\omega_0$. This is evident \nin Fig.2c) and also in eq.(\\ref{egd}) at $\\theta=0$. \nIn the regions (I) and (III) the \nout-of-phase oscillation is unfavorable due to the same reason \nas for the monopole mode: It loses the energy gained in the \nground state boson-fermion configuration. \nFor a weakly repulsive \n$h$ an interesting possibility arises: In the region (II)\nthe out-of-phase mode of the boson-fermion oscillation lies \nlower than the in-phase mode. Let us first note that at $h=0$ \nthe out-of-phase oscillation frequency becomes degenerate as the \nin-phase one because the bosonic and \nthe fermionic dipole modes are independent. One may note that at \n$h/g\\simeq 1$ again the degeneracy occurs. Here the potential term \nfor the fermion becomes almost linear to the fermion density \nitself (see, eq.(\\ref{tfeq})). This suggests that the fermion density is \ndetermined almost entirely by the chemical potentials and becomes \nnearly constant as far as the boson density is finite. A uniform dipole \nshift of fermions thus produces almost no effect on bosons, and results in \nthe degeneracy of the frequency. Between \n$h/g=0$ and 1, the boson-fermion repulsion is weaker for the out-of-phase \noscillation than the in-phase one (and hence the ground state) as \nreflected in the sign of $\\Omega$. \n\n\n\\vspace{2mm}\nIn the present paper, we studied collective oscillations\nof trapped boson-fermion mixed condensates using sum rule approach. \nWe introduced a mixing angle of bosonic and fermionic multipole \noperators so as to study if the in- or out-of-phase motion of \nthose particles is favored as a function of the boson-fermion interaction\nstrength. For the monopole and quadrupole cases, the coupling of the \nbose- and fermi-type oscillation is not large for moderate values of \nthe coupling strength $h$. At large values of $|h/g|$, the low-lying \nmodes become an in-phase oscillation of bosons and fermions. This is \nespecially so for the monopole oscillation at attractive boson-fermion \ninteraction: The excitation energy of this mode almost \nvanishes at the instability point of the ground state. In the case \nof the dipole mode, in contrast, the in-phase oscillation \nremains an exact eigenmode with a fixed energy for harmonic \noscillator potentials, while the average energy of the \nout-of-phase oscillation is strongly dependent on the \nboson-fermion interaction. We found that at weak repulsive values of \nthe interaction the out-of-phase motion stays lower than the \nin-phase oscillation. \n%%%%%%%%%%%%%%%%\nIn this paper we calculated also the frequencies of the high-lying modes \nfor each multipole, by adopting the operators orthogonal to the low-lying \nmodes. These modes, too, are collective in character and, in the present \nframework, their average frequencies showed rather strong dependences on \nthe boson-fermion coupling strength. \n%%%%%%%%%%%%%%%\nDeeper insight into the collective modes \nstudied in this paper will require a detailed investigation of \nthe solutions of, e.g., the self-consistent RPA type equations \nthat allow an arbitrary radial dependence of the excitation \noperators. Studies in this direction are now in progress. \n\n%\\newpage\n%\n%%%%%%%%%%%\n% References \n%%%%%%%%%%%\n%\n\\begin{references}\n\\bibitem{review} \nFor reviews, see: \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.\n{\\bf 71}, 463 (1999).\n%\n\\bibitem{Coll}\nD.~S.~Jin, J.~R.~Encher, M.~R.~Mathews, C.~E.~Wieman and E.~A.~Cornell, \nPhys.~Rev.~Lett.{\\bf 77},420(1996);\nM.-O.~Mewes, M.~A.~Andrews, N.~J.~van~Druten, D.~M.~Kurn, D.~S.~Durfee, \nC.~G.~Tounsend and W.~Ketterle, Phys.~Rev.~Lett.{\\bf 77},988(1996).\n%\n\\bibitem{Stringari}\nS.~Stringri, Phys.~Rev.~Lett. {\\bf 77}, 2360 (1996).\n%\n\\bibitem{Kimura}\nT.~Kimura, H.~Saito and M.~Ueda, J.~Phys.~Soc.~Jpn.{\\bf 68},1477(1999).\n%\n\\bibitem{TBEC}\nR.~Graham and D~Walls, Phys. Rev. {\\bf 57A}, 484 (1998);\nH~.Pu and N.~P.~Bigelow, Phys. Rev. Lett. {\\bf 80}, 1134 (1998);\nD.~Gordon and C.~M.~Sarvage, Phys. Rev. {\\bf 58A}, 1440 (1998).\n%\n\\bibitem{DeMarco}\nB.~DeMarco and D.~S.~Jin,Science {\\bf 285}, 1703 (1999)\n%\n\\bibitem{TFex}\nL.~Vichi and S.~Stringari, Phys.~Rev.{\\bf A60},4734(1999);\nG.~M.~Bruun and C.~W.~Clark, Phys.~Rev.~Lett.{\\bf 83},5415(1999);\nM.~Amoruso, I.~Meccoli, A.Minguzzi and \nM. P. Tosi, Eur.~Phys.~J. {\\bf D7},441(1999). \n%\n\\bibitem{symbf}\nE.~Timmermans and R.~C\\^{o}t\\'{e}, Phys.~Rev. ~Lett. {\\bf 80}, 3419 (1998);\nW.~Geist, L.~You, and T.~A.~B.~Kennedy, Phys.~Rev.{\\bf A59}, 1500 (1999);\nM.-O.~Mewes, G.~Ferrari, F.~Schreck, A.~Sinatra and C.~Salomon, physics/9909007\n%\n\\bibitem{hydro} \nI.~F.~Silvera, Physica {\\bf 109 \\& 110}, 1499 (1982); \nJ.~Oliva, Phys.~Rev. {\\bf B38}, 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. {\\bf A59}, 2974 (1999).\n%\n\\bibitem{Miyakawa}\nT.~Miyakawa, K.~Oda, T.~Suzuki and H.~Yabu ; cond-mat/9907009.\n%\n\\bibitem{Minniti} \nM.~Amoruso, C.~Minniti and M.~P.~Tosi, to be published in Eur.~Phys.~J. {\\bf D}.\n%\n\\bibitem{Instbf}\nT.~Miyakawa, T.~Suzuki and H.~Yabu, cond-mat/0002048.\n%\n\\bibitem{Amoruso} \nM.~Amoruso, A.~Minguzzi, S.~Stringari, M.~P.~Tosi and L.~Vichi,\nEur.~Phys.~J. {\\bf D4}, 261 (1998); \nL.~Vichi, M.~Inguscio, S.~Stringari and G.~M.~Tino, \nJ.~Phys. {\\bf B31}, L899 (1998). \n%\n\\bibitem{tsurumi}\nT.~Tsurumi and M.~Wadati, J.Phys.~Soc.~Jpn.{\\b 69} (2000) in print.\n%\n\\bibitem{Cote}\nR.~Cot\\'e, A.~Dalgarno, H.~Wang and W.~C.~Stwalley, \nPhys.~Rev. {\\bf A57}, R4118 (1998). \n%\n\\bibitem{Kscat1} \nJ.~L.~Bohn, J.~P. Burke,Jr, C.~H. Greene, H.~Wang, P.~L. Gould and \nW.~C. Stwalley, Phys. Rev. {\\bf A59}, 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{Fesh} \nE. Tiesinga, A.~J. Moerdijk, B.~J. Verhaar and H.~T.~C. Stoof,\nPhys.~Rev.{\\bf A46}, R1167 (1992); \nE. Tiesinga, B.~J. Verhaar, and H.~T.~C. Stoof, \nPhys.~Rev.{\\bf A47}, 4114 (1993); \nJ.~L. Bohn, cond-mat/9911132.\n%\n\\end{references}\n%\n\\newpage\n%\n%%%%%%%%%%%%\n% FIGURE CAPTIONS\n%%%%%%%%%%%%\n\\begin{figure}\n\\caption{(a) Ground state expectation values of the fermion kinetic \nenergy $\\Ekin^f$ (long dashed line), oscillator potential energies \n$\\Eho^b$ (short-dash-dotted line) and $\\Eho^f$ (long-dash-dotted line), \nboson-boson and boson-fermion interaction \nenergies $E_{bb}$ (dashed line) and $E_{bf}$ (dotted line), and the \nquantity $\\Delta$ (solid line) against the interaction \nstrength ratio $h/g$. The values are given in the unit of \n$N\\hbar\\omega_0$ and are dimensionless.\n(b) The quantity $\\Omega$ in the same unit.}\n\\end{figure}\n%\n\\begin{figure}\n\\caption{Excitation frequencies of collective oscillations\n( a) monopole, b) quadrupole, c) dipole )\nas functions of $h/g=\\tilde{h}/\\tilde{g}$, \ncalculated based on the eqs.(\\ref{egm})-(\\ref{egd}). The mixing angle \n$\\theta$ has been determined so as to minimize $\\omega$ for \neach multipole operator. The ordinates are given in the unit of \n$\\omega_0$ and are dimensionless. \nThe solid (dashed) lines are the lower (higher) energy mode\nfor each oscillation.} \n\\end{figure}\n\n\\begin{figure}\n\\caption{Mixing angles of in/out-of-phase excitation modes \nwhich are determined by minimizing excitation energies in Fig.3. \nThe solid line shows the monopole, the dashed line\nthe quadrupole, and the dotted line the dipole oscillations. \nShaded areas show mainly in-phase (hatches) and \nmainly out-of-phase (cross-hatches) regions in $\\theta$ \nAngles for pure bosonic- or fermionic-modes are also indicated.}\n\\end{figure}\n\\end{document}"
}
] |
[
{
"name": "cond-mat0002145.extracted_bib",
"string": "\\bibitem{review} \nFor reviews, see: \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.\n{\\bf 71}, 463 (1999).\n%\n\n\\bibitem{Coll}\nD.~S.~Jin, J.~R.~Encher, M.~R.~Mathews, C.~E.~Wieman and E.~A.~Cornell, \nPhys.~Rev.~Lett.{\\bf 77},420(1996);\nM.-O.~Mewes, M.~A.~Andrews, N.~J.~van~Druten, D.~M.~Kurn, D.~S.~Durfee, \nC.~G.~Tounsend and W.~Ketterle, Phys.~Rev.~Lett.{\\bf 77},988(1996).\n%\n\n\\bibitem{Stringari}\nS.~Stringri, Phys.~Rev.~Lett. {\\bf 77}, 2360 (1996).\n%\n\n\\bibitem{Kimura}\nT.~Kimura, H.~Saito and M.~Ueda, J.~Phys.~Soc.~Jpn.{\\bf 68},1477(1999).\n%\n\n\\bibitem{TBEC}\nR.~Graham and D~Walls, Phys. Rev. {\\bf 57A}, 484 (1998);\nH~.Pu and N.~P.~Bigelow, Phys. Rev. Lett. {\\bf 80}, 1134 (1998);\nD.~Gordon and C.~M.~Sarvage, Phys. Rev. {\\bf 58A}, 1440 (1998).\n%\n\n\\bibitem{DeMarco}\nB.~DeMarco and D.~S.~Jin,Science {\\bf 285}, 1703 (1999)\n%\n\n\\bibitem{TFex}\nL.~Vichi and S.~Stringari, Phys.~Rev.{\\bf A60},4734(1999);\nG.~M.~Bruun and C.~W.~Clark, Phys.~Rev.~Lett.{\\bf 83},5415(1999);\nM.~Amoruso, I.~Meccoli, A.Minguzzi and \nM. P. Tosi, Eur.~Phys.~J. {\\bf D7},441(1999). \n%\n\n\\bibitem{symbf}\nE.~Timmermans and R.~C\\^{o}t\\'{e}, Phys.~Rev. ~Lett. {\\bf 80}, 3419 (1998);\nW.~Geist, L.~You, and T.~A.~B.~Kennedy, Phys.~Rev.{\\bf A59}, 1500 (1999);\nM.-O.~Mewes, G.~Ferrari, F.~Schreck, A.~Sinatra and C.~Salomon, physics/9909007\n%\n\n\\bibitem{hydro} \nI.~F.~Silvera, Physica {\\bf 109 \\& 110}, 1499 (1982); \nJ.~Oliva, Phys.~Rev. {\\bf B38}, 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. {\\bf A59}, 2974 (1999).\n%\n\n\\bibitem{Miyakawa}\nT.~Miyakawa, K.~Oda, T.~Suzuki and H.~Yabu ; cond-mat/9907009.\n%\n\n\\bibitem{Minniti} \nM.~Amoruso, C.~Minniti and M.~P.~Tosi, to be published in Eur.~Phys.~J. {\\bf D}.\n%\n\n\\bibitem{Instbf}\nT.~Miyakawa, T.~Suzuki and H.~Yabu, cond-mat/0002048.\n%\n\n\\bibitem{Amoruso} \nM.~Amoruso, A.~Minguzzi, S.~Stringari, M.~P.~Tosi and L.~Vichi,\nEur.~Phys.~J. {\\bf D4}, 261 (1998); \nL.~Vichi, M.~Inguscio, S.~Stringari and G.~M.~Tino, \nJ.~Phys. {\\bf B31}, L899 (1998). \n%\n\n\\bibitem{tsurumi}\nT.~Tsurumi and M.~Wadati, J.Phys.~Soc.~Jpn.{\\b 69} (2000) in print.\n%\n\n\\bibitem{Cote}\nR.~Cot\\'e, A.~Dalgarno, H.~Wang and W.~C.~Stwalley, \nPhys.~Rev. {\\bf A57}, R4118 (1998). \n%\n\n\\bibitem{Kscat1} \nJ.~L.~Bohn, J.~P. Burke,Jr, C.~H. Greene, H.~Wang, P.~L. Gould and \nW.~C. Stwalley, Phys. Rev. {\\bf A59}, 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{Fesh} \nE. Tiesinga, A.~J. Moerdijk, B.~J. Verhaar and H.~T.~C. Stoof,\nPhys.~Rev.{\\bf A46}, R1167 (1992); \nE. Tiesinga, B.~J. Verhaar, and H.~T.~C. Stoof, \nPhys.~Rev.{\\bf A47}, 4114 (1993); \nJ.~L. Bohn, cond-mat/9911132.\n%\n"
}
] |
cond-mat0002146
|
Shape and Motion of Vortex Cores in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$
|
[
{
"author": "B.W.~Hoogenboom \\cite{email}"
},
{
"author": "M.~Kugler"
},
{
"author": "B.~Revaz \\cite{addressRevaz}"
},
{
"author": "I.~Maggio-Aprile"
},
{
"author": "and \\O.~Fischer"
}
] |
We present a detailed study on the behaviour of vortex cores in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ using scanning tunneling spectroscopy. The very irregular distribution and shape of the vortex cores imply a strong pinning of the vortices by defects and inhomogeneities. The observed vortex cores seem to consist of two or more randomly distributed smaller elements. Even more striking is the observation of vortex motion where the vortex cores are divided between two positions before totally moving from one position to the other. Both effects can be explained by quantum tunneling of vortices between different pinning centers.
|
[
{
"name": "move_vortex_prb2.tex",
"string": "% ****** Start of file move_vortex.tex ****** %\n%\n% Version 3.1 of REVTeX, July 1, 1996.\n%\n%\n%\\documentstyle[preprint,aps]{revtex}\n\\documentstyle[aps,prb,epsfig,multicol]{revtex}\n\\begin{document}\n\\draft\n\\title{Shape and Motion of Vortex Cores in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$}\n\\author{B.W.~Hoogenboom \\cite{email}, M.~Kugler, B.~Revaz \\cite{addressRevaz},\nI.~Maggio-Aprile, and \\O.~Fischer}\n\\address{DPMC, Universit\\'e de Gen\\`eve, 24 Quai Ernest-Ansermet,\n1211 Gen\\`eve 4, Switzerland}\n\\author{Ch.~Renner}\n\\address{NEC Research Institute, 4 Independence Way, Princeton,\nNew Jersey 08540, USA}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nWe present a detailed study on the behaviour of vortex cores in\nBi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$ using scanning tunneling\nspectroscopy. The very irregular distribution and shape of the\nvortex cores imply a strong pinning of the vortices by defects and\ninhomogeneities. The observed vortex cores seem to consist of two\nor more randomly distributed smaller elements. Even more striking\nis the observation of vortex motion where the vortex cores are\ndivided between two positions before totally moving from one\nposition to the other. Both effects can be explained by quantum\ntunneling of vortices between different pinning centers.\n\\end{abstract}\n\\pacs{PACS numbers: 74.50.+r, 74.60.Ec, 74.60.Ge, 74.72.Hs}\n\n\\begin{multicols}{2}\n\\narrowtext\n\n% body of paper here\n\n\\section{introduction}\n%%%%%%%%%%\nThe study of the vortex phases in high temperature superconductors\n(HTS's) has lead to both theoretical predictions of several novel\neffects and experiments accompanied by challenging\ninterpretations. The reasons are multiple. First, the\nunconventional symmetry of the order parameter --- most likely\n$d_{x^2-y^2}$ --- leads to the presence of low-lying quasiparticle\nexcitations near the gap nodes, which in turn has inspired the\npredictions of a nonlinear Meissner effect \\cite{Yip:1992}, of a\n$\\sqrt{H}$ dependence of the density of states at the Fermi level\nnear the vortex cores ($N(0,\\bf{r})$) \\cite{Volovik:1993}, and of\na four-fold symmetry of the vortices\n\\cite{Berlinsky:1995,Salkola:1996,Franz:1998}. However, the\nexperimental evidence for the first two effects is still\ncontroversial \\cite{Moler:1994,Maeda:1995,Revaz:1998,Amin:1998},\nand scanning tunneling spectroscopy (STS) measurements on vortex\ncores have not shown any clear signature of a $\\sqrt{H}$\ndependence of $N(0,\\bf{r})$ \\cite{Maggio:1995,Renner:1998}.\nConcerning the four-fold symmetry, a tendency of square vortices\nwas found in previous measurements \\cite{Renner:1998}, but\ninhomogeneities make it difficult to be decisive about it.\nInterestingly, a four-fold symmetry has been observed around\nsingle atom zinc impurities in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$\n(BSCCO), which are of a smaller size than the vortex cores\n\\cite{Pan:2000}.\n\nA second reason is related to the interaction of vortices with\npinning centers, responsible for the rich vortex phase diagram of\nthe HTS's \\cite{Blatter:1994}. The pinning of vortices is mainly\ndue to local fluctuations of the oxygen concentration\n\\cite{Tinkham:1988,Tinkham:1996,Li:1996,Erb:1999}, and facilitated\nby their highly 2D \"pancake\" character. In BSCCO the areal density\nof oxygen vacancies per Cu-O double layer is in fact surprisingly\nlarge: $10^{17}$~m$^{-2}$ \\ \\cite{Li:1996}, corresponding to an\naverage distance between the oxygen vacancies of the order of\n10~\\AA. This vortex pinning results for BSCCO in the absence of\nany regular flux line lattice at high fields, as demonstrated by\nboth neutron diffraction \\cite{Cubitt:1993} and STS\n\\cite{Renner:1998} experiments. Moreover, since the distance\nbetween the oxygen vacancies is of the order of the vortex core\nsize, one may expect that not only the vortex distribution, but\nalso the vortex core shape will be dominated by pinning effects,\nand {\\em not} by intrinsic symmetries like that of the order\nparameter. A detailed understanding of the interaction of vortices\nwith pinning centers will thus be of importance to explain the\nlack of correspondence between theoretical predictions about the\nvortex shape and STS measurements.\n\nThird, and again as a consequence of the anisotropy of the order\nparameter, low-energy quasiparticles are not truly localized in\nthe vortex core (contrarily to the situation in $s$-wave\nsuperconductors \\cite{Caroli:1964,Hess:1989,Renner:1991}). For\npure $d$-wave superconductors these quasiparticles should be able\nto escape along the nodes of the superconducting gap. Thus in the\nvortex core spectra one expects a broad zero-bias peak of\nspatially extended quasiparticle states \\cite{Franz:1998}.\nHowever, tunneling spectra of the vortex cores in\nYBa$_2$Cu$_3$O$_{7-\\delta}$ (YBCO) showed two clearly separated\nquasiparticle energy levels, which were interpreted as a signature\nof localized states \\cite{Maggio:1995}. In BSCCO two weak peaks\nhave been observed in the some vortex core spectra\n\\cite{Hoogenboom:2000,Pan:1999}, suggesting a certain similarity\nto the behaviour in YBCO. Another important characteristic of\nHTS's that follows from the STS studies mentioned above, is the\nextremely small size of the vortex cores in these materials. The\nlarge energy separation between the localized quasiparticle states\ndirectly implies that the vortex cores in YBCO are of such a size\nthat quantum effects dominate. This is even more true in BSCCO:\nnot only the in-plane dimensions of the vortex cores are smaller\nthan in YBCO and become of the order of the interatomic distances\n\\cite{Renner:1998}, but due to the extreme anisotropy of the\nmaterial also their out-of-plane size is strongly reduced. This\nhighly quantized character of vortices in HTS's is equally\ndemonstrated by the non-vanishing magnetic relaxation rate in the\nlimit of zero temperature, attributed to quantum tunneling of\nvortices through the energy barriers between subsequent pinning\ncenters \\cite{Blatter:1994}.\n\nIn this paper we present a detailed STS study of the shape of the\nvortices in BSCCO. We will show that this shape is influenced by\ninhomogeneities. The samples presented here, which we characterize\nas moderately homogeneous, are used to study the behaviour of\nvortex cores under these conditions. Apart from the vortex core\nshape, this also includes the evolution in time of the vortices.\nWe will show that both effects can be related to tunneling of\nvortices between different pinning centers. This is another\nindication of the possible extreme quantum behaviour of vortex\ncores in HTS's. A corollary of this paper is that only extremely\nhomogeneous samples will show intrinsic shapes of vortex cores.\n\n\n\\section{Experimental Details}\n%%%%%%%%%%\nThe tunneling spectroscopy was carried out using a scanning\ntunneling microscope (STM) with an Ir tip mounted perpendicularly\nto the (001) surface of a BSCCO single crystal, grown by the\nfloating zone method. The crystal was oxygen overdoped, with $T_c\n= 77$~K, and had a superconducting transition width of 1~K\n(determined by an AC susceptibility measurement). We cleaved {\\em\nin situ}, at a pressure $< 10^{-8}$~mbar, at room temperature,\njust before cooling down the STM with the sample. The sharpness of\nthe STM tip was verified by making topographic images with atomic\nresolution. Tunneling current and sample bias voltage were\ntypically 0.5~nA and 0.5~V, respectively. We performed the\nmeasurements at 4.2~K with a low temperature STM described in\nRef.~\\onlinecite{Renner:1990,Kent:1992}, and those at 2.5~K with a\nrecently constructed $^3$He STM \\cite{Kugler:2000}. A magnetic\nfield of 6~T parallel to the $c$-axis of the crystal was applied\nafter having cooled down the sample. The measurements presented\nhere were initiated 3 days after having switched on the field.\n\nThe $dI/dV$ spectra measured with the STM correspond to the\nquasiparticle local density of states (LDOS). In the\nsuperconducting state one observes two pronounced coherence peaks,\ncentered around the Fermi level, at energies $\\pm\\Delta_p$. The\ngap size $\\Delta_p$ varied from 30-50~meV. In the vortex cores the\nspectra are remarkably similar to those of the pseudogap in BSCCO\nmeasured above $T_c$ \\ \\cite{Renner:1998}, with a total\ndisappearance of the coherence peak at negative bias, a slight\nincrease of the zero bias conductivity, and a decrease and shift\nto higher energy of the coherence peak at positive bias. To map\nthe vortex cores we define a gray scale using the quotient of the\nconductivity $\\sigma(V_p)=dI/dV(V_p)$ at a negative sample voltage\n$V_p=-\\Delta_p/e$ and the zero bias conductivity\n$\\sigma(0)=dI/dV(0)$. Thus we obtain spectroscopic images, where\nvortex cores appear as dark spots. Since we measure variations of\nthe LDOS, which occur at a much smaller scale (the coherence\nlength $\\xi$) than the penetration depth $\\lambda$, we can get\nvortex images at high fields. A tunneling spectrum is taken on the\ntime scale of seconds, spectroscopic images typically take several\nhours (about 12~hours for the images of 100x100~nm$^2$ presented\nbelow). The images therefore necessarily reflect a time-averaged\nvortex density.\n\nIn all large-scale images we have suppressed short length-scale\nnoise by averaging each point over a disk of radius $\\sim 20$~\\AA.\nWhen zooming in to study the shape of individual vortices, we\nstrictly used raw data. Further experimental details can be found\nin previous publications\n\\cite{Renner:1998,Renner:1998a,Renner:1995}.\n\n\n\\section{Results}\n\n\\subsection{Vortex Distribution}\n%%%%%%%%%%\nIn Fig.~\\ref{greatimage} we show spectroscopic images of the\nsurface of a BSCCO crystal, at different magnetic field strengths.\nThe large dark structure, clearly visible at the right of\nFigs.~\\ref{greatimage}(b) and (c), corresponds to a degraded\nregion resulting from a large topographic structure, already\nobserved in the topographic image Figs.~\\ref{greatimage}(a). The\npresence of this structure allows an exact position determination\nthroughout the whole experimental run. As can be seen in\nFig.~\\ref{greatimage}, the number of vortices at 6 and at 2~T, in\nexactly the same region, scales very well with the total number of\nflux quanta ($\\Phi_0$) that one should expect at these field\nstrengths. This clearly proves that the observed dark spots are\ndirectly related to vortex cores, and not to inhomogeneities,\ndefects or any form of surface degradation. The large spot in the\nupper left corner of Fig.~\\ref{greatimage}(c) forms an exception:\nit appeared after a sudden noise on the tunnel current while we\nwere scanning on that position, showed semiconducting spectra\n(typical for degraded tunneling conditions) afterwards, and\nremained even after having set the external field to 0~T. One\nshould however not exclude that a vortex is pinned in this\ndegraded zone. Finally, the size and density of the vortices are\nfully consistent with previous measurements\n\\cite{Renner:1998,Pan:1999}.\n\nInstead of a well ordered vortex lattice, one observes patches of\nvarious sizes and shapes scattered over the surface. This clearly\nindicates the disordered nature of the vortex phase in BSCCO at\nhigh fields, again in consistency with previous STM studies\n\\cite{Renner:1998,Pan:1999} and neutron scattering data\n\\cite{Cubitt:1993}, and stressing the importance of pinning for\nthe vortex distribution.\n\n\n\\subsection{Vortex Shapes}\n%%%%%%%%%%\nAs a next step we increase the spatial resolution in order to\ninvestigate individual vortex cores. Some vortices appear with\nsquare shapes, but most vortices in this study have irregular\nshapes. Closer inspection of the tunneling spectra reveals small\nzones inside the vortex core that show superconducting behaviour.\nThat is, when scanning through a vortex core one often observes\n(slightly suppressed) coherence peaks (Fig.~\\ref{closeup}(a)),\ntypical for the superconducting state, at some spots {\\em inside}\nthe vortex core. The latter is generally characterized by the {\\em\nabsence} of these peaks. In some cases, the vortex cores are even\ntruly split into several smaller elements (Fig.~\\ref{closeup}(b)),\ntotally separated by small zones showing the rise of coherence\npeaks. This has been verified by measuring the full spectra along\nlines through the vortex core, as in Fig.~\\ref{closeup}(a).\n\nThe smaller elements of a split vortex core cannot be related to\nseparate vortices: first, the vortex-vortex repulsion makes it\nhighly improbable that several vortex cores are so close to each\nother; second, counting all these elements as a flux quantum in\nFig.~\\ref{greatimage}(b), one finds a total flux through the\nsurface that is far too large compared to the applied field. One\nshould note here that the magnetic size of a flux line is of the\norder of the penetration depth $\\lambda$, two orders of magnitude\nlarger than the vortex {\\em core} splitting observed here.\n\n\n\\subsection{Vortex Motion}\n%%%%%%%%%%\nWith subsequent spectroscopic images like\nFig.~\\ref{greatimage}(b), one can also study the vortex\ndistribution as a function of time. We expect the vortex motion to\nbe practically negligible, since we allowed the vortices to\nstabilize for more than 3 days \\cite{VanDalen:1996}. However, in\nFig.~\\ref{vortexcreep} one can see that many vortices still have\nnot reached totally stable positions. Many of them roughly stay on\nthe same positions over the time span of our measurement, but\nothers move to neighboring positions. Five different cases of\nmoving vortices are indicated by the ellipses and the rectangle in\nFig.~\\ref{vortexcreep}.\n\nIn the panels on the left side the precise intensity of each point\nis difficult to read out directly. In order to investigate more\nquantitatively the time evolution of the vortex distribution, from\none frame to the next, we show in the right part of\nFig.~\\ref{vortexcreep} 3D representations of the area that is\nmarked by the rectangle in the 2D spectroscopic images. They give\nan idea of the gray scale used in the 2D images, and provide a\ndetailed picture of the movement of the vortex core in front, from\nthe right in Fig.~\\ref{vortexcreep}(a) to the left in\nFig.~\\ref{vortexcreep}(c). The vortex core at the back does not\nmove, and serves as a reference for the intensity. We remind that\nthe intensity, or height in the 3D images, is a measure of the\nLDOS, which in a vortex core is different from the superconducting\nDOS. It is most interesting to see what happens in\nFig.~\\ref{vortexcreep}(b): the (moving) vortex core is {\\em\ndivided} between two positions. Thus, the vortex core moves from\none position to the other, passing through an intermediate state\nwhere the vortex splits up between the two positions. Note that\nthese two positions do not correspond to two vortices. In fact,\nthe split vortex is characterized by the lower intensity compared\nto the nearby (reference) vortex. This means that the coherence\npeak at negative voltage does not completely disappear, as it\nshould if we had a complete and stable vortex at each of these\npositions. Note also that the density of vortices around the\nrectangular area on the left side in Fig.~\\ref{vortexcreep} will\nclearly be too high if we count the mentioned positions and all\npositions in the ellipses as individual flux quanta. The split\nvortex discussed here is not a unique example. Similar behaviour\ncan be found for several other vortex cores, as indicated by the\nellipses in Fig.~\\ref{vortexcreep}. This gradual change of\nposition is in striking contrast to the STS observations of moving\nvortices in NbSe$_2$ \\ \\cite{Renner:1993,Troyanovski:1999} and\nYBCO \\cite{Maggio:1997}.\n\n\n\\subsection{Temperature Dependence}\n%%%%%%%%%%\nWe performed measurements both at 4.2 and at 2.5~K, on samples cut\nfrom the same batch of crystals. The data taken at 2.5~K (see also\nFig.~\\ref{closeup}) are fully consistent with the presented work\nat 4.2~K. In Fig.~\\ref{2K_data} we provide a general view of the\nvortex cores at 2.5~T, including an analogue of the moving vortex\ncore of Fig.~\\ref{vortexcreep}. Though it is hard to obtain any\nquantitative data, one can conclude that the vortex cores roughly\nhave the same size, similar irregular shapes, and examples of\nsplit vortex cores can be easily found.\n\n\n\\section{Discussion}\n\n\\subsection{Experimental Considerations}\n%%%%%%%%%%\nThe observation of such a highly irregular pattern of vortex\ncores, as presented above, requires a careful analysis of the\nexperimental setup. However, the fact that keeping exactly the\nsame experimental conditions the number of vortex cores scales\nwith the magnetic field, is a direct proof of the absence of\nartificial or noise-related structures in the spectroscopic\nimages. Furthermore, since topographic images showed atomic\nresolution, there is no doubt that the spatial resolution of the\nSTM is largely sufficient for the analysis of vortex core shapes.\n\nThe stability of the magnetic field can be verified by counting\nthe number of vortices in the subsequent images at 6~T\n(Fig.~\\ref{vortexcreep}). Since, excluding the split vortices\nmarked by the ellipses, this number is constant ($26\\pm3$), we can\nexclude any substantial long time-scale variation of the magnetic\nfield. Some variation in the total black area from one image to\nthe other can be related to the tunneling conditions: a little\nmore noise on the tunnel current will give a relatively large\nincrease of the small zero-bias conductance. Since we divide by\nthe zero-bias conductance to obtain the spectroscopic images, this\nmay lead to some small variations in the integrated black area of\nthe images.\n\n\n\\subsection{Delocalization}\n%%%%%%%%%%\nKeeping in mind the randomness of the vortex distribution at 6~T\ndue to pinning of vortices, we now relate both the split vortex\ncores (Fig.~\\ref{closeup}) and the intermediate state between two\npositions (Fig.~\\ref{vortexcreep} and Fig.~\\ref{2K_data}) to the\nsame phenomenon: the vortex cores appear to be delocalized between\ndifferent positions which correspond to pinning potential wells,\nand during the measurement hop back and forth with a frequency\nthat is too high to be resolved in this experiment. According to\nthis analysis not only the distribution, but also the observed\nshape of the vortex cores is strongly influenced by pinning.\n\nThe pinning sites most probably result from inhomogeneities in the\noxygen doping, which are thought to be responsible for the\nvariations of the gap size (see experimental details). The\ndistance over which the vortices are split corresponds to the\naverage spacing between oxygen vacancies ($10-100$~\\AA \\\n\\cite{Li:1996}). We did not observe any sign of resonant states\nrelated to impurities, as in recent STM experiments on BSCCO\n\\cite{Yazdani:1999,Hudson:1999}. The driving forces causing vortex\nmovements in Fig.~\\ref{vortexcreep} and Fig.~\\ref{2K_data} are\nmost probably due to a slow variation of the pinning potential,\nresulting from the overall rearrangement of vortices.\n\nThe vortex delocalization and movement presented here can directly\nbe connected to the vortex creep as measured in macroscopic\nexperiments, like magnetic relaxation \\cite{Blatter:1994}. The\nmain difference, of course, is that we do not observe whole\nbundles of vortices moving over relatively large distances, but\nonly {\\em single} vortex cores that are displaced over distances\nmuch smaller than the penetration depth $\\lambda$. That is, it\nwill not be necessary to displace whole groups of vortices, many\nof which might be pinned much stronger than the delocalized\nvortices we observe. A second difference is the absence of a\nuniform direction of the movements in the STM images, most\nprobably because the Lorentz driving forces have been reduced to\nan extremely small value (which also follows from the very gradual\nchanges in Fig.~\\ref{vortexcreep} and Fig.~\\ref{2K_data}).\n\n\n\\subsection{Thermal Fluctuations versus Quantum Tunneling}\n%%%%%%%%%%\nRegarding now the mechanism responsible for the vortex\ndelocalization, the main question is whether we are dealing with\nthermal fluctuations, or quantum tunneling between pinning\npotential wells. In fact magnetic relaxation measurements on BSCCO\nshow a crossover temperature from thermal to quantum creep of\n$2-5$~K \\cite{VanDalen:1996,Prost:1993,Aupke:1993,Monier:1998},\nwhich means that with these STM measurements we are on the limit\nbetween the two.\n\nIn the case of thermally induced motion, there is a finite\nprobability for the vortex to jump {\\em over} the energy barrier\nbetween the two potential wells. The vortex is continuously moving\nfrom one site to the other, with a frequency that is too high to\nbe resolved by our measurements. In the case of quantum tunneling,\nthe vortex is truly delocalized. That is, the vortex can tunnel\n{\\em through} the barrier, and one observes a combination of two\nbase states (i.e. positions), like in the quantum text book\nexample of the ammonia molecule \\cite{Feynman:1965}. Thermal\nfluctuations will lead to a continuous dissipative motion\n\\cite{Blatter:1994,Bardeen:1965} between the two sites; quantum\ntunneling gives a dissipationless state in which the vortex is\n{\\em divided} between two positions.\n\nAn instantaneous observation of several base states of a quantum\nobject would be impossible, since each measurement implies a\ncollapse of the quantum wave function into one state. However, the\nSTM gives only time averaged images, and with the tunneling\ncurrent in this experiment we typically detect one electron per\nnanosecond. If the vortex core relaxes back to its delocalized\nstate on a time scale smaller than nanoseconds, the vortex can\nappear delocalized in the STM images. Moreover, it should be clear\nthat the long time (~12 hours) between the subsequent images in\nFig.~\\ref{vortexcreep} and \\ref{2K_data} has nothing to do with\nthe vortex tunneling time; it is tunneling of the vortex that\nallows the intermediate state. The creep of vortices (either by\nquantum tunneling or by thermal fluctuations) is a slow phenomenon\nhere. At a given region the pinning potential due to\ninhomogenieties and interactions with other vortices evolves on a\ntime scale of hours, shifting the energetically most favorable\nposition from one site to the other. However, the tunneling occurs\nmuch faster ($<$ns) than this slow potential evolution, creating\nthe possibility of a superposition of both states (positions),\neach with a probability depending on the local value of the pining\npotential. Following the analogy of the ammonia molecule: a moving\nvortex core corresponds to an ammonia molecule submitted to an\nexternal external field (\"overall pinning potential\") that is\nslowly changing. Initially, due to this external field the state\nwith the hydrogen atom at the left is more favorable, then the\nfield changes such that left and right are equally stable (\"split\nvortex\"), and finally the right state is most probable: the\nhydrogen atom has moved from left to right on a long time scale,\nwhereas the tunneling itself occurs on a much faster time scale.\nBy the way, since the change of the local pinning potential\nresults from a rearrangement of all surrounding vortices (over a\ndistance $\\sim\\lambda$), the time scale of this change will still\nbe strongly dependent on the (short) tunneling time itself.\n\nIn order to discuss the possibility of quantum tunneling, one\nshould consider the tunneling time and the for our measurements\nnegligible temperature dependence\n\\cite{Blatter:1994,Blatter:1993}. The importance of the tunneling\ntime is two-fold: the tunneling rate is strongly dependent on the\ntime needed to pass through the pinning barriers (an effect which\nwill be extremely difficult to measure directly); and in our\nmeasurements the tunneling time must be faster than the probe\nresponse time (as explained in the previous paragraph). From\ncollective creep theory \\cite{Blatter:1994} one can get an\norder-of-magnitude estimate for the tunneling time\n$t_c\\sim\\hbar/U(S_E/\\hbar)\\sim10^{-11}$~s, with the Euclidian\naction for tunneling $S_E/\\hbar\\sim10^2$ and the effective pinning\nenergy $U\\sim10^2$~K derived from magnetic relaxation measurements\n\\cite{Blatter:1994,Li:1996,VanDalen:1996,Prost:1993,Aupke:1993,Monier:1998}.\nThis is clearly below the upper limit of 1~ns set by the probe\nresponse time.\n\nFor a discussion about the implications of the temperature\nindependence of quantum tunneling, one should first consider the\ntemperature dependence expected for vortex movements that result\nfrom thermal fluctuations. Thermally induced hopping between\ndifferent pinning sites should be proportional to $\\exp(U/k_BT)$,\nwhere $U$ is again the effective pinning energy\n\\cite{Blatter:1994}. From magnetic relaxation measurements on\nBSCCO one can derive a value of about $10-10^3$~K for this\nquantity\n\\cite{Li:1996,VanDalen:1996,Prost:1993,Aupke:1993,Monier:1998}.\nAssuming for the moment that this $U$ determines the hopping of\nindividual vortices, it should then be compared to the Euclidian\naction for quantum tunneling, which with magnetic relaxation\nmeasurements is estimated to be $S_E/\\hbar \\sim 10^2$ \\\n\\cite{Blatter:1994}, and plays a role like $U/k_BT$ in the\nBoltzman distribution. For measurements presented here, it is\nimportant to note again that they were taken more than 3 days\nafter having increased the field from 0 to 6~T. Since for $B =\n6$~T the induced current density $j$ relaxes back to less than\n$0.01$ of its initial value in about 10 seconds\n\\cite{VanDalen:1996}, we are clearly in the limit where $j$ and\nthus the Lorentz driving forces (which reduce the energy barrier\nfor vortex creep) approach zero. This means that the effective\npinning potential $U$ rises, if not to infinity like in isotropic\nmaterials, to a value which in principle is much higher than the\none which determines vortex creep in magnetic relaxation\nmeasurements at comparable field strengths\n\\cite{Blatter:1994,Tinkham:1996}. With nearly zero Lorentz forces\nthe tilt of the overall pinning potential will thus be small\ncompared to the pinning barriers, making thermal hopping {\\em\nover} the barriers highly improbable (at low temperatures).\nQuantum creep, in the limit of vanishing dissipation, is\nindependent of the collective aspect of $U$, while the probability\nfor thermal creep decreases as $\\exp(-U/k_BT)$ \\\n\\cite{Blatter:1994}. So one can expect quantum creep to become\nmore important than thermal creep when more time has passed after\nhaving changed the field. In other words, in spite of the fact\nthat the tilt of the overall pinning potential is small compared\nto the pinning barriers, the vortices can still move a little,\ni.e. disappearing and reappearing elsewhere.\n\nHowever, the collective $U$ may be higher than the pinning\nbarriers for the individual vortex movements observed in our\nexperiments. Thus, in order to find a lower bound for the latter,\nwe also estimate $U$ for the moving vortices from our microscopic\nmeasurement. First we calculate the magnetic energy of a vortex\ndue to the interaction with its nearest neighbors, using\n\\begin{equation}\n E_{int} = d {{\\Phi_0^2} \\over {8 \\pi^2 \\lambda^2}} \\sum_i\n \\{ln \\lgroup {\\lambda \\over {r_i}} \\rgroup +0.12\\},\n\\label{A}\n\\end{equation}\nwhere $d$ is the length of the vortex segment, $\\Phi_0$ is the\nflux quantum, $\\lambda$ the in-plane penetration depth and $r_i$\nthe distance to its $i$th neighbor \\cite{Tinkham:1996}. Parameters\nare conservatively chosen such to give a true minimum estimate for\n$E_{int}$ (and thus for $U$): we restrict the out-of-plane extent\nof the vortices to zero and thus only take $d = 15$~\\AA, the size\nof one double Cu-O layer \\cite{Harshman:1992} (\"pancake\nvortices\"), and for $\\lambda$ take the upper bound of different\nmeasurements, 2500~\\AA \\\n\\cite{max_lambda,Martinez:1992,Waldmann:1996}. Taking the vortex\nin Fig.~\\ref{vortexcreep}(b), and determining the positions\nbetween which it is divided as well as the positions of the\nneighboring vortices, one can find the difference between the\nmagnetic interaction energies of the delocalized vortex at its two\npositions. We obtain $E_{int} \\sim 120$~K. Now the absence of any\nvortex lattice indicates that the pinning potential wells are\ngenerally larger than the magnetic energy difference between the\nsubsequent vortex positions, and Fig.~\\ref{vortexcreep}(b)\nreflects a vortex state that is quite common in our measurements\n(Fig.~\\ref{greatimage}). Following these arguments one can safely\nassume that the effective potential well pinning the vortex in\nFig.~\\ref{vortexcreep} is larger than this difference: $U >\nE_{int} = 120$~K. in agreement with the estimates given above. So\nwe obtain $U/k_BT > 10-10^2$ for temperatures around 4~K. In the\nlimit of zero dissipation $S_E/\\hbar \\sim (k_F\\xi)^2$. On the\nbasis of STS experiments \\cite{Maggio:1995,Renner:1998} this can\nbe estimated to be $\\leq 10$. This value is smaller than the one\nquoted above, and suggests that quantum tunneling is dominant in\nour measurements.\n\nThe most direct evidence for quantum creep can be obtained from\nmeasurements at different temperatures. The hopping rate for\nthermally induced movements is given by $\\omega_0\\exp(-U/k_BT)$,\nwhere $U$ is the pinning potential, and $\\omega_0$ the\ncharacteristic frequency of thermal vortex vibration\n\\cite{Tinkham:1996}. Assuming $U = 100$~K, and a conservatively\nlarge estimate of $\\omega_0 \\sim 10^{11}$~s$^{-1}$, the hopping\nrate should drop from 1~s$^{-1}$ to 10$^{-7}$~s$^{-1}$ on cooling\nfrom 4.2 to 2.5~K. This gives a huge difference between the\nrespective measurements at these temperatures. However,\nspectroscopic images at 4.2 and 2.5~K show the same pattern of\nmoving and delocalized vortices. Following the same kind of\nestimations as above, the delocalized vortex at 2.5~K\n(Fig.~\\ref{2K_data}) gave $U > 210$~K, which makes thermal creep\neven more unlikely here. Even if the frequency of the individual\nthermal vortex movements were too high to be resolved by our\nmeasurements {\\em both} at 4.2 and at 2.5~K (this would mean a\nrather unrealistic characteristic frequency $\\omega_0 > 10^{15}$),\none would still expect to see a difference. As a matter of fact,\nthe driving force for the vortex movements results from an overall\nrearrangement of vortices. This means that the displacements of\nvortices will always depend on the hopping frequency, and that\neven for very high hopping rates one should observe a reduction of\nthe number of vortices that are displaced in our images, when the\nhopping rate is reduced by a factor 10$^7$.\n\n\n\\section{Conclusion}\n%%%%%%%%%%\nWe observed vortex cores that were delocalized over several pinning\npotential wells. Regardless of the exact mechanism (thermal\nhopping or quantum tunneling) responsible for this delocalization,\nour measurements point out that pinning effects not only dominate\nthe distribution of the vortex cores, but also their shape. As a\nconsequence intrinsic (four-fold?) symmetries of the vortex cores\nwill be obscured in microscopic measurements. The delocalization\nof the vortex cores implies that the vortex cores in this\nstudy appear larger than their actual --- unperturbed --- size,\nindicating a coherence length that is even smaller than was\nexpected on the base of previous studies \\cite{Renner:1998}.\n\nThe analysis given above strongly favors an interpretation in\nterms of quantum tunneling of vortex cores. This would not only\nmean the first microscopic signature of the vortex quantum\ntunneling as derived from magnetic relaxation measurements, it is\nalso a further indication \\cite{Arndt:1999} that objects of larger\nsize and complexity than one or several atoms can appear as a\nsuperposition of different quantum states.\n\n\\acknowledgements This work was supported by the Swiss National\nScience Foundation.\n\n\n% references\n%\\bibliographystyle{prsty}\n%\\bibliography{move_vortex_prb2}\n\\begin{thebibliography}{10}\n\n\\bibitem[*]{email}\nE-mail: Bart.Hoogenboom@physics.unige.ch.\n\n\\bibitem[\\dagger]{addressRevaz}\nPresent address: University of California at San Diego, Department\nof Physics, 9500 Gilman dr., La Jolla CA 92093, USA.\n\n\\bibitem{Yip:1992}\nS.K. Yip and J.A. Sauls, Phys. Rev. Lett. {\\bf 69}, 2264 (1992).\n\n\\bibitem{Volovik:1993}\nG.E. Volovik, JETP Lett. {\\bf 58}, 1174 (1993).\n\n\\bibitem{Berlinsky:1995}\nA.J. Berlinsky, A.L. Fletter, M. Franz, C. Kallin, and P.I.\nSoininen, Phys.\n Rev. Lett. {\\bf 75}, 2200 (1995).\n\n\\bibitem{Salkola:1996}\nM.I. Salkola, A.V. Balatsky, and D.J. Scalapino, Phys. Rev. Lett.\n{\\bf 77},\n 1841 (1996).\n\n\\bibitem{Franz:1998}\nM. Franz and Z. Te\\u{s}anovi\\'c, Phys. Rev. Lett. {\\bf 80}, 4763\n(1998).\n\n\\bibitem{Moler:1994}\nK.A. Moler, D.J. Baar, J.S. Urbach, R. Liang, W.N. Hardy, and A.\nKapitulnik,\n Phys. Rev. Lett. {\\bf 73}, 2744 (1994).\n\n\\bibitem{Maeda:1995}\nA. Maeda, Y. Lino, T. Hanaguri, N. Motohira, K. Kishio, and T.\nFukase, Phys.\n Rev. Lett. {\\bf 74}, 1202 (1995).\n\n\\bibitem{Revaz:1998}\nB. Revaz, J.-Y. Jenoud, A. Junod, K. Neumaier, A. Erb, and E.\nWalker, Phys.\n Rev. Lett. {\\bf 80}, 3364 (1998).\n\n\\bibitem{Amin:1998}\nM.H.S. Amin, I. Affleck, and M. Franz, Phys. Rev. B {\\bf 58},\n5848 (1998).\n\n\\bibitem{Maggio:1995}\nI. Maggio-Aprile, {Ch}. Renner, A. Erb, E. Walker, and {\\O}.\nFischer, Phys.\n Rev. Lett. {\\bf 75}, 2754 (1995).\n\n\\bibitem{Renner:1998}\n{Ch}. Renner, B. Revaz, K. Kadowaki, I. Maggio-Aprile, and {\\O}.\nFischer, Phys.\n Rev. Lett. {\\bf 80}, 3606 (1998).\n\n\\bibitem{Pan:2000}\nS.H. Pan, E.W. Hudson, K.M. Lang, H. Eisaki, S. Uchida, and J.C.\nDavis, Nature\n {\\bf 403}, 746 (2000).\n\n\\bibitem{Blatter:1994}\nG. Blatter, M.V. Feigel'man, V.B. Geschkenbein, A.I. Larkin, and\nV.M. Vinokur,\n Rev. Mod. Phys. {\\bf 66}, 1125 (1994), and references therein.\n\n\\bibitem{Tinkham:1988}\nM. Tinkham, Helv. Phys. Acta {\\bf 61}, 443 (1988).\n\n\\bibitem{Tinkham:1996}\nM. Tinkham, {\\em Introduction to Superconductivity}, 2nd. ed.\n(McGraw-Hill, New\n York, 1996).\n\n\\bibitem{Li:1996}\nT.W. Li, A.A. Menovsky, J.J.M. Franse, and P.H. Kes, Physica C\n{\\bf 257}, 179\n (1996).\n\n\\bibitem{Erb:1999}\nA. Erb, A.A. Manuel, M. Dhalle, F. Marti, J.-Y. Genoud, B. Revaz,\nA. Junod, D.\n Vasumathi, S. Ishibashi, A. Shukla, E. Walker, \\O. Fischer, R. Fl\\\"ukiger, R.\n Pozzi, M. Mali, and D. Brinkmann, Solid State Comm. {\\bf 112}, 245 (1999).\n\n\\bibitem{Cubitt:1993}\nR. Cubitt, E.M. Morgan, G. Yang, S.L. Lee, D.Mck. Paul, H.A. Mook,\nM. Yethiraj,\n P.H. Kes, T.W. Li, A.A. Menovsky, Z. Tarnawski, and K. Mortensen, Nature {\\bf\n 365}, 407 (1993).\n\n\\bibitem{Caroli:1964}\nC. Caroli, P.G. de~Gennes, and J. Matricon, Phys. Lett. {\\bf 9},\n307 (1964).\n\n\\bibitem{Hess:1989}\nH.F. Hess, R.B. Robinson, R.C. Dynes, J.M. Valles, and J.V.\nWaszczak, Phys.\n Rev. Lett. {\\bf 62}, 214 (1989).\n\n\\bibitem{Renner:1991}\n{Ch}. Renner, A.D. Kent, {Ph.} Niedermann, \\O. Fischer, and F.\nL\\'evy, Phys.\n Rev. Lett. {\\bf 67}, 1650 (1991).\n\n\\bibitem{Hoogenboom:2000}\nB.W. Hoogenboom, M. Kugler, Ch. Renner, B. Revaz, I.\nMaggio-Aprile, and \\O.\n Fischer, Physica C (2000).\n\n\\bibitem{Pan:1999}\nS.H. Pan, E.W. Hudson, A.J. Gupta, K.-W. Ng, and J.C. Davis,\npreprint.\n\n\\bibitem{Renner:1990}\n{Ch}. Renner, Ph. Niedermann, A.D. Kent, and {\\O}. Fischer, J.\nVac. Sci.\n Technol. A {\\bf 8}, 330 (1990).\n\n\\bibitem{Kent:1992}\nA.D. Kent, {Ch}. Renner, Ph. Niedermann, J.-G. Bosch, and {\\O}.\nFischer,\n Ultramicroscopy {\\bf 42-44}, 1632 (1992).\n\n\\bibitem{Kugler:2000}\nM. Kugler, {Ch}. Renner, V. Mikheev, G. Batey, and {\\O}. Fischer,\nRev. Sci.\n Instrum. {\\bf 71}, 1475 (2000).\n\n\\bibitem{Renner:1998a}\n{Ch}. Renner, B. Revaz, J.-Y. Genoud, K. Kadowaki, and {\\O}.\nFischer, Phys.\n Rev. Lett. {\\bf 80}, 149 (1998).\n\n\\bibitem{Renner:1995}\n{Ch}. Renner and {\\O}. Fischer, Phys. Rev. B {\\bf 51}, 9208\n(1995).\n\n\\bibitem{VanDalen:1996}\nA.J.J. van Dalen, R. Griessen, and M.R. Koblischka, Physica C {\\bf\n257}, 271\n (1996).\n\n\\bibitem{Renner:1993}\nCh. Renner, Ph.D. thesis, University of Geneva, 1993.\n\n\\bibitem{Troyanovski:1999}\nA.M. Troyanovski, J. Aarts, and P.H. Kes, Nature {\\bf 399}, 665\n(1999).\n\n\\bibitem{Maggio:1997}\nI. Maggio-Aprile, {Ch}. Renner, A. Erb, E. Walker, and {\\O}.\nFischer, Nature\n {\\bf 390}, 487 (1997).\n\n\\bibitem{Yazdani:1999}\nA. Yazdani, C.M. Howald, C.P. Lutz, A. Kapitulnik, and D.M.\nEigler, Phys. Rev.\n Lett. {\\bf 83}, 176 (1999).\n\n\\bibitem{Hudson:1999}\nE.W. Hudson, S.H. Pan, A.K. Gupta, K.-W. Ng., and J.C. Davis,\nScience {\\bf\n 285}, 88 (1999).\n\n\\bibitem{Prost:1993}\nD. Prost, L. Fruchter, and I.A. Campbell, Phys. Rev. B {\\bf 47},\n3457 (1993).\n\n\\bibitem{Aupke:1993}\nK. Aupke, T. Teruzzi, P. Visani, A. Amann, A.C. Mota, and V.N.\nZavaritsky,\n Physica C {\\bf 209}, 255 (1993).\n\n\\bibitem{Monier:1998}\nD. Monier and L. Fruchter, Phys. Rev. B {\\bf 58}, 8917 (1998).\n\n\\bibitem{Feynman:1965}\nR.P. Feynman, R.B. Leighton, and M. Sands, {\\em Lectures on\nPhysics}\n (Addison-Wesley, Reading, Massachusetts, 1965), Vol.~III.\n\n\\bibitem{Bardeen:1965}\nJ. Bardeen and M.J. Stephen, Phys. Rev. {\\bf 140}, A1197 (1965).\n\n\\bibitem{Blatter:1993}\nG. Blatter and V.B. Geschkenbein, Phys. Rev. B {\\bf 47}, 2725\n(1993).\n\n\\bibitem{Harshman:1992}\nD.R. Harshman and Jr. A.P.~Mills, Phys. Rev. B {\\bf 45}, 10684\n(1992), and\n references therein.\n\n\\bibitem{max_lambda}\nRef.~\\onlinecite{Harshman:1992} gives $\\lambda=2500$~\\AA, but for\na sample that\n has not been O$_2$ annealed. In Ref.~\\onlinecite{Waldmann:1996} it is shown\n that this indeed results in a higher $\\lambda$. Since the samples discussed\n here are overdoped (O$_2$ annealed), one can safely adopt the values for\n optimum doped or overdoped samples, considerably lower than 2500~\\AA \\\n \\cite{Martinez:1992,Waldmann:1996}.\n\n\\bibitem{Martinez:1992}\nJ.C. Mart\\'inez, S.H. Brongersma, A. Koshelev, B. Ivlev, P.H. Kes,\nR.P.\n Griessen, D.G. de~Groot, Z. Tarnavski, and A.A. Menovsky, Phys. Rev. Lett.\n {\\bf 69}, 2276 (1992).\n\n\\bibitem{Waldmann:1996}\nO. Waldmann, F. Steinmeyer, P. M\\\"uller, J.J. Neumeier, F.X.\nR\\'egi, H. Savary,\n and J. Schneck, Phys. Rev. B {\\bf 53}, 11825 (1996).\n\n\\bibitem{Arndt:1999}\nM. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van~der Zouw,\nand A.\n Zeilinger, Nature {\\bf 401}, 680 (1999).\n\n\\end{thebibliography}\n\n\n% figures\n\\begin{figure}\n \\epsfxsize=70mm\n \\centerline{\\epsffile{greatimage.eps}}\n \\caption{\n (a)~100x100~nm$^2$ topographic image of the BSCCO surface at 4.2~K and\n 6~T, taken at high bias voltage, $V_{bias}=0.4$~V, $I_t=0.6$~nA.\n The structure at the right gradually increases in height from\n 0~nm (lightest gray) to almost 5~nm (fully black at the right\n border), and is thus much larger than any atomic details. The\n surface roughness of the gray part is about 1~\\AA.\n (b)~Spectroscopic image of the same area, taken simultaneously with~(a);\n dark spots correspond to vortex cores, the dark region at the\n right corresponds to the degraded surface of the structure already observed\n in~(a). For the surface excluding this topographic structure one should\n expect 26 vortices, the image contains 27 (the circles around the vortex\n cores serve as a guide to the eye).\n (c)~Part of the same region, at 2~T, image taken after all measurements at\n 6~T. The number of vortices in the image\n again perfectly corresponds to what one should expect for the given surface.}\n \\label{greatimage}\n\\end{figure}\n\n\\begin{figure}\n \\epsfxsize=70mm\n \\centerline{\\epsffile{closeup.eps}}\n \\caption{(a) Spectra along a trace through a vortex core ($B = 6$~T,\n $T = 2.5$~K) reveal that in between regions with vortex core-like\n spectra (indicated by the two arrows) the superconducting coherence\n peaks come up again.\n (b) Image of a vortex core consisting of several separate elements.\n ($B = 6$~T, $T = 4.2$~K).}\n \\label{closeup}\n\\end{figure}\n\n\\begin{figure}\n \\epsfxsize=90mm\n \\centerline{\\epsffile{vortexcreep.eps}}\n \\caption{Sequence of images (each taking about 12~hours) to\n study the behaviour of the vortex cores in time, $B = 6$~T, $T =\n 4.2$~K. t corresponds to the starting time of each image.\n (a) t~=~0; (b) t~=~12h; (c) t~=~31h.\n Left: 2D representation. Right: 3D images of the zone marked by the\n rectangles in the 2D images. The vortex core seems to\n be split in (b), before it totally moves from one position in (a)\n to the other in (c).}\n \\label{vortexcreep}\n\\end{figure}\n\n\\begin{figure}\n \\epsfxsize=80mm\n \\centerline{\\epsffile{2K_data.eps}}\n \\caption{Subsequent images ((a) and (b)) at $B = 6$~T, $T = 2.5$~K. In (c) a\n 3D representation of the square marked in (a) and (b). At 2.5~K one observes\n the same phenomena as at 4.2~K in Fig.~\\ref{closeup} and\n Fig.~\\ref{vortexcreep}.}\n \\label{2K_data}\n\\end{figure}\n\n\n\\end{multicols}\n\\end{document}\n%\n% ****** End of file move_vortex.tex ******\n"
}
] |
[
{
"name": "cond-mat0002146.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem[*]{email}\nE-mail: Bart.Hoogenboom@physics.unige.ch.\n\n\\bibitem[\\dagger]{addressRevaz}\nPresent address: University of California at San Diego, Department\nof Physics, 9500 Gilman dr., La Jolla CA 92093, USA.\n\n\\bibitem{Yip:1992}\nS.K. Yip and J.A. Sauls, Phys. Rev. Lett. {\\bf 69}, 2264 (1992).\n\n\\bibitem{Volovik:1993}\nG.E. Volovik, JETP Lett. {\\bf 58}, 1174 (1993).\n\n\\bibitem{Berlinsky:1995}\nA.J. Berlinsky, A.L. Fletter, M. Franz, C. Kallin, and P.I.\nSoininen, Phys.\n Rev. Lett. {\\bf 75}, 2200 (1995).\n\n\\bibitem{Salkola:1996}\nM.I. Salkola, A.V. Balatsky, and D.J. Scalapino, Phys. Rev. Lett.\n{\\bf 77},\n 1841 (1996).\n\n\\bibitem{Franz:1998}\nM. Franz and Z. Te\\u{s}anovi\\'c, Phys. Rev. Lett. {\\bf 80}, 4763\n(1998).\n\n\\bibitem{Moler:1994}\nK.A. Moler, D.J. Baar, J.S. Urbach, R. Liang, W.N. Hardy, and A.\nKapitulnik,\n Phys. Rev. Lett. {\\bf 73}, 2744 (1994).\n\n\\bibitem{Maeda:1995}\nA. Maeda, Y. Lino, T. Hanaguri, N. Motohira, K. Kishio, and T.\nFukase, Phys.\n Rev. Lett. {\\bf 74}, 1202 (1995).\n\n\\bibitem{Revaz:1998}\nB. Revaz, J.-Y. Jenoud, A. Junod, K. Neumaier, A. Erb, and E.\nWalker, Phys.\n Rev. Lett. {\\bf 80}, 3364 (1998).\n\n\\bibitem{Amin:1998}\nM.H.S. Amin, I. Affleck, and M. Franz, Phys. Rev. B {\\bf 58},\n5848 (1998).\n\n\\bibitem{Maggio:1995}\nI. Maggio-Aprile, {Ch}. Renner, A. Erb, E. Walker, and {\\O}.\nFischer, Phys.\n Rev. Lett. {\\bf 75}, 2754 (1995).\n\n\\bibitem{Renner:1998}\n{Ch}. Renner, B. Revaz, K. Kadowaki, I. Maggio-Aprile, and {\\O}.\nFischer, Phys.\n Rev. Lett. {\\bf 80}, 3606 (1998).\n\n\\bibitem{Pan:2000}\nS.H. Pan, E.W. Hudson, K.M. Lang, H. Eisaki, S. Uchida, and J.C.\nDavis, Nature\n {\\bf 403}, 746 (2000).\n\n\\bibitem{Blatter:1994}\nG. Blatter, M.V. Feigel'man, V.B. Geschkenbein, A.I. Larkin, and\nV.M. Vinokur,\n Rev. Mod. Phys. {\\bf 66}, 1125 (1994), and references therein.\n\n\\bibitem{Tinkham:1988}\nM. Tinkham, Helv. Phys. Acta {\\bf 61}, 443 (1988).\n\n\\bibitem{Tinkham:1996}\nM. Tinkham, {\\em Introduction to Superconductivity}, 2nd. ed.\n(McGraw-Hill, New\n York, 1996).\n\n\\bibitem{Li:1996}\nT.W. Li, A.A. Menovsky, J.J.M. Franse, and P.H. Kes, Physica C\n{\\bf 257}, 179\n (1996).\n\n\\bibitem{Erb:1999}\nA. Erb, A.A. Manuel, M. Dhalle, F. Marti, J.-Y. Genoud, B. Revaz,\nA. Junod, D.\n Vasumathi, S. Ishibashi, A. Shukla, E. Walker, \\O. Fischer, R. Fl\\\"ukiger, R.\n Pozzi, M. Mali, and D. Brinkmann, Solid State Comm. {\\bf 112}, 245 (1999).\n\n\\bibitem{Cubitt:1993}\nR. Cubitt, E.M. Morgan, G. Yang, S.L. Lee, D.Mck. Paul, H.A. Mook,\nM. Yethiraj,\n P.H. Kes, T.W. Li, A.A. Menovsky, Z. Tarnawski, and K. Mortensen, Nature {\\bf\n 365}, 407 (1993).\n\n\\bibitem{Caroli:1964}\nC. Caroli, P.G. de~Gennes, and J. Matricon, Phys. Lett. {\\bf 9},\n307 (1964).\n\n\\bibitem{Hess:1989}\nH.F. Hess, R.B. Robinson, R.C. Dynes, J.M. Valles, and J.V.\nWaszczak, Phys.\n Rev. Lett. {\\bf 62}, 214 (1989).\n\n\\bibitem{Renner:1991}\n{Ch}. Renner, A.D. Kent, {Ph.} Niedermann, \\O. Fischer, and F.\nL\\'evy, Phys.\n Rev. Lett. {\\bf 67}, 1650 (1991).\n\n\\bibitem{Hoogenboom:2000}\nB.W. Hoogenboom, M. Kugler, Ch. Renner, B. Revaz, I.\nMaggio-Aprile, and \\O.\n Fischer, Physica C (2000).\n\n\\bibitem{Pan:1999}\nS.H. Pan, E.W. Hudson, A.J. Gupta, K.-W. Ng, and J.C. Davis,\npreprint.\n\n\\bibitem{Renner:1990}\n{Ch}. Renner, Ph. Niedermann, A.D. Kent, and {\\O}. Fischer, J.\nVac. Sci.\n Technol. A {\\bf 8}, 330 (1990).\n\n\\bibitem{Kent:1992}\nA.D. Kent, {Ch}. Renner, Ph. Niedermann, J.-G. Bosch, and {\\O}.\nFischer,\n Ultramicroscopy {\\bf 42-44}, 1632 (1992).\n\n\\bibitem{Kugler:2000}\nM. Kugler, {Ch}. Renner, V. Mikheev, G. Batey, and {\\O}. Fischer,\nRev. Sci.\n Instrum. {\\bf 71}, 1475 (2000).\n\n\\bibitem{Renner:1998a}\n{Ch}. Renner, B. Revaz, J.-Y. Genoud, K. Kadowaki, and {\\O}.\nFischer, Phys.\n Rev. Lett. {\\bf 80}, 149 (1998).\n\n\\bibitem{Renner:1995}\n{Ch}. Renner and {\\O}. Fischer, Phys. Rev. B {\\bf 51}, 9208\n(1995).\n\n\\bibitem{VanDalen:1996}\nA.J.J. van Dalen, R. Griessen, and M.R. Koblischka, Physica C {\\bf\n257}, 271\n (1996).\n\n\\bibitem{Renner:1993}\nCh. Renner, Ph.D. thesis, University of Geneva, 1993.\n\n\\bibitem{Troyanovski:1999}\nA.M. Troyanovski, J. Aarts, and P.H. Kes, Nature {\\bf 399}, 665\n(1999).\n\n\\bibitem{Maggio:1997}\nI. Maggio-Aprile, {Ch}. Renner, A. Erb, E. Walker, and {\\O}.\nFischer, Nature\n {\\bf 390}, 487 (1997).\n\n\\bibitem{Yazdani:1999}\nA. Yazdani, C.M. Howald, C.P. Lutz, A. Kapitulnik, and D.M.\nEigler, Phys. Rev.\n Lett. {\\bf 83}, 176 (1999).\n\n\\bibitem{Hudson:1999}\nE.W. Hudson, S.H. Pan, A.K. Gupta, K.-W. Ng., and J.C. Davis,\nScience {\\bf\n 285}, 88 (1999).\n\n\\bibitem{Prost:1993}\nD. Prost, L. Fruchter, and I.A. Campbell, Phys. Rev. B {\\bf 47},\n3457 (1993).\n\n\\bibitem{Aupke:1993}\nK. Aupke, T. Teruzzi, P. Visani, A. Amann, A.C. Mota, and V.N.\nZavaritsky,\n Physica C {\\bf 209}, 255 (1993).\n\n\\bibitem{Monier:1998}\nD. Monier and L. Fruchter, Phys. Rev. B {\\bf 58}, 8917 (1998).\n\n\\bibitem{Feynman:1965}\nR.P. Feynman, R.B. Leighton, and M. Sands, {\\em Lectures on\nPhysics}\n (Addison-Wesley, Reading, Massachusetts, 1965), Vol.~III.\n\n\\bibitem{Bardeen:1965}\nJ. Bardeen and M.J. Stephen, Phys. Rev. {\\bf 140}, A1197 (1965).\n\n\\bibitem{Blatter:1993}\nG. Blatter and V.B. Geschkenbein, Phys. Rev. B {\\bf 47}, 2725\n(1993).\n\n\\bibitem{Harshman:1992}\nD.R. Harshman and Jr. A.P.~Mills, Phys. Rev. B {\\bf 45}, 10684\n(1992), and\n references therein.\n\n\\bibitem{max_lambda}\nRef.~\\onlinecite{Harshman:1992} gives $\\lambda=2500$~\\AA, but for\na sample that\n has not been O$_2$ annealed. In Ref.~\\onlinecite{Waldmann:1996} it is shown\n that this indeed results in a higher $\\lambda$. Since the samples discussed\n here are overdoped (O$_2$ annealed), one can safely adopt the values for\n optimum doped or overdoped samples, considerably lower than 2500~\\AA \\\n \\cite{Martinez:1992,Waldmann:1996}.\n\n\\bibitem{Martinez:1992}\nJ.C. Mart\\'inez, S.H. Brongersma, A. Koshelev, B. Ivlev, P.H. Kes,\nR.P.\n Griessen, D.G. de~Groot, Z. Tarnavski, and A.A. Menovsky, Phys. Rev. Lett.\n {\\bf 69}, 2276 (1992).\n\n\\bibitem{Waldmann:1996}\nO. Waldmann, F. Steinmeyer, P. M\\\"uller, J.J. Neumeier, F.X.\nR\\'egi, H. Savary,\n and J. Schneck, Phys. Rev. B {\\bf 53}, 11825 (1996).\n\n\\bibitem{Arndt:1999}\nM. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van~der Zouw,\nand A.\n Zeilinger, Nature {\\bf 401}, 680 (1999).\n\n\\end{thebibliography}"
}
] |
cond-mat0002147
|
The ground state of Sr$_3$Ru$_2$O$_7$ revisited; Fermi liquid close to a ferromagnetic instability
|
[
{
"author": "Shin-Ichi Ikeda$^{1,2}$"
},
{
"author": "Yoshiteru Maeno$^{2,3}$"
},
{
"author": "Satoru Nakatsuji$^{2}$"
},
{
"author": "Masashi Kosaka$^4$"
},
{
"author": "and Yoshiya Uwatoko$^4$"
}
] |
We show that single-crystalline Sr$_3$Ru$_2$O$_7$ grown by a floating-zone technique is an isotropic paramagnet and a quasi-two dimensional metal as spin-triplet superconducting Sr$_2$RuO$_4$ is. The ground state is a Fermi liquid with very low residual resistivity ($\approx $3 $\mu \Omega $cm for in-plane currents) and a nearly ferromagnetic metal with the largest Wilson ratio $R_{W} \ge 10$ among paramagnets so far. This contrasts with the ferromagnetic order at $T_{C}$=104 K reported on single crystals grown by a flux method [Cao {et al.}, Phys. Rev. B ${55}$, R672 (1997)]. However, we have found a dramatic changeover from paramagnetism to ferromagnetism under applied pressure. This suggests the existence of a substantial ferromagnetic instability in the Fermi liquid state.
|
[
{
"name": "Sr327ikedaResub.tex",
"string": "%&latex209\n%\\documentstyle[eqsecnum,aps,psfig]{revtex}\n\\documentstyle[camera,prbplug,psfig]{article}\n%\\usepackage{ozfig}\n%\\documentstyle[preprint,eqsecnum,aps]{revtex}\n\\def\\btt#1{{\\tt$\\backslash$#1}}\n\\def\\BibTeX{\\rm B{\\sc ib}\\TeX}\n\\begin{document}\n\\draft\n%%%\\preprint{HEP/123-qed}\n\\title{The ground state of Sr$_3$Ru$_2$O$_7$ revisited; Fermi liquid\n close to a ferromagnetic instability }\n\\author{Shin-Ichi Ikeda$^{1,2}$, Yoshiteru Maeno$^{2,3}$, Satoru \nNakatsuji$^{2}$, \nMasashi Kosaka$^4$, and Yoshiya Uwatoko$^4$}\n\\address{$^1$Electrotechnical Laboratory, Tsukuba, Ibaraki 305-8568\n, Japan}\n\\address{$^2$Department of Physics, Kyoto University, Kyoto 606-8502,\n Japan}\n\\address{$^3$CREST, Japan Science and Technology Corporation, Kawaguchi, \nSaitama 332-0012, Japan}\n\\address{$^4$Department of Physics, Saitama University, Saitama 338-8570, \nJapan}\n\n\\maketitle\n\\begin{abstract}\nWe show that single-crystalline Sr$_3$Ru$_2$O$_7$ grown by a floating-zone \ntechnique is an isotropic paramagnet and a quasi-two dimensional metal \nas spin-triplet superconducting Sr$_2$RuO$_4$ is. The ground state is \na Fermi liquid with very low residual resistivity ($\\approx $3 $\\mu \\Omega \n$cm for in-plane currents) and a nearly ferromagnetic metal with the largest \nWilson ratio $R_{\\rm W} \\ge 10$ among paramagnets so far. This \ncontrasts with the ferromagnetic order at $T_{\\rm C}$=104 K reported \non single crystals grown by a flux method [Cao {\\it et al.}, Phys. Rev. B \n$\\bf{55}$, R672 (1997)]. However, we have found a dramatic changeover \nfrom\n paramagnetism to ferromagnetism under applied pressure. This suggests \nthe existence of a substantial ferromagnetic instability in the Fermi liquid state.\n\n\\end{abstract}\n\\pacs{PACS numbers: 75.40.-s, 71.27.+a, 75.30.Kz} \n\n%\\narrowtext\nThe discovery of superconductivity in the single-layered perovskite \nSr$_2$RuO$_4$ \\cite{maeno1} has motivated the search for new \nsuperconductors and anomalous metallic materials in Ruddlesden-Popper \n(R-P) type ruthenates (Sr,Ca)$_{n+1}$Ru$_n$O$_{3n+1}$. \n The recent determination of the spin-triplet pairing in its superconducting \nstate suggests that ferromagnetic (FM) correlations are quite important in \nSr$_2$RuO$_4$ \\cite{ishida1}, and the existence of enhanced spin fluctuations \nhas been suggested by nuclear magnetic resonance (NMR) \n\\cite{mukuda1,imai1}. On the other hand, the recent report has shown that \nenhanced magnetic excitations around $\\bf{q}$=0 is not detected but \nsizeable excitations have been seen around finite $\\bf{q}$ in Sr$_2$RuO$_4$ \nby inelastic neutron scattering\\cite{sidis}. This has stimulated debate on \nthe mechanism of the spin-triplet superconductivity, which had been naively \nbelieved to have a close relation to FM ($\\bf{q}$=0) spin excitations. Hence,\n it is desirable to investigate its related compounds as described below.\n\nThe simple perovskite (three dimensional) metallic SrRuO$_3$ ($n= \\infty $) \nhas been well known to order ferromagnetically below 160 K with a magnetic \nmoment $M=0.8 \\sim 1.0 \\mu_{\\rm B}/$Ru \\cite{kanb1,kiyama1}. FM \nperovskite oxides are relatively rare except for metallic manganites. For pure \nthin film SrRuO$_3$, analyses of quantum oscillations in the resistivity \nhave given good evidence for the Fermi liquid behavior \\cite{andy}. \n\nThe double layered perovskite Sr$_3$Ru$_2$O$_7$ ($n$=2) is regarded as \nhaving an intermediate dimensionality between the systems with $n=1$ and \n$n=\\infty$ \\cite{will1}. Investigations on polycrystalline \nSr$_3$Ru$_2$O$_7$ showed a magnetic-susceptibility maximum \naround 15 K with Curie-Weiss-like behavior above 100 K and a metallic \ntemperature dependence of the electrical resistivity \n\\cite{cava1,ikeda1}.\n\nIn the study presented here, we have for the first time succeeded in growing \nsingle crystals of Sr$_3$Ru$_2$O$_7$ by a floating-zone (FZ) method. \nThose single crystals (FZ crystals) do not contain any impurity phases \n(e.g. SrRuO$_3$) which was observed in polycrystals \\cite{ikeda1}. \n We report herein that the FZ crystal of Sr$_3$Ru$_2$O$_7$ is a nearly\nFM paramagnet (enhanced paramagnet) and a quasi-two dimensional\n metal with a strongly-correlated Fermi liquid state. In addition, we have \nperformed magnetization measurements under hydrostatic pressures up to \n1.1 GPa in order to confirm whether the FM instability is susceptible to \npressure. The results suggest that there is a changeover from paramagnetism \nto ferromagnetism, indicating a strong FM instability. Essential features \nof magnetism for FZ crystals as well as polycrystals are inconsistent\nwith the appearance of a FM ordering ($T_{\\rm c} = 104 {\\rm K}$) at\n ambient pressure for \nsingle crystals grown by a flux method \\cite{cao1} using SrCl$_2$ flux and Pt \ncrucibles. We will argue that FZ \ncrystals reflect the intrinsic behavior of Sr$_3$Ru$_2$O$_7$. \n\nDetails of the FZ crystal growth are explained elsewhere \\cite{ikeda2}. \nThe crystal structure of the samples at room temperature was characterized \nby powder x-ray diffraction. Electrical resistivity $\\rho (T)$ was measured \nby a standard four terminal dc-technique from 4.2 K to 300 K and by an ac \nmethod from 0.3K to 5K. Specific heat $C_{P}(T)$ was measured by a relaxation\nmethod from 1.8 K to 35 K (Quantum Design, PPMS). The temperature \ndependence of magnetic \nsusceptibility $\\chi (T) \\equiv M/H$ from 2 K to 320 K was measured \nusing a commercial SQUID magnetometer (Quantum Design, \nMPMS-5S). For magnetization measurements of FZ crystals at ambient \npressure, we performed sample rotation around the horizontal axis, \nnormal to the scan direction, using the rotator in MPMS-5S. We could align\n the \ncrystal axes exactly parallel to a field direction within 0.2 degree using this \ntechnique. For high pressures, we measured magnetization using a long-type \nhydrostatic pressure micro-cell \\cite{uwa} with the SQUID \nmagnetometer. Loaded pressures around 3 K were determined from the shift \nof superconducting transition temperature of Sn in the micro-cell in a 5 mT field . \n\nThe R-P type structure of $n$=2 for FZ crystals of Sr$_3$Ru$_2$O$_7$ \nwas confirmed by the powder x-ray diffraction patterns with crushed crystals, \nwhich indicated no impurity peaks. Recently, the crystal structure of \npolycrystalline Sr$_3$Ru$_2$O$_7$ has been refined by neutron powder \ndiffraction \\cite{huang,shaked}. Although they have concluded that symmetry \nof the structure is orthorhombic owing to the rotation of the RuO$_6$ octahedron \nabout the c-axis by about 7 degrees, we deduced lattice parameters at room \ntemperature by assuming tetragonal $I4/mmm$ symmetry as \n$a= 3.8872(4) {\\rm \\AA}$, and $c=20.732(3) {\\rm \\AA}$. These values \nare in good agreement with those of polycrystals obtained by neutron diffraction \n\\cite{huang,shaked} and x-ray diffraction \\cite{ikeda1}. \n\nThe temperature dependence of magnetic susceptibility $\\chi (T)=M/H$ in\n a field of 0.3 T is shown in Fig. 1. No hysteresis is observed between zero-field \ncooling (ZFC) and field cooling (FC) sequences, so we conclude that there is no \nferromagnetic ordering. Little magnetic anisotropy is observed in contrast to large \nanisotropy ($\\approx 10^2$) of flux-grown crystals \\cite{cao1}. The nearly \nisotropic susceptibility of Sr$_3$Ru$_2$O$_7$ is qualitatively similar to that of \nthe enhanced Pauli-paramagnetic susceptibility in Sr$_2$RuO$_4$ \n\\cite{maeno2}.\n For an applied field of 0.3 T, there is no in-plane anisotropy of the susceptibility \nfor the whole temperature range (2 K $\\le T \\le $ 300K), within the precision\n of our equipment (1$\\%$). \n\n\\begin{figure} \n\\centerline{\\psfig{file=Sr327fig1.eps,width=\\columnwidth}}\n\\caption{Magnetic susceptibility of FZ crystals of \nSr$_3$Ru$_2$O$_7$ under 0.3 T field above 2 K. \nThe inset shows the low temperature magnetic susceptibility against \ntemperature $T$.} \n\\end{figure} \n\n The susceptibility for both $H//$ab and $H//$c exhibits Curie-Weiss behavior \nabove 200 K. We have fitted the observed $\\chi (T)$ from 200 K to 320 K \nwith $\\chi (T)=\\chi_{\\rm 0}+\\chi_{\\rm CW}(T)$, where $\\chi_{\\rm 0}$ \nis the temperature independent term and $\\chi_{\\rm CW}(T) = \nC/(T-{\\it \\Theta}_{\\rm W})$ is the Curie-Weiss term. The effective Bohr \nmagneton numbers $p_{\\rm eff}$ deduced from $C$ are $p_{\\rm eff}=\n$ 2.52 (2.99) and ${\\it \\Theta}_{\\rm W}$ =$\\relbar $ 39 K ($\\relbar \n$ 45 K) for $H//ab (H//c)$. The negative values of ${\\it \\Theta}_{\\rm W}$ \nnormally indicate antiferromagnetic (AFM) correlations in the case of \nlocalized-spin systems. However, we cannot conclude that AFM \ncorrelations play an important role solely by the negative ${\\it \\Theta}_\n{\\rm W}$ in an metallic system like Sr$_3$Ru$_2$O$_7$ \\cite{yoshimura}. \n\nAround $T_{\\rm max}$ =16 K, $\\chi (T)$ shows a maximum for both \n$H//$ab and $H//$c. The maximum has been also observed in the \npolycrystals. The results of temperature dependence of specific heat, NMR \n\\cite{mukuda} and elastic neutron scattering \\cite{huang,shaked} for \npolycrystals indicate that there is no evidence for any long range order \nwith definite moments. The FZ crystal shows nearly isotropic $\\chi \n(T)$ for all crystal axes below $T_{\\rm max}$. Hence, the maximum \ncannot be accredited to the long range AFM order. Therefore, we conclude \nSr$_3$Ru$_2$O$_7$ to be a $paramagnet$. Concerning $\\chi (T)$ under higher\n fields, \n$T_{\\rm max}$ is suppressed down to temperatures below 5 K above 6 T\n\\cite{perry}. \nSuch a maximum in $\\chi (T)$ and a field dependent $T_{\\rm max}$ are\n often observed in a nearly ferromagnetic (enhanced paramagnetic) metal like \nTiBe$_2$ \\cite{jarl} or Pd \\cite{muell}. In addition, a similar \nbehavior in $\\chi (T)$ has been observed in (Ca,Sr)$_2$RuO$_4$ \n\\cite{c3po1} and MnSi \\cite{pfle}, which are recognized as examples \nof a critical behavior by spin fluctuations. Similar critical behavior, \noriginating especially from FM spin fluctuations, is also expected in \nSr$_3$Ru$_2$O$_7$. Nevertheless, we cannot rule out the possibility \nof AFM correlations as observed in Sr$_2$RuO$_4$, caused by the \nnesting of its Fermi surfaces with the vector $\\bf{Q}$ $= \n(\\pm 0.6 \\pi/a, \\pm 0.6 \\pi/a,0)$ \\cite{sidis}.\n\nAs shown in Fig.2, the specific heat coefficient of the FZ crystal of \nSr$_3$Ru$_2$O$_7$ is $\\gamma $ = 110 mJ/(K$^2$ Ru mol) \nsomewhat larger compared to other R-P type \nruthenates \\{$\\gamma = $80 mJ/(K$^2$ Ru mol) for CaRuO$_3$, 30 \nmJ/(K$^2$ Ru mol) for SrRuO$_3$ \\cite{kiyama1} and 38 \nmJ/(K$^2$ Ru mol) for Sr$_2$RuO$_4$ \\cite{maeno2,andy1}\\}.\n This suggests that Sr$_3$Ru$_2$O$_7$ is a strongly-correlated \n metallic oxide. For polycrystalline Sr$_3$Ru$_2$O$_7$, we obtained \nthe value $\\gamma = 63 $ mJ/(K$^2$ Ru mol) using an adiabatic method \n\\cite{ikeda1}.\n\n\\begin{figure} \n\\centerline{\\psfig{file=Sr327fig2.eps,width=\\columnwidth}}\n\\caption{Specific heat devided by temperature $C_P/T$ of FZ crystals of \nSr$_3$Ru$_2$O$_7$ above 2 K. $C_P/T$ is plotted against $T^2$.} \n\\end{figure} \n\nThe temperature dependence of the electrical resistivity $\\rho (T)$ is shown \nin Fig. 3 above 0.3 K. Both $\\rho_{\\rm ab}(T)$ and $\\rho_{\\rm c}(T)$\n are metallic ($d\\rho/dT >0$) in the whole region. The ratio of \n$\\rho_c$/$\\rho_{ab}$ is about 300 at 0.3 K and 40 at 300 K. This anisotropic \nresistivity is consistent with the quasi-two-dimensional Fermi surface sheets \nobtained from the band-structure calculations \\cite{hase1}. With lowering \ntemperature below 100 K, a remarkable decrease of $\\rho_{\\rm c}(T)$ is \nobserved around 50 K. This is probably due to the suppression of the thermal \nscattering with decreasing temperature between quasi-particles and phonons as \nobserved in Sr$_2$RuO$_4$ \\cite{yoshida,maeno2,andy2}. Thus, below 50K, \ninterlayer hopping propagations \nof the quasi-particle overcome the thermal scattering with phonons. This hopping \npicture for $\\rho_{\\rm c}(T)$ is well consistent with the large value of \n$\\rho_{\\rm c}(T)$ and nearly cylindrical Fermi surfaces.\nOn the other hand, $\\rho_{\\rm ab}(T)$ shows a \nchange of the slope around 20 K. Such a change in $\\rho _{\\rm ab}(T)$ has \nalso been reported for Sr$_2$RuO$_4$ under hydrostatic pressure \n($\\approx$ 3 GPa). That might be possibly due to the enhancement of \nferromagnetic spin fluctuations \\cite{yoshida}.\n\n\\begin{figure} \n\\centerline{\\psfig{file=Sr327fig3.eps,width=\\columnwidth}}\n\\caption{Electrical resistivity of FZ crystals of Sr$_3$Ru$_2$O$_7$ \nabove 0.3 K. Both $\\rho _{\\rm ab}$ and $\\rho _{\\rm c}$ are shown.\nThe inset shows the low temperature electrical resistivity against the square\n of temperature $T^{2}$.} \n\\end{figure} \n\nAs shown in the inset of Fig. 3, the resistivity yields a quadratic\n temperature dependence below 6 K for both $\\rho_{\\rm ab}(T)$ and \n$\\rho_{\\rm c}(T)$ , characteristic of a Fermi liquid as observed in \nSr$_2$RuO$_4$ \\cite{maeno2}. We fitted $\\rho_{\\rm ab}(T)$ by the \nformula $\\rho_{\\rm ab}(T)$=$\\rho_{\\rm 0}$+$AT^2$ below 6 K and \nobtained $\\rho_{\\rm 0}$=2.8 $\\mu \\Omega$ cm and $A$=0.075 $\\mu \n\\Omega $cm/K$^2$. Since the susceptibility is quite isotropic and \ntemperature independent below 6 K, the ground state of \nSr$_3$Ru$_2$O$_7$ \nis ascribable as a Fermi liquid. We now can estimate Kadowaki-Woods ratio \n$A/\\gamma ^2$. Assuming that electronic specific heat $\\gamma $ \n= 110 mJ/(K$^2$ Ru mol) is mainly due to the ab-plane component, we \nobtain $A/\\gamma ^{2} \\approx A_{\\rm ab}/\\gamma ^{2} = 0.6 \n\\times 10^{\\relbar 5} \\mu \\Omega$ cm/(mJ/K$^{2}$ Ru mol)$^{2}$ \nclose to that observed in heavy fermion compounds. \n\n Regarding to $\\chi(T)$ again, it is important to note that even at \ntemperatures much lower than $T_{\\rm max}$, $\\chi (T)$ remains \nquite large. It appears that the ground state maintains a highly enhanced \nvalue of $1.5 \\times 10^{-2}$ emu/Ru mol, comparable to that \nobtained for typical heavy fermion compounds. Considering that the \nobserved $\\chi$ is dominated by the renormalized quasi-particles, we \ncan estimate the Wilson ratio $R_{\\rm W} = 7.3 \\times 10^{4} \\times \n \\chi ({\\rm emu/mol})\\slash \\gamma ({\\rm mJ/(K^{2} mol})$) \nin the ground state. If we regard the observed values at $T$ = 2 K \nas that at $T$ = 0 K, we have $R_{\\rm W} =10 (18)$ using $\\gamma$ \nfor single crystals (polycrystals). Despite the difference in the $\\gamma$ \nvalue between polycrystals and single crystals, $R_{\\rm W}$ is much \ngreater than unity. This large value implies that FM correlations are \nstrongly enhanced in this compound, especially when compared with \nthe values of 12 for TiBe$_2$ and 6 for Pd \\cite{julian}. Therefore, the \nground state of Sr$_3$Ru$_2$O$_7$ is characterized by strongly-correlated \nFermi liquid behavior with enhanced FM spin fluctuations, \ni.e. Sr$_3$Ru$_2$O$_7$ is a strongly-correlated nearly FM metal.\n\nConcerning this FM correlations, it should be noted that, using single \ncrystals grown by a chlorine flux method with Pt crucibles \\cite{muller}, Cao \n{\\it et al.} have investigated remarkable magnetic and transport \nproperties of R-P ruthenates \\cite{cao1} prior to our crystal growth. \nThe ground state of Sr$_3$Ru$_2$O$_7$ was concluded to be an itinerant \n{\\it ferromagnet} with $T_{\\rm c} = 104$ ${\\rm K}$ and an ordered \nmoment $M=1.2$ $\\mu_{\\rm B}/$Ru. The flux-grown crystals were \nreported to have a residual resistivity ($\\rho_0=3$ m$\\Omega $ cm ) \n$10^3$ times greater than that of FZ crystals ($\\rho_0=3 \\mu \n\\Omega $ cm ) for in-plane transport. In addition, FZ crystals\nreveal $T$-square dependent resistivities at low temperatures as \nalready shown, which was not observed in flux-grown crystals. In general,\n the FZ method with great care can be impurity-free crystal growth, while\n the flux method tends to contaminate crystals due to impurity elements \nfrom both the flux and the crucible. This might be a main reason why \nthe resistivity is much higher for flux-grown crystals. Thus, we suppose \nwith assurance that the data from FZ crystals reflect the intrinsic nature of \nSr$_3$Ru$_2$O$_7$ better than those from flux-grown crystals.\n\nIn order to acquire the information of the magnetic instability in the FZ crystal \nof Sr$_3$Ru$_2$O$_7$, we have measured magnetization under hydrostatic \npressure up to 1.1 GPa. The temperature dependence of magnetization \n$M(T)$ is shown for several pressures under a 0.1 T field along c-axis in Fig. 4.\n Around 1 GPa, substantial increase is recognized below around 70 K with a \nclear FM component indicated by the difference between ZFC and FC \nsequences. Although the remanent moment at 2 K ($M\\approx 0.08$ \n$\\mu_{\\rm B}/$Ru) is much smaller than that expected for $S$=1 \nof Ru$^{4+}$, its susceptibility is quite large (0.4 emu/Ru mol). We infer \nthat this transition is a FM ordering of itinerant Ru$^{4+}$ spins. In Fig. 4, we \nalso show the field dependence \nof magnetization $M(H$//c) at 2 K for $P=0.1$ MPa and $P=1$ GPa. Obvious \nferromagnetic component appears at lower fields for $P=1$ GPa. Even at higher \nfields, increase in magnetization by pressure is also present as at lower fields.\n This feature endorses the drastic changeover from paramagnetism to \nferromagnetism induced by applied pressure. \nTo the best of our knowledge, this is the first example of the pressure-\ninduced changeover from Fermi liquid to ferromagnetism.\n\n\\begin{figure} \n\\centerline{\\psfig{file=Sr327fig4.eps,width=\\columnwidth}}\n\\caption{Pressure dependence of magnetization $M(T)$ for $H//$c. \nObvious ferromagnetic ordering appears at about 70 K under 1 GPa \npressure. The inset shows the field dependence of magnetization $M(H)$ \nunder 0.1 MPa and 1 GPa pressures.} \n\\end{figure}\n\nFor the purpose of understanding the observed behavior, we should begin \nwith Stoner theory. In the metallic state with correlated electrons, the \nferromagnetic order is driven by the Stoner criterion \n$U_{\\rm eff}N(E_{\\rm F})\\geq 1$, where $U_{\\rm eff}$ is an \neffective Coulomb repulsion energy. The systematics of band-width \n$W$ and the density of states $N(E_{\\rm F}$) in the R-P ruthenates \nis summarized by Maeno $et$ $al$ \\cite{maeno3}. In this system, \nincreasing $n$ from 1 to $\\infty$ causes enhancement of \n$N(E_{\\rm F}$) as well as $W$. This is opposite to the single band \npicture, i.e. increasing $N(E_{\\rm F})$ naively means decreasing $W$. \nIn the case of R-P ruthenates, the anomalous variation might be due to \nthe modifications of the degeneracy of three $t_{2g}$ orbitals for Ru-$4d$ \nelectrons. According to the summary \\cite{maeno3}, ferromagnetic \nSrRuO$_3$ is characterized by the highest $N(E_{\\rm F}$) and $W$ \namong them, satisfying the Stoner criterion. This implies that the \nenlargement of $N(E_{\\rm F}$) and $W$ reflects stronger three \ndimensionality in the R-P ruthenates. Hence, applying \npressure probably makes Sr$_3$Ru$_2$O$_7$ closer \nto SrRuO$_3$, leading to FM order. For further investigations, it is \nrequired that structural study, resistivity and specific heat under \n pressures will be performed.\n\nIn conclusion, by using the floating-zone method we have succeeded for \nthe first time in growing single crystals of Sr$_3$Ru$_2$O$_7$ with very \nlow residual resistivity in comparison with that of flux-grown crystals\n reported previously. The results of magnetization, resistivity and \nspecific heat measurements suggest that Sr$_3$Ru$_2$O$_7$ is \na strongly-correlated Fermi liquid with a nearly ferromagnetic \nground state, consistent with the observation of ferromagnetic ordering \nbelow 70 K under applied pressure ($P \\sim $1 GPa). As far as we \nknow, this is the first example of the pressure-induced changeover \nfrom Fermi liquid to ferromagnetism. This ferromagnetic ordering may \nguarantee the existence of the ferromagnetic spin fluctuations in \nSr$_3$Ru$_2$O$_7$. \n\nAuthors are very grateful to A. P. Mackenzie for his fruitful advice and critical \nreading of this manuscript. They thank T. Ishiguro, T. Fujita and K. Matsushige \nfor their helpful supports. They thank S. R. Julian, G.G. Lonzarich, G. Mori, \nD.M. Forsythe, R.S. Perry, K. Yamada, Y. Takahashi, and M. Sigrist for their \nuseful discussions and technical supports. They also thank N. Shirakawa for \nhis careful reading of this manuscript.\n\n%\\begin{references} \n\\begin{thebibliography} {99}\n%\\begin{thebibliography}\n\n\\bibitem{maeno1} Y. Maeno $et$ $al$., \nNature(London) $\\bf{372}$, 532 (1994). \n\\bibitem{ishida1} K. Ishida $et$ $al$., \nNature(London) $\\bf{396}$, 658 (1998).\n\\bibitem{mukuda1} H. Mukuda $et$ $al$.,\nJ. Phys. Soc. Jpn. $\\bf{67}$, 3945 (1998).\n\\bibitem{imai1} T. Imai, A. W. Hunt, K. R. Thurber, and F. C. Chou, \nPhys. Rev. Lett. $\\bf{81}$, 3006 (1998) \n\\bibitem{sidis} Y. Sidis $et$ $al$.,\nPhys. Rev. Lett. $\\bf{83}$ 3320 (1999).\n\\bibitem{kanb1} A. Kanbayasi, \nJ. Phys. Soc. 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Julian $et$ $al$., Physica B $\\bf{259-261}$,\n928 (1999).\n\\bibitem{muller} Hk. Muller-Buschbaum and J. Wilkens,\nZ. Anorg. Allg. Chem. $\\bf{591}$, 161 (1990).\n\\bibitem{maeno3} Y. Maeno, S. Nakatsuji, and S. Ikeda, \nMaterials Science and Engineering B $\\bf{63}$ 70 (1999).\n\\end{thebibliography}\n\n\\end{document}"
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{
"name": "cond-mat0002147.extracted_bib",
"string": "\\begin{thebibliography} {99}\n%\\begin{thebibliography}\n\n\\bibitem{maeno1} Y. Maeno $et$ $al$., \nNature(London) $\\bf{372}$, 532 (1994). \n\\bibitem{ishida1} K. Ishida $et$ $al$., \nNature(London) $\\bf{396}$, 658 (1998).\n\\bibitem{mukuda1} H. Mukuda $et$ $al$.,\nJ. Phys. Soc. Jpn. $\\bf{67}$, 3945 (1998).\n\\bibitem{imai1} T. Imai, A. W. Hunt, K. R. Thurber, and F. C. Chou, \nPhys. Rev. Lett. $\\bf{81}$, 3006 (1998) \n\\bibitem{sidis} Y. Sidis $et$ $al$.,\nPhys. Rev. Lett. $\\bf{83}$ 3320 (1999).\n\\bibitem{kanb1} A. Kanbayasi, \nJ. Phys. Soc. Jpn. $\\bf{44}$, 108 (1978).\n\\bibitem{kiyama1} T. Kiyama $et$ $al$.,\nJ. Phys. Soc. Jpn. $\\bf{67}$, 307 (1998). \n\\bibitem{andy} A. P. Mackenzie $et$ $al$., \nPhys. Rev. B $\\bf{58}$, R13318 (1998).\n\\bibitem{will1} T. Williams, F. Lichtenberg, A. Reller, and G. Bednorz, \nMat. Res. Bull. $\\bf{26}$, 763 (1991).\n\\bibitem{cava1} R. J. Cava $et$ $al$., \nJ. Solid State Chem. $\\bf{116}$, 141 (1995).\n\\bibitem{ikeda1} S. Ikeda, Y. Maeno and T. Fujita, Phys. Rev. B \n$\\bf{57}$, 978 (1998).\n\\bibitem{cao1} G. Cao, S. McCall, and J. E. Crow, \nPhys. Rev. B $\\bf{55}$, R672 (1997).\n\\bibitem{ikeda2} S.I. Ikeda, unpublished.\n\\bibitem{uwa} Y. Uwatoko $et$ $al$.,\nRev. High Pressure Sci. Technol. $\\bf{7}$, 1508 (1998).\n\\bibitem{huang} Q. Huang $et$ $al$., Phys. Rev. B $\\bf{58}$, 8515 (1998). \n\\bibitem{shaked} H. Shaked $et$ $al$., (unpublished).\n\\bibitem{maeno2} Y. Maeno $et$ $al$., J. Phys. Soc. Jpn. $\\bf{66}$,\n 1405 (1997).\n\\bibitem{yoshimura} K. Yoshimura $et$ $al$., Phys. Rev. Lett. \n$\\bf{83}$, 4397 (1999).\n\\bibitem{mukuda} H, Mukuda $et$ $al.$, (private communication).\n\\bibitem{perry} A.P. Mackenzie $et$ $al$., (unpublished). \n\\bibitem{jarl} T. Jarlborg and A. J. Freeman, Phys. Rev. B \n$\\bf{22}$, 2332 (1980).\n\\bibitem{muell} F. M. Mueller, A. J. Freeman, J. O. Dimmock, \nand A. M. Furdyna, Phys. Rev. B $\\bf{1}$, 4617 (1970). \n\\bibitem{c3po1} S. Nakatsuji and Y. Maeno, Phys. Rev. Lett. \n$\\bf{84}$, 2666 (2000).\n\\bibitem{pfle} C. Pfleiderer $et$ $al$., Phys. Rev. B $\\bf{55}$ 8330 (1997).\n\\bibitem{andy1} A.P. Mackenzie $et$ $al$., J. Phys. Soc. Jpn. $\\bf{67}$, \n385 (1998).\n\\bibitem{hase1} I. Hase and Y. Nishihara, J. Phys. Soc. Jpn. \n$\\bf{66}$, 3517 (1997).\n\\bibitem{yoshida} K. Yoshida $et$ $al$.,\nPhys. Rev. B $\\bf{58}$,15062 (1998).\n\\bibitem{andy2} A.P. Mackenzie $et$ $al$., Phys. Rev. Lett. $\\bf{76}$, \n3786 (1996).\n\\bibitem{julian} S.R. Julian $et$ $al$., Physica B $\\bf{259-261}$,\n928 (1999).\n\\bibitem{muller} Hk. Muller-Buschbaum and J. Wilkens,\nZ. Anorg. Allg. Chem. $\\bf{591}$, 161 (1990).\n\\bibitem{maeno3} Y. Maeno, S. Nakatsuji, and S. Ikeda, \nMaterials Science and Engineering B $\\bf{63}$ 70 (1999).\n\\end{thebibliography}"
}
] |
cond-mat0002148
|
Shape anisotropy and Voids
|
[
{
"author": "Gauri R. Pradhan"
}
] |
Numerical simulations on a 2-dimensional model system showed that voids are induced primarily due to shape anisotropy in binary mixtures of interacting disks. The results of such a simple model account for the key features seen in a variety of flux experiments using liposomes and biological membranes~\cite{Sita}.
|
[
{
"name": "paper.tex",
"string": "\\documentclass[prl,preprint,showpacs,byrevtex,superscriptaddress]{revtex4}\n\\begin{document}\n\\title {Shape anisotropy and Voids }\n\\author{Gauri R. Pradhan}\n\\email{gau@physics.unipune.ernet.in}\n\\affiliation{Department of Physics, University of Pune, Pune 411 007,\n India}\n\\author{Sagar A. Pandit}\n\\email{sagar@prl.ernet.in}\n\\affiliation{ Physical Research Laboratory, Ahmedabad 380 009, India}\n\\author{Anil D. Gangal}\n\\affiliation{Department of Physics, University of Pune, Pune 411 007,\n India}\n\\author{V. Sitaramam}\n\\email{sitaram@unipune.ernet.in}\n\\affiliation{Department of Biotechnology, University of Pune, Pune 411 \n 007, India} \n\\begin{abstract}\n\nNumerical simulations on a 2-dimensional model system showed that\nvoids are induced primarily due to shape anisotropy in binary mixtures\nof interacting disks. The results of such a simple model \naccount for the key features seen in a variety of flux\nexperiments using liposomes and biological membranes~\\cite{Sita}. \n\\end{abstract}\n\\pacs{87.16.Ac, 87.15.Aa, 05.65.+b, 87.16.Dg, 87.68.+z}\n\\maketitle\n\nA variety of lipid molecules contribute to the structure and barrier \nfunction of biological membranes. Part of each\nmolecule is hydrophilic (head) and part hydrophobic (tail). This\namphipathic nature, in the presence of water, leads to a bilayer\nstructure in which hydrophilic head groups \nhave maximum contact with water and hydrophobic tails have minimum\ncontact with water. The process of membrane formation is one\nof minimizing the free energy and maximizing the stability of the\nstructure~\\cite{Stein}. \n\n A major question has been whether the bilayer is to be viewed\nas an isotropic homogeneous phase or as a heterogeneous\nphase~\\cite{Lee}. If osmotic contraction\nof the bilayer vesicles leads to an altered hydraulic conductivity\n(water flux coefficient), one obviously favors a heterogeneous\nmembrane model. Otherwise a homogeneous, isotropic phase model would\nbe adequate, obviating the need to look for a fine structure within\nthe bilayer. Using the erythrocyte as an experimental system ( in\nwhich the area of the biconcave cell does not change when it is\nosmotically expanded to a spherical shape), it was concluded that\nhydraulic conductivity was stretch independent, i.e., in support of\nthe isotropic model~\\cite{Sita}. An alternative way to assess hydraulic\nconductivity is to use hydrogen peroxide as an analog of water: since\nmany experimental systems have catalase (an enzyme that degrades\nhydrogen peroxide to molecular oxygen and water) within the\nvesicle/cell, an assay of this occluded catalase directly permits one\nto measure the conductivity to exogenous hydrogen \nperoxide. Under equilibrium conditions of assay,\nthe rate of degradation would be same as the rate of permeation of the\nperoxide into the vesicle. Thus, one can directly assess the stretch\nsensitivity of the membrane by osmotic titrations with osmolites, using\nnon-electrolytes like hydrogen peroxide as probes of flux. In the course of\nthese experiments it was found that~\\cite{Sita}: (i) among all the\nlipid combinations tested, the phosphatidylcholine (PC) vesicles and\nintact erythrocytes, both, did not show a decrease in occluded catalase\nactivity on osmotic compression of the membrane (ii) on the other\nhand, all other membrane systems, such as \nperoxisomes, E.coli, macrophages showed stretch (osmotic) sensitivity\n(iii) so did liposomes made from these cells and organelles (iv)\nfurther, when binary mixtures were investigated, only cardiolipin and\ncerebrosides when added to PC (5 to 10$\\%$ of PC) conferred stretch\nsensitivity in liposomes (v) these binary mixtures also exhibited enhanced\nactivation volume(osmotic sensitivity) and diminished activation energy\nfor hydrogen peroxide flux (vi) further, glucose was readily\npermeable across these membranes of binary mixtures (vii) addition of\ncholesterol, which is abundantly found in erythrocytes, inhibited the stretch\nsensitivity to peroxide permeation (viii) evidence was also seen that\nthis diffusion increases with decrease in temperature, i.e., the\nprocess has a negative temperature coefficient.\n\nThese studies on biological membranes and liposomes, in which\ncomposition as well as dynamics considerably vary, prompted us to\nquestion as to what constitutes a minimal description to account for\nvariable permeability induced by doping across a liposomal membrane.\nFor instance, cardiolipin enhanced permeation of hydrogen\nperoxide and molecules as large as {\\it glucose}~\\cite{Sita}. Though possible descriptors \ncould be many, (composition, structure, dynamics in terms of inter\nand intra molecular potentials), we adopted a bare-bone \napproach to resolve this complex issue to arrive at a minimal\ndescription adequate to account for the observations. \n\n Structural changes in the membrane are best identified by\nnon-interactive molecules and therefore leaks across\nbilayers are commonly studied using non-electrolytes~\\cite{Stein}. The diagnostic\nfor non-specific permeation is size dependence such that, these\nhydrated solutes intercalate, penetrate and navigate through such\ninterstices, spaces or voids, stochastically or in files to reach the\nother side of the membrane~\\cite{Lee}. In order to capture the diverse features\nin a parsimonious manner, we restrict to a two dimensional\ncross-section of a three dimensional system. Such a restriction is reasonable since \nthe probe particle permeating across the membrane at any instant of time\nexperiences the effective \ncross section rather than the three dimensional obstruction. The \npermeation across the membrane depends primarily on the availability\nof free space or voids. Thus the problem reduces to the study of \npacking of 2-dimensional objects at the first instance. Then one needs\nto determine which factor(s) determine the appearance and \nsize-distribution of voids in such a 2-dimensional system. \n\nThe configuration space of this model system (membrane) is a\n2-dimensional box with \ntoroidal boundary conditions. The constituents of this two dimensional box\nare the circular disks (and/or the rigid combinations of the circular\ndisks as dopants) of unit radii. A circular disk simply represents\nthe hard core scattering cross section, seen by the passing\nparticle (a non-electrolyte which acts as a probe), across the\nthickness spanned by two lipid molecules, viewed somewhat as\ncylinders. A typical dopant is two or more circular disks rigidly joined\nin a\nprespecified geometry. These constituents are \nidentified by the position coordinates of their centers, the angle\nmade by the major axes with the side of the box ( in case of dopants)\nand the radii of the circular disks. \nIt is reasonable that the disks (which represent molecules,\nwith long range attractive interaction and hard core repulsion near\ncenter, contained within a structure) \ninteract pair wise via {\\it Lennard-Jones} potential (a measure of the\ninteraction energy), which has the form \n\\begin{eqnarray}\n%\\label{LJ}\nV_{LJ}(r_{ij}) = 4 \\epsilon \\sum_{i=1}^N \\sum_{j = i+1}^N \\Big( ({\\sigma\n\\over r_{ij}} )^{12} - ( { \\sigma \\over r_{ij}})^6 \\Big) \\nonumber\n\\end{eqnarray}\nwhere, $r_{ij}$ is the distance between the centers of the $i^{\\rm\nth}$ and $j^{\\rm th}$ disks, $\\sigma$ determines the range of hard\ncore part in the potential and the $\\epsilon$ signifies the depth of\nattractive part. \nWhile studying the binary mixtures, we consider different {\\it shape\n anisotropic} combinations ({\\it impurities} or {\\it dopants}) of\n$\\kappa$ number of circular disks. We treat these combinations as one\nunit. \ne.g. {\\it rod$_n$} denotes a single dopant made of $n$ unit circular disks rigidly\njoined one after another in a straight line. \nThe impurities interact with constituent circular disks via potential,\n\\begin{eqnarray}\nV(r_{ij}) = \\sum_{\\alpha = 1}^\\kappa V_{LJ} (r_{i_\\alpha j}) \\nonumber\n\\end{eqnarray}\nwhere, $r_{i_\\alpha j}$ is the distance between the centers of\n$\\alpha^{\\rm th}$ disk in $i^{\\rm th}$ impurity and the $j^{\\rm th}$\ncircular disk, and among themselves interact via\n\\begin{eqnarray}\nV(r_{ij}) = \\sum_{\\alpha = 1}^{\\kappa_1} \\sum_{\\beta = 1}^{\\kappa_2}\nV_{LJ}(r_{i_\\alpha j_\\beta})\\nonumber\n\\end{eqnarray}\nwhere, $r_{i_\\alpha j_\\beta}$ is the distance between the centers of\n$\\alpha^{\\rm th}$ disk in $i^{\\rm th}$ impurity and the $\\beta^{\\rm\nth}$ disk in $j^{\\rm th}$ impurity.\n\nAn $r$-void is defined as a closed area in a membrane\ndevoid of disks or impurities, and sufficient to accommodate a\ncircular disk of radius $r$~\\cite{Gauri}. Clearly, larger voids also\naccommodate smaller probes, i.e., an $r$-void is \nalso an $r^\\prime$-void if $r^\\prime < r$. Similarly, the\nvoids for the particle of size zero are the voids defined in the\nconventional sense, i.e., a measure of the net space unoccupied by the\ndisks.\n\nEquilibrium configurations of the model system are obtained by a Monte\nCarlo method (using the Metropolis algorithm) starting with a random\nplacement of the disks~\\cite{Gauri,Binder}\\footnote{The equilibrium\nconfigurations thus obtained are further confirmed by simulated\n annealing~\\cite{NR}.}. The box was filled with disks such that they\noccupy 70\\% area of the box,i.e.,loosely packed to facilitate the\nformation of voids. The temperature parameter, $T$, was so chosen\nthat the quantity $k_B T < 4 \\epsilon$. This ensured an approximate\nhexagonal arrangement of the disks and the presence of very few large voids in the\nabsence of dopants. (Fig. 1a; far too less r-voids for $r \\geq 0.5$\\footnote {It\nmay be recalled that glucose offers approximately half the radius of\nthe PC cross section, yielding a relevant definition for a larger void of\ninterest.}). Similar numerical simulations are performed \non the model system with dopants. The number of dopants is chosen to\nbe 10\\%~\\cite{Sita} of the number of circular disks with a\nconstraint that the total occupied area of the box is still 70\\% as\nthe focus was on the redistribution of void sizes. Fig. 1b illustrates\nthe formation of larger voids in the vicinity of the rod$_2$ impurities. \n\nThe variation in the number of $r$-voids as a function of the\nsize of the permeating particle(using the digitization algorithm\ndescribed in~\\cite{Gauri}) is shown in Fig. 2. When only circular \ndisks are present (dotted curve), hardy any large $r$-voids are seen. When\nmixed with the anisotropic impurities, say, rod$_2$, a distinct\nincrease in the number of large $r$-voids is seen with appropriate\nredistribution of smaller $r$-voids (solid curve). This result\nis consistent with the unexpected permeation of large\nmolecules such as glucose through the doped membrane, observed\nexperimentally~\\cite{Sita}.The difference curve showed the formation\nof a significant extent ($\\approx$ 30\\%) of $r$-voids of size $0.5$\nand above. \n\nIs the induction of large voids due to the anisotropy\nin potential of the impurities and, should the large voids form around\nthe rods, the centers of anisotropy? Firstly, we carried\nout simulations with large circular disks in place of rod$_2$ as\nimpurities. The radius of large disk was chosen in such a way that the\narea occupied by each of the large disk is same as that of a\nrod$_2$. Fig. 3 shows the result of such simulations. The curve (a)\nin Fig. 3 represents the difference curve of $r$-void distribution of\npure membrane and that of membrane doped with rod$_2$ impurity.\n The curve (b) represents the same when the membrane is doped with\nlarge circular disks. It can be clearly seen \nthat the number of larger $r$-voids is always less in the latter case,\nthus confirming the role of shape anisotropy in the induction\nof large $r$-voids.\nFurther, simulations are carried out with rod$_2$ of smaller size and\nrod$_4$ type impurities. The curves (c), (d) in Fig. 3 respectively\nshows the corresponding difference curves. It shows an interesting\nfeature that the peak of the difference curve shifts with the change\nin the type of anisotropy. This suggests a possible way of\nconstructing membrane with selective permeability properties. \nThe simulation regime adopted here, limited the exploration of ternary\nmixtures in yielding statistically significant results on\ntransport. However, by using rod$_2$ of 0.5 size (Fig.3, curve (c)) (an oval\napproximation of the small dopant cholesterol), we could\ndemonstrate a shift in void sizes to left in binary mixtures,\nconsistent with our experimental results in ternary mixtures with\ncholesterol~\\cite{Sita}. \n\nFurther, we considered rod$_n$ type\nimpurities. Fig. 4 represents the relation between the length\ndimension of rod$_n$ and the number of $r$-voids (for r=0.55). \nThe anisotropy in the potential of rod$_n$ increases with $n$, such\nthat the number of large $r$-voids should increase\nwith $n$. Fig. 4 indeed shows a jump when the rod$_2$ impurities are\nadded and afterwards, it shows a slow and almost linear increase with\nincrease in $n$.\n\nSince dopants induced voids, their influence is most likely to be seen \nin their own vicinity, enhancing the ``local transport''. As the\ndopants exhibit different potential in different directions, certain\npositions of the constituents are preferred from the point of view\nof energy minimization, eventually giving rise to voids in the vicinity of impurities. \nTo verify this, we calculate the {\\it local permeation probability}\nfor particle of size $r$, which is a ratio of the area of $r$-voids\nand the area of the local neighborhood. Fig. 5 shows the local\npermeation probability around ten randomly chosen impurities and\nten randomly chosen circular disks. The higher local permeation\nprobability is seen to be associated with rod$_2$.\n\nThe model is realistic in that, one can compute elastic properties (surface tension\nlike attribute) by stretching the membrane from one side and computing\nthe energy change, which yielded a change of $\\approx$ 13 dyn/cm which is of\nthe same order as the observed surface tension and changes there of in\nbilayers~\\cite{Jan}. The model is limited in relation to an investigation of\ntemperature effects which requires incorporation of multiple time\nscales. The model is really general because it not restricted by the\nparameter space of the components and therefore it is extendible to a\nvariety of phenomena including transport in weakly\nbound granular media, voids in polymers, modeling of zeolites which\nmay act like a sponge absorbing only desired species.\n\nIn summary, we proposed a two dimensional computational model system\ncomprising a mixture of objects interacting via Lennard-Jones\npotential to explain anomalous permeation seen in bilayers. The\nsignificance of these observations on permeation, in what is\nessentially a granular medium (with long range attraction as an added\nfeature), relates to development of large voids seeded by\nimpurities. Unlike shape anisotropy, the change in composition (via a change \nin $\\sigma$), lateral pressure/density, and presumably temperature (though\ndynamic simulations would need to be done) would all simply produce\nvoids within the bounds of a hexagonal array. Even among various\ndopants in shapes, X, L, Y, Z, T symmetrical or otherwise, all other factors being\nequal, size per size, rod$_n$ produce the largest voids~\\cite{Gauri}. \n\nThus the largest $r$-voids induced by anisotropy yield biological\nobservations of interest, without manifesting as ordered\ngeometric structures. In this sense these $r$-voids differ from the conventional\nresults obtained in studies on granular media (which also\ndo not usually incorporate long range interactions)~\\cite{Granular}. \nIt is increasingly becoming clear that voids play a pivotal role in\nrelating the dynamics of biopolymers to specific functional states~\\cite{Raj}.\n\nWe thank C. N. Madhavarao for discussions. The author (GRP) is\ngrateful to CSIR (India) for fellowship and (ADG) is grateful to DBT\n(India) for research grant. \n\\vskip 6in\n\\begin{thebibliography}{99}\n\\bibitem{Sita} J. C. Mathai, V. Sitaramam, J. Bio. Chem., {\\bf 269},\n 17784, (1994)\n\\bibitem{Stein} W. D. Stein, Transport and diffusion across cell\nmembranes, Academic Press, London, (1986). \n\\bibitem{Lee} A. G. Lee, Prog. Biophys. Molec.Biol., {\\bf 29}, 3-56, (1975).\n\\bibitem{Gauri} G. R. Pradhan, S. A. Pandit, A. D. Gangal,\n V. Sitaramam, Physica A, {\\bf 270}, 288, (1999)\n\\bibitem{Binder} K. Binder (Ed.), Monte Carlo methods for Statistical\n Physics, Springer, Berlin, 1979\n\\bibitem{NR} W. H. Press, S. A. Teukolsky, W. T. Vellerling,\n B. P. Flannery, Numerical Recipes in C (Second ed.), Cambridge\n University Press (1992).\n\\bibitem{Jan} A. Kotyk, K. Jan\\'{a}\\H{c}ek, J. Koryta, Biophysical\nChemistry of Membrane Functions, J. Wiley, New York, (1998).\n\\bibitem{Granular} P. Umbanhower, Nature, {\\bf 389}, 541, (1997).\n\\bibitem{Raj} K. Rajagopal, V. Sitaramam, J. Theor. Biol., {\\bf 195},\n245, (1998).\n\\end{thebibliography}\n\n\\newpage\n\\centerline{\\bf FIGURE CAPTIONS}\n\\begin{description}\n\\item[{\\bf Fig. 1}] Typical equilibrium configurations for interacting disks.\nThe parameter, $\\sigma$ in Lennard-Jones potential is 2 units. The size of the\nbox is 50$\\times$50 (a) A pure membrane with 556 unit circular disks. \n(b) A doped membrane with 464 circular disks and 46 rod$_2$(shown in\ngray).\n\n\\item[{\\bf Fig. 2}] Distribution of $r$-voids in two different\nconfigurations. The main graph shows the number of $r$-voids as a\nfunction of the relative size ($r$) of the probe particle. The vertical\nbars represent the error margins at the corresponding points. \nthe dotted curve gives the distribution in pure membrane while the\nsolid curve shows the same in a membrane doped with rod$_2$\n(10:1). Difference curve clearly demonstrates the presence of large voids in\nthe doped membrane. \n\n\\item[{\\bf Fig. 3}] Difference curves of distribution of $r$-voids. Curves are\nobtained by treating the void distribution for pure membrane as the base. \nDifference curves for membranes(a) doped with rod$_2$\n(b) doped with large circular disks (of radii $\\sqrt{2}$) which occupy \nthe same area as that of rod$_2$. These induce smaller number of\nlarge voids as compared to rod$_2$.\n(c) doped with small rods. Irrespective of their small size, they\ninduce large voids, but the peak shifts towards left.\n(d) doped with rod$_4$. Significantly large voids are induced. The\npeak shifts towards the right.\n\n\\item[{\\bf Fig. 4}] Dependence of the number of $r$-voids\non the length of the rod-shaped impurities. The graph shows a steady\nincrease in the number of $r$-voids (for $r=0.55$) with $n$. The\nfirst expected large jump in the number of voids because of the shape\nanisotropy is seen clearly when the configuration consists of\nmolecules in the shape of circular disks and rod$_2$.\n\n\\item[{\\bf Fig. 5}] Local permeation probability in a doped model\n system. The points show the {\\it local permeation probability}\naround ten randomly chosen unit disks and ten randomly chosen\nrod$_2$s. Further, as a guide line, averages are shown by the heights\nof the boxes, clearly indicating significantly more permeation in the\nneighborhood of rod$_2$.\n\\end{description}\n\\end{document}\n\n%%% Local Variables: \n%%% mode: latex\n%%% TeX-master: t\n%%% End: \n\n"
}
] |
[
{
"name": "cond-mat0002148.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{Sita} J. C. Mathai, V. Sitaramam, J. Bio. Chem., {\\bf 269},\n 17784, (1994)\n\\bibitem{Stein} W. D. Stein, Transport and diffusion across cell\nmembranes, Academic Press, London, (1986). \n\\bibitem{Lee} A. G. Lee, Prog. Biophys. Molec.Biol., {\\bf 29}, 3-56, (1975).\n\\bibitem{Gauri} G. R. Pradhan, S. A. Pandit, A. D. Gangal,\n V. Sitaramam, Physica A, {\\bf 270}, 288, (1999)\n\\bibitem{Binder} K. Binder (Ed.), Monte Carlo methods for Statistical\n Physics, Springer, Berlin, 1979\n\\bibitem{NR} W. H. Press, S. A. Teukolsky, W. T. Vellerling,\n B. P. Flannery, Numerical Recipes in C (Second ed.), Cambridge\n University Press (1992).\n\\bibitem{Jan} A. Kotyk, K. Jan\\'{a}\\H{c}ek, J. Koryta, Biophysical\nChemistry of Membrane Functions, J. Wiley, New York, (1998).\n\\bibitem{Granular} P. Umbanhower, Nature, {\\bf 389}, 541, (1997).\n\\bibitem{Raj} K. Rajagopal, V. Sitaramam, J. Theor. Biol., {\\bf 195},\n245, (1998).\n\\end{thebibliography}"
}
] |
cond-mat0002149
|
Critical behavior of n-vector model with quenched randomness
|
[
{
"author": "J.~Kaupu\\v{z}s"
}
] |
We consider the Ginzburg--Landau phase transition model with $O(n)$ symmetry (i.e., the $n$--vector model) which includes a quenched randomness, i.e., a random temperature disorder.~We have proven rigorously that within the diagrammatic perturbation theory the quenched randomness does not change the critical exponents at $n \to 0$, which is in contrast to the conventional point of view based on the perturbative renormalization group theory.
|
[
{
"name": "cond-mat0002149.tex",
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}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{document}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Journal identifier can be put here if required, e.g.\n%\\jl{14}\n\n\\title[]{Critical behavior of n-vector model with quenched\nrandomness}\n\n\\author{J.~Kaupu\\v{z}s}\n\n\\address{\\ Institute of Mathematics and Computer Science,\nUniversity of Latvia \\\\ Rainja Boulevard 29, LV-1459 Riga, Latvia\n\\\\ e--mail: \\; kaupuzs@latnet.lv}\n\n\\begin{abstract}\nWe consider the Ginzburg--Landau phase transition model with $O(n)$\nsymmetry (i.e., the $n$--vector model) which includes a quenched randomness,\ni.e., a random temperature disorder.~We have proven rigorously that within\nthe diagrammatic perturbation theory the quenched randomness does not change\nthe critical exponents at $n \\to 0$, which is in contrast to the\nconventional point of view based on the perturbative renormalization group\ntheory.\n\\end{abstract}\n\n\\pacs{64.70.-p \\\\\nKeywords: perturbation theory, critical phenomena, quenched randomness}\n\n% Uncomment for Submitted to journal title message\n%\\submitted\n\n% Comment out if separate title page not required\n%\\maketitle\n\n\\section{Introduction}\nThe phase transitions and critical phenomena is one of the most widely\ninvestigated topic in modern physics. Nevertheless, an eliminated number\nof exact and rigorous results are available, and they refer mainly to\nthe two--dimensional systems~\\cite{Baxter} and fractals~\\cite{ReMa}.\nRigorous results have been obtained also in four\ndimensions~\\cite{HaTa} based on an exact renormalization group (RG)\ntechnique~\\cite{GaKu}. The RG method, obviously, provides exact results\nat $d>4$ (where $d$ is the spatial dimensionality), but\nthis case is somewhat trivial in view of critical phenomena. In three\ndimensions, approximate methods are usually used based on perturbation\ntheory.\n\n Here we present a particular result obtained within the diagrammatic\nperturbation theory. The Ginzburg--Landau phase transition model with\n$O(n)$ symmetry (i.e., the $n$--vector model) is considered, which\nincludes a quenched random temperature disorder. The usual prediction\nof the perturbative RG field theory~\\cite{Wilson,Ma,Justin} is that, in\nthe case of the spatial dimensionality $d<4$ and small enough $n$\n(at $n=1$ and $n \\to 0$, in particular), the critical behavior\nof the $n$--component vector model is essentially changed by the quenched\nrandomness. Here we challenge this conventional point of view based on\na mathematical proof. We have proven rigorously that within the\ndiagrammatic perturbation theory the critical exponents in the actually\nconsidered model cannot be changed by the quenched randomness at\n$n \\to 0$.\n\n\\section{The model}\n We consider a model with the Ginzburg--Landau Hamiltonian\n\\begin{eqnarray} \\label{eq:Hr}\nH/T = \\int \\left[ \\left(r_0+ \\sqrt{u} \\, f({\\bf x}) \\right)\n\\varphi^2({\\bf x}) + c \\, (\\nabla \\varphi({\\bf x}) )^2 \\right] d{\\bf x}\n\\nonumber \\\\\n+ \\, uV^{-1} \\sum\\limits_{i,j,{\\bf k}_1,{\\bf k}_2,{\\bf k}_3 }\n\\varphi_i({\\bf k}_1) \\varphi_i({\\bf k}_2) \\, u_{{\\bf k}_1+{\\bf k}_2} \\,\n\\varphi_j({\\bf k}_3) \\varphi_j(-{\\bf k}_1-{\\bf k}_2-{\\bf k}_3)\n\\end{eqnarray}\nwhich includes a random temperature (or random mass) disorder\nrepresented by the term $\\sqrt{u}\\,f({\\bf x})\\,\\varphi^2({\\bf x})$.\nFor convenience, we call this model the random model. In Eq.~(\\ref{eq:Hr})\n$\\varphi({\\bf x})$ is an $n$--component vector (the order parameter\nfield) with components\n$\\varphi_i({\\bf x})=V^{-1/2} \\sum_{\\bf k} \\varphi_i({\\bf k}) e^{i{\\bf kx}}$,\ndepending on the coordinate ${\\bf x}$, and\n$f({\\bf x})=V^{-1/2} \\sum_{\\bf k} f_{\\bf k} e^{i{\\bf kx}}$ is a random\nvariable with the Fourier components\n$f_{\\bf k}=V^{-1/2} \\int f({\\bf x}) e^{-i{\\bf kx}} d{\\bf x}$.\nHere $V$ is the volume of the system, $T$ is the temperature, and\n$\\varphi_i({\\bf k})$ is the Fourier transform of $\\varphi_i({\\bf x})$.\nAn upper limit of the magnitude of wave vector $k_0$ is fixed. It means\nthat the only allowed configurations of the order parameter field are\nthose corresponding to $\\varphi_i({\\bf k})=0$ at $k>k_0$. This is the\nlimiting case $m \\to \\infty$ ($m$ is integer) of the model where all\nconfigurations of $\\varphi({\\bf x})$ are allowed, but Hamiltonian\n(\\ref{eq:Hr}) is completed by term\n$\\sum\\limits_{i,{\\bf k}} (k/k_0)^{2m} {\\mid \\varphi_i({\\bf k}) \\mid}^2$.\n\n The perturbation expansions of various physical quantities in powers\nof the coupling constant $u$ are of interest. In this case $n$ may be\nconsidered as a continuous parameter. In particular, the case\n$n \\to 0$ has a physical meaning\ndescribing the statistics of polymers~\\cite{Justin}.\n\n The system is characterized by the two--point correlation function\n$G_i({\\bf k})$ defined by the equation\n\\begin{equation}\n\\left< \\varphi_i({\\bf k}) \\varphi_j(-{\\bf k}) \\right>\n= \\delta_{i,j} \\, G_i({\\bf k}) \\;.\n\\end{equation}\nBecause of the $O(n)$ symmetry of the considered model, we have\n$G_i({\\bf k}) \\equiv G({\\bf k})$, i.e., the index $i$ may be omitted.\nIt is supposed that the averaging is performed over the $f({\\bf x})$\nconfigurations with a Gaussian distribution for the Fourier components\n$f_{\\bf k}$, i.e., the configuration $ \\{ f_{\\bf k} \\}$ is taken with\nthe weight function\n\\begin{equation} \\label{eq:P}\nP(\\{ f_{\\bf k} \\}) = Z_1^{-1} \\exp \\left( - \\sum\\limits_{\\bf k}\nb({\\bf k}) \\, {\\mid f_{\\bf k} \\mid}^2 \\right) \\;,\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:Z}\nZ_1 = \\int \\exp \\left( - \\sum\\limits_{\\bf k}\nb({\\bf k}) \\, {\\mid f_{\\bf k} \\mid}^2 \\right) \\, D(f) \\;,\n\\end{equation}\nand $b({\\bf k})$ is a positively defined function of $k$.\nEq.~(\\ref{eq:Hr}) defines the simplest random model considered\nin~\\cite{Ma} (according to the universality hypothesis, the factor\n$\\sqrt{u}$ does not make an important difference). Our random\nmodel describes a quenched randomness since the distribution over the\nconfigurations $\\{ f_{\\bf k} \\}$ of the random variable is given\n(by Eqs.~(\\ref{eq:P}) and (\\ref{eq:Z})) and does not depend neither on\ntemperature nor the configuration\n$\\{ \\varphi_i({\\bf k}) \\}$ of the order parameter field.\nMore precisely, the common distribution over the configurations\n$\\{ f_{\\bf k} \\}$ and $\\{ \\varphi_i({\\bf k}) \\}$ is given by\n\\begin{equation}\nP( \\{ f_{\\bf k} \\} , \\{ \\varphi_i({\\bf k}) \\} )\n= P( \\{ f_{\\bf k} \\} ) \\times Z_2^{-1}( \\{ f_{\\bf k} \\} ) \\,\n\\exp(-H/T) \\;,\n\\end{equation}\nwhere $Z_2( \\{ f_{\\bf k} \\} ) = \\int \\exp(-H/T) D(\\varphi)$ and $H$ is\ndefined by Eq.~(\\ref{eq:Hr}). For comparison, the common distribution\nis the equilibrium Gibbs distribution in a case of annealed randomness\nsometimes considered in literature.\n\n\\section{A basic theorem}\n We have proven the following theorem. \\bigskip\n\n{\\it Theorem}. \\, In the limit $n \\to 0$, the perturbation expansion\nof the correlation function $G({\\bf k})$ in $u$ power series for the\nrandom model with the Hamiltonian (\\ref{eq:Hr}) is identical to the\nperturbation expansion for the corresponding model with the Hamiltonian\n\\begin{eqnarray} \\label{eq:Hp}\nH/T = \\int \\left[ r_0 \\, \\varphi^2({\\bf x})\n+ c \\, (\\nabla \\varphi({\\bf x}) )^2 \\right] d{\\bf x} \\nonumber \\\\\n+ \\, uV^{-1} \\sum\\limits_{i,j,{\\bf k}_1,{\\bf k}_2,{\\bf k}_3 }\n\\varphi_i({\\bf k}_1) \\varphi_i({\\bf k}_2) \\, \\tilde u_{{\\bf k}_1+{\\bf k}_2} \\,\n\\varphi_j({\\bf k}_3) \\varphi_j(-{\\bf k}_1-{\\bf k}_2-{\\bf k}_3)\n\\end{eqnarray}\nwhere $\\tilde u_{\\bf k}=u_{\\bf k}- {1 \\over 2}\n\\left< {\\mid f_{\\bf k} \\mid}^2 \\right>$. \\medskip\n\n For convenience, we call the model without the term\n$\\sqrt{u} \\, f({\\bf x}) \\, \\varphi^2({\\bf x})$ the pure model, since this\nterm simulates the effect of random impurities~\\cite{Ma}.\n\\bigskip\n\n {\\it Proof of the theorem}. \\, According to the rules of the diagram\ntechnique, the formal expansion for $G({\\bf k})$ involves all connected\ndiagrams with two fixed outer solid lines. In the case of the pure model,\ndiagrams are constructed of the vertices {\\mbox \\zigv,} with factor\n$-uV^{-1} \\tilde u_{\\bf k}$ related to any zigzag line with wave vector\n${\\bf k}$.\nThe solid lines are related to the correlation function in the Gaussian\napproximation $G_0({\\bf k})=1/ \\left(2r_0+2ck^2 \\right)$. Summation over\nthe components $\\varphi_i({\\bf k})$ of the vector $\\varphi({\\bf k})$\nyields factor $n$ corresponding to each closed loop of solid lines in the\ndiagrams. According to this, the formal perturbation expansion is defined\nat arbitrary $n$. In the limit $n \\to 0$, all diagrams of $G({\\bf k})$\nvanish except those which do not contain the closed loops. In such a way,\nfor the pure model we obtain the expansion\n\\begin{equation} \\label{eq:expu}\nG({\\bf k})= \\Gausd + \\, \\zigd + \\, ... \\;.\n\\end{equation}\n In the case of the random model, the diagrams are constructed of\nthe vertices \\dshv and {\\mbox \\dotv.} Besides, it is important that only\nthose diagrams give a nonzero contribution where each dotted line is\ncoupled to another dotted line. The factors\n$uV^{-1} \\left< {\\mid f_{\\bf k} \\mid}^2 \\right>$ correspond to the\ncoupled dotted lines and the factors $-uV^{-1} u_{\\bf k}$ correspond to\nthe dashed lines. Thus, we have\n\\begin{equation} \\label{eq:exra}\nG({\\bf k}) = \\Gausd + \\left[ \\dashd + \\dotd \\right] + \\, ... \\;.\n\\end{equation}\nIn our notation the combinatorial factor corresponding to any specific\ndiagram is not given explicitly, but is implied in the diagram itself.\nIn the random model, first the correlation function $G({\\bf k})$\nis calculated at a fixed $\\{ f_{\\bf k} \\}$ according to the distribution\n$Z_2^{-1} \\left( \\{ f_{\\bf k} \\} \\right) \\, \\exp(-H/T)$\n(which corresponds to diagrams where solid lines are coupled, but the\ndotted lines with factors $-\\sqrt{u} \\,V^{-1/2} f_{\\bf k}$ are not coupled),\nperforming the averaging with the weight (\\ref{eq:P}) over the\nconfigurations of the random variable\n(i.e., the coupling of the dotted lines) afterwards.\nAccording to this procedure, the diagrams of the random model in\ngeneral (not only at $n \\to 0$) do not contain parts like {\\mbox \\bldot,}\n{\\mbox \\blddot,} {\\mbox \\bltdot,} etc. Such parts would appear after the\ncoupling of dotted lines only if unconnected (i.e., consisting of separate\nparts) diagrams with fixed $\\{f_{\\bf k}\\}$ would be considered.\n\n Thus, in the considered random model, the term of the\n$l$--th order in the perturbation expansion of $G({\\bf k})$\nin $u$ power series is represented by diagrams constructed of a number\n$M_1$ of vertices \\dshv and an even number $M_2$ of vertices \\dotv (i.e.,\n$M_2/2$ \\, double--vertices \\ddotv), such that $l=M_1+M_2/2$. In the pure\nmodel, defined by Eq.~(\\ref{eq:Hp}), this term is represented by diagrams\nconstructed of $l$ vertices {\\mbox \\zigv.} It is clear and evident from\nEqs.~(\\ref{eq:expu}) and (\\ref{eq:exra}) that all diagrams of the random\nmodel are obtained from those of the pure model if any of the zigzag\nlines is replaced either by a dashed or by a dotted line, performing\nsummation over all such possibilities. Note that such a method is valid in\nthe limit $n \\to 0$, but not in general. The problem is that, except\nthe case $n \\to 0$, the diagrams of the pure model contain parts like\n{\\mbox \\blzig,} {\\mbox \\bldzig,} {\\mbox \\bltzig,} etc. If all the depicted\nhere zigzag lines are replaced by the dotted lines, then we obtain diagrams\nwhich are not allowed in the random model, as it has been explained\nbefore. At $n \\to 0$, the only problem is to determine the\ncombinatorial factors for the diagrams obtained by the above replacements.\nFor a diagram constructed of $M_1$ vertices \\dshv and $M_2$ vertices \\dotv\nthe combinatorial factor is the number of possible different couplings of\nlines, corresponding to the given topological picture, divided by\n$M_1 ! M_2 !$.\n\n It is suitable to make some systematic grouping of the diagrams of\nthe random model. The following consideration is valid not only for\nthe diagrams of the two--point correlation function, but also for free\nenergy diagrams (connected diagrams without outer lines) and for the\ndiagrams of $2m$--point correlation function (i.e., the\ndiagrams with $2m$ fixed outer solid lines, containing no separate parts\nunconnected to these lines). It is supposed that at $n \\to 0$ the main terms\nare retained, which means that the free energy diagrams contain a single loop\nof solid lines. We define that all diagrams which can be obtained from the\n$i$--th diagram (i.e., the diagram of the $i$--th topology) of the pure\nmodel, belong to the $i$--th group. The sum of the diagrams of the\n$i$--th group can be found by the following algorithm.\n\n1. First, the $i$--th diagram of pure model is depicted in\nan a priori defined way.\n\n2. Each vertex \\zigv is replaced either by the vertex {\\mbox \\dshv,} or\nby the double-vertex {\\mbox \\ddotv,} performing the summation over all\npossibilities. Besides, all vertices \\dshv and \\dotv and all lines are\nnumbered before coupling, and all the distributions of the numbered\nvertices and lines over the numbered positions (arranged according to the\ngiven picture defined in step 1 and according to the actually considered\nchoice, defining which of the vertices \\zigv must be replaced by \\dshv\nand which of them must be replaced by \\ddotv) are counted as different.\nEach specific realization is summed over with the weight $1/(M_1!M_2!)$.\n\n3. To ensure that each specific diagram is counted with the correct\ncombinatorial factor, the result of summation in step 2 is divided by the\nnumber of independent symmetry transformations (including the identical\ntransformation) $S_i$ for the considered $i$--th diagram\nconstructed of vertices {\\mbox \\zigv,} where the symmetry transformation\nof a diagram is defined as any possible redistribution (such that the\nouter solid lines are fixed) of vertices and coupled lines not changing\nthe given picture. Really, the coupling of lines is not changed if any of\nthe symmetry transformations with any of the specific diagrams of the\n$i$--th group is performed, whereas, according to the algorithm of\nstep 2, original and transformed diagrams are counted as different.\n\n It is convenient to modify step 2 as follows. Choose any one replacement\nof the vertices \\zigv by \\dshv and {\\mbox \\ddotv,} and perform the summation\nover all such possibilities. For any specific choice we consider only one\nof the possible $M_1!M_2!$ distributions of the numbered $M_1$ vertices\n\\dshv and $M_2$ vertices \\dotv over the fixed numbered positions, and make\nthe summation with weight $1$ instead of the summation over\n$M_1!M_2!$ equivalent\n(i.e., equally contributing) distributions with the weight $1/(M_1!M_2!)$.\n\n Note that the location of any vertex \\dshv is defined by fixing the\nposition of dashed line, the orientation of which is not fixed. According\nto this, the summation over all possible distributions of lines (numbered\nbefore coupling) for one fixed location of vertices (as consistent with\nthe modified step 2) yields factor $8^{M_1}4^{M_2/2}$. The $i$--th\ndiagram of the pure model also can be calculated by such an algorithm.\nIn this case the summation over all\npossible line distributions yields a factor of $8^l$, where $l=M_1+M_2/2$\\,\nis the total number of vertices \\zigv in the $i$--th diagram. Obviously,\nthe summation of diagrams of the $i$--th group can be performed with\nfactors $8^l$ instead of $8^{M_1}4^{M_2/2}$, but in this case factors\n${1 \\over 2} uV^{-1} \\left< {\\mid f_{\\bf k} \\mid}^2 \\right>$\nmust be related to the coupled dotted lines instead of\n$uV^{-1} \\left< {\\mid f_{\\bf k} \\mid}^2 \\right>$. In this case the\nsummation over all possibilities where zigzag lines are replaced by dashed\nlines with factors $-uV^{-1}u_{\\bf k}$ and by dotted lines with factors\n${1 \\over 2} uV^{-1} \\left< {\\mid f_{\\bf k} \\mid}^2 \\right>$, obviously,\nyields a factor\n$uV^{-1} \\left(-u_{\\bf k}+{1 \\over 2} \\left< {\\mid f_{\\bf k} \\mid}^2\n\\right> \\right) \\equiv -uV^{-1} \\tilde u_{\\bf k}$ corresponding to each\nzigzag line with wave vector ${\\bf k}$. Thus, the sum over the diagrams of\nthe $i$--th group is identical to the $i$--th diagram of the pure model\ndefined by Eq.~(\\ref{eq:Hp}). By this the theorem has proved.\n\n\\section{Remarks and conclusions}\n The theorem has been formulated for the two--point correlation function,\nbut the proof, in fact, is more general, as regards the relation between\ndiagrams of random and pure models. Thus, the statement of the theorem is\ntrue also for free energy and for $2m$--point correlation function.\n\n According to the proven theorem and this remark, at $n \\to 0$\nthe considered pure and random models cannot be distinguished within the\ndiagrammatic perturbation theory. Thus, if, in principle, critical\nexponents can be determined from the diagrammatic perturbation theory at\n$n \\to 0$, then, in this limit, the critical exponents for the random\nmodel are the same as for the pure model. We think that in reality\ncorrect values of critical exponents can be determined from the\ndiagrammatic perturbation theory, therefore the quenched random\ntemperature disorder does\nnot change the universality class at $n \\to 0$. This our conclusion\ncorrelates with results of some other authors. In particular, there is a\ngood evidence that the universality class is not changed by the quenched\nrandomness at $n=1$. It has been shown by extensive Monte--Carlo\nsimulations of two--dimensional dilute Ising models \\cite{SzIg} that the\ncritical exponent of the defect magnetization is a continuous function of\nthe defect coupling. By analyzing the stability conditions, it has been\nconcluded in Ref.~\\cite{SzIg} that the critical exponent $\\nu$ of the\nbulk correlation length of the random Ising model does not depend on the\ndilution, i.e., $\\nu=1$ at $d=2$ both for diluted and not diluted Ising\nmodels. The standard (pertubative) RG method predicts\nthe change of the universality class by the quenched randomness.\nWe think, this is a false prediction. The fact that the standard\nRG method provides incorrect result is not surprising, since it has\nbeen demonstrated (in fact, proven) in Ref.~\\cite{Kaupuzs} that this\nmethod is not valid at $d<4$.\n\n\\References\n\\bibitem[1]{Baxter} Rodney J.~Baxter, {\\it Exactly Solved Models in\nStatistical Mechanics} (Academic Press, London etc.,1982)\n\\bibitem[2]{ReMa} Jose Arnaldo Redinz, Aglae C.~N.~de~Magelhaes,\n{\\it Phys. Rev. B} \\, {\\bf 51}, 2930 (1995)\n\\bibitem[3]{HaTa} T.~Hara, H.~Tasaki, {\\it J. Stat. Phys.}\n{\\bf 47}, 99 (1987)\n\\bibitem[4]{GaKu} K.~Gawedzki, A.~Kupiainen, {\\it Commun. Math. Phys.}\n{\\bf 99}, 197 (1985)\n\\bibitem[5]{Wilson} K.G.~Wilson, M.E.~Fisher, {\\it\nPhys.Rev.Lett.} {\\bf 28}, 240 (1972)\n\\bibitem[6]{Ma} Shang--Keng Ma, {\\it Modern Theory of Critical\nPhenomena} (W.A.~Benjamin, Inc., New York, 1976)\n\\bibitem[7]{Justin} J.~Zinn--Justin, {\\it Quantum Field Theory and\nCritical Phenomena} (Clarendon Press, Oxford, 1996)\n\\bibitem[8]{SzIg} F.~Szalma and F.~Igloi, {\\it Abstracts MECO 24, P69} \\\\\n(March 8-10, Lutterstadt Wittenberg, Germany, 1999)\n\\bibitem[9]{Kaupuzs} J.~Kaupu\\v{z}s, {\\it archived as cond-mat/0001414 of\nxxx.lanl.gov archive}, January 2000\n\\endrefs\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002149.extracted_bib",
"string": "\\bibitem[1]{Baxter} Rodney J.~Baxter, {\\it Exactly Solved Models in\nStatistical Mechanics} (Academic Press, London etc.,1982)\n\n\\bibitem[2]{ReMa} Jose Arnaldo Redinz, Aglae C.~N.~de~Magelhaes,\n{\\it Phys. Rev. B} \\, {\\bf 51}, 2930 (1995)\n\n\\bibitem[3]{HaTa} T.~Hara, H.~Tasaki, {\\it J. Stat. Phys.}\n{\\bf 47}, 99 (1987)\n\n\\bibitem[4]{GaKu} K.~Gawedzki, A.~Kupiainen, {\\it Commun. Math. Phys.}\n{\\bf 99}, 197 (1985)\n\n\\bibitem[5]{Wilson} K.G.~Wilson, M.E.~Fisher, {\\it\nPhys.Rev.Lett.} {\\bf 28}, 240 (1972)\n\n\\bibitem[6]{Ma} Shang--Keng Ma, {\\it Modern Theory of Critical\nPhenomena} (W.A.~Benjamin, Inc., New York, 1976)\n\n\\bibitem[7]{Justin} J.~Zinn--Justin, {\\it Quantum Field Theory and\nCritical Phenomena} (Clarendon Press, Oxford, 1996)\n\n\\bibitem[8]{SzIg} F.~Szalma and F.~Igloi, {\\it Abstracts MECO 24, P69} \\\\\n(March 8-10, Lutterstadt Wittenberg, Germany, 1999)\n\n\\bibitem[9]{Kaupuzs} J.~Kaupu\\v{z}s, {\\it archived as cond-mat/0001414 of\nxxx.lanl.gov archive}, January 2000\n\\endrefs\n"
}
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cond-mat0002150
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Semiflexible polymer on an anisotropic Bethe lattice
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[
{
"author": "J.F. Stilck\\cite{l}"
},
{
"author": "C.E. Cordeiro"
},
{
"author": "and R.L.P.G. do Amaral"
}
] |
The mean square end-to-end distance of a $N$-step polymer on a Bethe lattice is calculated. We consider semiflexible polymers placed on isotropic and anisotropic lattices. The distance on the Cayley tree is defined by embedding the tree on a sufficiently high dimensional Euclidean space considering that every bend of the polymer defines a direction orthogonal to all the previous ones. In the isotropic case, the result obtained for the mean square end-to-end distance turns out to be identical to the one obtained for {\em ideal} chains without immediate returns on an hypercubic lattice with the same coordination number of the Bethe lattice. For the general case, we obtain the asymptotic behavior in the semiflexible and also in the almost rigid limits.
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[
{
"name": "bethe3.tex",
"string": "\\documentstyle[preprint,aps]{revtex}\n\\tightenlines\n\\begin{document}\n\\draft\n\\title{\\bf Semiflexible polymer on an anisotropic Bethe lattice}\n\\author{J.F. Stilck\\cite{l}, C.E. Cordeiro, and R.L.P.G. do Amaral}\n\\address{\nInstituto de F\\'\\i sica,\\\\\nUniversidade Federal Fluminense\\\\\nAv. Litor\\^anea s/n\\\\\n24210-340, Niter\\'oi, R.J., Brazil\\\\}\n\\date{\\today }\n\\maketitle\n\\begin{abstract}\n\nThe mean square end-to-end distance of a $N$-step polymer on a Bethe\nlattice is calculated. We consider semiflexible polymers placed on\nisotropic and anisotropic lattices. The distance on the Cayley tree\nis defined by embedding the tree on a sufficiently high dimensional\nEuclidean space considering that every bend of the polymer defines a\ndirection orthogonal to all the previous ones. In the isotropic case,\nthe result obtained for the mean square end-to-end distance turns out\nto be identical to the one obtained for {\\em ideal} chains without\nimmediate returns on an hypercubic lattice with the same coordination\nnumber of the Bethe lattice. For the general case, we obtain the\nasymptotic behavior in the semiflexible and also in the almost rigid\nlimits.\n\\end{abstract}\n\\pacs{05.50.+q; 61.41+e}\n\n\\section{INTRODUCTION}\n\\label{s1}\nChain polymers are often approximated as self- and mutually avoiding\nwalks (SAW's) on a lattice, and much information about the behavior\nof polymers both in a melt or in solution has been understood\ntheoretically through this model \\cite{f53,dg79}. One of the\ncharacterizations of the conformations of a walk is through its\nmean-square end-to-end distance $\\langle R^{2}\\rangle $, where the mean is taken\nover all configurations of the $N$-step walk on the lattice. In the\nlimit $N\\rightarrow \\infty$ a scaling behavior $\\langle R^{2}\\rangle \\sim\nN^{2\\nu}$ is observed, where the exponent $\\nu$ exhibits universal\nbehavior, with a mean-field value $\\nu=1/2$ for random walks, which\ncorrespond to ideal chains, and $\\nu=3/4$ for SAW's on two-dimensional\nlattices \\cite{n82}, for example.\n\nAn interesting question arises if the chains are not considered to be\ntotally flexible, an energy being associated to bends of the chain.\nThis is often observed for real polymers. Let us, for simplicity,\nrestrict ourselves to SAW'S on hypercubic lattices. In this case,\nconsecutive steps of the walk are either in the same direction or\nperpendicular. So, a Boltzmann factor $z$ may be associated to each\npair of perpendicular consecutive steps of walk. This problem of\nsemiflexible polymers (also called persistent or biased walks) has\nbeen studied for some time\\cite{ts75,mn91,cfs92}, and there occurs a\ncrossover in the behavior of the walk between a rodlike behavior\n$\\nu_{r}=1$ for $z=0$, where the polymer is totally stiff, and the\nusual behavior with a different exponent $\\nu$ for nonzero values of\n$z$. Stating this point more precisely, the mean square end-to-end\ndistance displays scaling behavior in the limit $N\\rightarrow\n\\infty;z\\rightarrow0;N^{\\psi}z=constant$, which is given by\n\\begin{equation}\n\\langle R^{2}\\rangle \\sim N^{2\\nu_{r}}F(zN^{\\psi}),\n\\label{e1}\n\\end{equation}\nthe observed values being $\\nu_{r}=1$ and $\\psi=1$.\n\nThe scaling function has the behavior $F(x)\\sim x^{(2\\nu-2)/\\psi}$\nin the limit $x \\rightarrow \\infty$. This scaling form has been\nverified through several techniques, although in three dimensions a\nmean field exponent $\\nu=1/2$ was found for intermediate values of\nthe number of steps $N$, the crossover to the three-dimensional value\noccurring at rather high values of $N$\\cite{mn91}.\n\nIn this paper, we consider the problem of a semiflexible polymer on a\nBethe lattice\\cite{b82}, calculating exactly the mean square\nend-to-end distance of walks on the Cayley tree which start at the\ncentral site and have $N$ steps, supposing that the walks will never\nreach the surface of the Cayley tree, thus remaining in its core. We\ncalculate also the mean square end-to-end distance in the case when\nthe lattice is considered anisotropic, that is, when the edges of\nthe lattice are not equivalent with respect to their occupation by a\npolymer bond. The definition of the distance between two sites of the\nCayley tree is not obvious, and some possibilities exploring the fact\nthat the tree may be embedded in a hypersurface of a non-Euclidean\nspace have been given\\cite{m92}. In this paper, however, we used a\nsimpler definition, considering the Cayley tree in the thermodynamic\nlimit to be embedded in an infinite-dimensional Euclidean space. The\nresult for $\\langle R^{2}\\rangle (N,z)$, for the isotropic case has\nthe scaling form of Eq. \\ref{e1}. Not surprisingly the scaling\nfunction $F(x)$ is equal to the one obtained for random walks with no\nimmediate return on an hypercubic lattice with the same coordination\nnumber of the Bethe lattice considered. This might be expected since\nBethe lattice calculations lead to mean-field critical exponents.\nAlso, in the limit $N\\rightarrow\\infty$ for nonzero values of $z$ the\nscaling behavior $\\langle R^{2}\\rangle \\sim N^{2\\nu}$ with the\nclassical value $\\nu=1/2$ is verified in the expression for $\\langle\nR^{2}\\rangle (N,z)$. It should be mentioned that our proposal of\ndefining the Euclidean distance between two points of the Cayley tree\nis similar to earlier results in the literature relating this\ndistance to the chemical distance, measured along the chain\n\\cite{dq86}. However, the distinction between the chemical and \nthe Euclidean distances is not always properly considered \nin the literature, and this may lead to\ncontradicting results \\cite{hi98}, as we will discuss in more detail\nin the conclusion.\n\nIn section \\ref{s2} we define the model and calculate the mean square\nend-to-end distance recursively on the anisotropic Bethe lattice. The\nproblem is then reduced to finding the general term of a {\\em linear}\nmapping in six dimensions. In the particular case of an\nisotropic lattice, we find a closed expression for\n$\\langle R^2\\rangle $. In section \\ref{s3} the asymptotic behavior is\nstudied for the general case, based on the mapping. In section\n\\ref{s4} final comments and discussions may be found. Finally, we\npresent in the appendices a combinatorial calculation for $\\langle\nR^2\\rangle$ in the isotropic case.\n\n\n\\section{DEFINITION OF THE MODEL AND SOLUTION FOR THE ISOTROPIC\nLATTICE}\n\\label{s2}\nWe consider a Cayley tree of coordination number $q$ and place a\nchain on the tree starting at the central site. Each bond of the\ntree is supposed to be of unit length. Figure \\ref{f1} shows\na tree with $q=4$ and a polymer with $N=2$ steps placed on it. Since\nwe want the Cayley tree to be an approximation of a hypercubic\nlattice in $d$ dimensions, we will restrict ourselves to even\ncoordination numbers $q=2d$. As in the hypercubic lattice, the bonds\nincident on any site of the tree are in $d$ directions, orthogonal to\neach other. As may be seen in Figure \\ref{f1}, the central site of\nthe tree is connected to $q$ other sites, which belong to the first\ngeneration of sites. Each of the sites of the first generation is\nconnected to $(q-1)$ sites of the second generation, and this process\ncontinues until the surface of the tree is reached, after a number of\nsteps equal to the number of generations in the tree. Upon reaching a\nsite of the $i$'th generation coming from a site belonging to\ngeneration $(i-1)$, there are $(q-1)$ possibilities for the next step\nof the walk towards a site of generation $(i+1)$. One of them will be\nin the same direction as the previous step, while the remaining\n$(q-2)$ will be in directions orthogonal to {\\em all} previous steps.\nIn the second case, a statistical weight $z$ is associated to the\nelementary bend in the walk. Therefore, we admit that the $(q-2)$\nbonds which are orthogonal to the last step are also orthogonal to\n{\\em all} bonds of the lattice in earlier generations. Let us stress\ntwo consequences of this supposition: (i) A tree of coordination\nnumber $q$ with $N_{g}$ generations will be embedded in a space of\ndimension\n\\begin{equation}\nD=q/2+(N_{g}-1)(q/2-1).\n\\label{e3}\n\\end{equation}\nThe sites of the Cayley tree will all be sites of a hypercubical\nlattice in $D$ dimensions. This may be seen in Figure \\ref{f1} where\nthe sites of a tree with $q=4$ and $ N_{g}=2$ are sites of a cubic\n$(D=3)$ lattice. (ii) By construction, there will never be loops in\nthe tree, a property which is true for any Cayley tree. It is well\nknown\n\\cite{b82} that it may be shown by other means that the Cayley tree\nis a infinite-dimensional lattice in the thermodynamic limit\n$N_{g}\\rightarrow\\infty$. Finally, the anisotropy is introduced into\nthe model considering that {\\em bonds} of the chain in $s$ of the $q/2$\ndirections (we will call them special) at each lattice site \ncontribute with a factor $y$ to the partition function, while no\nadditional contribution comes from bonds in the remaining $t=q/2-s$\ndirections at each lattice site.\n\nUsually \\cite{b82}, the calculation of thermodynamic properties of\nmodels defined on the Bethe lattice is done in a recursive manner, so\nwe will follow a similar procedure in the calculation of the mean\nsquare end-to-end distance. We define a generalized partition (or\ngenerating) function for $N$-step chains\n\\begin{equation}\ng_N=\\sum z^m y^{N_e} p^{R^2},\n\\label{e4}\n\\end{equation}\nwhere the sum is over all configurations of the chain, $z$ is the\nstatistical weight of an elementary bend in the chain, $y$ is the\nstatistical weight of bonds in special directions and $p$ is a\nparameter associated to the square of the end-to-end distance of the\nchain. At the end of the calculation, we will take $p=1$. The numbers\nof elementary bends, bonds in special directions, and the square\nend-to-end distance of each chain are $m$, $N_e$, and $R^2$,\nrespectively. The mean\nsquare end-to-end distance may then be calculated through\n\\begin{equation}\n\\langle R^2 \\rangle _N=\\frac{1}{g_N}\\left.\\left(p\\frac{\\partial\ng_N}{\\partial p}\\right)\\right|_{p=1}.\n\\label{e5}\n\\end{equation}\n\nThe partition function may then be calculated in a recursive way if\nwe define partial partition functions $a_N^l$ and $b_N^l$ such that\nthe first ones include all $N$-bond chains whose last $l$ bonds are\ncollinear and in one of the special directions (there is necessarily a\nbend before the $l$ bonds, if $l<N$), while the last $l$ bonds of the\nchains contributing to $b_N^l$ are collinear and in one of the\nnon-special directions. The partition function may then be written as\n\\begin{equation}\ng_N=\\sum_{l=1}^{N} (a_N^l+b_N^l).\n\\label{e6}\n\\end{equation}\nDue to the fact that there are no closed loops on the Cayley tree, it\nis quite easy to write down recursion relations for the partial\npartition functions\n\\begin{mathletters}\n\\label{e7}\n\\begin{eqnarray}\na_{N+1}^1&=&2syzp \\sum_{l=1}^N b_N^l + 2(s-1)yzp \\sum_{l=1}^N\na_N^l,\\\\ \na_{N+1}^{l+1}&=&yp^{2l+1}a_N^l,\\\\\nb_{N+1}^1&=&2tzp\\sum_{l=1}^N a_N^l + 2(t-1)zp\\sum_{l=1}^N b_N^l,\\\\\nb_{N+1}^l&=&p^{2l+1}b_N^l,\n\\end{eqnarray}\n\\end{mathletters}\nwith the initial conditions\n\\begin{mathletters}\n\\label{e8}\n\\begin{eqnarray}\na_1^1&=&2syp,\\\\\nb_1^1&=&2tp.\n\\end{eqnarray}\n\\end{mathletters}\nFor example, in the first expression above, the new bond may be\npreceded by a bond in a special direction, with $2s$\npossibilities, or by a bond in a non-special direction, with $2(s-1)$\npossibilities. In both cases, a factor $p$ is present since the bond\nadded is in a direction perpendicular to all previous ones, and thus\n$R^2$ is increased by one unit. Finally, the inclusion of the new bond\nintroduces one bend in the chain, thus explaining the factor $z$, and\nsince the bond is in a special direction the factor $y$ is justified.\nIn the second expression, it should be mentioned that $R^2$ is\nincreased by $(l+1)^2-l^2$, thus explaining the exponent of $p$. If\nwe now define\n\\begin{mathletters}\n\\label{e9}\n\\begin{eqnarray}\na_N=\\sum_{l=1}^N a_N^l,\\\\\nb_N=\\sum_{l=1}^N b_N^l,\n\\end{eqnarray}\n\\end{mathletters}\nthe mean square end-to-end distance will be\n\\begin{equation}\n\\langle R^2 \\rangle_N=\\frac{1}{a_N+b_N}\\left.\\left[ p\\frac{\\partial}\n{\\partial p}(a_N+b_N) \\right] \\right|_{p=1} = \\frac{c_N+d_N}\n{a_N+b_N}. \n\\label{e10}\n\\end{equation}\nThe recursion relations for $a_N$ and $b_N$, as well as the ones for\nthe new variables $c_N$ and $d_N$ may be written, for $p=1$, as\n\\begin{mathletters}\n\\label{e11}\n\\begin{eqnarray}\na_{N+1}&=&2z\\sum_{l=0}^N y^{l+1}\\left[ (s-1)a_{N-l}+sb_{N-l} \\right],\\\\\nb_{N+1}&=&2z\\sum_{l=0}^N \\left[ ta_{N-l}+sb_{N-l} \\right],\\\\\nc_{N+1}&=&2z\\sum_{l=0}^N y^{l+1} \\left[ (s-1)(l+1)^2 a_{N-l}+s(l+1)^2\nb_{N-l} +(s-1)c_{N-l}+sd_{N-l} \\right],\\\\\nd_{N+1}&=&2z \\sum_{l=0}^N \\left[ t(l+1)^2 a_{N-l}+(t-1)(l+1)^2 b_{N-l}\n+ tc_{N-l}+(t-1)d_{N-l} \\right],\n\\end{eqnarray}\n\\end{mathletters}\nwith the initial conditions\n\\begin{mathletters}\n\\label{e12}\n\\begin{eqnarray}\na_0&=&\\frac{2s}{z(q-2)},\\\\\nb_0&=&\\frac{2t}{z(q-2)},\\\\\nc_0&=&d_0=0\n\\end{eqnarray}\n\\end{mathletters}\nAn undesirable feature of the recursion relations Eqs. \\ref{e11} is that\nthe new values of the iterating variables depend on all previous\nvalues. This dependence, however, is rather simple, and it is\npossible, introducing two more variables $e_N$ and $f_N$, to rewrite\nthe recursion relations as a mapping involving only one previous\nvalue of each variable, valid for $N \\ge 1$\n\\begin{mathletters}\n\\label{e13}\n\\begin{eqnarray}\na_{N+1}&=&ya_N+2zy\\left[ (s-1)a_N+sb_N \\right],\\\\\nb_{N+1}&=&b_N+2z \\left[ ta_N+(t-1)b_N \\right],\\\\\nc_{N+1}&=&yc_N+2zy\\left[ (s-1)(a_N+c_N)+s(b_N+d_N) \\right] +(2N+1)ya_N\n- 2ye_N,\\\\\nd_{N+1}&=&d_N+2z \\left[ t(a_N+c_N)+(t-1)(b_N+d_N) \\right] + (2N+1)b_N -\n2f_N,\\\\\ne_{N+1}&=&ye_N+2Nzy \\left[ (s-1)a_N+sb_N \\right],\\\\\nf_{N+1}&=&f_N+2Nz \\left[ ta_N+(t-1)b_N \\right],\n\\end{eqnarray}\n\\end{mathletters}\nwith the initial conditions\n\\begin{mathletters}\n\\label{e14}\n\\begin{eqnarray}\na_1&=&c_1=2sy,\\\\\nb_1&=&d_1=2t,\\\\\ne_1&=&f_1=0.\n\\end{eqnarray}\n\\end{mathletters}\nThe value for $\\langle R^2 \\rangle$ may be found iterating the\nmapping above through the Eq. \\ref{e10}. In principle, since the\nmapping is {\\em linear}, it is solvable. One starts finding the\ngeneral term of the first two equations, then solving the last two\nand finally solving the two remaining relations. A software for\nalgebraic computing is helpful, but we realized that the general\nanswer will be too large to be handled, and also the computer time\nand memory required are beyond the resources we have available. We\ntherefore restrict ourselves to a complete solution of the isotropic\ncase $y=1$ and to an exact study of the asymptotic properties of the\nsolution for the general case. It is worthwhile to observe in the\nmapping Eqs. \\ref{e13} that under a transformation\n\\begin{mathletters}\n\\label{e15}\n\\begin{eqnarray}\ns^{\\prime}=t,\\\\\nt^{\\prime}=s,\\\\\ny^{\\prime}=1/y,\n\\end{eqnarray}\n\\end{mathletters}\n$\\langle R^2 \\rangle$ will be invariant, as expected.\n\nFor the isotropic case ($y=1$) the mapping Eqs. \\ref{e13} is reduced to\nthree variables\n\\begin{mathletters}\n\\label{e16}\n\\begin{eqnarray}\n\\alpha_N&=&a_N+b_N,\\\\\n\\beta_N&=&c_N+d_N,\\\\\n\\gamma_N&=&e_N+f_N,\n\\end{eqnarray}\n\\end{mathletters}\nand it may be written as\n\\begin{mathletters}\n\\label{e17}\n\\begin{eqnarray}\n\\alpha_{N+1}&=&\\left[ 1+z(q-2) \\right] \\alpha_N,\\\\\n\\beta_{N+1}&=&\\left[ 1+z(q-2)\\right] \\beta_N+ \\left[ 2N+1+z(q-2)\n\\right] \\alpha_N - 2\\gamma_N,\\\\\n\\gamma_{N+1}&=& \\gamma_{N}+Nz(q-2)\\alpha_N.\n\\end{eqnarray}\n\\end{mathletters}\nThe initial conditions are\n\\begin{eqnarray}\n\\alpha_1&=&\\beta_1=q,\\\\\n\\gamma_1&=&0.\n\\label{e18}\n\\end{eqnarray}\nIt is easy to find the general solution for this mapping\n\\begin{mathletters}\n\\label{e19}\n\\begin{eqnarray}\n\\alpha_N&=&qk^{N-1},\\\\\n\\beta_N&=&\\frac{q}{(k-1)^2} \\left[ N(k^2-1)k^{N-1}+2-2k^N \\right],\\\\\n\\gamma_N&=&\\frac{q}{k-1} \\left[ N(k^2-1)k^{N-1}+1-k^N \\right],\n\\end{eqnarray}\n\\end{mathletters}\nwhere $k=1+(q-2)z$. The substitution of this solution in Eq. \\ref{e10}\nresults in\n\\begin{equation}\n\\langle R^{2}\\rangle\n=\\frac{2[1+a]}{a^{2}}\\left[Na-1+\\frac{1}{[1+a]^{N}}\\right]-N,\n\\label{e20}\n\\end{equation}\nwhere $a=k-1=(q-2)z$.\n\nThe properties of the mean square end-to-end distance Eq. \\ref{e20}\nin some limiting cases show that our result has the expected\nbehavior. First, we notice that when the bend statistical weight $z$\nvanishes we have\n\\begin{equation}\nlim_{z\\rightarrow0}\\langle R^{2}\\rangle =N^{2},\n\\label{e21}\n\\end{equation}\nfor any number of steps $N$. This rodlike behavior is expected, since\nno bend will be present in the walk. In the opposite limit of\ninfinite bending statistical weight the result is\n\\begin{equation}\nlim_{z\\rightarrow\\infty}\\langle R^{2}\\rangle =N,\n\\label{e22}\n\\end{equation}\nwhich is also an expected result, since in this limit there is a bend\nat every internal site of the chain, so that, according to the\ndefinition of the end-to-end distance we are using, the vector\n$\\vec{R}$ in this situation will have $N$ components, all of them\nbeing equal to 1.\n\nIn the limit of an infinite chain $N\\rightarrow\\infty$ we get, for\nnonzero $z$,\n\\begin{equation}\nlim_{N\\rightarrow\\infty}\\langle R^{2}\\rangle =\\frac{(2+a)N}{a},\n\\label{e23}\n\\end{equation}\nand we notice that the expected scaling behavior $\\langle\nR^{2}\\rangle \\sim N^{2\\nu}$ is obtained with the mean field\nexponent $\\nu=1/2$. The asymptotic behavior of $\\langle R^{2}\\rangle\n$ is different for zero and nonzero $a$, as may be appreciated\ncomparing Eqs. \\ref{e21} and\n\\ref{e23} respectively. So we may look for the crossover between both\nbehaviors in the limit of Eq. \\ref{e1}, getting the result\n\\begin{equation}\nlim_{N\\rightarrow\\infty;a\\rightarrow0;aN=x}\\langle R^{2}\\rangle =N^{2}F(x),\n\\label{e23p}\n\\end{equation}\nwith a scaling function\n\\begin{equation}\nF(x)=\\frac{2(x-1+\\exp(-x))}{x^{2}}.\n\\label{e24}\n\\end{equation}\nIt should be stressed that the square end-to-end distance given in\nEq. \\ref{e20} is the same obtained by adapting the general result of\nFlory for random walks without immediate return \\cite{f53} to\nhypercubic lattices. In general, it may be shown that the exact\nsolution of statistical models with first neighbor interactions on\nthe Bethe lattice is equivalent to the Bethe approximation on the\nBravais lattice with the same coordination number \\cite{b82}. The\nrandom walk without immediate returns corresponds to the Bethe\napproximation of the $n \\rightarrow 0$ model associated to the\nself-avoiding walk problem \\cite{sw87}, and here we show that the\nanalogy may be extended to the mean square end-to-end distance if we\ndefine distances on the Bethe lattice as was done above. Although the\nresults on the Bethe lattice as calculated here and the ones for\nideal chains without immediate return on a hypercubic lattice with\nthe same coordination number should have the same asymptotic\nbehaviors, it is at first surprising that they are actually\nidentical. However, it turns out that the mean value of the angle\nbetween successive bonds, as calculated by Flory in his original work\n\\cite{f53}, is actually {\\em exact} for chains on the Bethe lattice\nas we considered.\n\n\\section{ASYMPTOTIC BEHAVIOR IN THE GENERAL CASE}\n\\label{s3}\nIn this section we develop a study of the asymptotic solution of the\nmapping Eqs. \\ref{e13} for $N \\gg 1$. Let us\nreduce the dimension of the mapping by one defining new iteration\nvariables\n\\begin{mathletters}\n\\label{e25}\n\\begin{eqnarray}\nB_N=\\frac{b_N}{a_N},\\\\\nC_N=\\frac{c_N}{a_N},\\\\\nD_N=\\frac{d_N}{a_N},\\\\\nE_N=\\frac{e_N}{a_N},\\\\\nF_N=\\frac{f_N}{a_N}.\n\\end{eqnarray}\n\\end{mathletters}\nFrom Eqs. \\ref{e13} and the initial conditions Eqs. \\ref{e14} it is easy\nto write the recursion relations and initial conditions for the new\niterative variables in the mapping above. In the limit of large\nvalues of $N$, for fixed $z$ and $y$, the following asymptotic\nbehavior is observed\n\\begin{mathletters}\n\\label{e26}\n\\begin{eqnarray}\nB_N &\\sim& B^0,\\\\\nC_N &\\sim& C^0+C^1N,\\\\\nD_N &\\sim& D^0+D^1N,\\\\\nE_N &\\sim& E^0+E^1N,\\\\\nF_N &\\sim& F^0+F^1N.\n\\end{eqnarray}\n\\end{mathletters}\nThe substitution of the these expressions into the recursion\nrelations for the variables defined in Eqs. \\ref{e25}, obtained from the general\nmapping Eqs. \\ref{e13}, leads to the determination of the constants in the\nasymptotic behavior and thus we obtain\n\\begin{equation}\n\\langle R^2 \\rangle =\\frac{C_N+D_N}{1+B_N} \\sim \\frac{C^1+D^1}\n{1+B^0} N = C N,\n\\label{e27}\n\\end{equation}\nwhere the amplitude $C=C^1$ is given by\n\\begin{equation}\nC=\\frac{sy(B^0)^2 \\left[ \\frac{y(1+\\epsilon)+1}{y(1+\\epsilon)-1}\n\\right] +t\\left[ \\frac{2+\\epsilon}{\\epsilon}\\right] }{sy(B^0)^2+t},\n\\label{e28}\n\\end{equation}\nwhere\n\\begin{equation}\n\\epsilon=2z(s-1+sB^0),\n\\label{e29}\n\\end{equation}\nand $B^0$ is the positive root of\n\\begin{equation}\n2zsy(B^0)^2+\\left[ y-1+2z(sy-t-y+1)\\right] B^0-2zt=0.\n\\label{e30}\n\\end{equation}\nThe amplitude of the asymptotic behavior of $\\langle R^2 \\rangle$ thus\nmay be obtained exactly in the general case and, as may be seen in\nFigure \\ref{f2}, diverges as $z \\rightarrow 0$, as expected. Also,\nin the limit $y \\rightarrow \\infty$ the problem reduces to a walk on\nan isotropic lattice with coordination number equal to $2s$, and\nwe get\n\\begin{equation}\nC=\\frac{2z(s-1)+2}{2z(s-1)},\n\\label{e31}\n\\end{equation}\nwhich agrees with Eq. \\ref{e19} for the isotropic case.\n\nNow we will study the asymptotic behavior in the quasi-rigid limit $N\n\\rightarrow \\infty$, $z \\rightarrow 0$, and $N(q-2)z=x$. We thus expand\n$\\langle R^2 \\rangle$ for small values of $z$\n\\begin{equation}\n\\langle R^2 \\rangle (z,y,N) \\sim \\langle R^2 \\rangle (0,y,N) +\n\\left.\\frac{\\partial \\langle R^2 \\rangle}{\\partial z}\\right|_{z=0} z.\n\\label{e32}\n\\end{equation}\nFor $z=0$ the solution of the mapping Eqs. \\ref{e13} is\n\\begin{mathletters}\n\\label{e33}\n\\begin{eqnarray}\na_N&=&2sy^N,\\\\\nb_N&=&2t,\\\\\nc_N&=&2sy^NN^2,\\\\\nd_N&=&2tN^2,\\\\\ne_N&=&f_N=0;\n\\end{eqnarray}\n\\end{mathletters}\nand we have $\\langle R^2 \\rangle=N^2$, as expected. From the mapping\nEqs. \\ref{e13} the recursion relations for the derivatives of\nthe variables with respect to $z$ (at $z=0$) may be seen to be\n\\begin{mathletters}\n\\label{e34}\n\\begin{eqnarray}\na^\\prime_{N+1}&=&ya^\\prime_N+4ys\\left[(s-1)y^N+t\\right],\\\\\nb^\\prime_{N+1}&=&b^\\prime_N+4t(sy^N+t-1),\\\\\nc^\\prime_{N+1}&=&yc^\\prime_N+4ys(1+N^2)\\left[(s-1)y^N+t\\right] +y(2N+1)\na^\\prime_N-2ye^\\prime_N,\\\\\nd^\\prime_{N+1}&=&d^\\prime_N+4t(1+N^2)(sy^N+t-1)+(2N+1)b^\\prime_N -\n2f^\\prime_N,\\\\ \ne^\\prime_{N+1}&=&ye^\\prime_N+4Nys\\left[(s-1)y^N+t \\right],\\\\\nf^\\prime_{N+1}&=&f^\\prime_N+4Nt(sy^N+t-1),\n\\end{eqnarray}\n\\end{mathletters}\nwhere the values for the variables (Eqs. \\ref{e33}) have already been\nsubstituted and the initial conditions are\n$a^\\prime_1=b^\\prime_1=...= f^\\prime_1=0$. The general solution of\nthe recursion relations Eqs. \\ref{e34} is not difficult to obtain with the\naid of an algebra software. Considering the invariance described in Eqs. \\ref{e15} we\nwill restrict our discussion to the case $y>1$, without loss of\ngenerality. For large values of $N$, the dominant terms of the\nsolution of the mapping are\n\\begin{mathletters}\n\\label{e35}\n\\begin{eqnarray}\na_N+b_N &\\sim& 2sy^N,\\\\\nc_N+d_N &\\sim& 2sy^N N^2,\\\\\na^{\\prime}_N+b^{\\prime}_N &\\sim& \\left\\{\n\\begin{array}{ll}\n4s(s-1)y^N N & \\mbox{if $s>1$}\\\\\n\\frac{8ty^N}{y-1} & \\mbox{if $s=1$}\n\\end{array}\n\\right. ,\\\\\nc^{\\prime}_N+d^{\\prime}_N &\\sim& \\left\\{\n\\begin{array}{ll}\n\\frac{8}{3}s(s-1)y^N N^3 & \\mbox{if $s>1$}\\\\\n\\frac{8ty^N N^2}{y-1} & \\mbox{if $s=1$}\n\\end{array}\n\\right. .\n\\end{eqnarray}\n\\end{mathletters}\nThe leading term in the derivative of the mean-square end-to-end\ndistance will be\n\\begin{equation}\n\\left. \\frac{\\partial \\langle R^2 \\rangle}{\\partial z}\\right|_{z=0}\n\\sim -\\frac{2}{3}(s-1)N^3.\n\\label{e36}\n\\end{equation}\nTherefore, up to first order in $x$, the scaling function $F(x)$ in\nthe quasi-rigid limit is found to be $F(x) \\sim 1 - F_1(s,t)x$. \nConsidering the symmetry Eq. \\ref{e15} and\nthe solution for the isotropic case Eq. \\ref{e20}, we have\n\\begin{equation}\nF_1(s,t)=\\left\\{\n\\begin{array}{lll}\n\\frac{2(t-1)}{3(q-2)} & \\mbox{if $y<1$},\\\\\n\\frac{1}{3} & \\mbox{if $y=1$},\\\\\n\\frac{2(s-1)}{3(q-2)} & \\mbox{if $y>1$}.\n\\end{array}\n\\right. \n\\label{e37}\n\\end{equation}\nWe thus conclude that the scaling function in general displays a\ndiscontinuous derivative at $y=1$.\n\n\\section{CONCLUSION}\n\\label{s4}\nWe formulated the problem of the calculation of the mean square\nend-to-end distance of semiflexible polymers placed on a\n$q$-coordinated anisotropic Bethe lattice as a linear mapping, whose\ngeneral term may in principle be obtained. In the isotropic case, the\nmapping may easily be solved and leads to an expression for $\\langle\nR^2 \\rangle$ which is {\\em identical} to the one obtained for random\nwalks without immediate return on a hypercubic lattice with the same\ncoordination number \\cite{f53}. The identity between the two problems\nregarding thermodynamic properties derived from the free energy is\nwell known \\cite{sw87}, and is here extended for a thermodynamic\naverage of a geometric property. One point which should be stressed\nis that the definition of the Euclidean distance between two points\non the Bethe lattice is rather arbitrary. Here we defined the\ndistance by embedding the Cayley tree in a hypercubic lattice of\nsufficiently high dimensionality. In the thermodynamic limit the\ndimensionality of this lattice diverges, as expected \\cite{b82}.\nOther definitions of distance may be proposed \\cite{m92}. The simple\none we adopted here leads to meaningful conclusions. Since\ncalculations on the Bethe lattice are usually done recursively, and\none step in the recursion relations corresponds to adding another\ngeneration to the tree, it is tempting to define the distance between\ntwo sites on the tree as the difference between the numbers of the\ngenerations they belong to. This definition, although simple and\noperational, has serious drawbacks, however. This is quite clear for\nthe particular problem we looked at here, since it implies that\n$\\langle R^2 \\rangle$ for {\\em any} $N$-step chain is equal to $N^2$.\nWe would thus have $\\nu=1$, the one-dimensional value, and the\nidentity between the results for the Bethe lattice and for walks\nwithout immediate return on hypercubic lattices would break down.\nThis definition of distance was used recently in the exact\ncalculation of correlation functions for a general spin-$S$ magnetic\nmodel \\cite{hi98}, leading to $\\nu=1$, in opposition to the generally\naccepted mean-field value $\\nu=1/2$ \\cite{tm96}.\n\nThe fact that all walks we considered here have their initial site\nlocated at the central site of the Cayley tree is of course\nconvenient for the calculations and may be seen as a particular case.\nA closer consideration of this point, however, leads to the\nconclusion that our results are exact for any chains such that the\nassertion that at any bend the new direction is perpendicular to {\\em\nall} previous directions of bonds holds. Thus, it is clear that if\nthe whole chain is contained in one of the $q$ rooted sub-trees\nattached to the central site the results are still the same. If\nportions of the chain are located on two of these sub-trees the\ncalculation becomes more complicated since, as may be seen in figure\n\\ref{f1}, there are bonds in the same direction in different\nsub-trees. However, this problem may be easily avoided by enlarging\nthe dimension of the euclidean space in which the tree is embedded,\nthus assuring that any two bonds in the same direction are\nnecessarily connected by a walk without any bend. For such a tree,\nour results hold for any chain, regardless of the location of its\nendpoints. \n\nIn the general anisotropic case, we restricted ourselves to the\ndiscussion of the asymptotic behavior of $\\langle R^2 \\rangle$, which\nwas studied in the semiflexible case and also in the quasi-rigid\nlimit. The expected scaling behavior was obtained in both cases, and\na interesting discontinuity in the quasi-rigid limit amplitude is\nobserved as the isotropic value $y=1$ is crossed.\n\n\\acknowledgments\nWe acknowledge partial financial support from the\nBrazilian agencies CNPq and FINEP.\n\n\\appendix\n\\section{COMBINATORIAL SOLUTION IN THE ISOTROPIC CASE}\n\\label{a1}\n \nAny $N$-step walk on the Cayley tree will visit a subset of sites of\nthe D-dimensional hypercubic lattice defining a subspace whose\ndimensionality is between 1 and $N$. The limiting cases are the ones\nof a polymer without any bend (rod), which is one-dimensional, and a\npolymer where we have a bend at every internal site, and since at\neach bend the new bond is in a direction orthogonal to all precedent\nbonds of the polymer, the polymer is embedded in a $N$-dimensional\nsubspace. Since the initial site of the chain is supposed to be at\nthe central site of the tree, the end-to-end distance will be given\nby the modulus of the position vector of the final site, denoted by\n$\\vec{R}$. For a polymer with $m$ bends, the number of components of\nthis vector will be equal to $m+1$. For simplicity, we will admit\nthat each bond is of unit length, so that the components of $\\vec{R}$\nwill be integers. We want to compute the mean value of $\\vec{R}$ over\nall polymers with $N$ steps\n\\begin{equation}\n\\langle R^{2}\\rangle =\\frac{\\sum_{\\vec{R_{m}^{N}}}z^{m}R^{2}}{\\sum_{\\vec{R_{m}^{N}}}\nz^{m}},\n\\label{ea1}\n\\end{equation}\nwhere $m$ is the number of bends in the walk and the sum\nis over all configurations $\\vec{R_{m}^{N}}$ of polymers with $N$\nsteps. Besides the first and last components the values of the other\n$m-1$ components of $R$ are the numbers of steps between successive\nbends in the walk. We should remember that there are $q-2$\npossibilities for each bend. So we may rewrite Eq. \\ref{ea1}\n\\begin{equation}\n\\langle R^{2}\\rangle =\\frac{\\sum_{m=0}^{N-1}a^{m}B_{N,m}}{\\sum_{m=0}^{N-1}a^{m}A_{N,m}},\n\\label{ea2}\n\\end{equation}\nwhere $a=(q-2)z$ embodies all dependence on coordination number and\nstatistical weight as long as $q\\geq 4$, \n\\begin{equation}\nA_{N,m}=\\sum_{\\vec{R}_{m}^{N}}1,\n\\label{ea3}\n\\end{equation}\nand\n\\begin{equation}\nB_{N,m}=\\sum_{\\vec{R}_{m}^{N}}\\sum_{i=0}^{m+1}R_{i}^{2}.\n\\label{ea4}\n\\end{equation}\nNote that the effect of the bending energy can be described by introducing\nan effective coordination number $q^\\prime =a+2$ for an associated totally\nflexible polymer.\nThe sums in $A_{N,m}$ and $B_{N,m}$ are over all possible values for\n$\\vec{R}$ with $m+1$ components and subjected to the constraint of the total number\nof steps being equal to $N$, that is\n\\begin{equation}\n\\sum_{i=1}^{m+1}R_{i}=N.\n\\label{ea5}\n\\end{equation}\nThe sum in Eq. \\ref{ea3} is just the number of vectors $\\vec{R}$ with $m+1$\ncomponents which obey the constraint Eq. \\ref{ea5}. Since the minimum value\nof each component of $\\vec{R}$ is equal to 1, it is convenient to\ndefine $r_{i}=R_{i}-1$ and therefore $A_{N,m}$ is the number of ways\nto put the $N-m-1$ remaining steps into the $m+1$ components of $\\vec{R}$\n\\begin{equation}\nA_{N,m}=\\frac{(N-1)!}{m!(N-m-1)!}.\n\\label{ea6}\n\\end{equation}\nThe sum $B_{N,m}$ may then be rewritten as\n\\begin{equation}\nB_{N,m}=\\sum_{\\vec{r}^N_m}\\sum_{i=1}^{m+1}(1+r_{i})^{2},\n\\label{ea7}\n\\end{equation}\nwhere each components $r_{i}$ assumes values between 0 and $N-m-1$\nsubject to the constraint of Eq. \\ref{ea5}\n\\begin{equation}\n\\sum_{i=1}^{m+1}r_{i}=N-m-1.\n\\label{ea8}\n\\end{equation}\nThe calculation of $B_{N,m}$ is given in Appendix \\ref{a2}, and the result\nis\n\\begin{equation}\nB_{N,m}=\\frac{(m+1)(2N-m)N!}{(m+2)!(N-m-1)!}.\n\\label{ea9}\n\\end{equation}\nPerforming the sum in the denominator of Eq. \\ref{ea2} taking Eq.\n\\ref{ea6} into account, we have\n\\begin{equation}\n\\langle R^{2}\\rangle =\\frac{N}{[1+a]^{N-1}}\n\\left[2(N+1)\\sum_{m=0}^{N-1}\\left(\\begin{array}{c}\nN-1\\\\m\n\\end{array}\\right)\\frac{a^{m}}{m+2}-\\sum_{m=0}^{N-1}\\left(\\begin{array}{c}\nN-1\\\\m\n\\end{array}\\right)a^{m}\\right].\n\\label{ea10}\n\\end{equation}\nThe first sum may be calculated by noting that\n\\begin{equation}\n\\int_{0}^{A}x(1+x)^{N-1}dx=A^{2}\\sum_{m=0}^{N-1}\\left(\\begin{array}{c}\nN-1\\\\m\n\\end{array}\\right)\\frac{A^{m}}{m+2},\n\\label{ea11}\n\\end{equation}\nand therefore\n\\begin{equation}\n\\sum_{m=0}^{N-1}\\left(\\begin{array}{c}\nN-1\\\\m\n\\end{array}\\right)\\frac{a^{m}}{m+2}=\\frac{[1+a]^{N}[aN-1]+1}{N(N+1)a^{2}}.\n\\label{ea12}\n\\end{equation}\nSubstituting this result in Eq. \\ref{ea10} and performing the second sum we\nfinally get the expression\n\\begin{equation}\n\\langle R^{2}\\rangle\n=\\frac{2[1+a]}{a^{2}}\\left[Na-1+\\frac{1}{[1+a]^{N}}\\right]-N.\n\\label{ea13}\n\\end{equation}\n\n\\section{DERIVATION OF $B_{N,m}$}\n\\label{a2}\n\nIn this appendix we want to derive Eq. \\ref{ea9} for $B_{N,m}$.\nUsing Eq. \\ref{ea8} and defining for convenience ${\\cal N} = N-m-1$ the\nequation Eq. \\ref{ea7} is rewritten as\n\\begin {equation}\nB_{N,m}= (m+1) \\sum_{j=0}^{{\\cal N} +1}\\frac{ ({\\cal N} +m\n-j)!j^2}{({\\cal N} +1-j)! (m-1)!},\n\\label{eb1}\n\\end{equation}\nRedefining the summation variable with\n$i={\\cal N} +1-j$ this equation turns out to be\n\\begin {equation}\nB_{N,m}=(m+1)\\sum_{i=0}^{\\cal N} ( {\\cal N}+1-i)^2\\frac{(i+m-1)!}{i!(m-1)!}.\n\\label{eb2}\n\\end{equation}\nUsing the equality\n\\begin {equation}\n\\sum_{i=0}^{N}\\frac{(m+i)!}{m!i!}=\\frac{(m+N+1)!}{(m+1)!N!},\n\\label{eb3}\n\\end{equation}\nwe get after some manipulations\n\\begin {eqnarray}\nB_{N,m}&=&(m+1)\\left\\{({\\cal N}+1)\\frac{({\\cal N}+1+m)!}{{\\cal N}!m!} \n-2({\\cal N}+1)\\frac{m({\\cal N}+1+m)!}{{\\cal N}!(m+1)!} \\right.\\nonumber\\\\\n&&\\left. \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;+ \\sum_{i=1}\n^{{\\cal N}+1} \\frac{(i+m-1)!i}{(i-1)!(m-1)!}\\right\\}.\n\\label{eb4}\n\\end{eqnarray}\n\nThe last summation to be dealt with is just\n\\begin{equation}\n\\sum_{i=1}^{{\\cal N} +1} \\frac{(i+m-1)!i}{(i-1)!(m-1)!}.\n\\label{eb5}\n\\end{equation}\nDefining $j=i-1$ it follows that\n\\begin{equation}\n\\sum_{j=0}^{{\\cal N}} \\frac{(j+m)!(j+1)}{j!(m-1)!} = \\frac{m({\\cal N}\n+m +1)!}{{\\cal N} !(m+1)!} + m(m+1)\\frac{({\\cal N}+m+1)!}{({\\cal\nN}-1)!(m+2)!}.\n\\label {eb6}\n\\end {equation}\nSubstitution in Eq. \\ref{eb4} leads to\n\\begin{equation}\nB_{N,m}= \\frac {(m+1)({\\cal N} +m +1)! [2{\\cal N} +m +2]}{(m+2)!{\\cal\nN}!}.\n\\label{eb7}\n\\end{equation}\n\nSubstituting ${\\cal N}=N-m-1$, we get\n\\begin{equation}\nB_{N,m}=\\frac{(m+1)(2N-m)N!}{(m+2)!(N-m-1)!}.\n\\label{eb8}\n\\end{equation}\n\n\\begin{references}\n\n\\bibitem[*]{l}On a leave from Departamento de F\\'{\\i}sica, Universidade\nFederal de Santa Catarina.\n\n\\bibitem{f53}P.J.Flory, {\\it Principles of Polymer Chemistry}(Cornell\nUniversity, Ithaca, New York,1953).\n\n\\bibitem{dg79}P.G. de Gennes, {\\it Scaling Concepts in Polymer Physics}\n(Cornell University Press, Ithaca, New York,1979).\n\n\\bibitem{n82}B. Nienhuis, Phys. Rev. Lett.{\\bf 49},1062 (1982).\n\n\\bibitem{ts75}M.F. Thorpe and W.K.Scholl, J. Chem. Phys. {\\bf 75}, 5143\n(1981); W.Scholl and A.B. Thorpe, J. Chem. Phys. {\\bf76}, 6386\n(1982); J.W.Halley, H. Nakanishi, and R. Sundarajan, Phys.Rev.{\\bf\nB31}, 293 (1985); S. B. Lee and H. Nakanishi, Phys.Rev.{\\bf B33},\n1953 (1986); M.L. Glasser, V. Privman, and A. M. Szpilka, J. Phys.\n{\\bf A19}, L1185 (1986); V. Privman and S. Redner, Z.\nPhys,{\\bf B67},129 (1987); V. Privman and H.L.Frish, J. Chem. Phys.\n{\\bf 88}, 469 (1988); J.W.Halley, D. Atkatz, and H. Nakanishi, J.\nPhys. {\\bf A23}, 3297 (1990).\n\n\\bibitem{mn91}J. Moon and H. Nakanishi, Phys. Rev.{\\bf A44}, 6427 (1991).\n\n\\bibitem{cfs92}C.J.Camacho, M.E.Fisher, and J.P.Straley, Phys.Rev.\n{\\bf A46}, 6300 (1992).\n\n\\bibitem{b82}R.J.Baxter, {\\it Exactly Solved Models in Statistical\nMechanics} (Academic, london, 1982).\n\n\\bibitem{m92}F. Moraes, J. Physique {\\bf I2}, 1657 (1992); F. Moraes,\nMod. Phys. Lett {\\bf B8},909 (1994).\n\n\\bibitem{dq86}F. Peruggi, F. di Liberto, and G. Monroy, Physica\n{\\bf123A}, 175 (1984); S. L. A. de Queiroz, J. Phys. A {\\bf19}, L433,\n(1986).\n\n\\bibitem{hi98}C.-K. Hu and N. Sh. Izmailian, Phys. Rev. E {\\bf 58},\n1644 (1998).\n\n\\bibitem{sw87}J. F. Stilck and J. C. Wheeler, J. Stat. Phys. {\\bf\n46}, 1 (1987).\n\n\\bibitem{tm96}C. Tsallis and A. C. N. de Magalh\\~aes, Phys. Rep. {\\bf\n268}, 305 (1996).\n\\end{references}\n\n\\begin{figure}\n\\caption{A four-coordinated Cayley tree with a 2-step polymer\nplaced on it. The tree has $N_g=2$ generations, and is embedded \non a cubic lattice. For the polymer shown $R^2=2$.}\n\\label{f1}\n\\end{figure}\n\n\\begin{figure}\n\\caption{The amplitude of $\\langle R^2 \\rangle$ as a function of $z$\nand $y$ for a lattice with $s=1$ and $t=2$ ($q=6$). As expected, the\namplitude diverges as $z \\rightarrow 0$. Since $s=1$, a divergence is\nalso observed as $y \\rightarrow \\infty$.}\n\\label{f2}\n\\end{figure}\n\n\\end{document}\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002150.extracted_bib",
"string": "\\bibitem[*]{l}On a leave from Departamento de F\\'{\\i}sica, Universidade\nFederal de Santa Catarina.\n\n\n\\bibitem{f53}P.J.Flory, {\\it Principles of Polymer Chemistry}(Cornell\nUniversity, Ithaca, New York,1953).\n\n\n\\bibitem{dg79}P.G. de Gennes, {\\it Scaling Concepts in Polymer Physics}\n(Cornell University Press, Ithaca, New York,1979).\n\n\n\\bibitem{n82}B. Nienhuis, Phys. Rev. Lett.{\\bf 49},1062 (1982).\n\n\n\\bibitem{ts75}M.F. Thorpe and W.K.Scholl, J. Chem. Phys. {\\bf 75}, 5143\n(1981); W.Scholl and A.B. Thorpe, J. Chem. Phys. {\\bf76}, 6386\n(1982); J.W.Halley, H. Nakanishi, and R. Sundarajan, Phys.Rev.{\\bf\nB31}, 293 (1985); S. B. Lee and H. Nakanishi, Phys.Rev.{\\bf B33},\n1953 (1986); M.L. Glasser, V. Privman, and A. M. Szpilka, J. Phys.\n{\\bf A19}, L1185 (1986); V. Privman and S. Redner, Z.\nPhys,{\\bf B67},129 (1987); V. Privman and H.L.Frish, J. Chem. Phys.\n{\\bf 88}, 469 (1988); J.W.Halley, D. Atkatz, and H. Nakanishi, J.\nPhys. {\\bf A23}, 3297 (1990).\n\n\n\\bibitem{mn91}J. Moon and H. Nakanishi, Phys. Rev.{\\bf A44}, 6427 (1991).\n\n\n\\bibitem{cfs92}C.J.Camacho, M.E.Fisher, and J.P.Straley, Phys.Rev.\n{\\bf A46}, 6300 (1992).\n\n\n\\bibitem{b82}R.J.Baxter, {\\it Exactly Solved Models in Statistical\nMechanics} (Academic, london, 1982).\n\n\n\\bibitem{m92}F. Moraes, J. Physique {\\bf I2}, 1657 (1992); F. Moraes,\nMod. Phys. Lett {\\bf B8},909 (1994).\n\n\n\\bibitem{dq86}F. Peruggi, F. di Liberto, and G. Monroy, Physica\n{\\bf123A}, 175 (1984); S. L. A. de Queiroz, J. Phys. A {\\bf19}, L433,\n(1986).\n\n\n\\bibitem{hi98}C.-K. Hu and N. Sh. Izmailian, Phys. Rev. E {\\bf 58},\n1644 (1998).\n\n\n\\bibitem{sw87}J. F. Stilck and J. C. Wheeler, J. Stat. Phys. {\\bf\n46}, 1 (1987).\n\n\n\\bibitem{tm96}C. Tsallis and A. C. N. de Magalh\\~aes, Phys. Rep. {\\bf\n268}, 305 (1996).\n"
}
] |
cond-mat0002151
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Cryptoferromagnetic state in superconductor-ferromagnet multilayers
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[
{
"author": "F.S. Bergeret$^{(1)}$"
},
{
"author": "K.B. Efetov$^{(1,2)}$"
},
{
"author": "A.I. Larkin$% ^{(3,2,1)} $"
}
] |
We study a possibility of a non-homogeneous magnetic order (cryptoferromagnetic state) in heterostructures consisting of a bulk superconductor and a ferromagnetic thin layer that can be due to the influence of the superconductor. The exchange field in the ferromagnet may be strong and exceed the inverse mean free time. A new approach based on solving the Eilenberger equations in the ferromagnet and the Usadel equations in the superconductor is developed. We derive a phase diagram between the cryptoferromagnetic and ferromagnetic states and discuss the possibility of an experimental observation of the CF state in different materials. PACS: 74.80.Dm,74.50.+r, 75.10.-b
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[
{
"name": "paper.tex",
"string": "%\\documentstyle[aps,prl,multicol,epsf]{revtex}\n\n\n\\documentstyle[prl,twocolumn,aps,epsf]{revtex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%TCIDATA{OutputFilter=Latex.dll}\n%TCIDATA{LastRevised=Mon Feb 07 10:16:12 2000}\n%TCIDATA{<META NAME=\"GraphicsSave\" CONTENT=\"32\">}\n%TCIDATA{Language=American English}\n%TCIDATA{CSTFile=revtex.cst}\n\n\\begin{document}\n\\title{Cryptoferromagnetic state in superconductor-ferromagnet multilayers}\n\\author{F.S. Bergeret$^{(1)}$, K.B. Efetov$^{(1,2)}$, A.I. Larkin$%\n^{(3,2,1)} $}\n\\address{$^{(1)}$Theoretische Physik III,\\\\\nRuhr-Universit\\\"{a}t Bochum, 44780 Bochum, Germany\\\\\n$^{(2)}$L.D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia \n\\\\\n$^{(3)}$Theoretical Physics Institute, University of\\\\\nMinnesota, Minneapolis, MN 55455, USA}\n\\maketitle\n\n\\begin{abstract}\nWe study a possibility of a non-homogeneous magnetic order\n(cryptoferromagnetic state) in heterostructures consisting of a bulk\nsuperconductor and a ferromagnetic thin layer that can be due to the influence\nof the superconductor. The exchange field in the ferromagnet may be strong\nand exceed the inverse mean free time. A new approach based on solving the\nEilenberger equations in the ferromagnet and the Usadel equations in the\nsuperconductor is developed. We derive a phase diagram between the\ncryptoferromagnetic and ferromagnetic states and discuss the\npossibility of an experimental observation of the CF state in different\nmaterials.\n\nPACS: 74.80.Dm,74.50.+r, 75.10.-b \n\\end{abstract}\n\n%\\begin{multicols}{2}\n\nIn the last years, the interest in experiments on\nsuperconducting-ferromagnet ($S/F$) hybrid structures has grown rapidly.\nSuch structures show the coexistence of these two antagonistic orderings but\ntheir mutual influence is still a controversial point \\cite\n{koorevaar,strunk,jiang,zabel,verbanck,aarts}. In these experiments, the\nmultilayers contained strong ferromagnets like $Fe$ or $Gd$ with the Curie\ntemperature up to $1000K$ and superconductors with transition\ntemperatures not exceeding $10K$, like $Nb$ or $V$. \n\nNaturally, in most theoretical works only the influence of the ferromagnet\non the superconductivity of $S/F$ systems was considered \\cite\n{radovic,sipr,melo}. One may argue that a modification of the magnetic\nordering would need energies of the order of the Curie, which is much larger than the superconducting transition\ntemperature $T_{c}$. Therefore, any change of the ferromagnetic order would\nbe less energetically favorable than the destruction of the\nsuperconductivity in the vicinity of the ferromagnet.\n\nThis simple argument was questioned in a recent experimental work \\cite\n{muhge}, where $Nb/Fe$ bilayers were studied using different experimental\ntechniques. Direct measurements using the ferromagnetic resonance \nshowed that in several samples with thin ferromagnetic layers $(10- 15$%\n\\AA $)$ the average magnetic moment started to decay at the superconducting\ntransition temperature $T_{c}$. The measurements were possible only in a\nlimited range of the temperatures below $T_{c}$ and the decrease of the\nmagnetic moment in this interval reached $10\\%$ without any sign of a\nsaturation. As a possible explanation of the effect, it was assumed in Ref. \\cite{muhge} that the superconductivity affected the magnetic order causing a\ndomain-like structure.\n\nA possibility of a domain-like magnetic structure in presence of\nsuperconductivity has been first suggested by Anderson and Suhl long ago \n\\cite{suhl}. They argued that a weak ferromagnetism of localized electrons\nshould not destroy the superconductivity in the conduction band. Instead, it\nmay become more favorable energetically to build a domain structure called {\\em cryptoferromagnetic state} \\cite{suhl}. Later this state was\ninvestigated both theoretically and experimentally in detail (for\nreview see, e.g.\\cite{review}).\n\nIn this paper, we investigate theoretically the possibility of a\ncryptoferromagnetic-like (CF) state in $S/F$ bilayers with parameters\ncorresponding to the structures used in the experiments \\cite\n{koorevaar,strunk,jiang,zabel,verbanck,aarts,muhge}. Such a study is very\nimportant because it may allow to clarify the question about the\ncryptoferromagnetic state in the experiment \\cite{muhge} and to make\npredictions for other $S/F$ multilayers. From the theoretical point of\nview, large magnetic energies involved make the problem quite non-trivial\nand demand development of new approaches.\n\nTo the best of our knowledge the possibility of a non-homogeneous magnetic\norder in multilayers was considered only in Ref. \\cite{buzdin}. However,\nalthough the authors of Ref. \\cite{buzdin} came to the conclusion that the\ndomain-like structure due to the interaction with the superconductor was\npossible, the results obtained can hardly be used for quantitative\nestimates. For example, they assumed that the period of the structure $b$\nhad to be not only much smaller than the size of the Cooper pair $\\bar{\\xi}$%\n, but also than $\\bar{\\xi}\\sqrt{T_{c}/h}$, where $h$ is the energy of interaction\nof conduction electrons (CEs) with the localized magnetic moments (LMs). In addition, a very rough\nboundary condition at the $S/F$ boundary was used.\n\nIn contrast, we present here a microscopic derivation of the phase diagram\nvalid for realistic parameters of the problem involved. We will show that\nthe phase transition between the CF and ferromagnetic (F) phases is\ncontinuous and the period of the structure $b$ goes to infinity at the\ncritical point. The only restrictions we use are \n\\begin{equation}\nd\\ll \\xi _{F}=v_{0}/h,\\qquad T_{c}\\ll h\\ll \\epsilon _{0} \\label{a1}\n\\end{equation}\nwhere $d$ is the thickness of the ferromagnetic layer, $v_{0}$ and $%\n\\varepsilon _{0}$ are the Fermi-velocity and Fermi-energy.\n\nEven in the such strong ferromagnet as iron, $\\xi _{F}$ is of the order $10$%\n\\AA . For weaker ferromagnets like $Gd$, $\\xi _{F\\text{ \\ }}$is considerably\nlarger and the inequalities (\\ref{a1}) can be fulfilled rather easily.\n\nWe assume that the superconductor occupies the half-space $x>0$ while the\nferromagnetic film occupies the region $-d<x<0$ and write the Hamiltonian as \n\\begin{equation}\nH\\!\\!\\!=\\!H_{BCS}+\\!\\!\\gamma \\!\\!\\int d{\\bf r}\\Psi _{\\alpha }^{+}({\\bf r})\\left[ \n{\\bf h}({\\bf r}){\\bf \\sigma }\\right] _{\\alpha \\beta }\\Psi _{\\beta }({\\bf r}%\n)+H_{M} \\label{hamiltonian}\n\\end{equation}\nwhere $H_{BCS}$ is the usual BCS Hamiltonian (in the presence of\nnon-magnetic impurities) describing the superconducting state in the $S$\nlayer, $\\gamma$ is a constant which will be put to 1 at the end. The second term in Eq. (\\ref{hamiltonian}) stands for the\ninteraction between the LMs of the ferromagnet and the\n CEs, where $\\!{\\bf h}$ is the exchange field and $%\n{\\bf \\sigma }$ is the vector containing the Pauli matrices as components. We neglect the influence of the\nLMs on the orbital motion of the CEs since the exchange interaction is the\ndominant Cooper pair breaking mechanism \\cite{review} for the problem\ninvolved. The term $H_{M}$ describes the interaction between the LM in the ferromagnet.\n\nOur aim is to obtain an expression for the free\nenergy of the system for different magnetic structures in the F layer. To\ndetermine the contribution of an inhomogeneous alignment of magnetic spins\nto the total energy we use the limit of a {\\em continuous} material and\nreplace the spins by classical vectors. We assume that the anisotropy energy\nof the ferromagnet is smaller than the exchange energy and hence there is no\neasy axis of magnetization. This can definitely be a good approximation for\niron with a cubic lattice used in the work \\cite{muhge}. The energy $H_{M}$\nof a non-homogeneous structure can be written in the continuum limit as \n\\begin{equation}\nH_{M}=\\int J\\left[ \\left( {\\bf \\nabla }S_{x}\\right) ^{2}+\\left( {\\bf \\nabla }%\nS_{y}\\right) ^{2}+\\left( {\\bf \\nabla }S_{z}\\right) ^{2}\\right] dV,\n\\label{exchange-energy}\n\\end{equation}\nwhere the magnetic stiffness $J$ characterizes the strength of the coupling\nbetween LMs in the F layer and $S_{i}$'s are the components of a unit vector. Writing ${\\bf S}=\\left( 0,-\\sin \\Theta , \\cos \\Theta \\right)$\nand minimizing the energy $H_{M}$ we obtain the equation $\\Delta \\Theta=0$. We consider only the solutions of this equation that are of interest for us:\n\\begin{equation}\na)\\text{ }\\Theta =0,\\text{ \\ \\ \\ \\ \\ \\ }b)\\text{\\ }\\Theta =Qy \\label{a3}\n\\end{equation}\nThe solution a) in Eq. (\\ref{a3}) corresponds to the F state, whereas the solution b) describes a CF state with a homogeneously rotating magnetic moment. The wave\nvector of this rotation is denoted by $Q$. The magnetization is chosen to be\nparallel to the FS interface, {\\em i.e}. to the $yz-$plane. This allows to\nneglect Meissner currents in the superconductor. With all this assumptions the \n magnetic energy $\\Omega _{M}$ (per unit surface area)\nis given by \n\\begin{equation}\n\\Omega _{M}=JdQ^{2} \\label{magnetic}\n\\end{equation}\nThe corresponding energy of the F state equals zero.\n\nThe superconducting part of the energy can be calculated deriving from Eq. (%\n\\ref{hamiltonian}) proper Eilenberger equations \\cite{eilen} for the\nsuperconductor and the ferromagnet, solving these equations and then\nmatching the solutions. In practice, this is difficult and we simplify the\nproblem considering the ``dirty limit'' $l\\ll \\xi _{0}$, where $l$ is the\nmean free path and $\\xi _{0}=v/T_{c}$ is the coherence length of the\nsuperconductor in the clean limit, which allows to use the more simple Usadel\nequations \\cite{usadel}. If we assume that $|\\tau |\\ll 1$, $\\tau =\\left(\nT-T_{c}\\right) /T_{c}$, the Usadel equations together with the\nself-consistency equation can be further reduced to the Ginzburg-Landau (GL)\nequation\\cite{agd,degennes,abrikosov}. However, the latter equation can be\nused only sufficiently far from the $S/F$ boundary at distances exceeding $%\n\\bar{\\xi}\\sim \\sqrt{\\xi _{0}l}$. At the distances of the order of $\\bar{\\xi}$\none should write again the Usadel equations but in the limit $|\\tau |\\ll 1$\nthey can be linearized. This is a conventional scheme of calculation for\ninterfaces between superconductors and normal metals or ferromagnets.\n\nWriting the Usadel equations in the ferromagnet may not be a good\napproximation because the exchange energy $h$ in realistic cases is not\nnecessarily smaller than $1/\\tau _{tr}$, where $\\tau _{tr}$ the mean free\ntime, and so one should write in this region the Eilenberger equations. At\nthe end one should match the solutions of all the equations.\n\nNow we start the calculations following this program. The loss of the\nsuperconducting energy due to the suppression of the superconductivity in\nthe $S$-layer can be found from the solution of the GL equation for the\norder parameter $\\Delta ({\\bf r})$. At distances $x\\gg \\bar{\\xi}$, the\nproper solution is \\cite{agd,degennes,abrikosov} \n\\begin{equation}\n\\Delta (x)=\\Delta (T)\\tanh \\left( \\frac{x}{\\sqrt{2}\\xi (T)}+C\\right) ,\n\\label{orderparameter}\n\\end{equation}\nwhere $\\Delta (T)=\\sqrt{\\frac{8\\pi ^{2}}{7\\zeta (3)}|\\tau |}T_{c}\\equiv\n\\Delta _{0}\\tau ^{1/2}$ is the value of the order parameter in the bulk\nsuperconductor, $\\xi (T)=\\sqrt{\\frac{\\pi D}{8T_{c}}}|\\tau |^{-1/2}$ is the\ncharacteristic scale of the spacial variation of $\\Delta \\left( {\\bf r}%\n\\right) $, $D$ is the diffusion coefficient in the superconductor, and $C$\nis a constant. Substituting $\\Delta \\left( x\\right) $, Eq. (\\ref\n{orderparameter}), into the GL free energy functional one can evaluate the\nloss of the superconducting energy at the $F/S$ interface per unit surface\narea as function of $C$\\cite{degennes} \n\\begin{equation}\n\\Omega _{S}=\\frac{\\sqrt{\\pi }}{6\\sqrt{2}}|\\tau |^{3/2}\\left( 2+K\\right)\n(1-K)^{2} \\label{energysuper}\n\\end{equation}\nwhere $K=\\tanh C$. The influence of the ferromagnet on the superconductivity\nis determined by the parameter $K$ that will be found by minimizing the\ntotal energy.\n\nThe contribution $\\Omega _{M/S}$ of the second term in (\\ref{hamiltonian})\nto the total energy has still to be determined. First, we write the\nEilenberger equation for the magnetic moment ${\\bf h}\\left( {\\bf r}\\right) $\ndepending on coordinates. Introducing the quasiclassical matrix Green\nfunction $\\check{g}_{\\omega }({\\bf r},{\\bf p}_{0})$ \n\\[\n\\check{g}=\\left( \n\\begin{array}{cc}\n\\hat{g} & -\\hat{f} \\\\ \n\\hat{f}^{+} & -\\hat{g}^{+}\n\\end{array}\n\\right) \n\\]\none derives in the standard way the Eilenberger equation in the spin$\\otimes \n$particle-hole space \n\\begin{equation}\n\\left[ \\left\\{ \\omega \\check{\\tau}_{3}-i\\check{\\Delta}+i\\gamma \\check{V}+i%\n\\check{\\Sigma}{\\it _{imp}}\\right\\} ,\\check{g}\\right] +{\\bf v}_{0}\\nabla _{%\n{\\bf r}}\\check{g}=0\\,. \\label{eilenbergercompleta}\n\\end{equation}\nwhere ${\\bf p}_{0}$ and ${\\bf v}_{0}$ are the momentum and velocity at the\nFermi-surface.\n\nIn Eq. (\\ref{eilenbergercompleta}), $\\check{\\tau}_{i}$, $i=1,2,3$, are Pauli\nmatrices in the particle-hole $\\!$ space, $\\check{\\Delta}\\!\\!=\\!\\!\\check{\\tau}_{1}\\otimes\ni\\sigma _{y}\\Delta ({\\bf r})$, $\\check{V}={\\rm Re}\\left( {\\bf h}({\\bf r})%\n{\\bf \\sigma }\\right) \\otimes \\check{1}+{\\rm Im}\\left( {\\bf h}({\\bf r}){\\bf %\n\\sigma }\\right) \\otimes \\check{\\tau}_{3}$, and $\\Delta $ should be\ndetermined self-consistently \n\\begin{equation}\n\\Delta ({\\bf r})=-\\frac{i}{2}\\pi \\nu \\lambda _{0}T\\sum_{n}<f_{12}({\\bf r},%\n{\\bf p}_{0},\\omega _{n})>_{0}\\,, \\label{selbstconsist}\n\\end{equation}\nwhere $\\left\\langle ...\\right\\rangle _{0}$ denotes averaging over the Fermi\nvelocity and $\\lambda _{0}$ is the constant of the electron-electron\ninteraction, $\\nu $ is the density of states. We assume for simplicity that $%\n\\lambda _{0}=0$ and hence $\\Delta =0$ in the ferromagnet. At the same time, $%\nh=0$ in the superconductor. The term $i\\check{\\Sigma}_{{\\it imp}}$ describes\nscattering by impurities. For a short range interaction, $\\check{\\Sigma}_{%\n{\\it imp}}=-\\frac{i}{2\\tau }\\left\\langle \\check{g}\\right\\rangle _{0}$. Eq. (%\n\\ref{eilenbergercompleta}) is complemented by the normalization condition $%\n\\check{g}^{2}=\\check{1}$. Once we know $\\hat{g}$, we can determine $\\Omega\n_{M/S}$ using the expression \\cite{agd}: \n\\begin{equation}\n\\Omega _{M/S}=-i\\pi T\\nu _{0}\\sum_{\\omega }\\int_{0}^{1}d\\gamma \\int d^{3}%\n{\\bf r}({\\bf h\\sigma })_{\\alpha \\beta }\\left\\langle g_{\\beta \\alpha\n}\\right\\rangle _{0}\\, \\label{energiaMS}\n\\end{equation}\nNear $T_{c}$, the anomalous functions $\\hat{f}$ and $\\hat{f}^{+}$ are small\nand $\\hat{g}\\approx sgn\\left( \\omega \\right) $. Then, in the limit $T_{c}\\ll\nh$ the off-diagonal component (1,2) in particle-hole space of the equation (%\n\\ref{eilenbergercompleta}) in the region $-d<x<0$ is\n\n\\begin{eqnarray}\n{\\bf v}_{0}\\nabla \\hat{f}\\!\\! &=&\\!\\!-i\\hat{V}\\hat{f}^{(F)}+i\\hat{f}^{(F)}%\n\\hat{V}^{\\ast }\\!\\!-\\frac{sgn\\left( \\omega \\right) }{\\tau }(\\hat{f}^{(F)}-<\\!%\n\\hat{f}^{(F)}\\!>) \\label{eilenberger12} \\\\\n\\hat{V} &=&h(x)\\sigma _{z}\\exp (iQy\\sigma _{x}) \\nonumber\n\\end{eqnarray}\n$h$ is the strength of the exchange field in the $F$-layer.\n\nAssuming that $d\\ll v_{0}/h$ we can relate the values of the function $\\hat{f%\n}^{(F)}({\\bf v}_{{\\bf 0}},{\\bf r})$ at the interface, {\\em i.e.} at $x=0^{-}$\nto the values at the boundary to the vacuum at $x=-d$ using the Taylor\nexpansion: \n\\begin{equation}\n\\hat{f}^{(F)}({\\bf v}_{0},{{\\bf r}_{0}}-{\\bf r}_{{\\bf d}})\\approx \\hat{f}%\n^{(F)}({\\bf v}_{0},{{\\bf r}_{0}})-d\\partial _{x}\\hat{f}^{(F)}({\\bf v}_{0},%\n{\\bf r}_{0})\\,, \\label{Taylor}\n\\end{equation}\nwhere ${{\\bf r}_{0}}=(0,y,z)$ and ${\\bf r}_{{\\bf d}}=(-d,y,z)$ . Applying\ngeneral boundary conditions \\cite{zaitsev} to the problem involved we\nconclude that for a perfectly transparent interface the function $\\hat{f}$\nis continuous at the interface. At the boundary with the vacuum ($x=-d$) the\nfunction $\\hat{f}$ satisfies \n\\begin{equation}\n\\hat{f}^{(F)}(v_{x},{\\bf r}_{0}-{\\bf r}_{{\\bf d}})=\\hat{f}^{(F)}(-v_{x},{\\bf %\nr}_{0}-{\\bf r}_{{\\bf d}}) \\label{boundaryvacuum}\n\\end{equation}\nUsing Eqs. (\\ref{eilenberger12}, \\ref{Taylor}, \\ref{boundaryvacuum}) and the\ncontinuity of $\\hat{f}$ at ${\\bf r}={\\bf r}_{0}$ the problem is reduced to the solving of the Usadel\nequation in the superconductor with the following effective boundary\ncondition at the interface \n\\begin{equation}\n\\eta D(\\partial _{x}+d\\partial _{y}^{2})\\hat{f}_{0}({\\bf r}_{0})+isgn\\left(\n\\omega \\right) \\left( -\\hat{V}\\hat{f}_{0}+\\hat{f}_{0}\\hat{V}^{\\ast }\\right)\n_{\\!\\!{\\bf r}_{0}}\\!\\!\\!\\!=\\!\\!0 \\label{boundaryinterface}\n\\end{equation}\nwhere $\\eta =v_{0}^{F}/v_{0}^{S}$ and $\\hat{f}_{0}$ is the zero harmonics of\nthe function $\\hat{f}$ in the superconductor. When deriving Eq. (\\ref\n{boundaryinterface}) we used the fact that the Usadel equation \nis applicable in the $S$- layer at distances down to the mean free path $l$\nand extrapolated its solution to the interface. Only first two spherical\nharmonics $\\hat{f}^{(s)}\\approx \\hat{f}_{0}+{\\bf v}_{{\\bf 0}}\\hat{{\\bf f}}%\n_{1}$ were kept in the derivation.\n\nThe linearized Usadel equation for the superconductor can be written in the\nstandard form \n\\begin{equation}\nD\\nabla ^{2}\\hat{f}_{0}-2|\\omega |\\hat{f}_{0}-2\\Delta (x)\\sigma _{y}=0\n\\label{a10}\n\\end{equation}\n\n The general solution of Eq. (\\ref{a10}) with the boundary condition, Eq. (\\ref{boundaryinterface}), and $\\hat{V}$ from Eq. (\\ref\n{eilenberger12}) can be written as\n\\begin{equation}\n\\hat{f}_{0}({\\bf r},\\omega )=\\alpha _{\\omega }(x)\\sigma _{x}e^{-i\\sigma\n_{x}Qy}+\\beta _{\\omega }(x)i\\sigma _{y\\,,} \\label{ansatz2}\n\\end{equation}\nwhere $\\alpha _{\\omega }(x)=C_{\\omega }\\exp \\left( -\\sqrt{Q^{2}+\\frac{%\n2|\\omega |}{D}}x\\right) $ and $\\beta _{\\omega }(x)=-i\\frac{\\Delta (x)}{%\n|\\omega |}+B_{\\omega }\\exp \\left( -\\sqrt{\\frac{2|\\omega |}{D}}x\\right) $.\nEq. (\\ref{ansatz2}) is applicable at distances much smaller than $\\xi \\left(\nT\\right) $, where the solution for $\\Delta $\ncan be approximated by a linear function. One can check using the\nself-consistency Eq. (\\ref{selbstconsist}) that the relative correction to $%\n\\Delta $ coming from the exponentially decaying part of Eq. (%\n\\ref{ansatz2}), is of the order $\\left( \\ln \\frac{\\omega _{D}}{T_{c}}\\right)\n^{-1}$, where $\\omega _{D}$ is the Debye frequency, and we neglected it. The\ncoefficients $C_{\\omega }$ and $B_{\\omega }$ can be now determined from Eq. (%\n\\ref{boundaryinterface}). Using the condition $\\check{g}^{2}=1$ and Eq. (\\ref\n{energiaMS}) we can find the energy $\\Omega _{M/S}$. Introducing the\ndimensionless parameters: \n\\begin{equation}\na^{2}\\equiv \\frac{2h^{2}d^{2}}{DT_{c}\\eta ^{2}}\\text{, \\ }q^{2}\\equiv \\frac{%\nDQ^{2}}{2T_{c}}\\text{, \\ \\ }\\widetilde{\\Omega }\\equiv \\frac{\\Omega }{\\nu\n_{F}\\Delta _{0}^{2}}\\sqrt{\\frac{2T_{c}}{D}} \\label{a100}\n\\end{equation}\nand using Eq. (\\ref{orderparameter}) one obtains \n\\begin{eqnarray}\n\\widetilde{\\Omega }_{M/S} &=&\\frac{\\pi }{2}F_{3/2,1}K^{2}|\\tau |+\\sqrt{2}%\nF_{2,1}K\\left( 1-K^{2}\\right) |\\tau |^{3/2} \\nonumber \\\\\n&&+\\pi ^{-1}F_{5/2,1}\\left( 1-K^{2}\\right) ^{2}|\\tau |^{2}, \\label{a11} \\\\\nF_{m,l} &=&\\eta \\frac{4a^{2}}{\\pi ^{3/2-m}}\\sum_{n>0}\\alpha _{n}^{-m}\\left[ \n\\sqrt{\\alpha _{n}\\left( \\alpha _{n}+q^{2}\\right) }+a^{2}\\right] ^{-l} \n\\nonumber\n\\end{eqnarray}\nwhere $\\alpha _{n}=\\pi (2n+1)$ and $\\nu _{F}$ is the density of states in\nthe ferromagnet.The total energy is given by $\\widetilde{\\Omega }=\\widetilde{\\Omega }_{M}+%\n\\widetilde{\\Omega }_{S}+\\widetilde{\\Omega }_{M/S}$, Eqs. (\\ref{magnetic}, \\ref{energysuper}, \\ref{a11}) and is a functions of two parameters, $K$ and $q$, that should be\ndetermined from the conditions $\\partial \\widetilde{\\Omega }/\\partial\nK=\\partial \\widetilde{\\Omega }/\\partial q=0$. The parameter $q$ is in fact\nthe order parameter for the CF state. Close to the CF-F transition this\nparameter is small and one can expand the energy $\\widetilde{\\Omega }_{M/S}$%\n, Eq. (\\ref{a11}), in $q^{2}$. As concerns the value $K_{0}$ at the minimum,\nit can be found near the transition minimizing $\\widetilde{\\Omega }_{M/S}$\nat $q=0$. As a result, the first terms of the expansion of the energy $%\n\\widetilde{\\Omega }$ in $q^{2}$ near the CF-F transition can be written as\n\n\\begin{equation}\n\\scriptstyle \n\\begin{array}{l}\n\\widetilde{\\Omega }\\approx \\widetilde{\\Omega }_{s}(K_{0})+\\widetilde{\\Omega }%\n_{M/S}(K_{0},q=0)- \\\\ \n-\\frac{q^{2}}{2}\\left[ \\frac{\\pi }{2}F_{3/2,2}K_{0}^{2}|\\tau |+\\sqrt{2}%\nF_{2,2}K_{0}(1-K_{0}^{2})|\\tau |^{\\frac{3}{2}}+\\right. \\\\ \n\\left. +\\pi ^{-1}F_{5/2,2}\\left( 1-K_{0}^{2}\\right) ^{2}|\\tau |^{2}-2\\lambda \n\\right] _{q=0}\n\\end{array}\n\\label{a13}\n\\end{equation}\n\nOne can check that the term proportional to $q^{4}$ is positive, which means\nthat the CF-F transition is of the second order. This is in contrast to the\nconclusion of Ref. \\cite{buzdin}. The parameter $\\lambda $ in Eq. (\\ref{a13}%\n) is \n\\begin{equation}\n\\lambda \\equiv \\frac{Jd}{\\nu \\sqrt{2T_{c}D^{3}}}\\frac{7\\zeta (3)}{2\\pi ^{2}}\n\\label{a14}\n\\end{equation}\nAccording to the Landau theory of phase transitions the transition from the\nF state ($q=0$) to the CF state ($q\\neq 0$) should occur when the\ncoefficient in the second-order term turns to zero. The phase diagram for\nthe variables $h$ and $J $, Eqs. (\\ref{a100}, \\ref{a14}), is\nrepresented in Fig.1. The curves are plotted for different values of $|\\tau |\n$. The function $\\widetilde{\\Omega }(q)$ has only one minimum at $q_{0}$ continuously going to zero as the system approaches the transition point.This demonstrates that the transition is of second order. Not close to the transition point $Q\\sim \\bar{\\xi}^{-1}$. \n\nThe stiffness $J$ for materials like $Fe$ and $Ni$ is $\\approx 60K/$\\AA .\nUsing the data for Nb $T_{c}\\!=\\!10$\\AA , $v_{F}\\!=\\!1,37.10^{8}$cm/s, setting $%\nl\\!=\\!100 $\\AA , $d\\!=\\!10$\\AA , and $h\\!=\\!10^{4}$K, which is proper for iron, and\nassuming that the Fermi velocities and energies of the ferromagnet and\nsuperconductor are close to each other we obtain $a\\approx 25$ and $\\lambda\n\\sim 6.10^{-3}$. It is clear from Fig.1 that the CF state is hardly possible in the $Fe/Nb$ structure studied in \n\\cite{muhge}.\n\nHow can one explain the decay of the average magnetic moment below $T_{c}$\nobserved in that work? This can be understood if one assumes that there were\n``islands'' in the magnetic layers with smaller values of $J$ and/or $h$. A\nreduction of these parameters in the multilayers $Fe/Nb$ is not unrealistic\nbecause proximity to $Nb$ leads to formation of non-magnetic ``dead'' layers \n\\cite{zabel}, and can affect the parameters of the ferromagnetic layers,\ntoo. If the CF state were realized only on the islands, the average magnetic\nmoment would be reduced but remain finite, which would correlate with the\nexperiment \\cite{muhge}. One can also imagine islands very\nweakly connected to the rest of the layer, which would lead to smaller\nenergies of a non-homogeneous state.\n\nAnother possibility to observe the CF state would be to use\nmultilayers with a weaker ferromagnet. A good candidate for this purpose\nmight be $Gd/Nb$. The exchange energy $h$ in $Gd$ is $h\\approx 10^{3}K$ and\nthe Curie temperature and, hence, the stiffness $J$ is $3$ times smaller\nthan in $Fe$. So, one can expect $a\\approx 2.5$ and $\\lambda \\approx\n2.10^{-3}$. Using Fig.1 we see that the CF phase is possible for these\nparameters. One can also considerably reduce the exchange energy $h$ in $%\nV_{1-x}Fe_{x}/V$ multilayers \\cite{aarts} varying the alloy composition.\nHopefully, the measurements that would allow to check the existence of the CF\nphase in these multilayers will be performed in the nearest future.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{figure}\n\\epsfysize = 5.5cm\n\\vspace{0.2cm}\n\\centerline{\\epsfbox{finalfig1.eps\n }}\n\\vspace{0.2cm}\n\\caption{Phase diagrams ($\\protect\\lambda ,a$). The area above (below) the\ncurves corresponds to the F (CF) state }\n\\end{figure}\n\n\nIn conclusion, we studied a possibility of the CF state in ($S/F$)\n multilayers. We derived a phase diagram that\nallows to make definite predictions for real materials.\n\nWe are grateful to I.A. Garifullin for numerous discussion of experiments and to D. Taras-Semchuck and F.W.J. Hekking for helpful discussions.\nF.S.B. and K.B.E. thank SFB 491 {\\it Magnetische Heterostrukturen }for a\nsupport. The work of A.I.L. was supported by the NSF grant DMR-9812340 and\nthe A.v. Humboldt Foundation.\n\n\\begin{references}\n\\bibitem{koorevaar} P.Koorevaar {\\it et al.}, Phys. Rev. B {\\bf 49}, 441 (1994)\n\n\\bibitem{strunk} C. Strunk {\\it et al.}, Phys. Rev. B {\\bf 49}, 4053 (1994)\n\n\\bibitem{jiang} J.S. Jiang {\\it et al.}, Phys.Rev. Lett. {\\bf 74}, 314 (1995)\n\n\\bibitem{zabel} Th. M\\\"{u}hge {\\it et al.}, Phys. Rev. Lett. {\\bf 77}, 1857 (1996)\n\n\\bibitem{verbanck} G. Verbanck {\\it et al.}, Phys. Rev. B {\\bf 57}, 6029 (1998)\n\n\\bibitem{aarts} J. Aarts {\\it et al.}, Phys. Rev. B {\\bf 56, }2779 (1997)\n\n\\bibitem{radovic} Z. Radovic {\\it et al.}, Phys. Rev. B {\\bf 38}, 2388 (1988); Z. Radovic {\\it et al.}, Phys. Rev. B{\\bf 44},\n759 (1991); A. I. Buzdin {\\it et al.}, Physica (Amsterdam), {\\bf 185C}, 2025 (1991)\n\n\\bibitem{sipr} O. Sipr {\\it et al.}, J.Phys. Cond.Matt. {\\bf 7}, 5239\n(1995)\n\n\\bibitem{melo} C.A.R. Sa de Melo, Phys. Rev. Lett. {\\bf 79}, 1933 (1997)\n\n\\bibitem{muhge} Th. M\\\"{u}hge {\\it et al.}, Physica C, {\\bf %\n296}, 325 (1998)\n\n\\bibitem{suhl} P.W.Anderson and H. Suhl, Phys. Rev. {\\bf 116}, 898 (1959)\n\n\\bibitem{review} L.N.Bulaevskii{\\it et al.}, Advances in Physics {\\bf 34}, 175 (1985)\n\n\\bibitem{buzdin} A.I.Buzdin and L.N. Bulaevskii, Sov. Phys. JETP {\\bf 67}, 576 (1988)\n\n\\bibitem{eilen} G. Eilenberger, Z. Phys. {\\bf 214},195 (1968); A.I. Larkin\nand Yu.N. Ovchinnikov, Sov. Phys. JETP {\\bf 28},1200 (1969)\n\n\\bibitem{usadel} K.D.Usadel, Phys. Rev. Lett. {\\bf 25}, 507 (1970)\n\n\\bibitem{agd} A. Abrikosov, L. Gorkov and I. Dzyaloshinski{\\em , Methods of\nQuantum Field Theory in Statistical Physics} (Dover Publications, N.Y.,1963)\n\n\\bibitem{degennes} P.G. de Gennes, {\\em Superconductivity of Metals and\nAlloys} (Benjamin, 1966)\n\n\\bibitem{abrikosov} A.A.Abrikosov, {\\em Fundamentals of the Theory of Metals%\n} (North-Holland, Amsterdam,1988)\n\n\\bibitem{zaitsev} A.V.Zaitsev, Sov. Phys. JETP {\\bf 59},1015 (1985)\n\\end{references}\n\n%\\end{multicols}\n\n\\end{document}\n"
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[
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"name": "cond-mat0002151.extracted_bib",
"string": "\\bibitem{koorevaar} P.Koorevaar {\\it et al.}, Phys. Rev. B {\\bf 49}, 441 (1994)\n\n\n\\bibitem{strunk} C. Strunk {\\it et al.}, Phys. Rev. B {\\bf 49}, 4053 (1994)\n\n\n\\bibitem{jiang} J.S. Jiang {\\it et al.}, Phys.Rev. Lett. {\\bf 74}, 314 (1995)\n\n\n\\bibitem{zabel} Th. M\\\"{u}hge {\\it et al.}, Phys. Rev. Lett. {\\bf 77}, 1857 (1996)\n\n\n\\bibitem{verbanck} G. Verbanck {\\it et al.}, Phys. Rev. B {\\bf 57}, 6029 (1998)\n\n\n\\bibitem{aarts} J. Aarts {\\it et al.}, Phys. Rev. B {\\bf 56, }2779 (1997)\n\n\n\\bibitem{radovic} Z. Radovic {\\it et al.}, Phys. Rev. B {\\bf 38}, 2388 (1988); Z. Radovic {\\it et al.}, Phys. Rev. B{\\bf 44},\n759 (1991); A. I. Buzdin {\\it et al.}, Physica (Amsterdam), {\\bf 185C}, 2025 (1991)\n\n\n\\bibitem{sipr} O. Sipr {\\it et al.}, J.Phys. Cond.Matt. {\\bf 7}, 5239\n(1995)\n\n\n\\bibitem{melo} C.A.R. Sa de Melo, Phys. Rev. Lett. {\\bf 79}, 1933 (1997)\n\n\n\\bibitem{muhge} Th. M\\\"{u}hge {\\it et al.}, Physica C, {\\bf %\n296}, 325 (1998)\n\n\n\\bibitem{suhl} P.W.Anderson and H. Suhl, Phys. Rev. {\\bf 116}, 898 (1959)\n\n\n\\bibitem{review} L.N.Bulaevskii{\\it et al.}, Advances in Physics {\\bf 34}, 175 (1985)\n\n\n\\bibitem{buzdin} A.I.Buzdin and L.N. Bulaevskii, Sov. Phys. JETP {\\bf 67}, 576 (1988)\n\n\n\\bibitem{eilen} G. Eilenberger, Z. Phys. {\\bf 214},195 (1968); A.I. Larkin\nand Yu.N. Ovchinnikov, Sov. Phys. JETP {\\bf 28},1200 (1969)\n\n\n\\bibitem{usadel} K.D.Usadel, Phys. Rev. Lett. {\\bf 25}, 507 (1970)\n\n\n\\bibitem{agd} A. Abrikosov, L. Gorkov and I. Dzyaloshinski{\\em , Methods of\nQuantum Field Theory in Statistical Physics} (Dover Publications, N.Y.,1963)\n\n\n\\bibitem{degennes} P.G. de Gennes, {\\em Superconductivity of Metals and\nAlloys} (Benjamin, 1966)\n\n\n\\bibitem{abrikosov} A.A.Abrikosov, {\\em Fundamentals of the Theory of Metals%\n} (North-Holland, Amsterdam,1988)\n\n\n\\bibitem{zaitsev} A.V.Zaitsev, Sov. Phys. JETP {\\bf 59},1015 (1985)\n"
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cond-mat0002152
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Magnetic field of an in-plane vortex \\ inside and outside a layered superconducting film
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"author": "Edson Sardella"
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In the present work we study an anisotropic layered superconducting film of finite thickness. The film surfaces are considered parallel to the $bc$ face of the crystal. The vortex lines are oriented perpendicular to the film surfaces and parallel to the superconducting planes. We calculate the local field and the London free energy for this geometry. Our calculation is a generalization of previous works where the sample is taken as a semi-infinite superconductor. As an application of this theory we investigate the flux spreading at the superconducting surface.
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"name": "streamp2.TEX",
"string": "\\documentstyle[prb,preprint,aps]{revtex}\n\\begin{document}\n\\draft\n\\title{Magnetic field of an in-plane vortex \\\\\ninside and outside a layered superconducting film}\n\\author{Edson Sardella}\n\\address{Departamento de F\\'{\\i}sica,\nFaculdade de Ci\\^encias, Universidade Estadual Paulista \\\\\nCaixa Postal 473, 17033-360, Bauru-SP, Brazil}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nIn the present work we study an anisotropic\nlayered superconducting film of finite thickness. The\nfilm surfaces are\nconsidered parallel to the $bc$ face of the crystal. The\nvortex lines are oriented perpendicular to the film surfaces\nand parallel to the superconducting planes.\nWe calculate the local field and the London free energy for this\ngeometry. Our calculation is a generalization of previous works\nwhere the sample is taken as a semi-infinite superconductor.\nAs an application of this theory we investigate the flux spreading\nat the superconducting surface.\n\\end{abstract}\n\n\\pacs{PACS numbers: 74.60.Ec, 74.60.Ge}\n\n%\\tightenlines\n\n%\\section{Introduction}\\label{sec1}\nScanning superconducting quantum interference device (SQUID)\nmicroscope has been used to image interlayer Josephson vortices\ntrapped between the planes of layered superconductors. This\ntechnique has been used to measure the out-plane London\npenetration depth that gives the distance over which the\ninterlayer current $j_c$ changes as a function of in-plane\ncoordinates.\\cite{moler98,kirtley98} These measurements have been\nimportant to test the interlayer tunneling model as a candidate\nto explain the mechanism of superconductivity for the high-$T_c$\nsuperconductors.\\cite{anderson98,chakravarty98}\n\nRecently, Kirtley, Kogan, Clem, and Moler\\cite{kirtley99} have\nfound expressions for the local magnetic field emerging from a\nsuperconductor with the vortex lines parallel to the planes, and\nnormal to a crystal face. Their geometry consists of a\nsemi-infinite anisotropic superconductor. Furthermore, they have\nused these expressions to fit the experimental data at the surface\nin order to obtain an estimate of the value of the out-plane\npenetration depth $\\lambda_c$. They have shown that, neglecting\nthe vortex spreading at the surface may overestimate $\\lambda_c$\nas much as 30\\%.\n\nIn the present paper we extend the work of Ref.~\\onlinecite{kirtley99}\nto an anisotropic layered superconducting film of finite thickness\nand of infinite extent in the $bc$ face of the crystal. We will\nshow that, if the thickness of the film is of order or smaller\nthan $\\lambda_c$, the magnetic field distribution\nis even more affected by flux spreading.\n\nLet us first formulate the problem to be solved. The geometry\nwe consider is illustrated in Fig.~\\ref{fig1}.\nWe suppose that the vortex line is perpendicular to the film. We\nwill calculate the local field inside the film using the London\nequation. For this geometry this equation is given by\n\n\\begin{equation}\n\\mbox{\\boldmath $\\nabla$}\\times\n[\\stackrel{\\leftrightarrow}{\\mbox{\\boldmath $\\nabla$}}\n\\times{\\bf h}]+{\\bf h}=\\hat{\\bf z}\\Phi_0\\delta({\\bf r})\\;,\n\\label{eq_london}\n\\end{equation}\nwhere $\\stackrel{\\leftrightarrow}{\\mbox{\\boldmath $\\nabla$}}$\nis the London\n(tensor) penetration depth. This tensor is diagonal and its\ncomponents are given by\n$\\Lambda_{xx}=\\Lambda_{yy}=\\lambda_c^2$, $\\Lambda_{zz}=\n\\lambda_{ab}^2$;\nhere $\\lambda_{ab}$ and $\\lambda_c$ are the in- and\nout-plane penetration\ndepth respectively; $\\Phi_0$ is the quantum flux. The film is\nanisotropic along the $c$ direction.\n\nOutside the sample, the local field satisfies the equation\n\n\\begin{equation}\n\\nabla^2{\\bf h}=0\\;.\\label{eq_laplace}\n\\end{equation}\n\nAlthough we will consider the case of a single vortex, the\ngeneralization to the case of $N$ vortices is straightforward. To\nproceed is more convenient to Fourier transform\nEqs.~(\\ref{eq_london}) and (\\ref{eq_laplace}). For $|z| < d/2$,\nusing the Maxwell equation $\\mbox{\\boldmath $\\nabla$}\\cdot{\\bf\nh}=0$, we obtain a set of three coupled differential equations\nfor the two dimensional Fourier transform of the local magnetic\nfield ${\\bf h}({\\bf k},z)=\\int\\,d^2r\\,e^{-\\i{\\bf k}\\cdot{\\bf\nr}}\\,{\\bf h}({\\bf r},z)\\;$,\n\n\\begin{equation}\n\\left [ 1+\\lambda_{ab}^2k^2-\\lambda_{ab}^2\\frac{\\partial^2}{\\partial z^2}\\right ]\nh_x\n = 0\\;,\\label{eq_hx}\n\\end{equation}\n\\begin{equation}\n\\left [ 1+\\lambda_{ab}^2k^2-\\lambda_c^2\\frac{\\partial^2}\n{\\partial z^2}\\right ] h_y\n+(\\lambda_c^2-\\lambda_{ab}^2)ik_y\\frac{\\partial h_z}{\\partial z}\n = 0 \\;,\\label{eq_hy}\n\\end{equation}\n\\begin{equation}\n\\left [ 1+\\lambda_{ab}^2k_x^2+\\lambda_c^2k^2_y-\n\\lambda_{ab}^2\\frac{\\partial^2}{\\partial z^2}\\right ] h_z\n+(\\lambda_c^2-\\lambda_{ab}^2)ik_y\\frac{\\partial h_y}{\\partial z}\n=\\Phi_0\\;.\\label{eq_hz}\n\\end{equation}\n\nFor $|z| > d/2$ one has\n\n\\begin{equation}\n(\\frac{\\partial^2}{\\partial z^2}-k^2){\\bf h}=0\\;.\n\\label{eq_laplace_fourier}\n\\end{equation}\n\nAt the vacuum-superconductor interfaces $z=\\pm d/2$ the field\ncomponents are continuous and the component of the current\nperpendicular to both film surfaces vanishes. One has\n\n\\begin{equation}\n{\\bf h}_<({\\bf k},-d/2)={\\bf h}_m({\\bf k},-d/2)\\;,\\label{boundary<}\n\\end{equation}\n\\begin{equation}\n{\\bf h}_m({\\bf k},d/2)={\\bf h}_>({\\bf k},d/2)\\;,\\label{boundary>}\n\\end{equation}\n\\begin{equation}\n\\hat{\\bf z}\\cdot\\left [ {\\bf D}_z({\\bf k})\n\\times{\\bf h}_m\\right ] _{z=\\pm d/2}=0\\;,\\label{boundary_current}\n\\end{equation}\n\\begin{equation}\n{\\bf D}_z({\\bf k})\\cdot{\\bf h}=0\\;,\\label{eq_maxwell}\n\\end{equation}\nwhere the operator\n${\\bf D}_z({\\bf k})=i{\\bf k}+\\hat{\\bf z}\\frac{\\partial}{\\partial z}$. The\nsubscripts ($<,>$) stand for below the surface $z=-d/2$ and\nabove the surface $z=d/2$, respectively, whereas the subscript $m$\nis meant for the field inside the sample.\n\nWe start by solving first Eq.~(\\ref{eq_laplace_fourier}).\nThe solution which satisfies the boundary condition of\nEq.~(\\ref{eq_maxwell}) takes the form\n\n\\begin{eqnarray}\n{\\bf h}_>({\\bf k},z) & = & (-i{\\bf k}+\\hat{\\bf z}k)\n\\varphi({\\bf k})e^{-k(z-d/2)}\\;,\\label{h>} \\\\\n{\\bf h}_<({\\bf k},z) & = & (i{\\bf k}+\\hat{\\bf z}k)\n\\varphi({\\bf k})e^{k(z+d/2)}\\;,\\label{h<}\n\\end{eqnarray}\nwhere $\\varphi({\\bf k})$ is a scalar function which will\nbe determined by using the boundary condition either of\nEq.~(\\ref{boundary<}) or (\\ref{boundary>}).\n\nEq.~(\\ref{eq_hx}) can also be easily solved. We have\n\n\\begin{equation}\nh_{m,x}({\\bf k},z)=W_1e^{\\alpha z}+W_2e^{-\\alpha z}\\;,\\label{sol_hx}\n\\end{equation}\nwhere\n\n\\begin{equation}\n\\alpha=\\sqrt{\\frac{1+\\lambda_{ab}^2k^2}{\\lambda_{ab}^2}}\\;,\n\\end{equation}\nand the $W$'s are two constants to be determined by using\nthe boundary conditions.\n\nThe other two components of the local field can be determined\nby decoupling Eqs.~(\\ref{eq_hy}) and (\\ref{eq_hz}). This can be done\nby calculating the determinant of the matrix\nformed by the coefficients of Eqs.~(\\ref{eq_hy})\nand Eq.~(\\ref{eq_hz}). This yields the following equation\nfor $h_{m,y}$\n\n\\begin{equation}\n\\left [ 1+\\lambda_{ab}^2k^2-\\lambda_{ab}^2\\frac{\\partial^2}{\\partial z^2}\\right ]\n\\left [\n1+\\lambda_{ab}^2k_x^2+\\lambda_c^2k_y^2-\\lambda_c^2\n\\frac{\\partial^2}{\\partial z^2}\n\\right ] h_{m,y}=0\\;.\n\\end{equation}\n\nThe solution for this equation is given by\n\n\\begin{equation}\nh_{y,m}({\\bf k},z)=W_3e^{\\alpha z}+W_4e^{-\\alpha z}\n+W_5e^{\\gamma z}+W_6e^{-\\gamma z}\\;,\\label{sol_hy}\n\\end{equation}\nwhere the $W$'s are constants to be determined\nby using the boundary conditions and\n\n\\begin{equation}\n\\gamma=\\sqrt{\\frac{1+\\lambda_{ab}^2k_x^2+\\lambda_c^2k_y^2}\n{\\lambda_c^2}}\\;.\n\\end{equation}\n\nThe solution for $h_{m,z}$ can be found by inserting Eq.~(\\ref{sol_hy})\nback into Eq.~(\\ref{eq_hy}) or (\\ref{eq_hz}).One has\n\n\\begin{equation}\nh_{m,z}({\\bf k},z)=\\frac{\\Phi_0}{\\lambda_c^2\\gamma^2}\n+\\frac{\\alpha}{ik_y}(W_3e^{\\alpha z}-W_4e^{-\\alpha z})\n-\\frac{ik_y}{\\gamma}(W_5e^{\\gamma z}-W_6e^{-\\gamma z})\\;.\n\\label{sol_hz}\n\\end{equation}\n\nThe determination of the constants $W_i$ is very cumbersome and\nwe omit it here. We just present the main steps of the complete\nsolution. First of all, we use the Maxwell\nequation (\\ref{eq_maxwell}). This allows us\nto write $W_3$ and $W_4$ in terms of $W_1$ and $W_2$.\nSecondly, we use the boundary condition of\nEq.~ (\\ref{boundary_current}) in both faces of the film. This leads\nus to the solution of $W_5$ and $W_6$ in terms of\n$W_1$ and $W_2$. Then, we are left only with three constants\nto determine, namely, $W_1$, $W_2$, and $\\varphi$. Thirdly, we use\nthe continuity of the local field at the film surfaces\n[either Eq.~(\\ref{boundary<}) or (\\ref{boundary>}); both of them\nyields the same solution to these constants]. One obtains,\n\n\\begin{eqnarray}\nW_1 & = & -ik_x\\frac{\\varphi}\n{2\\sinh\\left ( \\frac{\\alpha d}{2} \\right )}\n\\;, \\label{w1_final} \\\\\nW_2 & = & ik_x\\frac{\\varphi}\n{2\\sinh\\left ( \\frac{\\alpha d}{2} \\right )}\n\\;, \\label{w2_final} \\\\\nW_3 & = & \\frac{\\lambda_{ab}^2k_xk_y}{1+\\lambda_{ab}^2k_x^2}W_1\\;,\n\\label{w3_final} \\\\\nW_4 & = & \\frac{\\lambda_{ab}^2k_xk_y}{1+\\lambda_{ab}^2k_x^2}W_2\\;,\n\\label{w4_final} \\\\\nW_5 & = & \\frac{k_y}{k_x(1+\\lambda_{ab}^2k_x^2)\\sinh (\\gamma d)}\n\\left \\{ W_1\\sinh\\left [ \\left (\n\\gamma+\\alpha \\right ) \\frac{d}{2}\\right ]\n+W_2\\sinh\\left [ \\left ( \\gamma-\\alpha \\right )\n\\frac{d}{2}\\right ]\\right \\} \\;, \\label{w5_final} \\\\\nW_6 & = & \\frac{k_y}{k_x(1+\\lambda_{ab}^2k_x^2)\\sinh (\\gamma d)}\n\\left \\{ W_1\\sinh\\left [ \\left (\n\\gamma-\\alpha \\right ) \\frac{d}{2}\\right ]\n+W_2\\sinh\\left [ \\left ( \\alpha+\\gamma \\right )\n\\frac{d}{2}\\right ]\\right \\}\\;, \\label{w6_final} \\\\\n\\varphi({\\bf k}) & = & \\frac{\\Phi_0}{\\lambda_c^2\\gamma^2}\n\\Delta({\\bf k})\\;, \\label{phi_final}\n\\end{eqnarray}\nwhere\n\n\\begin{equation}\n\\Delta({\\bf k})=\\left[ k+\\frac{\n\\lambda_{ab}^2k_x^2\\alpha\\coth\\left ( \\frac{\\alpha d}{2} \\right ) +\n\\frac{k_y^2}{\\gamma}\\coth\\left ( \\frac{\\gamma d}{2} \\right )\n}{1+\\lambda_{ab}^2k_x^2}\\right]^{-1}\\;.\\label{Delta}\n\\end{equation}\n\nFinally, upon substituting Eqs.~(\\ref{w1_final}-\\ref{phi_final})\ninto Eqs.~(\\ref{sol_hx}), (\\ref{sol_hy}) and (\\ref{sol_hz}), we\nfind for the local magnetic field inside the film\n\n\\begin{eqnarray}\nh_{m,x}({\\bf k},z) & = & -ik_x\\varphi({\\bf k})\\frac{\\sinh(\\alpha z)}\n{\\sinh\\left( \\frac{\\alpha d}{2} \\right)}\\;,\\label{hmx_final} \\\\\nh_{m,y}({\\bf k},z) & = & -ik_y\\frac{\\varphi({\\bf k})}{1+\\lambda_{ab}^2k_x^2}\n\\left [ \\lambda_{ab}^2k_x^2\\frac{\\sinh(\\alpha z)}\n{\\sinh\\left( \\frac{\\alpha d}{2} \\right)}+\n\\frac{\\sinh(\\gamma z)}{\\sinh\\left( \\frac{\\gamma d}{2} \\right)}\n\\right ]\\;,\\label{hmy_final} \\\\\nh_{m,z}({\\bf k},z) & = & \\frac{\\Phi_0}{\\lambda_c^2\\gamma^2}\n-\\frac{\\varphi({\\bf k})}{1+\\lambda_{ab}^2k_x^2}\n\\left [\n\\lambda_{ab}^2k_x^2\\alpha\\frac{\\cosh(\\alpha z)}\n{\\sinh\\left( \\frac{\\alpha d}{2} \\right)}\n+\\frac{k_y^2}{\\gamma}\\frac{\\cosh(\\gamma z)}\n{\\sinh\\left( \\frac{\\gamma\n d}{2} \\right)}\n\\right ]\\;.\\label{hmz_final}\n\\end{eqnarray}\n\nWe would like to point out that these results could not\nbe obtained from those of Ref.~\\onlinecite{kirtley99} without\nsolving the problem. In fact, the solution of the\nLondon equation for a superconducting film is different and\nmore difficult than for a semi-infinite superconductor.\n\n%\\section{Free Energy}\nLet us turn our discussion to the\ncalculation of the London free energy. The energy of the\nvortex system is given by\n$F=F_V+F_S$, where $F_V$ is the field energy in the vacuum\nand $F_S$ is the energy inside the superconductor. One has\n\n\\begin{eqnarray}\nF_V & = & \\frac{1}{8\\pi}\\,\\int\\,\\frac{d^2k}{(2\\pi)^2}\\,\n\\left \\{\n\\int_{d/2}^{\\infty}\\,dz\\,|h_>({\\bf k},z)|^2\n+\\int_{-\\infty}^{-d/2}\\,dz\\,|h_<({\\bf k},z)|^2\n\\right \\} \\;,\\label{freev} \\\\\nF_S & = & \\frac{1}{8\\pi}\\,\\int\\,\\frac{d^2k}{(2\\pi)^2}\\,\n\\int_{-d/2}^{d/2}\\,dz\\,\\left \\{ |h_m({\\bf k},z)|^2 \\right .\n\\nonumber \\\\\n& & +\\left . \\left [ {\\bf D}_z({\\bf k})\\times{\\bf h}_m({\\bf k},z)\n\\right ] \\cdot\n\\stackrel{\\leftrightarrow}{\\mbox{\\boldmath $\\nabla$}} \\cdot\n\\left [ {\\bf D}_z(-{\\bf k})\\times{\\bf h}_m(-{\\bf k},z)\\right ]\n\\right \\} \\;.\\label{frees}\n\\end{eqnarray}\n\nBy substituting the appropriate expressions of the local magnetic\nfield inside Eqs.~(\\ref{freev}) and (\\ref{frees}), after\na length algebra, we obtain\n\n\\begin{equation}\nF=\\frac{\\Phi_0^2}{8\\pi}\\,\\int\\,\\frac{d^2k}{(2\\pi)^2}\\,\n\\frac{1}{\\lambda_c^2\\gamma^2}\\left [\nd+2 \\frac{\\Delta({\\bf k})}{\\lambda_c^2\\gamma^2}\n\\right ]\\;.\\label{free_final}\n\\end{equation}\n\nThe free energy can be generalized to an ensemble of\n$N$ interacting vortex lines upon multiplying the\nintegrand of Eq.~(\\ref{free_final}) by\n$|S({\\bf k})|^2$ where the structure factor is\ngiven by\n\n\\begin{equation}\nS({\\bf k})=\\sum_{i}\\,e^{i{\\bf k}\\cdot{\\bf R}_i}\\;.\n\\end{equation}\n\nHere ${\\bf R}_i$ is the position of the $i$-vortex line.\nNote that this extended result should be valid for an ensemble\nof distorted vortices, that is, the positions of the\nvortices do not necessarily correspond to the equilibrium\nconfiguration.\nThe first term inside Eq.~(\\ref{free_final}) represents\nthe interaction energy of the vortex lines as if\nthe surfaces were absent.\nThe second term represents the surface energy\nassociated to the magnetic energy of the stray field\nat the superconductor-vacuum interface. Notice that for\n$k$ small (large $r$), $\\gamma^2\\sim 1/\\lambda_c^2$,\nand $\\Delta({\\bf k})\\sim 1/k$. Thus, the surface\nenergy goes as $\\Phi_0^2/8\\pi^2r$. Consequently, the\ninteraction on the surface depends neither on\nthe film thickness nor on the anisotropy. This is the Pearl result\nfor vortices emerging from a semi-infinite\nisotropic supercondutor.\\cite{pearl66}\nAnother interesting particular case of Eq.~(\\ref{free_final}) is the\nlimit of a very thin film $d\\rightarrow 0$, and $k$ small.\nIn this limit, from Eq.~(\\ref{Delta}) it is straightforward\nto show that\n$\\Delta({\\bf k})=1/(k+2(\\lambda_{ab}^2k_x^2+\\lambda_c^2k_y^2)/d)$.\nTherefore, from Eq.~(\\ref{free_final}) we obtain\n\n\\begin{equation}\nF=E_0\\,\\int\\,\\frac{dk^2}{(2\\pi)^2}\\,\\frac{2\\pi d}\n{k\\Lambda^{-1}+(k_x^2+\\Gamma k_y^2)}\\;,\\label{energy_film}\n\\end{equation}\nwhere $E_0=(\\Phi_0/4\\pi\\lambda_{ab})^2$, $\\Lambda=2\\lambda_{ab}^2/d$,\nand $\\Gamma=\\lambda_c^2/\\lambda_{ab}^2$ is the anisotropy parameter.\nThis is precisely the energy of a single vortex in very thin film\nfirst obtained by Pearl.\\cite{pearl64,kogan93}\n\n%\\section{Distribution of Magnetic Field}\nNow we will turn our attention to the streamlines\nof the integrated field over $x$.\nThe distribution of magnetic field emerging on the surface\ncan be probed with a SQUID pickup loop. If the\nSQUID probe is oriented in the $xy$ plane, the\ntotal magnetic flux will be nearly equal to the pickup loop\nsize times\\cite{kirtley99}\n\n\\begin{equation}\n{\\cal H}_z(y,z)=\\int_{-\\infty}^{\\infty}\\,h_z(x,y,z)\\,dx=\n\\int_{-\\infty}^{\\infty}\\,\\frac{dk_y}{2\\pi}\\,h_z(0,k_y,z)e^{ik_yy}\\;,\n\\label{hzcal}\n\\end{equation}\nwhereas, if the SQUID probe is oriented along the $xz$ plane,\nthe total magnetic flux is measured through the pickup loop\nsize times\n\n\\begin{equation}\n{\\cal H}_y(y,z)=\\int_{-\\infty}^{\\infty}\\,h_y(x,y,z)\\,dx=\n\\int_{-\\infty}^{\\infty}\\,\\frac{dk_y}{2\\pi}\\,h_y(0,k_y,z)e^{ik_yy}\\;.\n\\label{hycal}\n\\end{equation}\n\nIn order to compare our results with the results of\nRef.~\\onlinecite{kirtley99}, we will replace the\nvacuum-superconductor surfaces at $z=0$ and $z=-d$. This can be\ndone through the translation $z\\rightarrow z+d/2$. From Eqs.~\n(\\ref{h>}), (\\ref{h<}), and (\\ref{hmx_final}-\\ref{hmz_final})\nwe obtain,\n\n\\begin{eqnarray}\n{\\bf h}_>({\\bf k},z) & = & (-i{\\bf k}+\\hat{\\bf z}k)\n\\varphi({\\bf k})e^{-kz}\\;, \\\\\n{\\bf h}_<({\\bf k},z) & = & (i{\\bf k}+\\hat{\\bf z}k)\n\\varphi({\\bf k})e^{k(z+d)}\\;,\n\\end{eqnarray}\n\n\\begin{eqnarray}\nh_{m,x}({\\bf k},z) & = & -ik_x\\varphi({\\bf k})\\frac{\\sinh\\left [\n\\alpha \\left ( z+\\frac{d}{2}\\right ) \\right ]}\n{\\sinh\\left( \\frac{\\alpha d}{2} \\right)}\\;, \\\\\nh_{m,y}({\\bf k},z)& =& -ik_y\\frac{\\varphi({\\bf k})}{1+\\lambda_{ab}^2k_x^2}\n\\left \\{ \\lambda_{ab}^2k_x^2\\frac{\\sinh\\left [\n\\alpha \\left ( z+\\frac{d}{2}\\right ) \\right ]}\n{\\sinh\\left( \\frac{\\alpha d}{2} \\right)}\n+\\frac{\\sinh\\left [\n\\gamma \\left ( z+\\frac{d}{2}\\right ) \\right ]}\n{\\sinh\\left( \\frac{\\gamma d}{2} \\right)}\n\\right \\}\\;, \\\\\nh_{m,z}({\\bf k},z) & = & \\frac{\\Phi_0}{\\lambda_c^2\\gamma^2}\n-\\frac{\\varphi({\\bf k})}{1+\\lambda_{ab}^2k_x^2}\n\\left \\{\n\\lambda_{ab}^2k_x^2\\alpha\\frac{\\cosh\\left [\n\\alpha \\left ( z+\\frac{d}{2}\\right ) \\right ]}\n{\\sinh\\left( \\frac{\\alpha d}{2} \\right)}\n+\\frac{k_y^2}{\\gamma}\\frac{\\cosh\\left [\n\\gamma \\left ( z+\\frac{d}{2}\\right ) \\right ]}\n{\\sinh\\left( \\frac{\\gamma\n d}{2} \\right)}\n\\right \\}\\;.\n\\end{eqnarray}\n\nThe substitution of the appropriate expressions into\nEqs.~(\\ref{hzcal}) and (\\ref{hycal}) yields for the $z$\ncomponent of the $\\vec{\\cal H}$ field,\n\n\\begin{eqnarray}\n{\\cal H}^z_>(y,z) & = & \\frac{\\Phi_0}{\\pi\\lambda_c}\\,\n\\int_0^{\\infty}\\,du\\,\\frac{\\cos(y^{\\prime}\\sinh u)\n\\,e^{-z^{\\prime}\\sinh u}}\n{\\cosh u + \\sinh u \\coth \\left ( \\frac{d}{2\\lambda_c}\n\\cosh u \\right )} \\;, \\\\\n{\\cal H}_{m,z}(y,z) & = & \\frac{\\Phi_0}{\\pi\\lambda_c}\\left \\{\n\\frac{\\pi}{2}e^{-|y^{\\prime}|}-\\int_0^{\\infty}\\,du\\,\n\\tanh u \\right . \\nonumber \\\\\n& & \\left .\n\\times\\frac{\\cos(y^{\\prime}\\sinh u)}\n{\\cosh u + \\sinh u \\coth \\left ( \\frac{d}{2\\lambda_c}\n\\cosh u \\right )}\\frac{\\cosh\\left [\n\\left ( z^{\\prime}+\\frac{d}{2\\lambda_c} \\right )\\cosh u\n\\right ] }{\\sinh \\left ( \\frac{d}{2\\lambda_c}\\cosh u \\right )}\n\\right \\}\\;, \\\\\n{\\cal H}^z_<(y,z) & = & \\frac{\\Phi_0}{\\pi\\lambda_c}\\,\n\\int_0^{\\infty}\\,du\\,\\frac{\\cos(y^{\\prime}\\sinh u)\n\\,e^{(z^{\\prime}+d/\\lambda_c)\\sinh u}}\n{\\cosh u + \\sinh u \\coth \\left ( \\frac{d}{2\\lambda_c}\n\\cosh u \\right )} \\;,\n\\end{eqnarray}\nwhere $y^{\\prime}=y/\\lambda_c$ and $z^{\\prime}=z/\\lambda_c$.\n\nThe $y$ component takes the form\n\n\\begin{eqnarray}\n{\\cal H}^y_>(y,z) & = & \\frac{\\Phi_0}{\\pi\\lambda_c}\\,\n\\int_0^{\\infty}\\,du\\,\\frac{\\sin(y^{\\prime}\\sinh u)\n\\,e^{-z^{\\prime}\\sinh u}}\n{\\cosh u + \\sinh u \\coth \\left ( \\frac{d}{2\\lambda_c}\n\\cosh u \\right )} \\;, \\\\\n{\\cal H}_{m,y}(y,z) & = & \\frac{\\Phi_0}{\\pi\\lambda_c}\\,\n\\int_0^{\\infty}\\,du\\,\n\\frac{\\sin(y^{\\prime}\\sinh u)}\n{\\cosh u + \\sinh u \\coth \\left ( \\frac{d}{2\\lambda_c}\n\\cosh u \\right )}\n\\frac{\\sinh\\left [\n\\left ( z^{\\prime}+\\frac{d}{2\\lambda_c} \\right )\\cosh u\n\\right ] }{\\sinh \\left ( \\frac{d}{2\\lambda_c}\\cosh u \\right )}\n\\;, \\\\\n{\\cal H}^y_<(y,z) & = & -\\frac{\\Phi_0}{\\pi\\lambda_c}\\,\n\\int_0^{\\infty}\\,du\\,\\frac{\\sin(y^{\\prime}\\sinh u)\n\\,e^{(z^{\\prime}+d/\\lambda_c)\\sinh u}}\n{\\cosh u + \\sinh u \\coth \\left ( \\frac{d}{2\\lambda_c}\n\\cosh u \\right )} \\;.\n\\end{eqnarray}\n\nNote that in the limit of $d \\rightarrow \\infty$, our results\nare exactly the same as those of Ref.~\\onlinecite{kirtley99}.\n\nThe results for the $\\vec{\\cal H}$ field presented above should\nbe useful to interpret the experimental data obtained by using\nscanning SQUID micorscopy. Unfortunately, the experiments\nhave been performed in samples of large thickness.\n\\cite{moler98,kirtley98} This renders\nthe test of the theory impracticable. In fact, vortices have been\nmagnetically imaged in films, but for a different geometry, that is,\nthe superconducing planes are taken parallel to the surfaces of\nthe film and the\nvortex lines are considered perpendicular to the film surfaces.\n\\cite{kirtley99b}\nIn this case, we can extract the in-plane penetration depth\n$\\lambda_{ab}$ rather than $\\lambda_c$, from the fitting\nof the experimental data. So, we will restrict our\nanalysis only to the theoretical expressions.\n\nFig.~\\ref{fig2} shows the streamlines of the $\\vec{\\cal H}(y,z)$\nfield for a single interlayer vortex centered at $x=0$, $y=0$. The\nstreamlines were generated as sketched in\nRef.~\\onlinecite{kirtley99}. We used various values of the\nfilm thickness. Note that as the thickness of the film grows,\nthe flux spreading\nis important only near the surface, whereas deep inside\nthe thinner film the streamlines are still very distorted,\nexcept those close to the center of the vortex.\n\nTo see how important the flux spreading inside\na superconducting film is, we calculated numerically\n$\\pi \\lambda_c{\\cal H}_z(y,z)/\\Phi_0$ as function of $y/\\lambda_c$\nfor three different values of $d$ at $z=0$. As can be seen from\nFig.~\\ref{fig3}, the full width at half maximum of the\nflux contour is $1.87\\lambda_c$ for the case $d=5\\lambda_c$, while\nit is $1.65\\lambda_c$ for $d=\\lambda_c$. Thus, if the flux spreading\ninside the film is not taken into account, the value of\n$\\lambda_c$ could be underestimated by 10\\%. This error grows\nas the film thickness decreases.\n\nFinally, we would like to point out that the present\nresults agree with their isotropic counterpart. If we set\n$\\lambda_{ab}=\\lambda_c=\\lambda$ in Eq.~(\\ref{phi_final}) and\n(\\ref{Delta}), we obtain the same result as in Ref.~\n\\onlinecite{kirtley99b}.\nApparently, our results are different of those found in\nRef.~\\onlinecite{gilson00}, but they show very similar streamlines.\n\nIn summary, we have calculated the field distribution of\na single vortex inside and\noutside a layered superconducting film of arbitrary thickness.\nWe also calculated the London free energy of an ensemble of vortices.\nFrom the expression for the energy one can recover\nthe interaction potential between vortices for a very\nthin film\\cite{pearl64,kogan93} and the vortices\nemerging from a semi-infinite superconductor.\\cite{kirtley99,pearl66}\nIn addition, we have shown that flux spreading inside\na superconducting film of order or smaller than $\\lambda_c$\naffects substantially the full width at half maximum of the\nflux contour.\n\n\n\\acknowledgments\nThe author thanks the Brazilian Agencies FAPESP and CNPq\nfor financial support.\n\n\\begin{references}\n\\bibitem{moler98}K.\\ A.\\ Moler, J.\\ K.\\ Kirtley, D.\\ G. Hinks,\nT.\\ W.\\ Li, and M. Xu, Science {\\bf 279}, 1193 (1998).\n\\bibitem{kirtley98}J.\\ R.\\ Kirtley, K.\\ A.\\ Moler, G.\\ Villard,\nand A.\\ Maigman, \\prl {\\bf 81}, 2140 (1998).\n\\bibitem{anderson98}P.\\ A.\\ Anderson, Science {\\bf 279},\n1196 (1998).\n\\bibitem{chakravarty98}S.\\ Chakravarty, Eur.\\ Phys.\\ J. B {\\bf 5},\n337 (1998).\n\\bibitem{kirtley99} J.\\ R.\\ Kirtley, V.\\ G.\\ Kogan, J.\\ R.\\ Clem\nK. A. Moler, \\prb {\\bf 59}, 4343 (1999).\n\\bibitem{pearl66}J.\\ Pearl, J.\\ Appl.\\ Phys.\\ {\\bf 37}, 4139 (1966).\n\\bibitem{pearl64}J.\\ Pearl, Appl.\\ Phys.\\ Lett.\\ {\\bf 5}, 65 (1964).\n\\bibitem{kogan93}V.\\ G.\\ Kogan, A. Yu.\\ Simonov, and M.\\ Ledvij,\nPhys.\\ Rev.\\ B {\\bf 48}, 392 (1993). In this reference,\nthe anisotropic version of Pearl's vortex interaction\nfor very thin film has been found. Their result is identical\nto Eq.~(\\ref{energy_film}).\n\\bibitem{kirtley99b}J.\\ R.\\ Kirtley, C.\\ C.\\ Tsuei,\nK. A.\\ Moler, V.\\ G.\\ Kogan, J.\\ R.\\ Clem, and A. J. Turberfield,\nAppl.\\ Phys.\\ Lett.\\ {\\bf 74}, 4011 (1999).\n\\bibitem{gilson00}G.\\ Carneiro and E.\\ H.\\ Brandt, \\prb {\\bf 61},\n6370 (2000); J.\\ C.\\ Wei and T.\\ J.\\ Yang, Jpn.\\ J.\\ Appl.\\\nPhys.\\, Part 1 {\\bf 35}, 5696 (1996).\n\\end{references}\n\n\\begin{figure}\n\\caption{Geometry of the film used in this work. The vortex\nlines are oriented perpendicular to the $bc$ face of the crystal.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Streamline mapping of the integrated field\n$\\vec{\\cal H}$, for an anisotropic superconducting film.\nThe spacing of the streamlines is proportional to\n$(\\partial {\\cal H}_z/\\partial y)^{-1}$ at $z=-(d+0.5\\lambda_c)$.\n\\emph{(a)} shows the streamlines for a film of thickness\n$d=\\lambda_c$, \\emph{(b)} for $d=2.5\\lambda_c$, and\n\\emph{(c)} for $d=5\\lambda_c$.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\caption{The $z$ component of the integrated field as a function of\n$y$ at $z=0$ for three different values of the\nfilm thickness: \\emph{(a)}\n$d=5\\lambda_c$ (dashed line); \\emph{(b)} $d=1.5\\lambda_c$\n(dot-dashed line); \\emph{(c)} $d=\\lambda_c$ (continuous line)\n}\\label{fig3}\n\\end{figure}\n\n\\newpage\n\n\\begin{center}\n\\setlength{\\unitlength}{1cm}\n\\begin{picture}(8,6.5)(0,11)\n\\thicklines\n\\multiput(1.5,11)(0,-0.3){15}{\\line(1,0){6}}\n\\multiput(7.5,11)(0,-0.3){15}{\\line(2,3){2}}\n\\put(1.5,11){\\line(2,3){2}}\n\\put(5.2,12){\\vector(0,1){1.5}}\\put(5.2,13.8){\\makebox(0,0){$c,x$}}\n\\put(5.2,12){\\vector(1,0){1.5}}\\put(7.25,12){\\makebox(0,0){$a,z$}}\n\\put(5.2,12){\\vector(2,3){0.8}}\\put(6.1,13.5){\\makebox(0,0){$b,y$}}\n\\put(1.5,6.5){\\line(0,1){0.1}}\\put(1.5,6.5){\\line(0,-1){0.1}}\n\\put(7.5,6.5){\\line(0,1){0.1}}\\put(7.5,6.5){\\line(0,-1){0.1}}\n\\put(4.3,6.5){\\vector(-1,0){2.7}}\n\\put(4.5,6.5){\\makebox(0,0){$d$}}\n\\put(4.7,6.5){\\vector(1,0){2.7}}\n\\end{picture}\n\\end{center}\n\\vspace*{8cm}\n\\begin{center}\n{\\Large Fig. 1/Sardella}\n\\end{center}\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002152.extracted_bib",
"string": "\\bibitem{moler98}K.\\ A.\\ Moler, J.\\ K.\\ Kirtley, D.\\ G. Hinks,\nT.\\ W.\\ Li, and M. Xu, Science {\\bf 279}, 1193 (1998).\n\n\\bibitem{kirtley98}J.\\ R.\\ Kirtley, K.\\ A.\\ Moler, G.\\ Villard,\nand A.\\ Maigman, \\prl {\\bf 81}, 2140 (1998).\n\n\\bibitem{anderson98}P.\\ A.\\ Anderson, Science {\\bf 279},\n1196 (1998).\n\n\\bibitem{chakravarty98}S.\\ Chakravarty, Eur.\\ Phys.\\ J. B {\\bf 5},\n337 (1998).\n\n\\bibitem{kirtley99} J.\\ R.\\ Kirtley, V.\\ G.\\ Kogan, J.\\ R.\\ Clem\nK. A. Moler, \\prb {\\bf 59}, 4343 (1999).\n\n\\bibitem{pearl66}J.\\ Pearl, J.\\ Appl.\\ Phys.\\ {\\bf 37}, 4139 (1966).\n\n\\bibitem{pearl64}J.\\ Pearl, Appl.\\ Phys.\\ Lett.\\ {\\bf 5}, 65 (1964).\n\n\\bibitem{kogan93}V.\\ G.\\ Kogan, A. Yu.\\ Simonov, and M.\\ Ledvij,\nPhys.\\ Rev.\\ B {\\bf 48}, 392 (1993). In this reference,\nthe anisotropic version of Pearl's vortex interaction\nfor very thin film has been found. Their result is identical\nto Eq.~(\\ref{energy_film}).\n\n\\bibitem{kirtley99b}J.\\ R.\\ Kirtley, C.\\ C.\\ Tsuei,\nK. A.\\ Moler, V.\\ G.\\ Kogan, J.\\ R.\\ Clem, and A. J. Turberfield,\nAppl.\\ Phys.\\ Lett.\\ {\\bf 74}, 4011 (1999).\n\n\\bibitem{gilson00}G.\\ Carneiro and E.\\ H.\\ Brandt, \\prb {\\bf 61},\n6370 (2000); J.\\ C.\\ Wei and T.\\ J.\\ Yang, Jpn.\\ J.\\ Appl.\\\nPhys.\\, Part 1 {\\bf 35}, 5696 (1996).\n"
}
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cond-mat0002153
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Johnson-Nyquist noise in films and narrow wires
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[
{
"author": "Misha Turlakov"
}
] |
The Johnson-Nyquist noise in narrow wires having a transverse size smaller than the screening length is shown to be white up to the frequency $D/L^2$ and to decay at higher frequencies as $\omega^{-\frac{1}{2}}$. In two-dimensional films having a thickness smaller than the screening length, the Johnson-Nyquist noise is predicted to be frequency independent up to the frequency $\sigma_{2D}/L$ and to have a {universal} $1/\omega$ spectrum at higher frequencies. These results are contrasted with the conventional noise spectra in neutral and three-dimensional charged liquids.
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[
{
"name": "cond-mat0002153.tex",
"string": "\\documentstyle[aps,prb,multicol]{revtex}\n\n\\newcommand{\\beq}{\\begin{eqnarray}}\n\\newcommand{\\eeq}{\\end{eqnarray}}\n\\newcommand{\\n}{\\nonumber}\n\\newcommand{\\epi}{\\epsilon_\\infty}\n\\begin{document}\n\\draft\n\\title\n{Johnson-Nyquist noise in films and narrow wires}\n\\author{ Misha Turlakov}\n\\address{ Department of Physics, University of Illinois, 1110 W. Green Street, Urbana, IL 61801}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\n\nThe Johnson-Nyquist noise in narrow wires having a transverse size smaller than\nthe screening length is shown to be white up to the frequency $D/L^2$ and to decay\nat higher frequencies as $\\omega^{-\\frac{1}{2}}$. \nIn two-dimensional films having a thickness smaller than the screening length, \nthe Johnson-Nyquist noise\nis predicted to be frequency independent up to the frequency $\\sigma_{2D}/L$ \nand to have a {\\it universal} $1/\\omega$ spectrum \nat higher frequencies.\nThese results are contrasted\nwith the conventional noise spectra in neutral and three-dimensional charged liquids.\n\n\\end{abstract}\n\\pacs{PACS numbers: 71.10.Ca, 72.70.+m, 72.30.+q, 05.40.Ca}\n% \\vspace{.1cm}\n\n\\begin{multicols}{2}\n\n% Introduction\n\n It is interesting to analyze the differences between charged and neutral systems\ndue to the long-range nature of the Coulomb interaction.\nThe role of the Coulomb interaction \n depends crucially on the effective dimensionality\nof the charged system. For instance, \n in three-dimensional systems Coulomb interaction transforms \nthe gapless density excitations of neutral liquids\n(acoustic phonons) to gapped plasmons. Nevertheless, in one- and two-dimensional\nsystems plasmons remain gapless. \nHere I examine the noise spectrum as another aspect \nof the singular role of Coulomb interaction on collective phenomena\n(plasmons, noise) which depends critically on the dimensionality.\n\n\nThe noise spectrum is quite different in charged and neutral liquids. The equilibrium Johnson-Nyquist\nnoise (JNN)\\cite{Nyquist}\n in an electrical conductor (with a screening length smaller than any size of the conductor)\n is independent of frequency (white noise) up to the very high frequency\n(the smaller of the elastic scattering rate $1/\\tau$ and \nthe Maxwell relaxation frequency $4\\pi\\sigma$)\\cite{plasma};\nwhile in neutral liquids, the noise becomes frequency-dependent above the ``Thouless'' frequency\n$D/L^2$ ($D$ is a diffusion constant and $L$ is the distance between points). The difference \nis due to the screening in charged liquids and depends on the dimensionality of\nthe conductor. I show here that\nfor electrical wires having a transverse size ($a$) smaller than the screening \nlength $\\lambda_D$ (here referred to as ``narrow'' wires), the JNN decays \nas $\\omega^{-\\frac{1}{2}}$ above the ``Thouless''\nfrequency.\nSimilarly, for two-dimensional films having a thickness smaller\nthan the screening length the JNN decreases as $1/\\omega$ above the characteristic \nfrequency $\\sigma_{2D}/L$.\n\n\n% main derivation for narrow wire: diffusion and induced potential\n\n\nTo calculate the fluctuations of the electrochemical potential,\n we need to relate it to the coupled\nfluctuations of charge density and currents. We start by writing the continuity equation and\nthe equation for the current valid in the hydrodynamic limit\\cite{Nozieres}:\n\n\\beq\n\\frac{\\partial \\rho}{\\partial t} + div(\\vec{j})=0;~~\\vec{j}=\n\\sigma \\vec{E}^{tot} - D \\vec{\\nabla} \\rho. \\n\n\\eeq\n\nFor self-consistency, we need to account for the potential induced by the fluctuation of charge density:\n$\\phi_{q,\\omega}^{ind}= u_1(q) \\rho_{q,\\omega}$ (Coulomb's law). \nIf we consider a conductor\nwith transverse dimensions ($a$) smaller \nthan the screening length $\\lambda_D$, then $u_1(q)= 2 ln\\frac{1}{qa}$ is\na one-dimensional Coulomb potential ($q$ is a wave vector along the one-dimensional conductor).\nThe total potential driving current\nis the sum of the external and induced potentials. Thus the full system of equations is: \n\n\\beq\ni\\omega \\rho_{q,\\omega}+iq j_{q,\\omega}=0,~~j_{q,\\omega}=(iq)\\sigma \\phi_{q,\\omega}^{tot}\n+(iq)D\\rho_{q,\\omega}, \\n \\\\\n\\phi_{q,\\omega}^{tot}=\\phi_{q,\\omega}^{ext}+\\phi_{q,\\omega}^{ind},\n~~\\phi_{q,\\omega}^{ind}= 2 ln\\frac{1}{qa} \\rho_{q,\\omega}. \\n\n\\eeq\n\nFinally, after some elementary algebra, we can use the above equations to relate the charge density \nvariation to the external potential:\n\n\\beq\n\\rho_{q,\\omega}=-\\frac{\\sigma_1 q^2}{-i\\omega+\n(D+2 \\sigma_1 ln(1/qa))q^2}\\phi_{q,\\omega}^{ext}, \\n \n\\eeq \nwhere $\\sigma_1=\\sigma a^2$ is a one-dimensional conductivity.\nThe conductivity $\\sigma$ and the diffusion constant $D$ are related by the Einstein formula\n$\\sigma=D\\chi_0$, where\nthe static charge compressibility $\\chi_0$ can be expressed through the Debye screening (or Thomas-Fermi)\nlength $\\chi_0=1/4\\pi\\lambda_D^2$.\nThe density-density response function $\\chi_{q,\\omega}$ is\n\n\\beq\n\\chi_{q,\\omega}\\equiv \\frac{\\rho_{q,\\omega}}{\\phi_{q,\\omega}^{ext}}\n = - \\frac{D\\chi_0 a^2 q^2}{-i\\omega+Dq^2(1 +2a^2\\chi_0 ln(1/qa)) }. \\n\n\\eeq\n\nWe can now apply the fluctuation dissipation theorem (FDT) to calculate the density fluctuation\nspectrum (assuming classical fluctuations, $\\hbar \\omega \\ll kT$)\\cite{classical}:\n\n\\beq\n<\\mid\\delta \\rho_{q,\\omega} \\mid^2>=\\frac{\\hbar}{\\pi} Im\\chi_{q,\\omega} \ncoth(\\frac{\\hbar \\omega}{2kT}) \\cong\n\\frac{2 kT}{\\pi \\omega} Im\\chi_{q,\\omega}. \\n\n\\eeq\n\nThe induced potential fluctuations can be expressed through the charge density fluctuations:\n\n\\beq\n<\\mid \\phi_{q,\\omega}^{ind} \\mid^2>= (u_1(q))^2 <\\mid\\delta \\rho_{q,\\omega} \\mid^2>, \\n \\\\\n<\\mid \\phi_{q,\\omega}^{ind} \\mid^2>= \n \\frac{2 kT}{\\pi} \n\\frac{\\sigma_1 q^2(2ln\\frac{1}{qa})^2}\n{\\omega^2 +D^2(1+\\frac{a^2}{2\\pi\\lambda_D^2}ln\\frac{1}{qa})^2q^4}.\n\\label{eq:1d} \n\\eeq\n\n% compare with neutral liquid\n% compare with charged liquid\n\nWe can compare this expression for the spectral density of noise in a 1D wire (Eqn.\\ref{eq:1d}) \nwith the spectral density of potential fluctuations\nin bulk three-dimensional charged and neutral liquids.\nIn the case of a three-dimensional charged liquid, we need to\nuse the three-dimensional Coulomb potential \n$\\phi_{q,\\omega}^{ind}=u_3(q)\\rho_{q,\\omega} =\\frac{4 \\pi}{ q^2} \\rho_{q,\\omega}$.\nFollowing the above simple derivation, we get the expression for voltage fluctuations\n(it is sufficient for our purposes to consider only longitudinal fluctuations) in a three-dimensional\nconductor\\cite{2-der}:\n\n\\beq\n<\\mid \\phi_{q,\\omega}^{ind(3d)} \\mid^2>= \n\\frac{2 kT}{\\pi q^2}\\frac{16\\pi^2 \\sigma}{\\omega^2 +(Dq^2+4 \\pi\\sigma)^2}. \\label{eq:charged}\n\\eeq\n\n\n In the case of a neutral liquid, there is no\nlong-range induced potential; therefore, we get the standard density-density response function\nand potential fluctuations describing diffusion:\n\n\\beq\n<\\mid \\phi_{q,\\omega}^{(n)} \\mid^2>= \\frac{<\\mid\\delta \\rho_{q,\\omega} \\mid^2>}{\\chi_0^2}=\n\\frac{2T}{\\pi \\chi_0^2} \\frac{D |\\chi_0| q^2}{\\omega^2 +(Dq^2)^2}. \\label{eq:neutral}\n\\eeq\n\nWe can now use the spectral densities (Eqns. \\ref{eq:1d}-\\ref{eq:neutral}) to calculate the\nexperimentally measured differential noise between the two ends of the sample, averaged\nover transverse modes:\n\n\\beq\n<\\mid \\phi_{12}(\\omega) \\mid^2>= \n\\sum_{q_x} 4 \\frac{ sin^2(q_x a/2)}{q_x^2 a^2}\n\\sum_{q_y} 4 \\frac{ sin^2(q_y a/2)}{q_y^2 a^2} \\n \\\\\n\\int \\frac{dq_z}{2\\pi}~ 4sin^2 \\left( \\frac{q_zL}{2} \\right) \n<\\mid \\phi (q,\\omega) \\mid^2>. \\label{eq:ends} \n\\eeq\n\nThe Johnson-Nyquist noise in a three-dimensional conductor (\\ref{eq:charged}) can be easily calculated,\nbecause the dominant contribution to the sums comes from the transverse zero mode\n($q_x=q_y=0$, corresponding to the uniform density of the liquid along the transverse directions):\n\n\\beq\n<\\mid \\phi_{12}^{3d}(\\omega) \\mid^2> = \\frac{2kT}{\\pi} \\frac{L}{\\sigma S} \n\\frac{1}{1+ (\\frac{\\omega}{4\\pi\\sigma})^2 }. \\label{eq:3D} \n\\eeq\n\nSuch noise is readily interpreted as the noise from a conductor \n having an internal resistance $R=\\frac{L}{\\sigma S}$ and an\ninternal capacitance $C=\\frac{ S}{4\\pi L}$ connected in parallel\\cite{Rytov,quasi}:\n$ R(\\omega)=R/(1+(RC\\omega)^2)$. Remarkably, the Johnson-Nyquist noise is white up to the frequency\n$4\\pi\\sigma$, which is independent of the length of the wire. \nIt is important to point out\nthat the noise can depend on frequency through the frequency dependence \nof the conductivity $\\sigma(\\omega)$.\nFor the Drude model of conductivity, the characteristic frequency for fall-off of the conductivity\n$\\sigma(\\omega)$ is then the elastic scattering rate $1/\\tau$. \n\nIn the case of a ``one-dimensional'' wire $a<\\lambda_D$, we can take into account only one\n``zero mode''($q_x=q_y=0$), \nsince higher harmonics contribute at frequencies of order $D/a^2 > \\sigma$.\nIf we approximate the weak logarithmic dependence in Eqn.(\\ref{eq:1d}) by a constant \n$ln\\frac{1}{qa} \\rightarrow ln\\frac{L}{a}$,\nwe get an expression similar to Eqn.(\\ref{eq:neutral}) with the renormalized diffusion\ncoefficient $D'\\equiv D(1+\\frac{a^2}{2\\pi\\lambda_D^2} ln(L/a))$.\n Thus the frequency dependence of noise\nfor a ``one-dimensional'' wire is the same as for a neutral liquid.\n This result is to be expected, since the screening is not efficient in one dimension. \nThe integral over wave vector $q_z$ can be evaluated explicitly assuming for simplicity\n$a^2 ln(L/a) \\gg 2\\pi \\lambda_D^2$.\nThe Johnson-Nyquist noise for a narrow wire is \n\n\\begin{eqnarray}\n&&<\\mid \\phi_{12}^{1d}(\\omega) \\mid^2> = 2kT \\frac{ L}{\\sigma_1 \\theta}\n(1-e^{-\\theta}(cos\\theta-sin\\theta)), \\label{eq:1d-neutral} \n% &&\\Gamma\\equiv \\left( 1 +\\frac{2\\pi\\lambda_D^2}{a^2ln\\frac{L}{a}} \\right)^{-2}, \\n\n\\end{eqnarray}\nwhere \n% $\\Gamma\\equiv \\left( 1 +2\\pi\\lambda_D^2/(a^2ln\\frac{L}{a}) \\right)^{-2}$,\n$\\theta=(\\omega/2\\omega_0)^{1/2}$ and $\\omega_0=D'/L^2$ is the natural diffusion frequency.\nThe expression in the bracket of Eqn. \\ref{eq:1d-neutral} is always positive as it must be, \nand the oscillating nature of the second term\nis due to the ``resonant'' contributions of the longitudinal ``diffusion modes''.\nFrom the above expression for noise in a one-dimensional wire, we see that it is approximately white\nup to the ``Thouless'' frequency $\\omega_0$ and equal to $4 kT \\frac{L}{\\sigma_1}$.\nIt decays above this frequency as $1/\\sqrt{\\omega}$.\nThe same frequency dependence (with $D' \\equiv D$) is expected \nfor the fluctuations of the chemical potential\nbetween two points in a narrow vessel ($\\omega \\ll D/a^2$) of neutral liquid.\nIn fact, it is the classical result for any quantity (such as temperature, density) obeying\na diffusion process that does not have long-range correlations.\\cite{voss} \n\n%-----------begin of 2D addition-----------------------------------\n\nThe noise in a two-dimensional film ($a<\\lambda_D$, $L_z, L_\\perp \\gg \\lambda_D$)\ncan be calculated likewise using the above formalism. \nThe noise is measured along the $z$ direction, $L_\\perp$ and $x$ are the transverse width and \nthe transverse coordinate of\nthe film, respectively, and $a$ is the thickness.\nUsing the corresponding expressions for 2D Coulomb potential \n$u_2(q)=\\frac{2\\pi}{ q}$ and the two-dimensional conductivity $\\sigma_{2D}=\\sigma a$,\nthe spectral density of 2D noise is\n\n\\beq\n&&<\\mid \\phi_{q,\\omega}^{ind(2d)} \\mid^2>= \n\\frac{2kT}{\\pi} \\frac{4 \\pi^2 \\sigma_{2D}}{\\omega^2 +(Dq^2+2 \\pi\\sigma_{2D}q)^2}. \n\\label{eq:2D-charged}\n\\eeq\n\nSince the in-plane dimensions of the film are much larger than the screening length,\nwe can simplify the denominator of the above equation by neglecting the term $Dq^2$\n(since $q \\ll \\frac{\\sigma_{2D}}{D}=\\frac{a}{4\\pi\\lambda_D}\n\\frac{1}{\\lambda_D}$, then $Dq^2 \\ll \\sigma_{2D} q$).\nThe differential noise between the two ends of the two-dimensional strip, \naveraged over the transverse modes, is\n\n\\begin{eqnarray}\n<\\mid \\phi_{12}^{(2d)}(\\omega) \\mid^2>=\n\\int \\frac{dq_z}{2\\pi}~ 4sin^2 \\left( \\frac{q_zL_z}{2} \\right) \\n\\\\ \n\\sum_{q_x} 4 \\frac{ sin^2(q_x L_\\perp/2)}{q_x^2 L_\\perp^2} \n<\\mid \\phi_{q,\\omega}^{ind(2d)} \\mid^2>. \\label{eq:2D-ends} \n\\end{eqnarray}\n\nThe integration (if the sum can be approximated by the integral)\n over the transverse dimension $x$ can be done exactly, and\nwe get the expression:\n\n\\begin{eqnarray}\n<\\mid \\phi_{12}^{(2d)}&&(\\omega) \\mid^2>=\n\\frac{2kT}{\\pi \\sigma_{2D}}\n\\int \\frac{dq_z}{2\\pi}~ 4sin^2 \\left( \\frac{q_zL_z}{2} \\right) \\n\\\\ \n&&\\int \\frac{dq_x}{2\\pi}~ 4\\frac{ sin^2(q_x L_\\perp/2)}{q_x^2 L_\\perp^2}\n\\frac{1}{(b^2+q_z^2)+q_x^2}= \\n\\\\\n&&=\\frac{2kT}{\\pi \\sigma_{2D}}\n\\int \\frac{dq_z}{2\\pi}~ 4sin^2 \\left( \\frac{q_zL_z}{2} \\right) F(q_z). \\n\n\\end{eqnarray}\n\\beq \nF(q_z)\\equiv \\frac{1}{2L_\\perp^2(q_z^2+b^2)} \n\\left(2L_\\perp-\\frac{1-exp(-2L_\\perp\\sqrt{q_z^2+b^2})}\n{\\sqrt{q_z^2+b^2}} \\right),\\n\n\\eeq\nwhere $b\\equiv\\frac{\\omega}{2\\pi\\sigma_{2D}}$ is the inverse of the characteristic length scale of the\nproblem. The limiting expressions for the function $F(q_z)$ are:\n\n\n\\[ F(q_z)\\simeq \\left\\{ \\begin{array}{ll}\n \\frac{1}{L_\\perp(q_z^2+b^2)} &\\mbox{if $\\sqrt{q_z^2+b^2}L_\\perp \\gg 1$}\\\\\n \\frac{1}{\\sqrt{q_z^2+b^2}} &\\mbox{if $\\sqrt{q_z^2+b^2}L_\\perp \\ll 1$ }\n \\end{array}\n \\right. \\]\n\nThe integral over wave vector $q_z$ can be taken in such limiting cases.\nBut a careful analysis shows that for the case $\\sqrt{q_z^2+b^2}L_\\perp \\ll 1$ the summation over\nthe transverse modes $q_x$ cannot be approximated by the integral. The main contribution\nactually comes from the ``zero mode'' $q_x=0$ in spite of the condition $L_\\perp \\gg \\lambda_D$.\nTaking the above considerations into account, the answer for the noise\nacross two-dimensional film is given below. \n\n\\[<\\mid \\phi_{12}^{(2d)}(\\omega) \\mid^2> \\simeq \\frac{2kT}{\\pi\\sigma_{2D}}* \\left\\{ \\begin{array}{ll}\n \\frac{\\sigma_{2D}}{\\omega L_\\perp} &\\mbox{if $\\omega \\gg \\frac{2\\pi\\sigma_{2D}}{L_z},\n \\frac{\\sigma_{2D}}{L_\\perp} $}\\\\\n \\frac{L_z}{L_\\perp} &\\mbox{if $\\omega \\ll \\frac{\\sigma_{2D}}{max(L_z,L_\\perp)} $ }\\\\\n \\end{array}\n \\right. \\]\n\nIn some cases, the integral can be expressed through Bessel functions, but only the final asymptotic\nexpressions are of interest here. As pointed above \nwhen $L_\\perp \\ll \\frac{\\sigma_{2D}}{\\omega}$, the main contribution \n(after the integration over $q_z$) calculated from the ``zero mode'' $q_x=0$\nis proportional to $L_z/L_\\perp$, while the estimate of the integral over the higher harmonics of $q_x$\nis smaller and proportional to $ln\\frac{L_z}{\\lambda_D}$ (if $L_z \\gg L_\\perp$).\nAt the frequencies of the order of the 3D Maxwell frequency, the film cannot be considered as\ntwo-dimensional and transverse harmonics other than the ``zero-frequency'' one ($q_y=0$) \nmust be taken into account.\n\n%------discussion of 2D noise\n\nIt is very interesting that for frequencies above $\\frac{\\sigma_{2D}}{L_{\\perp,z}}=\n\\sigma\\frac{a}{L_{\\perp,z}}$ frequency and below the Maxwell relaxation frequency $4\\pi\\sigma$\nthe noise is {\\it universal}\n (independent of the material specific conductivity $\\sigma_{2D}$\nand dependent only on the transverse size $L_\\perp$) and equal to $\\frac{4kT}{\\omega L_\\perp}$. \n\nBy the FDT the noise is proportional\nto the total dissipation which is the product of the dissipation per unit length and the \ncharacteristic dissipative length scale. The dissipation per unit length is inversely proportional\nto the material specific conductivity $\\sigma_{2D}$.\nAt the low frequencies, the dissipative length scale is set by the longitudinal size $L_z$ of the sample,\ntherefore the noise is proportional to $L_z/\\sigma_{2D}$. \nAt the high frequencies, as soon as\nthe length scale $\\sigma_{2D}/\\omega$ becomes smaller than the longitudinal size $L_z$, the\ndissipative length scale is set by this length $\\sigma_{2D}/\\omega$. \nTherefore the high frequency noise becomes independent of the conductivity.\n{\\it The universality of the noise} at high frequencies is special to the two-dimensional\nsituation and is due to dimensional reasons. Only in two dimensions the length scale is \ngiven by the simple ratio $\\sigma_{2D}/\\omega$\nof the conductivity $\\sigma_{2D}$ (or the conductance) and the frequency $\\omega$.\n\n\nAt the low frequencies ($\\omega \\rightarrow 0$),\nthe noise \nhas the standard form consistent with the fluctuation-dissipation theorem\\cite{callen} \napplied to the whole sample\n\n\\beq\n<\\mid \\phi_{12}^{(2d)} \\mid^2> \\simeq \\frac{2}{\\pi} kT R, \\label{eq:2D}\n\\eeq \nwhere $R=\\frac{L_z}{\\sigma_{2D}L_\\perp}$ is the dc resistance of the film. \n%\nThe noise in the 3D wire (Eqn. \\ref{eq:3D})\n and the 1D wire (Eqn. \\ref{eq:1d-neutral})\nis consistent with the FDT as well.\n\n%It appears very important to understand the limits of validity of fluctuation-dissipation theorem\n%applied to a whole sample (as a relation between noise spectra and properly defined ac resistance)\n%in the situation of restricted geometries and the presence of long-range interactions. \n\nThe fluctuation-dissipation theorem applied to the whole sample relates the voltage noise\nbetween the ends of the sample to the real part of the impedance $Z(\\omega)$ of the sample. \nAt zero frequency\nthe capacitive part of the impedance is always short-circuited by the dissipative part\n(the resistance).\nThe resistance of the wire in all considered cases \nis expressed through the geometrical sizes, as it can be seen from \nthe Eqns. (\\ref{eq:3D},\\ref{eq:1d-neutral},\\ref{eq:2D}).\nIf the resistance is measured\nfrom the zero frequency expression of the noise, then the effective capacitance $C$ of the sample\ncan be measured from the high-frequency ($\\omega RC \\gg 1$) expression of the noise:\n$ Re Z(\\omega)\\simeq\\frac{R}{(\\omega RC)^2}$.\nWe can use this equation to interpret the high-frequency noise in the 1D wire\nand the 2D strip and to write the expressions for the effective capacitances\nof the corresponding wires. \nIn case of the 3D wire (with the well screened Coulomb interaction), the sample\nhas a constant (frequency independent) capacitance $C=\\frac{S}{4\\pi L}$. \n% (see the discussion after the Eqn.\\ref{eq:3D}). \nThe effective capacitance of the 2D strip at the high frequencies \n($\\sigma \\gg \\omega \\gg \\sigma_{2D}/L_z$) is\n\n\\beq\nC_{2D}\\simeq L_\\perp \\sqrt{\\frac{\\sigma_{2D}}{\\omega L_z}}=\n\\frac{L_\\perp a}{L_z} \\sqrt{\\frac{\\sigma L_z}{\\omega a}}\\gg \\frac{L_\\perp a}{L_z}. \n\\eeq\nThe effective capacitance of the 1D wire at the frequencies ($\\sigma \\gg \\omega \\gg D/L^2$) is\n\n\\beq\n&&C_{1D}\\simeq \\left(\\frac{\\sigma}{\\omega} \\right)^{3/4} \\frac{a^2}{2\\pi^{3/4} \\lambda_D^{1/2} L^{1/2}}=\n\\n \\\\\n&&=\\frac{1}{2\\pi^{3/4}} \\left(\\frac{L}{\\lambda_D} \\right)^{1/2} \\left(\\frac{\\sigma}{\\omega} \\right)^{3/4}\n\\frac{a^2}{L} \\gg \\frac{a^2}{L}.\n\\eeq\nIn the both 1D and 2D wires, the effective capacitances are frequency dependent and \nmuch larger than the standard geometrical\ncapacitances, because the Coulomb interaction is not completely screened and non-local.\n\n\n% -----------end of 2D addition------------------------------------\n\nSince the noise has a frequency dependent form, by FDT it implies the same frequency dependence\nof the real part of the impedance $Z(\\omega)$. The measurement of the complex impedance \ncan be more straightforward way to access the predicted frequency dependencies of noise\nthan a direct measurement of noise.\n\n%discussion, conclusion\n\nThe nature of the relaxation of a random potential fluctuation is quite different in charged\nand neutral liquids. In charged three-dimensional liquids, it is essentially \nthe fast process of screening, and\nin neutral liquids it is the process of diffusion. \nThe appropriate physical picture\nof fluctuations in a three-dimensional electrical conductor is \nthat charge fluctuations relax on a very fast\ntime scale $1/4\\pi\\sigma$, producing quasi-homogeneous current fluctuations. \nIn one-dimensional systems such as narrow wires,\nthe Coulomb interaction does not cause long-range correlations; therefore, the noise in a narrow conductor\nis similar to the noise spectra in neutral systems. \nThe difference\nin the chemical potential between two points is relaxed through diffusion on a characteristic time scale\n$L^2/D$\nquadratically dependent on the length $L$ between points.\nThe situation of two-dimensional noise is intermediate, and\nthe characteristic time scale $L/\\sigma_{2D}$ of the relaxation of \nthe voltage fluctuation difference between\ntwo points is linearly dependent on the distance $L$. \n\nThe noise spectrum and the spectrum of collective modes are closely related. Since the\nspectrum of collective modes (plasmons) is given by the zeros of \nthe dielectric constant $\\epsilon(q,\\omega)$,\nthey give rise to the dominant contribution to the noise spectrum \nwhich is proportional to the $Im\\frac{1}{\\epsilon(q,\\omega)}$ (see\nthe comment\\cite{2-der}). At the end, both the frequency dependence of the noise and the dispersion\nof the collective modes depend essentially only on the effective dimensionality of the Coulomb \ninteraction.\n\nThe experimental observation of the predicted noise properties is feasible \nin semiconducting materials\nhaving a low carrier concentration.\\cite{validity} The screening length \n$\\lambda_D$ in such materials\n\\cite{epi} \ncan be as large as $10^{-4} cm$.\nIn metals, both the elastic rate $1/\\tau$ and the Maxwell frequency $4\\pi\\sigma$ are\nhigh and difficult to observe,\nwhile the typical screening length for a metal \nis of order of $10^{-8}cm$.\nIn fact, with current experimental techniques \n(see Reference\\cite{Naveh} for a review of experiments), even the high Maxwell relaxation frequency\ncrossover $4\\pi\\sigma$\ncan be observed in ``wide'' wires ($a \\gg \\lambda_D$, the situation almost always encountered)\n with poor conductivity.\nBy a convenient choice of the mobilities of the semiconductor materials and\ntheir size $L$, the ``Thouless frequency'' $D/L^2$ and the two-dimensional ``relaxation frequency'' \n$\\sigma_{2D}/L$ should be accessible. Several other experimental low-dimensional systems\nsatisfying the condition of the absence of screening can be suggested.\n\nThe contacts to the external leads (and associated boundary conditions) are not considered\nexplicitly in this paper. It is assumed that the main source of noise is the bulk of a wire\nor a film, and the contacts have a resistance much lower than a bulk system.\n\nThe question of the frequency dependence of equilibrium and ``shot'' noise \nwas raised recently by Y. Naveh {\\it et al}.\\cite{Naveh} Special geometries \nwith external screening\nwere suggested to observe the Maxwell and Thouless crossover frequencies. The above calculation shows\nthat the crossover at the Maxwell relaxation frequency is a general property of Coulomb systems\n and should be observed independently of geometry and length $L$ for ``wide'' wires. \nMoreover, for ``narrow'' wires ($a < \\lambda_D$)\nthe Thouless frequency crossover should be seen independently of ``external screening'' by\nelectrodes or the ground plane.\n\nIn conclusion, the noise in narrow wires ($a<\\lambda_D$) becomes frequency-dependent\nstarting from the low frequency $D/L^2$ (quite similar to simple diffusion systems), although\nin wide conductors, the noise is white up to the smaller of the frequencies $4\\pi\\sigma$ or $1/\\tau$.\nIn two-dimensional films, the Johnson-Nyquist noise has a universal $1/\\omega$ spectrum\nin the wide range of frequencies $\\sigma \\gg \\omega \\gg \\sigma\\frac{a}{L}$.\n\nThis work was supported by the National Science Foundation through the Science and Technology\nCenter for Superconductivity (Grant No. DMR-91-20000).\nI thank A. Leggett and M. Weissman for helpful discussions. I am grateful to A. Leggett,\nR. Ramazashvili and H. Westfahl\nfor the useful remarks and the careful reading of the manuscript.\n%----------------------------------------------------------\n\\begin{references}\n\n\\bibitem{Nyquist}\nM.B. Johnson, Phys. Rev. {\\bf 29}, 367 (1927), H. Nyquist, Phys. Rev. {\\bf 32}, 110 (1928).\n\n\\bibitem{plasma}\nThe Maxwell frequency $4\\pi\\sigma$ can be expressed through the three-dimensional plasma\nfrequency $\\omega_p$ and the elastic scattering time $\\tau$: $4\\pi\\sigma=\\omega_p^2\\tau$.\nThe threshold frequencies $D/L^2$ and $\\sigma_{2D}/L$ are much smaller than the three-dimensional\nMaxwell frequency $4\\pi\\sigma$: $4\\pi\\sigma \\gg D/L^2=4\\pi\\sigma (\\lambda_D^2/L^2)$ (1D),\n$4\\pi\\sigma \\gg \\sigma_{2D}/L =4\\pi\\sigma (a/(4\\pi L))$ (2D). \nThese threshold frequency $D/L^2$ and $\\sigma_{2D}/L$ can be made arbitrarily \nsmall by increasing the separation between two points $L$. \n\n\n\\bibitem{Nozieres}\nD. Pines, Ph. Nozi\\'{e}res, {\\it The theory of quantum liquids} (Benjamin, New York, 1966). For brevity,\nwe omit the sign $\\delta$ of small variations in all expressions.\n\n\\bibitem{classical}\nExcept ultra-low temperatures, the Johnson-Nyquist noise is classical at the frequencies\nof interest (e.g. a temperature $kT=0.1 K$ corresponds to a frequency $1.6 GHz$).\n\n\\bibitem{2-der}\nEquation (\\ref{eq:charged}) can be also derived in two other equivalent ways, elucidating\nthe meaning of charge and current fluctuations in a charged liquid. \nOne way is to introduce\na stochastic current source $j^{ac}$ % probably due to ``external'' circuit\ninto the current equation\\cite{kogan}:\n$j_{q,\\omega}=\\sigma(iq) \\phi_{q,\\omega}\n+D(iq)\\rho_{q,\\omega} + j^{ac} $.\nThis derivation illustrates the role of quasi-homogeneous current fluctuations effectively\nscreening charge fluctuations. Yet a third derivation explicitly demonstrates the role\nof screening. Since\n$ \\delta \\phi^{tot}_{q,\\omega}= \\frac{4\\pi}{\\epsilon(q,\\omega)\\epsilon_0 q^2} \\delta \\rho_{q,\\omega} $,\n the fluctuations of $\\delta \\phi^{tot}_{q,\\omega}$ due to the FDT are given by the expression\n $<|\\delta \\phi^{tot}_{q,\\omega}|^2>=\\frac{\\hbar}{\\pi} Im\\frac{u_3 (q)}{\\epsilon(q,\\omega)} \ncoth\\frac{\\omega}{2kT} $.\nThe dielectric constant $\\epsilon(q,\\omega)$ can be represented\nin the hydrodynamic limit\\cite{Nozieres} as:\n $ \\epsilon(q,\\omega)= 1-4\\pi \\chi^{(n)}(q,\\omega)/q^2 =\n1+4 \\pi \\sigma/(i\\omega +D q^2).$ \n\n\\bibitem{Rytov}\nS.M. Rytov, Yu.A. Kravtsov, and V.I. Tatarskii, ~~{\\it Principles of Statistical Radiophysics}, v.3\n(Springer-Verlag, Berlin, 1976), p. 167.\n\n\\bibitem{quasi}\nAll calculations here are done in the assumption of a quasi-stationary condition\n( the length of a conductor $L$ is much smaller than the electromagnetic wavelength \n$\\lambda=c\\frac{2\\pi}{\\omega}$). This condition implies the absence of the retardation\neffects for an electromagnetic field profile inside a conductor (it does not imply the\nhomogeneity of electric field inside a conductor due to screening and diffusion processes).\n\n\\bibitem{kogan}\nSh. Kogan,~~{\\it Electronic noise and fluctuations in solids}, (Cambridge University Press, Cambridge, 1996);\nB.L. Altshuler, A.G. Aronov, and D.E. Khmelnitsky, J. Phys. C {\\bf 15}, 7367-86 (1982).\n% \\\\\n%see also thermodynamic original argument by Nyquist.\n%\\bibitem{landau}\n%L.D. Landau,E.M. Lifshits,~~ {\\it Statistical Physics, Part 1,2}\n\n\\bibitem{voss}\nR. Voss, J. Clark, Phys. Rev. B {\\bf 13}, 556-573 (1976).\n\n\\bibitem{callen}\nH.B. Callen, T.A. Welton, Phys. Rev. B {\\bf 83}, 34-40 (1951).\n\n\\bibitem{validity}\nIt is important to point out that the condition $a<\\lambda_D$ can be satisfied,\nwhile ``weak localization'' and Luttinger-liquid physics are not relevant\nfor the problem discussed here. If the electron Fermi-wavelength\n$\\lambda_F$ is much smaller than the diameter $a$ (many transverse channels),\nthe Luttinger liquid physics is not relevant. If either \n$\\sigma_1=\\frac{D}{4\\pi\\lambda_D^2}\\frac{\\pi a^2}{L} \\gg e^2/\\hbar$ \n(or equivalently $L \\ll L_{loc}=\\frac{\\hbar}{e^2} D\\frac{a^2}{2 \\lambda_D^2}$)\nor $L_\\phi , L_T \\ll a$ (the phase coherence length $L_\\phi$, the thermal length $L_T$), than\nthe ``weak localization'' corrections are small. \n\n\n\\bibitem{epi}\nThe large high-frequency dielectric constant $\\epi$ of a material is favorable for the experimental\nobservation, since the screening length $\\tilde{\\lambda_D}$ becomes larger \n($\\tilde{\\lambda_D}^2=\\epi \\lambda_D^2$). The threshold frequencies become smaller \n(e.g. $\\tilde{\\omega_p}^2 =\\omega_p^2/\\epi$).\n\n\\bibitem{Naveh}\nY. Naveh, D. Averin, and K. Likharev, Phys. Rev. Lett. {\\bf 79}, 3482 (1997), \n Phys. Rev. B {\\bf 59}, 2848-60 (1999).\n\n\\end{references}\n\n\n\\end{multicols}\n\n\\end{document}"
}
] |
[
{
"name": "cond-mat0002153.extracted_bib",
"string": "\\bibitem{Nyquist}\nM.B. Johnson, Phys. Rev. {\\bf 29}, 367 (1927), H. Nyquist, Phys. Rev. {\\bf 32}, 110 (1928).\n\n\n\\bibitem{plasma}\nThe Maxwell frequency $4\\pi\\sigma$ can be expressed through the three-dimensional plasma\nfrequency $\\omega_p$ and the elastic scattering time $\\tau$: $4\\pi\\sigma=\\omega_p^2\\tau$.\nThe threshold frequencies $D/L^2$ and $\\sigma_{2D}/L$ are much smaller than the three-dimensional\nMaxwell frequency $4\\pi\\sigma$: $4\\pi\\sigma \\gg D/L^2=4\\pi\\sigma (\\lambda_D^2/L^2)$ (1D),\n$4\\pi\\sigma \\gg \\sigma_{2D}/L =4\\pi\\sigma (a/(4\\pi L))$ (2D). \nThese threshold frequency $D/L^2$ and $\\sigma_{2D}/L$ can be made arbitrarily \nsmall by increasing the separation between two points $L$. \n\n\n\n\\bibitem{Nozieres}\nD. Pines, Ph. Nozi\\'{e}res, {\\it The theory of quantum liquids} (Benjamin, New York, 1966). For brevity,\nwe omit the sign $\\delta$ of small variations in all expressions.\n\n\n\\bibitem{classical}\nExcept ultra-low temperatures, the Johnson-Nyquist noise is classical at the frequencies\nof interest (e.g. a temperature $kT=0.1 K$ corresponds to a frequency $1.6 GHz$).\n\n\n\\bibitem{2-der}\nEquation (\\ref{eq:charged}) can be also derived in two other equivalent ways, elucidating\nthe meaning of charge and current fluctuations in a charged liquid. \nOne way is to introduce\na stochastic current source $j^{ac}$ % probably due to ``external'' circuit\ninto the current equation\\cite{kogan}:\n$j_{q,\\omega}=\\sigma(iq) \\phi_{q,\\omega}\n+D(iq)\\rho_{q,\\omega} + j^{ac} $.\nThis derivation illustrates the role of quasi-homogeneous current fluctuations effectively\nscreening charge fluctuations. Yet a third derivation explicitly demonstrates the role\nof screening. Since\n$ \\delta \\phi^{tot}_{q,\\omega}= \\frac{4\\pi}{\\epsilon(q,\\omega)\\epsilon_0 q^2} \\delta \\rho_{q,\\omega} $,\n the fluctuations of $\\delta \\phi^{tot}_{q,\\omega}$ due to the FDT are given by the expression\n $<|\\delta \\phi^{tot}_{q,\\omega}|^2>=\\frac{\\hbar}{\\pi} Im\\frac{u_3 (q)}{\\epsilon(q,\\omega)} \ncoth\\frac{\\omega}{2kT} $.\nThe dielectric constant $\\epsilon(q,\\omega)$ can be represented\nin the hydrodynamic limit\\cite{Nozieres} as:\n $ \\epsilon(q,\\omega)= 1-4\\pi \\chi^{(n)}(q,\\omega)/q^2 =\n1+4 \\pi \\sigma/(i\\omega +D q^2).$ \n\n\n\\bibitem{Rytov}\nS.M. Rytov, Yu.A. Kravtsov, and V.I. Tatarskii, ~~{\\it Principles of Statistical Radiophysics}, v.3\n(Springer-Verlag, Berlin, 1976), p. 167.\n\n\n\\bibitem{quasi}\nAll calculations here are done in the assumption of a quasi-stationary condition\n( the length of a conductor $L$ is much smaller than the electromagnetic wavelength \n$\\lambda=c\\frac{2\\pi}{\\omega}$). This condition implies the absence of the retardation\neffects for an electromagnetic field profile inside a conductor (it does not imply the\nhomogeneity of electric field inside a conductor due to screening and diffusion processes).\n\n\n\\bibitem{kogan}\nSh. Kogan,~~{\\it Electronic noise and fluctuations in solids}, (Cambridge University Press, Cambridge, 1996);\nB.L. Altshuler, A.G. Aronov, and D.E. Khmelnitsky, J. Phys. C {\\bf 15}, 7367-86 (1982).\n% \\\\\n%see also thermodynamic original argument by Nyquist.\n%\n\\bibitem{landau}\n%L.D. Landau,E.M. Lifshits,~~ {\\it Statistical Physics, Part 1,2}\n\n\n\\bibitem{voss}\nR. Voss, J. Clark, Phys. Rev. B {\\bf 13}, 556-573 (1976).\n\n\n\\bibitem{callen}\nH.B. Callen, T.A. Welton, Phys. Rev. B {\\bf 83}, 34-40 (1951).\n\n\n\\bibitem{validity}\nIt is important to point out that the condition $a<\\lambda_D$ can be satisfied,\nwhile ``weak localization'' and Luttinger-liquid physics are not relevant\nfor the problem discussed here. If the electron Fermi-wavelength\n$\\lambda_F$ is much smaller than the diameter $a$ (many transverse channels),\nthe Luttinger liquid physics is not relevant. If either \n$\\sigma_1=\\frac{D}{4\\pi\\lambda_D^2}\\frac{\\pi a^2}{L} \\gg e^2/\\hbar$ \n(or equivalently $L \\ll L_{loc}=\\frac{\\hbar}{e^2} D\\frac{a^2}{2 \\lambda_D^2}$)\nor $L_\\phi , L_T \\ll a$ (the phase coherence length $L_\\phi$, the thermal length $L_T$), than\nthe ``weak localization'' corrections are small. \n\n\n\n\\bibitem{epi}\nThe large high-frequency dielectric constant $\\epi$ of a material is favorable for the experimental\nobservation, since the screening length $\\tilde{\\lambda_D}$ becomes larger \n($\\tilde{\\lambda_D}^2=\\epi \\lambda_D^2$). The threshold frequencies become smaller \n(e.g. $\\tilde{\\omega_p}^2 =\\omega_p^2/\\epi$).\n\n\n\\bibitem{Naveh}\nY. Naveh, D. Averin, and K. Likharev, Phys. Rev. Lett. {\\bf 79}, 3482 (1997), \n Phys. Rev. B {\\bf 59}, 2848-60 (1999).\n\n"
}
] |
cond-mat0002154
|
Viscosity critical behaviour at the gel point in a $3d$ lattice model
|
[
{
"author": "Emanuela Del Gado"
},
{
"author": "$^{a}$ Lucilla de Arcangelis"
},
{
"author": "$^{b}$ and Antonio Coniglio $^{a}$"
}
] |
Within a recently introduced model based on the bond-fluctuation dynamics we study the viscoelastic behaviour of a polymer solution at the gelation threshold. We here present the results of the numerical simulation of the model on a cubic lattice: the percolation transition, the diffusion properties and the time autocorrelation functions have been studied. From both the diffusion coefficients and the relaxation times critical behaviour a critical exponent $k$ for the viscosity coefficient has been extracted: the two results are comparable within the errors giving $k \sim 1.3$, in close agreement with the Rouse model prediction and with some experimental results. In the critical region below the transition threshold the time autocorrelation functions show a long time tail which is well fitted by a stretched exponential decay.\\ PACS: 05.20.-y, 82.70.Gg, 83.10.Nn
|
[
{
"name": "art3rev_n.tex",
"string": "%Dear Editor European Physical Journal E,\n%\n%Enclosed please find a revtex copy of the revised version of the manuscript\n%\"Viscosity critical behaviour at the gel point in a 3d lattice model\"\n%by E.Del Gado, L.de Arcangelis and A.Coniglio, \n%ref.E9040.\n%The figures 1(a), 1(b), 2, 3, 4, 5, 6, 7, 8, 9 are sent in the next e-mail\n%in a EPS version.\n%We have taken into account both the revisions suggested by the second referee.\n%More precisely at the bottom of page 3 and in the conclusion we have made a \n%distinction \n%between the de Gennes' analogy and the result for the conductivity critical \n%exponent in the random superconducting network obtained by Kertesz (1983);\n%at page 9 we have omitted to say that the characteristic time linked to the \n%viscosity is the longest relaxation time in the system and for a more detailed\n%discussion on the distribution of relaxation times and the frequency \n%dependence of the viscoelastic properties we have referred to the \n%works of Adam et al.(1987,1992) and Daoud (1992). \n%We hope that the paper is now suitable for the publication on the \n%European Physical Journal E.\n%Thank you for your consideration,\n%\n%Yours sincerely,\n% \n%Emanuela Del Gado (corresponding author)\n% \n%Dipartimento di Scienze Fisiche, Universita' di Napoli \"Federico II\"\n%Mostra d'Oltremare, pad.20\n%80125 Napoli, Italy\n%\n%e-mail: delgado@na.infn.it\n%fax: +39 0812394508\n\n\\documentstyle[amstex,preprint,aps]{revtex}\n%\n\\title{Viscosity critical behaviour at the gel point in a $3d$ lattice model}\n\\author{Emanuela Del Gado,$^{a}$ Lucilla de Arcangelis, $^{b}$ and\n Antonio Coniglio $^{a}$}\n\\address{${}^a$Dipartimento di Scienze Fisiche, Universit\\`a di Napoli\n \"Federico II\",\\\\ INFM Sezione di Napoli, Mostra d'Oltremare\n pad. 19, 80125 Napoli, Italy}\n\\address{${}^b$Dipartimento di Ingegneria dell'Informazione, \n Seconda Universit\\`a di Napoli,\\\\ INFM Sezione di Napoli, via Roma 29,\n 81031 Aversa (Caserta), \n Italy}\n%\n%\n\\begin{document}\n%\n\\maketitle\n\\begin{abstract}\nWithin a recently introduced model based on the bond-fluctuation \ndynamics we study the viscoelastic behaviour of a polymer solution at the \ngelation threshold.\nWe here present the results of the numerical \nsimulation of the model \non a cubic lattice: the percolation transition, the diffusion properties and \nthe time autocorrelation functions have been studied.\nFrom both the diffusion coefficients and the relaxation times critical behaviour\na critical exponent $k$ for the viscosity coefficient has been extracted: the \ntwo results are comparable within the errors giving $k \\sim 1.3$, in close \nagreement with the Rouse model prediction and \nwith some experimental results. In the \ncritical region below the transition threshold the time autocorrelation \nfunctions show a long time tail which is well fitted by a stretched exponential \ndecay.\\\\\nPACS: 05.20.-y, 82.70.Gg, 83.10.Nn\n\\end{abstract}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\n\\newpage\nPolymeric materials are characterized by a rich and complex phenomenology, \nwhich goes from non newtonian dynamic behaviour in polymer liquids to \nviscoelastic properties of polymer gels and the glass transition of polymer \nmelts, intensively investigated in the last decades. The interest \nin such systems has been recently further increased by several \npossibilities of technological applications in many different fields but still \nthe non trivial viscoelastic behaviour of polymeric systems does not \nhave a completely satisfying description.\\\\ \nThe gelation transition which transforms the polymeric solution, i.e. the sol, \nin a polymeric gel is characterized by a dramatic change in the viscoelastic \nproperties: this is usually described in terms of the divergence of the \nviscosity coefficient and the appearance of an elastic modulus which \ncharacterizes the gel phase \\cite{flo,degl}. \nIn the experiments both the viscosity coefficient and the elastic modulus \ndependence on the \npolymerization reaction extent are well fitted by a power law \nbut the experimental determination of these critical exponents is quite \ncontroversial: the results are in fact scattered, probably because of the \npractical difficulties in obtaining the gelation transition in a reproducible \nmanner, and could only be interpreted on the basis of a better \ncomprehension of the relevant mechanisms in the transition. Recent experimental \nmeasurements of the viscosity critical exponent $k$ give values ranging from \n$0.7$ in diisocyanate/triol to $1.5$ in epoxy resins whereas for the elastic \nmodulus critical exponent $t$ the \nvalues are even more scattered, ranging from $1.9$ in diisocianate/triol to \n$3.0$ in polyesters (see references \n\\cite{ad97,ad91,ad92,mawi,col1,col2,nico,malla}).\\\\\nWith simple statistical mechanics models it is possible to analyse the \nessential aspects of the transition and its\ncritical properties. From the Flory model this has led to the description \nof the gelation process in terms of a percolation transition. The percolation \nmodel has turned out to be the \nsatisfactory model for the sol-gel transition \\cite{adconst,staul}, it is \nable to describe the role \nof connectivity and gives the critical \nexponents for all the geometrical properties, which are in perfect agreement \nwith the experimental results. \nOn the other hand the viscoelastic dynamic\nbehaviour is not simply obtained in terms of the connectivity transition.\\\\ \nThe difficulties in studying the viscosity critical behaviour at the gel point \ncome from the determination of the viscosity of a very complex medium, \nwhich is the sol at \nthe transition threshold, a highly polydisperse polymeric solution at high \nconcentration. The complex polymer dynamics is characterized by the relaxation \nprocesses over many different time scales and compete with the increasing \nconnectivity to produce the observed viscoelastic behaviour. The simplest \napproach consists in \nconsidering the sol as a polydisperse \nsuspension of \nsolid spheres neglecting the cluster-cluster interactions and generalizing the \nEinstein formula for the \nviscosity of a monodisperse suspension of solid spheres \n\\cite{adconst}, which corresponds to a highly diluted regime. \nWithin the Flory classical theory of gelation the viscosity \nremains finite \\cite{flo} or diverges at most logarithmically. \nUsing instead the Rouse model for the polymer dynamics \\cite{doed}, which neglects \nthe entanglement effects and the hydrodynamics interactions, the \nviscosity in a solution of polymeric clusters, expressed \nin terms of the \nmacroscopic relaxation time, grows like $<R^{2}>$ as the cluster radius \n$R$ grows in the gelation process. The \ncontribution of the $n_{s}$ molecules of size $s$ and gyration radius $R_{s}$ \nto the average $<>$ \nis of the order of $sn_{s}R^{2}_{s}$ leading to the critical exponent \n$k = 2\\nu-\\beta$ \\cite{deg78}, where $\\nu$ is the critical exponent \nof the correlation length diverging at the gel point and $\\beta$ is \nthe critical exponent describing the growth of the gel phase. With the \nrandom percolation exponents the value $k \\sim 1.35$ is found, that \nagrees quite well with the experimental measurements for silica gels of\nref.\\cite{mawi,col1}. Actually this Rouse exponent could be \nconsidered as an upper limit due to \nthe complete screening of the hydrodynamic interactions and the entanglement \neffects.\nThe Zimm approach \\cite{doed}, where the monomer correlation due to \nthe hydrodynamic interactions are not completely screened, would give a \nsmaller exponent.\\\\\nAnother approach has been proposed by de Gennes \\cite{degl,deg78} using an \nanalogy between the viscosity at the gelation \nthreshold and the diverging conductivity in the {\\em random superconducting \nnetwork} model. Following this analogy an exponent $k \\sim 0.7$ is obtained \nin $3d$, according to the determination of the conductivity critical exponent \nin the {\\em random superconducting model} \\cite{ker}. This result is in \ngood agreement with the values experimentally obtained in gelling solutions of \npolystyrene/divinylbenzene and diisocyanate/triol \n\\cite{ad91,ad92}.\\\\ \nOur approach consists in directly investigating the viscoelastic properties at \nthe sol-gel transition, introducing within the random percolation \nmodel the bond fluctuation ($BF$) \ndynamics which is able to take into account the polymer \nconformational changes \\cite{carkr1,carkr2}. We study a \nsolution of tetrafunctional monomers at concentration $p$ and with a \nprobability $p_{b}$ of \nbonds formation. In terms of these two parameters one has different cluster \nsize distributions and eventually a percolation transition. Monomers interact \nvia excluded volume interactions and can diffuse with local random \nmovements. The monomer diffusion process produces a variation of the bond \nvectors and is constrained by the excluded volume \ninteraction and the SAW condition for polymer clusters: these two requirements\ncan be satisfied if the bond lengths vary within an allowed range. \nThis dynamics results to take into account the main \ndynamic features of polymer molecules.\\\\ \nWe have performed numerical simulations of the model on the cubic lattice of \ndifferent lattice sizes ($L=24,32,40$) with periodic boundary conditions. \nThe eight sites which are the vertices of a lattice \nelementary cell are simultaneously occupied by a monomer, with the constraint \nthat two nearest neighbour ($nn$) \nmonomers are always separated by an empty elementary cell, i.e. two \noccupied cells cannot have common sites.\nThe lattice of cells, with double lattice spacing, has been occupied with \nprobability $p$, which coincides with the monomer concentration on the \nmain lattice in the thermodynamic limit \\cite{dedecon1}. \nMonomers are randomly distributed on the main lattice via a diffusion \nprocess, then between two $nn$ or next nearest neighbour ($nnn$) monomers bonds \nare instantaneously created with probability $p_{b}$ along lattice directions. \nSince most of the experimental data on the gelation transition refer to \npolymers with monomer functionality $f=4$ we have considered this case \nallowing the formation of at most four bonds per monomer.\\\\\nFirst, a qualitative phase diagram has been determined, studying the \nonset of the gel phase varying $p$ and $p_{b}$: on this basis we have then \nfixed $p_{b}=1$ and let $p$ vary in the interesting range from the sol to \nthe gel phase. \nThe percolation transition has thus been studied via the percolation \nprobability $P$ and the mean cluster size $\\chi$ on lattices of different \nsize. From their finite size scaling \nbehaviours we have obtained the percolation threshold \n$p_{c} \\simeq 0.718 \\pm 0.005$, the critical exponents $\\nu \\simeq 0.89 \n\\pm 0.01$ for \nthe percolation correlation length $\\xi$ and \n$\\gamma \\simeq 1.8 \\pm 0.05$ for the mean cluster size $\\chi$ \n(fig.(\\ref{fig1})). \nThese results do agree with the \nrandom percolation \ncritical exponents \\cite{staul}. \n\\footnote{The same agreement with the random percolation critical exponents has \nalready been obtained in the $d=2$ version of the model \\cite{dedecon1}. Here \nit is worth to mention that this case on a cubic lattice with monomer \nfunctionality \n$f=4$ is rather the problem of a {\\em restricted valence percolation}. This is \na percolation on a lattice where the number of bonds emanating from the same \nsite is restricted or no site may have more than a fixed number of {\\em nn}. \nIt does reproduce the occurrence of valence saturation for monomers in the \ngelation process. If the number of allowed bonds per site is greater than $2$ \nthis problem is expected to belong to the same universality class as \nthe random percolation \\cite{adconst}. As the restriction on the number of \nbonds per monomer clearly introduces a \ncorrelation effect in the process of bond formation we have optimized our \nalgorithm to minimize this effect and actually obtained a good agreement with \nthe random percolation exponents.}\\\\ \nThe system evolves according to the bond fluctuation dynamics, i.e. the \nmonomers diffuse with random local movements along lattice directions within \nthe excluded volume constraint and produce the bond length fluctuation among \nthe allowed values. In fact the $BF$ dynamics can be easily expressed in terms \nof a lattice algorithm and on a cubic lattice it can be shown that the bond \nlengths which guarantee the SAW condition are \n$l=2,\\sqrt{5},\\sqrt{6},3,\\sqrt{10}$ in lattice spacing units \\cite{carkr2}. \nIn fig.(\\ref{fig2}) a simple example of time evolution of a cluster is \nshown.\\\\ \nWe remind that the percolation properties are not modified during the dynamic \nevolution of the system, as it is not possible to break or form bonds, \nwhich would change the cluster size distribution.\\\\ \nIn order to determine the viscosity critical behaviour we use two independent \nways, based respectively on the diffusion behaviour of the clusters and \non the relaxation times. Within the study of diffusion properties in the \nsystem an interesting picture is obtained with a simple scaling argument on the \ndiffusion coefficients as presented in \\cite{madwi}: the sol at the sol-gel \ntransition is a heteregenous medium formed by the \nsolvent and all the other clusters of different sizes, with the mean cluster \nsize rapidly growing near the percolation threshold. \nA cluster with gyration radius $R$ can be seen as a probe diffusing in \nsuch a medium: as long as its radius is much greater than the value of \nthe percolation correlation \nlength in the sol the Stokes-Einstein relation is expected to be valid, \nand the diffusion \ncoefficient $D(R)$ of the probe will decrease proportionally to the inverse \nof the \nviscosity coefficient of the medium, $D(R) \\sim 1/R^{d-2} \\eta$. Then the \ngeneric probe of size $R$ diffuses in a medium with a \nviscosity coefficient depending on \n$R$, $\\eta(R)$, and a Stokes-Einstein {\\em generalized} relation $D(R) \n\\simeq 1/R^{d-2} \\eta(R)$ should be expected to hold.\\\\\nAs the percolation threshold is approached the probe diffuses in a medium \nwhere a spanning cluster appears with a self-similar structure of \nholes at any length scale. At the \npercolation threshold the viscosity coefficient of the sol (the {\\em bulk} \nviscosity coefficient) diverges as $\\eta \\sim (p_{c} - p) ^{-k}$ and for the \nviscosity coefficient depending on $R$ the scaling behaviour \n$\\eta(R) \\sim R^{k/\\nu}$ should be expected. When $R$ is of the order of the \ncorrelation length then $\\eta(R) \\sim \\eta$. Following this \nscaling argument $D(R) \\sim 1/R^{d-2+ k/\\nu}$ at $p = p_{c}$.\\\\\nWithin this description the use of the Rouse model would consist in taking \n$D(s)\\sim 1/s$ for a cluster of size $s$.\nThen for large enough cluster sizes $s$ in percolation $s \\sim \nR^{d_{f}}$ where $d_{f}$ is the fractal dimensionality of the percolating \ncluster. Taking $D(R) \\sim 1/R^{d_{f}}$ leads to $k\\sim (d_{f}+2-d)\\nu$, \nwhich again gives $k = 2\\nu -\\beta$, the Rouse exponent given in \n\\cite{deg78}.\\\\ \nWe then study the diffusion of monomers and clusters via the mean square \ndisplacement of the center of mass. For a cluster of size $s$ (an \n$s$-cluster) this quantity is \ncalculated from the coordinates of its center of mass $\\vec{R}_{s}(t)$\nas\n\\begin{equation}\n \\langle \\Delta R^{2}_{s}(t) \\rangle = \\frac{1}{N_{s}} \\sum_{\\alpha=1}^{N_{s}}\n (\\vec{R}^{\\alpha}_{s}(t) - \\vec{R}^{\\alpha}_{s}(0))^{2}\n\\end{equation}\nwhere the index alpha refers to the $\\alpha$th $s$-cluster and $N_{s}$ is the \nnumber of $s$-clusters so that this quantity is averaged over \nall the $s$-clusters. All the data refer to a lattice size $L=32$ and the \ncalculated \nquantities have been averaged over $\\sim 30$ different site and bond \nconfigurations of the same $(p,p_{b})$ values \\cite{dedecon1}.\\\\\nOn the basis of the theory of Brownian motion and a simple Rouse model \napproach \\cite{doed} the center of mass of a polymeric molecule is expected \nto behave as a \nBrownian particle after a sufficiently long time. Indeed we find a linear \ndependence on time in the long \ntime behaviour of $\\langle \\Delta R^{2}_{s}(t) \\rangle$ and determine the \ndiffusion coefficient of an $s$-cluster in the environment formed by the \nsolvent \nand the other clusters. \nIn fig.(\\ref{fig3}) it is shown how the asymptotic diffusive behaviour \nis reached after a time which increases with the cluster size $s$, due to the \nmore complicated relaxation mechanism linked to the inner degrees of freedom in \nthe molecule. As $p$ increases towards $p_{c}$ the $s$-clusters move in a \nmedium which is more viscous and whose structure is more and more \ncomplex. As a consequence, we observe an immediate increase in the time \nnecessary to \nreach the asymptotic diffusive behaviour of the center of mass motion for all \nthe cluster sizes.\\\\\nAt $p=p_{c}$ we have calculated the diffusion coefficients of clusters of \nsize $s$ and radius of gyration $R$ in order to obtain the \ndependence $D(R)$. \nThe diffusion coefficients decrease with the increasing size of the \ncluster, as it is expected, but after a gradual decrease their values \ndramatically go to zero. This behaviour which is due to the block of \nthe diffusion for finite \ncluster size in a finite system has allowed to consider only cluster size up \nto $s=30$. \nIn fig.(\\ref{fig4}) we have plotted $D(R)$ for different cluster sizes \n($s=5$ to $30$): the data results to be well fitted by a power law behaviour. \nUsing this scaling argument which gives the prediction $D(R) \\sim \n1/R^{1+k/\\nu}$ we obtain for the viscosity coefficient a critical exponent \n$k \\sim 1.3 \\pm 0.1$.\\\\\nWe here briefly mention that on the other side the diffusion coefficients of \nvery small clusters are not expected to be linked to the macroscopic \nviscosity, and in fact the diffusion coefficient of \nmonomers does not go to zero at $p_{c}$, but has a definitively non-zero \nvalue for $p > p_{c}$ (fig.(\\ref{fig5})). It is also interesting to notice that \nthe data seems to agree with a dependence $ e^{-1/1-p}$, suggesting \nsome cooperative mechanism in the diffusion process.\\\\ \nIn order to study the viscosity in the system independently from the \nscaling hypothesis given before, we \nhave studied the relaxation times via the density time autocorrelation \nfunctions. We \nhave calculated the time autocorrelation function $g(t)$ of the number of \npairs of $nn$ monomers $\\varepsilon(t)$, defined as\n\\begin{equation}\ng(t)= \\langle \\frac{ \\overline{\\varepsilon(t')\\varepsilon(t'+t)} - \\overline{\n \\varepsilon(t')}^{2}}\n{\\overline{ \\varepsilon(t')^{2}} - \\overline{\\varepsilon(t')}^{2}} \\rangle\n\\label{autc}\n\\end{equation}\nwhere the bar indicates the average over $t'$ (of the order of $10^3$ time \nintervals) and the brackets indicate\nthe average over about $30$ different initial site and bond configurations.\\\\\nAt different $p$ values in the critical region after a fast transient \n$g(t)$ decays to zero but can not be fitted by a simple time exponential \nbehaviour. This is a sign of the existence of a distribution of relaxation \ntimes which \ncannot be related to a single time and this behaviour had already been \nobserved in the $d=2$ study of the model. It is a typical feature of polymeric \nsystems where \nthe relaxation process always envolves the rearrangement of the system over \nmany different length scales \\cite{ferry,daoud}. This idea of a \ncomplex relaxation behaviour is further confirmed by the good fit of the \nlong-time decay of the $g(t)$ with a stretched exponential law \n(fig.(\\ref{fig6})). This \nbehaviour of the relaxation functions is considered typical of complex \nmaterials and usually interpreted in terms of a very broad distribution of \nrelaxation times or eventually an infinite number of them and it is in fact \nexperimentally \nobserved in a sol in the gelation critical regime \\cite{mawio,adprl}. The \npicture we obtain via the density time autocorrelation functions is coherent \nand very close to the experimental characterization of the sol at the gelation \nthreshold.\\\\\nFor $p \\sim p_{c}$ we have then fitted the \n$g(t)$ with a stretched exponential \nbehaviour $e^{(-t/\\tau_{0})^{\\beta}}$ (fig.(\\ref{fig7})), \nwhere $\\beta \\sim 0.3$. This value is quite lower than the ones experimentally \nobtained in ref.\\cite{mawio} for a gelling solution or analitically predicted \nin ref.\\cite{chen} for randomly branched polymers: this discrepancy could be \ndue to the fact that our data refer to a quite narrow region near the \ngelation threshold where $\\beta$ is expected to assume the lowest value. On \nthe other hand this value of $\\beta$ agrees with the experimental results in \nref.\\cite{adprl} and with the asymptotic value of the stretched exponential \nexponent $\\beta$ in both Ising spin glasses and polymer melts close to the \nfreezing point in ref.\\cite{campb}.\\\\\nThe characteristic time $\\tau_{0}$ varies with $p$ and increases as \n$p_{c}$ is approached. Plotting $\\tau_{0}$ as a function of $(p_{c} - p)$ the \ndata can be fitted by a power law dependence, with a critical exponent $\\sim \n1.27 \\pm\\ 0.05$ (fig.(\\ref{fig8})), very close to the viscosity critical \nexponent obtained from the diffusion properties. This characteristic time \nextracted from the long time relaxation behaviour diverges at the percolation \nthreshold because of the passage between two different \nviscoelastic regimes.\\\\ \nThe most immediate way to characterize the distribution of the relaxation \ntimes in the system is the average characteristic time defined as\n\\begin{equation}\n\\tau(p) = \\frac{\\int_{0}^{t} t' g(t') dt'}{\\int_{0}^{t} g(t') dt'}\n\\label{time}\n\\end{equation}\nwhich is a typical macroscopic relaxation time and can be directly linked to \nthe viscosity coefficient.\\\\\nNumerically in the eq.(\\ref{time}) $t$ has been chosen by the condition \n$g(t') \\leq 0.001$ for $t'\\geq t$.\nThis characteristic time grows with $p$ and diverges at\nthe percolation threshold according to the critical behaviour\n\\begin{equation}\n\\tau \\propto (p_{c}-p)^{-k}\n\\label{tau}\n\\end{equation}\nwith an exponent $k \\simeq 1.3 \\pm 0.03$ (fig.\\ref{fig9}), \nwhich gives the critical exponent for the viscosity at the sol-gel \ntransition. From an immediate comparison between $k$ and the critical \nexponent for $\\tau_{o}$ there seems to be a unique power law characterizing \nthe divergence of the relaxation times in the system approaching from below \nthe transition \nthreshold.\\\\\nFor a more detailed description of the distribution of relaxation times at the\nsol-gel transition together with a study of the frequency dependence of the \nviscoelastic\nproperties see references\n\\cite{ad92,adfr,daoud,mawio}.\\\\ \nThe critical exponent obtained for the viscosity critical behaviour is then \n$k \\sim 1.3$ and it well agrees with the value of $k$ previously given, \nalthough independently obtained. This value is quite close to the one \nexperimentally measured in silica gels \\cite{col1,madwi} and interestingly \nthese systems are characterized by \na polyfunctional condensation mechanism with tetrafunctional monomers, \nwhich is actually very similar to the case simulated here. \nThis result is also close to the value obtained by recent accurate \nmeasurements in PDMS \\cite{ad97}.\\\\\nIt does not agree with the {\\em random superconducting network} exponent $k=0.7$\\cite{ker} \naccording to the de Gennes' analogy, whereas \nit is quite close to the Rouse exponent discussed above. We have already \nmentioned that a Rouse-like description of a polymer solution corresponds to a \ncomplete screening of the entanglement effects and the hydrodynamic \ninteractions and is usually considered not realistic enough. Actually \nthe entanglement effects \ncould be not so important in the relaxation mechanism in the sol on the \nmacroscopic relaxation time scale: due to the fractal structure of the gel \nphase the system is in fact quite fluid, probably \nthere is no blocking entanglement yet and such temporary entanglements relaxe\non a smaller time scale, not really affecting the macroscopic relaxation time \n\\cite{zip}. Furthermore the screening effect of the hydrodynamic \ninteractions in a polymeric solution in the semidiluted regime can be quite \nstrong, drastically reducing the range of the interactions so that the Rouse \nmodel results to be in fact very satisfactory \\cite{doed}. This could \nreasonably be the case of the sol at the gelation threshold too, and the \ndeviation of the real critical exponent from the Rouse value would turn out \nto be actually very small.\\\\\nThe numerical simulations have been performed on the parallel Cray-t3e system \nof CINECA (Bologna,Italy), taking about 15000 hours of CPU time. The authors \naknowledge partial support from the European TMR Network-Fractals \nc.n.FMRXCT980183 and from the MURST grant (PRIN 97). \nThis work was also supported by the INFM Parallel Computing and by the European \nSocial Fund.\\\\ \n%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\n\\begin{references}\n\\bibitem{flo} Flory,P.J., {\\em Principles of polymer chemistry} \n Cornell Univ.Press, Ithaca 1953\n\\bibitem{degl} De Gennes P.G. {\\em Scaling concepts in polymer physics} \n Cornell Univ. Press, Ithaca 1980 \n\\bibitem{ad97} Adam M., Lairez D., Karpasas M., Gottlieb M. \n {\\em Macromolecules } 30 5920(1997)\n\\bibitem{ad91} Adam M. {\\em Makromol.Chem.,Macromol.Symp.} 45 (1991)\n\\bibitem{ad92} Lairez D., Adam M., Raspaud E., Emery J.R., Durand D. {\\em\n Prog.Colloid Polym.Sci. } 90 (1992)\n\\bibitem{mawi} Martin J.E., Wilcoxon J.P. {\\em Phys.Rev.A}\n 39, 252(1989)\n\\bibitem{col1} Colby R.H., Coltrain B.K., Salva J.M., Melpolder S.M. in \n {\\em Fractal Aspects of Materials: Disordered Systems}, \n edited by A.J.Hurd, D.A.Weitz, and B.B.Mandelbrot \n (Materials Research Society, Pittsburgh, PA, 1987)\n\\bibitem{col2} Colby R.H., Gillmor J.R., Rubinstein M.,{\\em Phys.Rev.E} \n 48, 3712 (1993)\n\\bibitem{nico} Nicolai T., Randrianantoandro H., Prochazka F., Durand D.\n\t {\\em Macromolecules} 30, 5897 (1997)\n\\bibitem{malla} Mallamace F., Chen S., Tartaglia P. {\\em to be published}\n\\bibitem{adconst} Adam M., Coniglio A., Stauffer D. {\\em Adv.in Polymer Sci.} \n Springer-Verlag, Berlin (1982) \n\\bibitem{staul} Aharony A., Stauffer D. \n {\\em Introduction to percolation theory} \n Taylor and Francis, London (1994)\n\\bibitem{doed} Doi M., Edwards S.F. {\\em The Theory of polymer Dynamics} \n Clarendon Press, Oxford (1986)\n\\bibitem{deg78} de Gennes P.G. {\\em C.R.Acad.Sc.Paris B} 131 (1978)\n\\bibitem{ker} Kertesz J. {\\em J.Phys.A} 16, L471 (1983)\n\\bibitem{carkr1} Carmesin I., Kremer K. {\\em Macromolecules} 21 \n 2819 (1988) \n\\bibitem{carkr2} Carmesin I., Kremer K. {\\em J.Phys.(Paris)} 51 950(1990)\n\\bibitem{dedecon1} Del Gado E., de Arcangelis L., Coniglio A. {\\em J.Phys.A} 31 \n 1901(1998)\n\\bibitem{madwi} Martin J.E., Douglas A., Wilcoxon J.P. {\\em Phys.Rev.Lett.}\n 61, 2620(1988)\n\\bibitem{dedecon2} Del Gado E., de Arcangelis L., Coniglio A. {\\em \n Europhys.Lett.} 46 288(1999)\n\\bibitem{ferry} Ferry J. {\\em Viscoelastic properties of polymers} Wiley \n New York 1980 \n\\bibitem{adfr} Durand D., Delsanti M., Adam M., Luck J.M. \n {\\em Europhysics Letters} 3, 297 (1987)\n\\bibitem{daoud} Daoud M. {\\em J.Phys. A} 21 L973 (1988)\n\\bibitem{mawio} Martin J.E., Wilcoxon J.P., Odinek J. {\\em Phys.Rev.A}\n 43, 858(1991)\n\\bibitem{adprl} Adam M., Delsanti M., Munch J.P., Durand D. {\\em \n Phys.Rev.Lett.} 61 706 (1988)\n\\bibitem{chen} Kemp J.P., Chen Z.Y. {\\em cond-mat/9810108}\n\\bibitem{campb} Mari P.O., Campbell I.A., Alegr{\\'i}a A., Colmenero J.\n {\\em Physica A} 257, 21(1998)\n\\bibitem{zip} Broderix K., Lowe H., Muller P., Zippelius A. {\\em cond-mat/ \n 9811388} 1998\n\\end{references}\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%% FIGURE CAPTIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\caption{Data collapse for lattice sizes $L=24,32,40$ for the \npercolation probability $P$ $(a)$ and for the mean cluster size \n$\\chi$ $(b)$: the best collapse is obtained with \n$p_{c}=0.718 \\pm 0.05$, $\\nu = 0.89 \\pm 0.01$, $\\gamma = 1.8 \\pm 0.05$.}\n\\label{fig1}\n\\end{figure}\n\\begin{figure}\n\\caption{Four different possible configurations for a cluster formed by four \nmonomers in the time evolution according to the bond fluctuation dynamics. We \nconsider the subsequent monomer movements: \nin $a$ \nthe bond lengths are (starting from the upper center bond and clockwise) \n$l=\\protect\\sqrt{5},3,3,2$; moving forward the upper left monomer \nwe have $b$ \nwith $l=2,3,3,\\protect\\sqrt{5}$; from this configuration moving \nright the other left monomer we have $c$ with $l=2,3,\n\\protect\\sqrt{6},\\protect\\sqrt{6}$; \nthen moving right the front \nmonomer on the right leads to $d$ and $l=2,\n\\protect\\sqrt{10},\\protect\\sqrt{6},\\protect\\sqrt{6}$.\n}\n\\label{fig2}\n\\end{figure}\n\\begin{figure}\n\\caption{ $\\langle \\Delta R^{2}_{s}(t) \\rangle /t$ as a function of time for \n$p=0.69$ for different cluster sizes; from top to bottom $s=1,2,3,4,5,8,10$. \nThe \nbroken lines correspond to the asymptotic values, i.e. the values of the \ndiffusion coefficients. The data have been averaged over $32$ different \nconfigurations.}\n\\label{fig3}\n\\end{figure}\n\\begin{figure}\n\\caption{The diffusion coefficient at $p_{c}$ averaged \nover $32$ different configurations for different cluster size as a \nfunction of the cluster radius of gyration $R$: supposing the scaling \nbehaviour $D(R) \\sim 1/R^{1+k/\\nu}$, gives $k \\sim 1.3 \\pm0.02$.}\n\\label{fig4}\n\\end{figure}\n\\begin{figure}\n\\caption{The monomer self-diffusion coefficient as function of $p$. The data \nshow how the self-diffusion coefficient of monomers is not linked to the \nmacroscopic viscosity diverging at the gel point. $D(s=1)$ becomes \nnumerically undistinguishable from zero only at $p > p_{c}$ and the data are \nwell fitted with the dependence $\\sim e^{-1/1-p}$ (the continuous line).}\n\\label{fig5}\n\\end{figure}\n\\begin{figure}\n\\caption{Time autocorrelation function $g(t)$ of the density of $nn$ monomers \ndefined in eq.(\\ref{autc}) as function of time for various $p$. From top to \nbottom \n$p=0.705, 0.7, 0.69, 0.66, 0.5$ (data averaged over $\\sim 30$ different \nconfigurations).}\n\\label{fig6}\n\\end{figure}\n\\begin{figure}\n\\caption{Stretched exponential behaviour $e^{-(t/\\tau_{0})\\protect ^\\beta}$ of \nthe long time tail of the time \nautocorrelation function $g(t)$ for $p=0.66$ (bottom) and $p=0.69$ (top).}\n\\label{fig7}\n\\end{figure}\n\\begin{figure}\n\\caption{The characteristic time $\\tau_{0}$ obtained from the stretched \nexponential fit of the time autocorrelation function long time behaviour as a \nfunction of $(p_{c} - p)$. The data are well fitted by a power law\nwith an exponent $1.29 \\pm 0.03$.}\n\\label{fig8}\n\\end{figure}\n\\begin{figure}\n\\caption{The characteristic integral time $\\tau$ calculated according to \neq.(\\ref{time}) as a function of $(p_{c} - p)$. The data are well fitted by a \npower law with a critical exponent $k \\sim 1.31 \\pm 0.05$.}\n\\label{fig9}\n\\end{figure}\n%\n\\end{document}\n%\n"
}
] |
[
{
"name": "cond-mat0002154.extracted_bib",
"string": "\\bibitem{flo} Flory,P.J., {\\em Principles of polymer chemistry} \n Cornell Univ.Press, Ithaca 1953\n\n\\bibitem{degl} De Gennes P.G. {\\em Scaling concepts in polymer physics} \n Cornell Univ. Press, Ithaca 1980 \n\n\\bibitem{ad97} Adam M., Lairez D., Karpasas M., Gottlieb M. \n {\\em Macromolecules } 30 5920(1997)\n\n\\bibitem{ad91} Adam M. {\\em Makromol.Chem.,Macromol.Symp.} 45 (1991)\n\n\\bibitem{ad92} Lairez D., Adam M., Raspaud E., Emery J.R., Durand D. {\\em\n Prog.Colloid Polym.Sci. } 90 (1992)\n\n\\bibitem{mawi} Martin J.E., Wilcoxon J.P. {\\em Phys.Rev.A}\n 39, 252(1989)\n\n\\bibitem{col1} Colby R.H., Coltrain B.K., Salva J.M., Melpolder S.M. in \n {\\em Fractal Aspects of Materials: Disordered Systems}, \n edited by A.J.Hurd, D.A.Weitz, and B.B.Mandelbrot \n (Materials Research Society, Pittsburgh, PA, 1987)\n\n\\bibitem{col2} Colby R.H., Gillmor J.R., Rubinstein M.,{\\em Phys.Rev.E} \n 48, 3712 (1993)\n\n\\bibitem{nico} Nicolai T., Randrianantoandro H., Prochazka F., Durand D.\n\t {\\em Macromolecules} 30, 5897 (1997)\n\n\\bibitem{malla} Mallamace F., Chen S., Tartaglia P. {\\em to be published}\n\n\\bibitem{adconst} Adam M., Coniglio A., Stauffer D. {\\em Adv.in Polymer Sci.} \n Springer-Verlag, Berlin (1982) \n\n\\bibitem{staul} Aharony A., Stauffer D. \n {\\em Introduction to percolation theory} \n Taylor and Francis, London (1994)\n\n\\bibitem{doed} Doi M., Edwards S.F. {\\em The Theory of polymer Dynamics} \n Clarendon Press, Oxford (1986)\n\n\\bibitem{deg78} de Gennes P.G. {\\em C.R.Acad.Sc.Paris B} 131 (1978)\n\n\\bibitem{ker} Kertesz J. {\\em J.Phys.A} 16, L471 (1983)\n\n\\bibitem{carkr1} Carmesin I., Kremer K. {\\em Macromolecules} 21 \n 2819 (1988) \n\n\\bibitem{carkr2} Carmesin I., Kremer K. {\\em J.Phys.(Paris)} 51 950(1990)\n\n\\bibitem{dedecon1} Del Gado E., de Arcangelis L., Coniglio A. {\\em J.Phys.A} 31 \n 1901(1998)\n\n\\bibitem{madwi} Martin J.E., Douglas A., Wilcoxon J.P. {\\em Phys.Rev.Lett.}\n 61, 2620(1988)\n\n\\bibitem{dedecon2} Del Gado E., de Arcangelis L., Coniglio A. {\\em \n Europhys.Lett.} 46 288(1999)\n\n\\bibitem{ferry} Ferry J. {\\em Viscoelastic properties of polymers} Wiley \n New York 1980 \n\n\\bibitem{adfr} Durand D., Delsanti M., Adam M., Luck J.M. \n {\\em Europhysics Letters} 3, 297 (1987)\n\n\\bibitem{daoud} Daoud M. {\\em J.Phys. A} 21 L973 (1988)\n\n\\bibitem{mawio} Martin J.E., Wilcoxon J.P., Odinek J. {\\em Phys.Rev.A}\n 43, 858(1991)\n\n\\bibitem{adprl} Adam M., Delsanti M., Munch J.P., Durand D. {\\em \n Phys.Rev.Lett.} 61 706 (1988)\n\n\\bibitem{chen} Kemp J.P., Chen Z.Y. {\\em cond-mat/9810108}\n\n\\bibitem{campb} Mari P.O., Campbell I.A., Alegr{\\'i}a A., Colmenero J.\n {\\em Physica A} 257, 21(1998)\n\n\\bibitem{zip} Broderix K., Lowe H., Muller P., Zippelius A. {\\em cond-mat/ \n 9811388} 1998\n"
}
] |
cond-mat0002155
|
Solution of the Schr\"odinger Equation for Quantum Dot Lattices with Coulomb Interaction between the Dots
|
[
{
"author": "by M. Taut"
},
{
"author": "Institut f\\\"ur Festk\\\"orper-- und Werkstoff-- Forschung Dresden"
},
{
"author": "Postfach 270016"
},
{
"author": "01171 Dresden"
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"author": "Germany"
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The Schr\"odinger equation for quantum dot lattices with non-cubic, non-Bravais lattices built up from elliptical dots is investigated. The Coulomb interaction between the dots is considered in dipole approximation. Then only the center of mass (c.m.) coordinates of different dots couple with each other. This c.m. subsystem can be solved exactly and provides magneto-- phonon like {\em collective excitations}. The inter--dot interaction is involved only through a single interaction parameter. The relative coordinates of individual dots form decoupled subsystems giving rise to {\em intra--dot excitation} modes. As an example, the latter are calculated exactly for two--electron dots.\\ Emphasis is layed on {\em qualitative} effects like: i) Influence of the magnetic field on the lattice instability due to inter--dot interaction, ii) Closing of the gap between the lower and the upper c.m. mode at B=0 for elliptical dots due to dot interaction, and iii) Kinks in the intra dot excitation energies (versus magnetic field) due to change of ground state angular momentum. It is shown that for obtaining striking qualitative effects one should go beyond simple cubic lattices with circular dots. In particular, for observing effects of electron-- electron interaction between the dots in FIR spectra (breaking Kohn's Theorem) one has to consider dot lattices with at least two dot species with different confinement tensors.
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"name": "PRstyle.tex",
"string": "\\documentstyle[psfig,prb,aps]{revtex}\n\\draft\n\n\\title{Solution of the Schr\\\"odinger Equation for Quantum Dot Lattices\nwith Coulomb Interaction between the Dots}\n\\author{by M. Taut\\\\Institut f\\\"ur Festk\\\"orper-- und Werkstoff-- Forschung\nDresden\\\\ Postfach 270016,\n01171 Dresden, Germany\\\\\nemail: m.taut@ifw-dresden.de\\\\}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nThe Schr\\\"odinger equation for quantum dot lattices with non-cubic,\nnon-Bravais lattices built up from elliptical dots is investigated.\nThe Coulomb interaction between the dots is considered in dipole approximation.\nThen only the center of mass (c.m.) coordinates of different dots couple\nwith each other.\nThis c.m. subsystem can be solved exactly\nand provides magneto-- phonon like {\\em collective excitations}.\nThe inter--dot interaction is involved only through\n a single interaction parameter.\nThe relative coordinates of individual dots form decoupled subsystems\ngiving rise to {\\em intra--dot excitation} modes.\nAs an example, the latter are calculated exactly for two--electron dots.\\\\\nEmphasis is layed on {\\em qualitative} effects like:\ni) Influence of the magnetic field on the lattice instability\ndue to inter--dot interaction, ii) Closing of the gap between\nthe lower and the upper c.m. mode at B=0 for elliptical\ndots due to dot interaction, and iii) Kinks in the intra dot excitation\nenergies\n(versus magnetic field) due to change of ground state angular momentum.\nIt is shown that\nfor obtaining striking qualitative effects one should\ngo beyond simple cubic lattices with circular dots.\nIn particular, \nfor observing effects\nof electron-- electron interaction between the dots in FIR \nspectra (breaking Kohn's Theorem) one has to consider dot lattices\nwith at least two dot species with different confinement tensors.\n\\end{abstract}\n\\pacs{PACS: 73.20.D (Quantum dots), \n 73.20.Mf (Collective Excitations)} \n\n\\section{Introduction}\nQuantum dots have been in the focus of intensive research already for at least\na decade which lead to a countless number of \npublications \\footnote{Therefore we will refer here only to papers which\nare directly connected to the scope of this work} \n(for a recent book see Ref.\\onlinecite{Hawrylak-Buch}). \nAlthough almost all experiments are performed at dot lattices,\nin the vast majority of theoretical investigations the interaction between dots\nis neglected. This is for the following reasons: \ni) Because the confinement frequency $\\omega_0$ is a parameter, which\nis mainly extracted from optical properties, it is difficult \nto tell the influence of dot interaction\napart from the intrinsic single-- dot value. (Possibilities to\novercome this problem are discussed in the present work.)\nii) The theory of Raman spectra, \nwhich can in principle monitor \nthe dispersion (wave number dependence) of excitation energies as a\ndirect consequence of interdot-- interaction,\nis not yet advanced enough to extract the dispersion.\niii) The lattice constant of dot arrays produced with current technologies\nis so large ($>2000${\\AA}) that large \nelectron numbers $N$ per dot are necessary\nto obtain a seizable amount of shift. For these N, however,\n reliable first principle calculations are not possible.\nWith the advent of self-- assembled dot arrays the last item might change.\\\\\nThe scope of this paper is to investigate conditions, which lead to\n{\\em qualitative} and observable effects of interdot-- interaction\non excitation spectra and the phase transition found in Ref.\\onlinecite{Broido}.\nUnlike in Ref.\\onlinecite{Broido}, a magnetic field {\\bf B}\n is explicitly taken into account and a microscopic theory is applied. \nOur approach is purely microscopic, i.e. we solve the Schr\\\"odinger\nequation of a model system {\\em exactly}. Our model comprises the following\napproximations: i) The dot confinement is strictly parabolic in \nradial direction, but with anisotropic \nconfinement frequencies $\\omega_i\\; (i=1,2)$ and independent of $N$ and {\\bf B}.\nii) Overlap of wave functions between different dots is neglected (no hopping).\niii) The Coulomb interaction of the electrons in different dots is \ntreated in dipole approximation (second order in dot diameter over lattice constant).\nOur model is similar to that in Ref.\\onlinecite{Halperin}, but allows more complicated\ndots and lattice structures. Besides we calculate \nalso the intra dot excitations \n(apart from the collective center-- of-- mass excitations)\nfor $N=2$ explicitly and discuss the instability \\cite{Broido} in this\nmicroscopic model. Our results on the lateral dot dimer are compared\nwith a former paper \\cite{Chakraborty}, \nwhich uses a high magnetic field approach, in Sect. III. \\\\\nThe plan of this paper is as follows. For further reference, \nwe briefly summarize in Sect. I \nsome relevant results for one single dot, or for dot lattices, where the\ndistance between the dots is very large. This is important,\nbecause all exact solutions in the center-- of-- mass subsystem \nare traced back (by special transformations) to the solution of this\none-- electron Hamiltonian. This is analogous to ordinary\nmolecular and lattice dynamics.\n After this, we consider a dot dimer,\nwhich mimics a lattice, where the dots are pairwise close to each other.\nThis model can give an idea of the effects expected in\ndot lattices with a basis. Next we consider a rectangular, but primitive\nlattice in order to obtain the dispersion in the spectra. \nFinally, the intra-dot excitations of the Hamiltonian in the\nrelative coordinates are calculated numerically. The paper ends with a summary.\nIn the Appendix we give a short and elementary proof for the fact\n that the Generalized Kohn Theorem holds even for arbitrary\narrays of identical non-circular quantum dots \nwith Coulomb interaction (between the dots) in an homogeneous magnetic field.\n\n\n\\section{Single Dot}\nThe Hamiltonian considered here reads (in atomic units $\\hbar=m=e=1$)\n\\begin{equation}\nH=\\sum^N_{i=1}\n\\left\\{ \n\\frac{1}{2 m^*}\n\\left[ \n{\\bf p}_i+\n{1\\over c} {\\bf A}({\\bf r}_i) \n\\right]^2 +\n\\frac{1}{2} \\;{\\bf r}_i \\cdot \n{\\bf C }\n\\cdot {\\bf r}_i \n\\right\\} \n+ \\frac{1}{2} \\sum_{i\\ne k}\\;\\frac{\\beta}{|{\\bf r}_i-{\\bf r}_k|}\n\\end{equation}\nwhere $m^*$ is the effective mass (in units of the bare\nelectron mass $m$), $\\beta$ the inverse dielectric \nconstant of the background, and\n$\\bf C$ a symmetric tensor. In case \nof a single dot, $\\bf C$ is given by the confinement potential\nand we define ${\\bf C}={\\bf \\Omega}$.\nIt is always possible to find a coordinate system where\n$\\Omega_{12}=\\Omega_{21}=0$ and $\\Omega_{ii}=\\omega_i^2=m^*\\omega_i^{*2}$.\nWe use the symmetric gauge ${\\bf A}=\\frac{1}{2}\\;{\\bf B}\\times {\\bf r}$\nthroughout.\nThe Zeeman term\nin $H$ is disregarded at the moment.\\\\\nFor $N=1$, the Hamiltonian\n\\begin{equation}\nH=\n\\frac{1}{2 m^*}\n\\left[\n{\\bf p}+\n{1\\over c} {\\bf A}({\\bf r})\n\\right]^2 +\n\\frac{1}{2}\\; {\\bf r} \\cdot\n{\\bf C }\n\\cdot {\\bf r}\n\\label{H-N=1}\n\\end{equation}\ncan be diagonalized exactly.\nLater on we will see that also the case of interacting dots\ncan be traced back to the solution of type (\\ref{H-N=1}).\n(Therefore, we kept the off diagonal elements of $\\bf C$ in the results given below\nbecause the dynamical matrix, which also contributes to $\\bf C$, is\ngenerally non--diagonal and we want to use the same coordinate system for all\n$\\bf q$ values.)\nAfter transforming the\noperators ${\\bf r}_i$ and ${\\bf p}_i$ to creation-- annihilation operators\n(see e.g. Ref.\\onlinecite{Hawrylak-Buch}) and using the procedure described by\nTsallis \\cite{Tsallis},\nwe obtain for the eigenvalues\n\\begin{equation}\nE(n_+,n_-)= (n_+ +\\frac{1}{2})\\; \\omega_+ + (n_- +\\frac{1}{2})\\; \\omega_- ~;\n~~~n_\\pm=0,1,2,...\n\\label{E-N=1}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\omega_\\pm &=& \\sqrt{\n\\frac{\\omega_c^{*2}}{2}+\\tilde\\omega_0^2 \\pm \n\\sqrt{\\frac{\\omega_c^{*4}}{4}+\\omega_c^{*2}\\;\\tilde\\omega_0^2+\\frac{\\Delta^2}{4}\n+C_{12}^2}}\n\\label{omega-pm-1}\\\\\n&=&\\sqrt{\n\\left[\n\\frac{1}{2} \\sqrt{\\omega_c^{*2}+4\\; \\tilde\\omega_0^2+\n\\frac{(\\Delta^2+4\\;C_{12}^2)}{\\omega_c^{*2}} }\n\\pm \\frac{\\omega_c^*}{2}\n\\right]^2\n-\\frac{(\\Delta^2+4\\;C_{12}^2)}{4\\;\\omega_c^{*2}} }\n\\label{omega-pm-2}\n\\end{eqnarray}\n\\begin{equation}\n\\tilde\\omega_0^2=\\frac{1}{2}(C_{11}+C_{22})~;~~~~~\n\\Delta=C_{11}-C_{22}\n\\end{equation}\nand $\\omega_c^*=\\frac{B}{m^* c}$ is the cyclotron frequency \nwith the effective mass.\n(The results for the special case $C_{12}=0$ can also be found in Ref. \n\\onlinecite{other}.)\nThe optical selection rules are the same as in the circular case, i.e.,\nthere are two possible types of excitations \n\\begin{equation}\n(\\Delta n_+=\\pm 1 ~~~\\mbox{and}~~~\\Delta n_-=0)~~~\\mbox{or}~~~\n(\\Delta n_-=\\pm 1 ~~~\\mbox{and}~~~\\Delta n_+=0)\n\\end{equation}\nleading to the excitation energies $\\Delta E=\\omega_+$ and $\\omega_-$. \nIt is easily seen that the form (\\ref{omega-pm-2}) reduces\nto the familiar formula in the circular case, where $\\Delta=0$ and\n$C_{12}=0$. By inspection of (\\ref{omega-pm-1}) we find\nthat a {\\em soft mode} $\\omega_-(B)=0$ can only occur if\n$C_{11}\\cdot C_{22}=C_{12}^2$. For a diagonal $\\bf C$ this means that\n$min(C_{11},C_{22})=0$. The last condition is of importance for\ninteracting dots considered in the next Sections.\\\\\nIn the limiting case $B = 0$ we obtain from (\\ref{omega-pm-1})\n\\begin{equation}\n\\omega_\\pm(B=0)=\\sqrt{\\frac{(C_{11}+C_{22})}{2}\n\\pm \\sqrt{\\frac{(C_{11}-C_{22})^2}{4}+C_{12}^2}}\n\\label{B=0}\n\\end{equation}\nWe see that {\\em degeneracy} $\\omega_+(B=0)=\\omega_-(B=0)$ can only happen if\n$C_{12}=0$ {\\em and} $C_{11}=C_{22}$.\nFor a diagonal confinement tensor with $C_{12}=0$ we obtain\n$\\omega_+(B=0)=\\mbox{max}(\\omega_1,\\omega_2)$ and \n$\\omega_-(B=0)=\\mbox{min}(\\omega_1,\\omega_2)$. \nAs to be expected, we observe a gap\nbetween the two excitation curves $\\omega_+(B)$ and $\\omega_-(B)$ at\n$B=0$, if the two confinement frequencies do not agree.\\\\\nAlternatively we can introduce the quantum numbers\n\\begin{equation}\nk=\\frac{(n_+ + n_-)-|n_+ + n_-|}{2}~;~~~m_z=n_+ - n_-\n\\end{equation}\nwhere $k$ is the node number and $m_z$ turns in the circular limit \ninto the angular momentum quantum number.\\\\\n\nFor arbitrary $N$, the center of mass (c.m.) \n${\\bf R}=\\frac{1}{N}\\sum_i {\\bf r}_i$\ncan be separated $H=H_{c.m.}+H_{rel.}$ with\n\\begin{equation}\nH_{c.m.}=\\frac{1}{N}\n\\left\\{\n\\frac{1}{2 m^*}\n\\left[\n{\\bf P}+\n{N\\over c} {\\bf A}({\\bf R})\n\\right]^2 +\n\\frac{N^2}{2} {\\bf R} \\cdot\n{\\bf C }\n\\cdot {\\bf R}\n\\right\\}\n\\label{H-cm}\n\\end{equation}\nwhere ${\\bf P}=-i \\nabla _{\\bf R}$ (see Appendix). As well known, $H_{c.m.}$\ndoes not contain the electron-- electron-- interaction. $H_{c.m.}$ \ncan be obtained from the one-- electron Hamiltonian (\\ref{H-N=1}) by \nthe substitution:\n$B \\rightarrow N B$,~${\\bf C} \\rightarrow N^2 {\\bf C}$ and\n$H \\rightarrow \\frac{1}{N} H$. If we make the same \nsubstitution in the eigenvalues (\\ref{E-N=1}), we obtain \n$$E_{c.m.}(n_+,n_-)=E_{N=1}(n_+,n_-)$$\ni.e., the eigenvalues of the c.m. Hamiltonian are independent of $N$.\nIn other words, in $H$ there are excitations, in which the pair correlation \nfunction is not changed, or classically speaking, where the charge distribution\noscillates rigidly.\n Because FIR radiation (in the limit\n$\\lambda \\rightarrow \\infty$) can excite only the c.m. subspace, all\nwe see in FIR spectra is the c.m. modes.\n\n\\section{Dot Dimer}\nWe consider two {\\em identical} elliptical dots centered at \n$\\mbox{\\boldmath $a$}_1=(-a/2,0)$\nand $\\mbox{\\boldmath $a$}_2=(+a/2,0)$.\nWe expand the Coulomb interaction between electrons\nin {\\em different} dots in a multi-pole series and restrict ourselves\nto the dipole approximation.\nBy introduction\nof c.m. and relative coordinates within each dot, the c.m. coordinates and the\nrelative coordinates decouple \n\\footnote{ The tilde\nindicates that in this preliminary Hamiltonian a common gauge\ncenter for both dots is used.}\n\\begin{equation}\n\\tilde H=\\tilde H_{c.m.}({\\bf R}_1,{\\bf R}_2) +\n \\sum_\\alpha^{1,2}\\;\\tilde H_\\alpha\\left(\\{ {\\bf r} \\}_\\alpha^{(N-1)}\\right)\n\\label{separation}\n\\end{equation}\n$\\{ {\\bf r} \\}_\\alpha^{(N-1)}$ symbolizes $(N-1)$ relative coordinates in\nthe $\\alpha^{th}$ dot.\nThis means, we have 3 decoupled Hamiltonians: the c.m. Hamiltonian and two\nHamiltonians in the relative coordinates of either dot. \nThis leads to two types of excitations:\\\\\n i) {\\em Collective excitations }\nfrom $\\tilde H_{c.m.}$ which involve \nthe c.m. coordinates of both dots simultaneously.\nBecause of the harmonic form (in the dipole approximation), there are exactly\n two modes per dot, thus a total of four.\n Each excitation can be classically visualized as \nvibrations of rigidly moving charge distributions of both dots.\\\\\nii) {\\em Intra dot excitations} which are doubly degenerate \nfor two identical dots. Because \n$\\tilde H_\\alpha(\\{ {\\bf r} \\}_\\alpha^{(N-1)})$ \nis not\nharmonic (it includes the exact Coulomb interaction between the electrons\nwithin each dot, which is not harmonic), this spectrum is very complex. It\nis the excitation spectrum of a single dot {\\em in a modified\nconfinement potential} where the c.m. coordinate is\nfixed. The extra term in the modified confinement potential\ncomes from the dipole contribution of the interdot Coulomb interaction.\\\\\nIn this Section we consider only the c.m. Hamiltonian and focus our attention\nto the the effects of ellipticity in the dot confinement potential. The \nrelative Hamiltonian for $N=2$ is explicitly given in the last Section\nand solved for circular dots. For the elliptical confinement potential\nconsidered in this Section, the relative Hamiltonian cannot be solved\neasily, even if we restrict ourselves to $N=2$, because the elliptic\nconfinement potential breaks the circular symmetry of the\nrest of the relative Hamiltonian.\n\n\\subsection{ Center of Mass Hamiltonian of the Dimer}\nThe c.m. Hamiltonian in the dipole approximation reads \n\\begin{equation}\n\\tilde H_{c.m.}=\\frac{1}{N}\n\\left\\{ \\sum_\\alpha ^{1,2}\n\\frac{1}{2 m^*}\n\\left[\n{\\bf P_\\alpha}+\n{N\\over c} {\\bf A}({\\bf U_\\alpha} + \\mbox{\\boldmath $a$}_\\alpha)\n\\right]^2 +\n\\frac{N^2}{2} \\sum_{\\alpha,\\alpha'}{\\bf U}_\\alpha \\cdot\n{\\bf C}_{\\alpha,\\alpha'}\n\\cdot {\\bf U}_{\\alpha'}\n\\right\\}\n\\label{H-cm-dimer}\n\\end{equation}\n where the small elongation ${\\bf U}_\\alpha$ is defined\nby ${\\bf R}_\\alpha=\\mbox{\\boldmath $a$}_\\alpha+ {\\bf U}_\\alpha$ and\n${\\bf P}=-i \\nabla_{\\bf R}=-i \\nabla_{\\bf U}$.\nThe tensor $\\bf C$ is\n\\begin{eqnarray}\nC_{\\alpha,\\alpha}&=& {\\bf \\Omega} + \\beta N \\sum_{\\alpha'(\\ne \\alpha)}\n{\\bf T}(\\mbox{\\boldmath $a$}_{\\alpha, \\alpha'}) \\label{C1}\\\\\nC_{\\alpha,\\alpha'}&=&\n-\\beta N \\;{\\bf T}(\\mbox{\\boldmath $a$}_{\\alpha, \\alpha'})~~~\n\\mbox{for} ~~~\\alpha \\ne \\alpha'\n\\label{C2}\n\\end{eqnarray}\nwhere $\\mbox{\\boldmath $a$}_{\\alpha, \\alpha'}=\n\\mbox{\\boldmath $a$}_{\\alpha}-\\mbox{\\boldmath $a$}_{\\alpha '}$ ,\nand the dipole tensor is\n\\begin{equation}\n{\\bf T}(\\mbox{\\boldmath $a$})=\\frac{1}{a^5}\\;\\left [ 3 \\; \\mbox{\\boldmath $a$}\n\\circ \\mbox{\\boldmath $a$} - a^2\\;\n{\\bf I} \\right ]\n\\label{dipole-tensor}\n\\end{equation}\ncontaining a dyad product ($\\circ$) and the unit tensor $\\bf I$.\nAs in the c.m. system of a single dot, the explicit $N$-- dependence in\n(\\ref{H-cm-dimer}) cancels in the eigenvalues.\nWhat is left is only the $N$-- dependence in the\ndipole contribution of the dot interaction appearing\nin (\\ref{C1}) and (\\ref{C2}). This means,\nthat the c.m. spectrum of interacting dots\nis no longer independent of $N$.\\\\\nThe term $\\mbox{\\boldmath $a$}_\\alpha$ in the argument of the\nvector potential in (\\ref{H-cm-dimer}) causes trouble in finding the\neigenvalues. This shift is a consequence of the fact that we\nhave to adopt a common gauge center for both dots \n(we chose the middle between both dots). This problem can be\nsolved by applying the following unitary transformation\n\\begin{equation}\n H_{c.m.} = Q^{-1}\\; \\tilde H_{c.m.} \\; Q~;~~~\nQ=\\prod_\\alpha^{1,2}\\;\ne^{-i\\frac{N}{2c}\\;({\\bf B}\\times\n \\mbox{\\boldmath $a$}_\\alpha)\\cdot {\\bf U}_\\alpha}\n\\label{gauge-shift}\n\\end{equation}\nIn other words, $H_{c.m.}$ agrees with $\\tilde H_{c.m.}$ except for\nthe missing shift in the argument of the vector potential.\\\\\nThe 4 modes inherent in $H_{c.m.}$ are not yet explicitly known, because the\n4 degrees of freedom are coupled. Decoupling into two oscillator \nproblems of type (\\ref{H-N=1}) can be achieved by the following transformation:\n\\begin{equation}\n{\\bf U}^{(+)}=\\frac{1}{2}({\\bf U}_2+{\\bf U}_1)~;~~~~~\n{\\bf U}^{(-)}={\\bf U}_2-{\\bf U}_1\n\\label{cm-diff-coord}\n\\end{equation}\nThis results in \n\\begin{equation}\nH_{c.m.}=\\frac{1}{2} H^{(+)}\\; + \\; 2 H^{(-)}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nH^{(+)}&=& \\frac{1}{N} \\left\\{\n\\frac{1}{2 m^*}\n\\left[\n{\\bf P}^{(+)}+\n{2 N\\over c} {\\bf A}\\left({\\bf U}^{(+)}\\right)\n\\right]^2 +\n\\frac{N^2}{2} {\\bf U}^{(+)} \\cdot\n\\Bigg(\n4 {\\bf \\Omega}\n\\Bigg)\n\\cdot {\\bf U}^{(+)} \\right\\}\n\\label{H+}\\\\\nH^{(-)}&=& \\frac{1}{N} \\left\\{\n\\frac{1}{2 m^*}\n\\left[\n{\\bf P}^{(-)}+\n{N\\over 2 c} {\\bf A}\\left({\\bf U}^{(-)}\\right)\n\\right]^2 +\n\\frac{N^2}{2} {\\bf U}^{(-)} \\cdot\n\\Bigg(\n\\frac{1}{4} {\\bf \\Omega} +\\frac{N}{2} \\beta \\; {\\bf T}(\\mbox{\\boldmath $a$}) \n\\Bigg)\n\\cdot {\\bf U}^{(-)}) \\right\\}\n\\label{H-}\n\\end{eqnarray}\nand $\\mbox{\\boldmath $a$}$ is a vector pointing from one dot center \nto the other. Then ${\\bf T}(\\mbox{\\boldmath $a$})$ has the following components\n\\begin{equation}\nT_{11}=\\frac{2}{a^3}~;~~~T_{22}=-\\frac{1}{a^3}~;~~~\nT_{12}=T_{21}=0\n\\end{equation}\nNow we assume that the principle axes of the confinement potentials\nare in x-y-direction. This means\n\\begin{equation}\n\\Omega_{11}=\\omega_1^2~;~~~\\Omega_{22}=\\omega_2^2~;~~~ \n\\Omega_{12}=\\Omega_{21}=0\n\\end{equation}\nThe eigenvalues of $H^{(+)}$ \ncan be obtained from (\\ref{E-N=1}) and (\\ref{omega-pm-1})\nwith\n\\begin{equation}\n\\tilde\\omega_0^2=\\frac{1}{2}(\\omega_1^2+\\omega_2^2)~;~~\n~\\Delta=(\\omega_1^2-\\omega_2^2)\n\\label{input1}\n\\end{equation}\nand for $H^{(-)}$ with \n\\begin{equation}\n\\tilde\\omega_0^2=\\frac{1}{2}(\\omega_1^2+\\omega_2^2)+\\frac{1}{2}\\;p~;~~\n~\\Delta=(\\omega_1^2-\\omega_2^2)+3\\; p\n\\label{input2}\n\\end{equation}\nwhere the interaction parameter is defined by\n\\begin{equation}\np=\\frac{2N\\beta}{a^3}\n\\label{p}\n\\end{equation}\nObserve that the dependence on $N$ cancels,\nexcept that included in $p$ (see discussion following (\\ref{H-cm})).\n\n It is important that the dot interaction influences\nthe result only through a single parameter. This conclusion\nagrees with the semi-- phenomenological theory in Ref.\\onlinecite{Broido}.\\\\\n\nIn all our figures\nwe express frequencies in units of the average confinement frequency\n$\\omega_0=\\frac{1}{2}(\\omega_1+\\omega_2)$, and $\\Delta$ and $p$ in units\n$\\omega_0^2$.\nThen, all\nsystems can be characterized by the two parameters:\n $\\omega_1/\\omega_2$ and $p$.\nIn other words, all systems having the $\\omega_1/\\omega_2$ ratio \nindicated in the figures \nare represented by the family of curves with the $p$ values shown.\nThe only exception we made is the cyclotron frequency $\\omega^*_c$.\n$\\omega^*_c/\\omega_0$ would be a good parameter in this sense, but we chose\nto use the magnetic field in $Tesla$ instead for better physical\nintuition. The conversion between both scales is given by\n$\\omega^*_c[a.u.^*]=\\frac{0.9134\\cdot10^{-2}}{m^*}\\; B[Tesla]$ or,\n$\\omega_c^*[\\omega_0]=\\frac{0.9134\\cdot10^{-2}}{m^*\\;\\omega_0[a.u.^*]}B[Tesla]$\nIn this paper we used $\\omega_0=0.2 \\;a.u.^*= 2.53\\; meV$\nand $m^*$ of GaAs. (We want to stress\nthat this choice effects only the magnetic field scale\n and not the qualitative features of the figures.)\nFor easy comparison with experimental parameters we add the definition of\neffective atomic units ($a.u.^*$) \nin GaAs ($m^*=0.067, \\;\\beta=1/12$) for the energy: \n$1\\;a.u.^*=4.65\\cdot10^{-4}\\; double\\; Rydberg=12.64\\;meV$ ,\n and for\nthe length: $1\\;a.u.^*=1.791\\cdot10^{2}\\; Bohr\\; radii\n=0.9477\\cdot10^{2}\\;\\AA$.\\\\\n\n\nBecause $\\bf U^{(+)}$ agrees with the\n{\\em total} c.m. ${\\bf R}=\\frac{1}{2}({\\bf R}_1+{\\bf R}_2)$ of the system,\n$H^{(+)}$ is the total c.m. Hamiltonian. \nFor $B=0$, the eigenmodes can be visualized by classical oscillations.\nThe two eigenmodes of $H^{(+)}$ are (rigid) in-- phase\n oscillations of the dots in x and y direction, respectively. \nBecause of the Kohn theorem (see Appendix), the independence of\n$H^{(+)}$ on the Coulomb interaction does not only hold in the dipole \napproximation, but it is rigorous. This shows also that the dipole\napproximation is consistent with the Kohn theorem, which is not guaranteed\nfor single particle approaches. Because FIR radiation excites \n(in the dipole approximation) only\nthe c.m. modes, it is only the $p$--independent \neigenmodes of $H^{(+)}$ which are seen in \nFIR absorption experiments.\\\\\nThis statement is in contradiction\nto Ref.\\onlinecite{Chakraborty}. They performed numerical diagonalizations\nfor a lateral pair of circular dots \nconfining the set of basis\nfunctions to the lowest Landau level and considering parallel spin \nconfigurations only.\nThis is justified in the limit of high magnetic fields. They found a splitting\nof the two dipole allowed modes at $B=0$ due to dot interaction and\nsome anti-crossing structures in the upper mode, whereas the lower mode is\nalways close to the single particle mode. This fact is already a strong\n indication that\nthe missing higher Landau levels cause both spurious effects. (Observe that the\nlifting of the degeneracy at $B=0$ in the dipole allowed excitations \nin Fig.1 is due to the ellipticity\nof the intrinsic confinement and not due to dot interaction.)\\\\\nThe eigenvalues of $H^{(-)}$\ndo depend on $p$ because the dots oscillate (rigidly)\nin its two eigenmodes\nin opposite phase, one mode in $x$ and one mode in $y$ direction. \nThis leads to a change in the Coulomb energy. The two eigenmodes of $H^{(-)}$\ncan also be described as a breathing mode (in x direction) \nand a shear mode (in y direction).\\\\\n\n\\newpage\n\n\n\\begin{figure}[th]\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=Fig.1a.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n{\\psfig{figure=Fig.1b.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[]{\nExcitation energies $\\Delta E^{(\\pm)}_\\pm =\\omega^{(\\pm)}_\\pm$\nfor a dimer of elliptical dots as a\nfunction of the magnetic field for some discrete values of the interaction\nparameter $p$ ($p$ values in the inset are in units $\\omega_0^2$).\nThe ratio of the oscillator frequencies in the direction of the\ncapital axes $\\omega_1/\\omega_2$ is a) $3/2$ and b) $2/3$.\nThe dipole allowed excitations $\\Delta E^{(+)}$ of $H^{(+)}$ (thick full line)\nare not influenced by the dot interaction and therefore independent of $p$.\n}\n\\label{Fig1}\n\\end{center}\n\\end{figure}\n\n\\subsection{Special Features of the Excitation Spectrum}\nIn Fig.1a and 1b, the four excitation frequencies of the dimer are shown\nwith $p$ as a parameter.\n\nFor $p=0$, the two modes $\\omega^{(-)}_\\pm$ agree with \nthe two modes $\\omega^{(+)}_\\pm$. \nIn all symbols, the superscript sign refers to the system $H^{(+)}$\nand $H^{(-)}$ (c.m. or relative\ncoordinate), and the subscript sign discriminates the two modes of the same \nsystem.\nThe two modes $\\omega^{(+)}_\\pm$ are independent of $p$.\nThere are two qualitatively different cases.\n(Consider that $\\omega_1$ is the oscillator frequency parallel to the\nline, which connects the two dot centers, and $\\omega_2$ is the \noscillator frequency perpendicular to this line.)\nIf $\\omega_1 \\ge \\omega_2$\n (Fig.1a), the gap between \n$\\omega^{(-)}_+$ and $\\omega^{(-)}_-$ at $B=0$ increases steadily\nwith increasing $p$ until, for a critical $p_{cr}=\\omega_2^2$\n(in our numerical case: $p_{cr}[\\omega_0^2]=16/25=0.64$)\nthe lower mode $\\omega^{(-)}_-$\nbecomes soft. This transition is {\\em independent} of $B$. For \n$\\omega_1 \\le \\omega_2$ the gap \nbetween $\\omega^{(-)}_+$ and $\\omega^{(-)}_-$ at $B=0$\n first decreases with increasing $p$\nuntil it vanishes for $p=\\frac{1}{3}(\\omega_2^2-\\omega_1^2)$\n (in our numerical case: $p[\\omega_0^2]=4/15=0.27$). Afterwards,\nit increases until the lattice becomes soft at $p_{cr}=\\omega_2^2$\n(in our numerical case: $p_{cr}[\\omega_0^2]=36/25=1.44$).\nThe dependence of the two excitation energies\n$\\omega^{(-)}_+$ and $\\omega^{(-)}_-$ on $p$ for $B=0$ in the second case\nis shown in Fig.2. Comparison of Fig.s 2a and 2b demonstrates \nthat the dot architecture\nin Fig.2a is much more sensitive to interdot interaction than that in\nFig.2b. Thus, if we want to observe or use the instability,\n this event happens in case 2a for for smaller $p$\n(or larger lattice constants) than in case 2b. Additionally, the assumption of\nnon--overlapping dot wave functions (for a given lattice constant)\nis better fulfilled in case 2a than in case 2b. \\\\[.5cm]\n\n\\begin{figure}[!th]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=Fig.2.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nExcitation energies $\\Delta E^{(-)}_\\pm =\\omega^{(-)}_\\pm$ of the\nHamiltonian $H^{(-)}$ for $B=0$ and $\\omega_1/\\omega_2=2/3$ as a function\nof the interaction parameter $p$.\n}\n\\label{Fig2}\n\\end{center}\n\\end{figure}\n\n\nFor GaAs as a typical substance, (\\ref{p}) can be\nrewritten in more convenient units as\n\\begin{equation}\np[\\omega_0^2]=\\frac{2.26 \\cdot10^7 \\; N}{\\big(a[\\AA\\big])^3\\;\n\\big(\\omega_0[meV]\\big)^2}\n\\label{p-GaAs}\n\\end{equation}\nObviously, we need large dots (large $N$, small $\\omega_0$-- which means\nlarge polarizability), and a small\ndot distance $a$ for a seizable interaction effect.\nOn the other hand, the dot radius\nfor $N=1$ is of order of the effective magnetic length \\cite{Hawrylak-Buch}\n$l_0=\\bigg( (2\\omega_0)^2+(\\omega_c^*)^2 \\bigg)^{-1/4}$,\nwhich reads for GaAs\n\\begin{equation}\nl_0[\\AA]=\\frac{238}{\\bigg(\\big(\\omega_0[meV]\\big)^2+0.739 \\;\n\\big(B[Tesla]\\big)^2\\bigg) ^\\frac{1}{4}}\n\\label{l0-GaAs}\n\\end{equation}\nand we need small dots and high magnetic fields for small overlap.\nConsequently, a magnetic field helps avoiding overlap of the dots, although\ne.g. the critical $p$ for soft modes is independent of $B$.\nThe question is, if there exists a window between these two (partly)\nconflicting demands. For an order-- of-- maigntude estimate, \nlet us consider GaAs with $\\omega_0$\nas chosen above and the worst case $N=1$.\nThen (\\ref{p-GaAs}) with a typical $p[\\omega_0^2]=0.1$\n(which seems to be the minimum for any observable effect)\nprovides a dot distance of $a[\\AA]=327$ and (\\ref{l0-GaAs}) gives\nfor $B=0$ a radius of $l_0[\\AA]=150$ and for $B[Tesla]=10$ a radius of\n$l_0[\\AA]=80$. Consequently, \nthe constraint $l_0<a/2$ for our model can be fulfilled. \nFor obtaining larger interaction effects the parameters have to be optimized.\n\nThe next question is what happens in \n{\\em mode softening} physically? \nFirstly, it is the antisymmetric shear mode $\\omega^{(-)}_-$\n which has the lowest frequency\nand which becomes soft.\nIf the interaction parameter is strong enough ($p>p_{cr}$),\nthe {\\em de}crease in interdot-- Coulomb energy\nwith increasing elongation of the dots\nbecomes larger than the {\\em in}crease of confinement potential energy. Because\n{\\em in the harmonic model} \nboth energies depend quadratically on elongation, \nthe dimer {\\em would} be ionized, i.e.\nstripped of the electrons. Clearly, in this case we have to\ngo beyond the dipole approximation for the interdot interaction and \nbeyond the harmonic approximation for the confinement potential.\n\nIn order to obtain a hand-waving picture of what happens,\nthe confinement potential of the system for shear mode oscillations \nis supplemented by a $4^{th}$ order term in the following way:\n$\nV_{conf.}=2 N \\left[ \\frac{1}{2} \\omega_2^2\\; U^2-A \\;U^4\\right]\n$\nwith $(A > 0)$, \nand the Coulomb interaction in $4^{th}$ order reads:\n$\nV_{int}=-p N\\; U^2+ (3pN/a^2)\\; U^4\n$\nwhere $p=2 N \\beta/a^3$ as above. Then, the stability condition reads\n\\begin{equation}\n\\frac{V_{tot}}{N}=(\\omega_2^2-p)\\; U^2+(\\frac{3p}{a^2}-2A)\\; U^4 \\ge 0\n\\end{equation}\nThe condition for the existence of a bound state\nis that the $U^4$- term is positive: $3 p/a^2>2A$.\nFor a positive $U^2$- term ( $p<\\omega_2^2$ ), the \nequilibrium position is $U_0=0$. If the $U^2$- term becomes \nnegative ( $p>\\omega_2^2$ ),\nthe system finds a new equilibrium at a finite elongation\n\\begin{equation}\nU_0=\\pm \\sqrt{\\frac{(p-\\omega_2^2)}{2(\\frac{3p}{a^2}-2 A)}}\n\\end{equation}\nThis new ground state is doubly degenerate:\n${\\bf U}_1=(-a,+U_0)~,~{\\bf U}_2=(+a,-U_0)$ and\n${\\bf U}_1=(-a,-U_0)~,~{\\bf U}_2=(+a,+U_0)$ have the same energy.\nIn short, at $p_{cr}=\\omega_2^2$ there is an electronic phase transition to\na polarised state, where the equilibrium position of the c.m. is no more in\nthe middle of the dots.\nAt the end we want to stress that all these stability considerations\nare only valid if the confinement potential is not changed under\nelongation of the c.m. of the dots. Secondly, it is not rigorous\nto include the fourth order terms {\\em after} separation of c.m. and\nrelative coordinates, because in fourth order these two coordinates \ndo not decouple exactly.\n\n\\section{Dot Lattice}\n\nWe consider a periodic lattice of {\\em equal} quantum dots at lattice sites\n${\\bf R}_{n,\\alpha}^{(0)}={\\bf R}_n^{(0)}+\\mbox{\\boldmath $a$}_\\alpha$.\nThe vectors ${\\bf R}_n^{(0)}$ \nform a Bravais lattice and $\\mbox{\\boldmath $a$}_\\alpha$\nruns over all sites within an unit cell.\nIn developing a theory for these lattices we have to repeat all steps\nin Sect. II from (\\ref{separation}) to (\\ref{gauge-shift}) just by supplementing\nthe index $\\alpha$ by the index $n$ for the unit cell.\n\n\\subsection{Center of Mass Hamiltonian of the Dot Lattice}\nThe c.m. Hamiltonian in the dipole approximation then reads\n\\begin{eqnarray}\nH_{c.m.}&=&\\frac{1}{N}\n\\bigg\\{ \\sum_{n,\\alpha} \n\\frac{1}{2 m^*}\n\\left[\n{\\bf P}_{n,\\alpha}+\n{N\\over c} {\\bf A}\\left({\\bf U}_{n,\\alpha} \\right)\n\\right]^2 \\nonumber \\\\\n && + \\frac{N^2}{2} \\sum_{n,\\alpha\\atop n',\\alpha'}\\; {\\bf U}_{n,\\alpha} \\cdot\n{\\bf C}_{n,\\alpha;\\, n',\\alpha'}\n\\cdot {\\bf U}_{n',\\alpha'}\n\\bigg\\}\n\\label{H-cm-latt}\n\\end{eqnarray}\nwhere ${\\bf U}_{n,\\alpha}={\\bf R}_{n,\\alpha}-{\\bf R}_{n,\\alpha}^{(0)}$ \nis the elongation of the c.m. \nat lattice site $(n,\\alpha)$ and the force constant \ntensor $\\bf C$ is defined in analogy to (\\ref{C1}) and (\\ref{C2}).\n\nThe Hamiltonian (\\ref{H-cm-latt}) \nis a phonon Hamiltonian in an additional homogeneous magnetic field.\nThe first stage of decoupling can be achieved by the usual \nphonon transformation\n\\begin{eqnarray}\n{\\bf U}_{n,\\alpha}&=&\\frac{1}{\\sqrt{N_c}} \\sum_{\\bf q}^{BZ}\ne^{-i{\\bf q}\\cdot R_n^{(0)}}\\;{\\bf U}_{{\\bf q},\\alpha}\\\\\n{\\bf P}_{n,\\alpha}&=&\\frac{1}{\\sqrt{N_c}} \\sum_{\\bf q}^{BZ}\ne^{+i{\\bf q}\\cdot R_n^{(0)}}\\;{\\bf P}_{{\\bf q},\\alpha}\n\\label{phonon-trafo}\n\\end{eqnarray}\nwhere $N_c$ is the number of unit cells and the \ntransformed coordinates have the following properties\n${\\bf U}_{-{\\bf q},\\alpha}={\\bf U}_{{\\bf q},\\alpha}^*\n={\\bf U}_{{\\bf q},\\alpha}^\\dagger$ and\n${\\bf P}_{-{\\bf q},\\alpha}={\\bf P}_{{\\bf q},\\alpha}^\\dagger$. \nThe Hamiltonian in the new coordinates\nis a sum of $N_c$ subsystems of dimension $2\\times$ number of dots per unit \ncell: \n$H_{c.m.}= \\sum_{\\bf q}\\;H_{\\bf q}$, where\n\\begin{eqnarray}\nH_{\\bf q}&=& \\frac{1}{N} \\bigg\\{ \\sum_\\alpha\\; \\frac{1}{2 m^*} \n\\left[\n{\\bf P}_{{\\bf q},\\alpha}+\\frac{N}{c}{\\bf A}({\\bf U}_{{\\bf q},\\alpha}^*)\n\\right]^\\dagger \\cdot \n\\left[\n{\\bf P}_{{\\bf q},\\alpha}+\\frac{N}{c}{\\bf A}({\\bf U}_{{\\bf q},\\alpha}^*)\n\\right] \\nonumber\\\\\n& &+ \\frac{N^2}{2} \\sum_{\\alpha,\\alpha'}\n{\\bf U}_{{\\bf q},\\alpha}^* \\cdot\n{\\bf C}_{{\\bf q};\\alpha,\\alpha'}\n\\cdot {\\bf U}_{{\\bf q},\\alpha'} \\bigg\\}\n\\label{H-phonon}\n\\end{eqnarray}\nThe dynamical matrix is defined by\n\\begin{equation}\n{\\bf C}_{{\\bf q};\\alpha,\\alpha'}=\\sum_n\\;\ne^{i{\\bf q}\\cdot {\\bf R}_n^{(0)}}\\;\n{\\bf C}_{\\alpha,\\alpha'}\\left({\\bf R}_n^{(0)}\\right)\n~;~~~~{\\bf C}_{\\alpha,\\alpha'}\\left({\\bf R}_n^{(0)}\\right)\n= {\\bf C}_{n,\\alpha;\\,0,\\alpha'}\n\\label{dyn-mat-full}\n\\end{equation}\nand it is hermitean ${\\bf C}_{{\\bf q};\\alpha',\\alpha}=\n{\\bf C}^*_{{\\bf q};\\alpha,\\alpha'}={\\bf C}_{-{\\bf q};\\alpha,\\alpha'}$.\\\\\nNext we want to recover the limiting case considered in Sect. III.\nIf the dots in a given unit cell are far away from those in neighboring\ncells, then in (\\ref{dyn-mat-full}) only the term with ${\\bf R}_n^{(0)}=0$\ncontributes, $\\bf C$ does not depend on $\\bf q$, consequently the index\n$\\bf q$ is redundant, and (\\ref{H-phonon}) agrees with (\\ref{H-cm-dimer}).\\\\\nOur preliminary result \n(\\ref{H-phonon}) is not yet diagonal in $\\alpha,\\alpha'$. \nIn some special cases (see e.g. two identical dots\nper unit cell considered in Sect. III) this can be \nachieved by an unitary transformation\n\\begin{equation}\n{\\bf U}_{{\\bf q},\\alpha}=\\sum_{\\alpha'}\\;Q_{{\\bf q};\\alpha,\\alpha'}\\cdot\n\\tilde {\\bf U}_{{\\bf q},\\alpha'}~;~~~~~Q^*_{{\\bf q};\\alpha',\\alpha}=\nQ^{-1}_{{\\bf q};\\alpha,\\alpha'}\n\\end{equation}\nunder which the one-- particle term in (\\ref{H-phonon}) is invariant\nand the transformed interaction term \n\\begin{equation}\n\\frac{1}{2} \\sum_{\\alpha,\\alpha'}\n\\tilde{\\bf U}_{{\\bf q},\\alpha}^* \\cdot\n\\tilde{\\bf C}_{{\\bf q};\\alpha,\\alpha'}\n\\cdot \\tilde{\\bf U}_{{\\bf q},\\alpha'}~~~\\mbox{with}~~~\n\\tilde{\\bf C}_{{\\bf q};\\alpha,\\alpha'}= \\sum_{\\alpha_1,\\alpha_2}\nQ^{-1}_{\\alpha,\\alpha_1}\\;{\\bf C}_{{\\bf q};\\alpha_1,\\alpha_2}\\;\nQ_{\\alpha_2,\\alpha'}\n\\end{equation}\ncan be made diagonal $\\tilde{\\bf C}_{{\\bf q};\\alpha,\\alpha'}=\n\\tilde{\\bf C}_{{\\bf q};\\alpha}\\;\\delta_{\\alpha,\\alpha'}$ by \na proper choice of $Q_{\\alpha,\\alpha'}$. Now, (\\ref{H-phonon}) reads \n$H_{\\bf q}=\\sum_{\\alpha}\\;H_{{\\bf q},\\alpha}$, where\n\\begin{eqnarray}\nH_{{\\bf q},\\alpha}&=& \\frac{1}{N} \\bigg\\{ \\frac{1}{2 m^*}\n\\left[\n\\tilde{\\bf P}_{{\\bf q},\\alpha}+\\frac{N}{c}{\\bf A} (\n\\tilde{\\bf U}_{{\\bf q},\\alpha}^*)\n\\right]^\\dagger \\cdot\n\\left[\n\\tilde{\\bf P}_{{\\bf q},\\alpha}+\\frac{N}{c}{\\bf A} (\n\\tilde{\\bf U}_{{\\bf q},\\alpha}^*)\n\\right] \\nonumber\\\\\n&&+ \\frac{N^2}{2} \n\\tilde{\\bf U}_{{\\bf q},\\alpha}^* \\cdot\n\\tilde{\\bf C}_{{\\bf q};\\alpha}\n\\cdot \\tilde{\\bf U}_{{\\bf q},\\alpha} \\bigg\\}\n\\label{H-phonon-diag}\n\\end{eqnarray}\nThe eigenvalues of (\\ref{H-phonon-diag}) can be obtained from those\nof (\\ref{H-N=1}) because corresponding quantities have the same\ncommutation rules. Such an unitary transformation does not\nexist, e.g., for two different dots per cell. Then (\\ref{H-phonon})\nhas to be solved directly using the method described in Ref. \n\\onlinecite{Tsallis}.\\\\\n\n\n\\subsection{Dynamical Matrix for Bravais lattices}\nFrom now on we consider {\\em Bravais lattices} what means that we can forget\nthe indices $\\alpha$ in the first part of this Section. \nThen the dynamical matrix \n\\begin{equation}\n{\\bf C}_{\\bf q}={\\bf \\Omega}+\\beta N \\;\\sum_{{\\bf R}_n^{(0)}\\ne0}\n\\left( 1-e^{i{\\bf q}\\cdot {\\bf R}_n^{(0)}} \\right) \\;\n{\\bf T}\\left({\\bf R}_n^{(0)}\\right)\n\\label{dyn-mat}\n\\end{equation}\nis real and symmetric, but generally not diagonal, even if $\\bf \\Omega$\nis diagonal.\nA very important conclusion is apparent in (\\ref{dyn-mat}). {\\em In the limit\n${\\bf q}\\rightarrow 0$, the inter-- dot interaction (represented by $\\beta$) \nhas no influence on ${\\bf C}_{\\bf q}$\nand therefore on the spectrum. This means, that the excitation spectrum\nobserved by FIR spectroscopy is not influenced by inter-- dot interaction\nand agrees with the one-- electron result (as in the single dot)}.\nThis statement is rigorous for parabolic confinement (see Appendix).\nIt can also be understood intuitively,\n because a $q=0$-- excitation is connected with\nhomogeneous in-- phase elongations of the dots which do not change\nthe distance between the electrons. We want to mention that\nthis conclusion seems to be in contradiction with the experimental\nwork in Ref.\\onlinecite{Kotthaus}. They found a splitting of \nthe upper and lower excitation branch at $B=0$ and $q=0$ \nfor circular dots in a rectangular\nlattice, which they interpreted within a\nphenomenological model of interacting dipoles as \na consequence of lattice interaction. However, they use mesoscopic dots\nwith a diameter of $370 000 \\AA$ and lattice periods of\n$400 000$ and $800 000 \\AA$. These dots are clearly\nbeyond our microscopic quantum mechanical model, which rests on a \nparabolic confinement.\\\\\n\nFor the {\\em rectangular lattices} considered in our numerical examples\nwe define\n${\\bf R}^{(0)}=N_1 a_1 {\\bf e}_1+N_2 a_2 {\\bf e}_2$ and\n${\\bf q}=q_1\\frac{2 \\pi}{a_1} {\\bf e}_1+q_2\\frac{2 \\pi}{a_2} {\\bf e}_2$\nwith the lattice constants $a_1$ and $a_2$ and integers $N_1$ and $N_2$\ncharacterizing the lattice sites.\nThe components of ${\\bf q}$ vary in the Brillouin zone (BZ) in the range\n$[-1/2,+1/2]$.\nThe dipole tensor (\\ref{dipole-tensor}) reads\n\\begin{equation}\n{\\bf T}(N_1,N_2)=\\frac{1}{(N_1^2 a_1^2+N_2^2 a_2^2)^{5/2}}\n\\left[\n\\begin{array} {cc}(2N_1^2 a_1^2-N_2^2 a_2^2)&\n3N_1N_2a_1a_2\\\\ 3N_1N_2a_1a_2&(2N_2^2 a_2^2-N_1^2 a_1^2) \\end{array}\n\\right]\n\\end{equation}\nAlthough for all figures the exact dynamical matrix is used, it\nis useful to consider the results with {\\em nearest neighbor} (n.n.)\nlattice sums in (\\ref{dyn-mat}) separately.\nThis provides simple formulas for order-- of-- \nmagnitude estimates. \n\\begin{eqnarray}\nC_{11}&=&\\omega_1^2+2\\;p_1\\;[1-cos(2\\pi q_1)]-p_2\\;[1-cos(2\\pi q_2)]\n\\nonumber\\\\\nC_{22}&=&\\omega_2^2+2\\;p_2\\;[1-cos(2\\pi q_2)]-p_1\\;[1-cos(2\\pi q_1)]\n\\label{C-nn}\\\\\nC_{12}&=&\\Omega_{12}\n\\nonumber\n\\end{eqnarray}\nwhere we introduced the interaction parameters \n$p_i=\\frac{2\\beta N}{a_i^3}$.\\\\[1cm]\n\n\\begin{figure}[!th]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=Fig.3.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nConvergency of the lattice sums $S_{ik}$ as defined in\n(\\ref{S_ik}) with increasing\nnumber of cubic shells $N_{max}$ for $\\bf q$ in the middle of the\nBrillouin zone.\n}\n\\label{Fig3}\n\\end{center}\n\\end{figure}\n\n\nThe convergence of the lattice sums $S_{ik}$\nin the dynamical matrix is shown Fig.3.\n$S_{ik}$ is defined by\n\\begin{equation}\nC_{ik}=\\Omega_{ik}+p_2\\; S_{ik} \n\\label{S_ik}\n\\end{equation}\nand depends only on $\\bf q$ and the ratio\n$a_1/a_2$. Apart from the off-diagonal elements, which vanish in n.n.\napproximation, the error of the n.n. approximation is less than $30\\%$.\n\n\n\\subsection{Special Features of the Magneto-- Phonon Spectrum}\nFig.s 4-6 show the magneto-- phonon \n\\footnote{ The term {\\em magneto-- phonon} is attributed to the fact that\nthe there is no exchange and there are harmonic forces \nbetween the oscillating individuals. One could also call them\n{\\em magneto-- plasmons}, if one wants to emphasize that it is only\nelectrons which oscillate, and no nuclei.}\nspectrum for circular dots\non a rectangular lattice with $a_1=2 a_2$.\nBecause the two interaction parameters have a fixed ratio, it\nsuffices to use one of them for characterizing the interaction strength.\nWe chose the larger one $p_2=p$.\nFor $B=0$ and isolated dots $(p=0)$, the two excitation modes \nare degenerate. If we tune up the interaction strength represented by $p$,\na $\\bf q$ dependent splitting appears (see Fig. 4). \\\\[1cm]\n\n\\begin{figure}[!th]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=Fig.4.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nMagneto-- phonon dispersion at $B=0$ for several interaction parameter\nvalues and $\\bf q$ on symmetry lines of the Brillouin zone.\n($p=p_2$ values in the inset are in units $\\omega_0^2$.)\nThick (lower) and thin (upper) lines indicate $\\Delta E_-$ and $\\Delta E_+$,\nrespectively.\n}\n\\label{Fig4}\n\\end{center}\n\\end{figure}\n\nThis splitting is a manifestation of the dot interaction.\nFor a certain critical\n$p_{cr}$ the lower mode becomes soft. This feature will be discussed below.\nThere are points in the BZ, however, where the {\\em degeneracy}\nfor finite $p$ remains. These\npoints will be investigated now. We demonstrated in Sect.II after formula\n(\\ref{B=0}) that {\\em necessary} \nfor degeneracy is $C_{12}=0$, i.e., the dynamical \nmatrix must be diagonal. Then the points with degeneracy are defined\nby the condition $C_{11}=C_{22}$.\nAs seen in (\\ref{dyn-mat}),\nfor circular dots $\\omega_1=\\omega_2=\\omega_0$ this happens\nin the center of the BZ ${\\bf q}=0$. \nThe next question to be discussed is if there are other points with\ndegeneracy. The first condition $C_{12}=0$, \nis fulfilled for all points on the surface of the BZ. The second condition\nmust be investigated for special cases. We find, that for quadratic\nlattices $a_1=a_2$ with circular dots $\\omega_1=\\omega_2$ both modes\nare degenerate at the point ${\\bf q}=(1/2,1/2)$. In the case shown in Fig.4\nthis point is somewhere between $(1/2,1/2)$ and $(1/2,0)$. \\\\\nIn n.n. approximation (\\ref{C-nn}), however, this equation is even \nfulfilled on full curves in the BZ\ndefined by \\mbox{$p_1\\;[1-cos(2\\pi q_1)]=p_2\\;[1-cos(2\\pi q_2)]$}.\nIn a cubic lattice, this is the straight lines $q_2=\\pm q_1$.\nThe contributions beyond n.n.s remove the exact degeneracy on this curve\nin the interior of the BZ,\nbut leave a kind of anti-crossing behavior of the two branches.\\\\\n\nAn important parameter, which characterizes \nthe influence of the dot interaction in circular dots,\nis the {\\em band width} at $B=0$, i.e. the maximum\nsplitting of the two branches due to dot interaction. \n(Remember that this splitting vanishes for noninteracting \ncircular dots.)\nAssume $a_1> a_2$. Then the largest splitting for circular dots\nin n.n. approximation\n appears at ${\\bf q}=(0,1/2)$\nand has the amount \n\\begin{equation}\nW=max(\\Delta E_+ - \\Delta E_-)=max(\\omega_+-\\omega_-)=\\sqrt{\\omega_0^2+4\\;p_2}-\n \\sqrt{\\omega_0^2-2\\;p_2}\n\\end{equation}\nFor small dot interaction and in units $\\omega_0$, this is \nproportional to the interaction parameter\n$ \\frac{W}{\\omega_0} \\rightarrow 3 p_2 $.\\\\\n\nWe next discuss the appearance of {\\em soft modes}. \nThe question is, for which $\\bf q$, $B$ and interaction parameter $p$\nthis happens.\nThe general condition\nfor vanishing of the lowest mode is \n$C_{11}\\cdot C_{22}=C_{12}^2$ (see Sect. II). \nIn this condition the magnetic field does not appear.\nFor circular dots and with the definition \n(\\ref{S_ik}) this equation reads\n\\begin{equation}\n[\\omega_0^2+p_2\\;S_{11}][\\omega_0^2+p_2\\;S_{11}]=p_2^2\\;S_{12}^2\n\\end{equation}\nAfter introducing a dimensionless critical interaction parameter\n$P_{cr}=p_2/\\omega_0^2$, we obtain a quadratic equation for $P_{cr}$\nwhich has the solution\n\\begin{equation}\nP_{cr}=-\\frac{1}{2}\\frac{tr}{det}\n\\pm \\sqrt{\\left( \\frac{1}{2}\\frac{tr}{det}\\right)^2-\\frac{1}{det}}\n\\end{equation}\nwhere $det=S_{11}S_{22}-S_{12}^2$ and $tr=S_{11}+S_{22}$. For \nour numerical case $a_1=2\\;a_2$ and n.n. interaction for $S_{ik}$\nthe lowest mode becomes soft at ${\\bf q}=(0,1/2)$ and the critical interaction\nparameter is $P_{cr}=1/2$. Inclusion of lattice contributions beyond n.n.\nshifts $P_{cr}$ to 0.7543. The most important result of this paragraph is \nthat lattice softening is {\\em independent}\nof $B$ (see also Fig.s 5 and 6). \nThe latter conclusion is {\\em exact} within the range of validity of\n the Hamiltonian (\\ref{H-cm-latt}) and no consequence of any\nsubsequent approximation or specialization.\\\\[1cm]\n\n\\begin{figure}[!th]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=Fig.5.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nThe same as Fig.4, but for $p=p_{cr}$ and several magnetic fields.\n}\n\\label{Fig5}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!th]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=Fig.6.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nMagneto-- phonon excitations as a function of $B$ for the\nsymmetry points in the Brillouin zone. The upper abscissa is independent\nof the effective mass, the lower one applies to GaAs.\n}\n\\label{Fig6}\n\\end{center}\n\\end{figure}\n\n\n\\section{Intra-dot-- Excitations for N=2}\nIntra-dot excitations for circular dots\nin a cubic lattice and for $N=2$ can be calculated easily.\nWe define the relative coordinate ${\\bf r}={\\bf r}_2-{\\bf r}_1$,\nand assume that all dots are equivalent (also with respect to\ntheir environment). Then the indexes $(n,\\alpha)$ can be chosen as $(0,0)$\nand omitted.\nThe relative Hamiltonian reads \n\\begin{equation}\nH_{rel}=2 \\bigg\\{ \n\\frac{1}{2 m^*}\n\\left[\n{\\bf p}+\n{1\\over 2 c} {\\bf A}({\\bf r})\n\\right]^2 +\n\\frac{1}{2}\\; {\\bf r} \\cdot {\\bf D } \\cdot {\\bf r}\n+\\frac{\\beta}{2\\;r}\n\\bigg\\}\n\\label{H-rel}\n\\end{equation}\nwhere\n${\\bf p}=-i \\nabla_{\\bf r}$ and\n\\begin{equation}\n{\\bf D}=\\frac{1}{4}{\\bf \\Omega}+\\frac{\\beta}{2}\n{\\bf T}_0~;~~~~\n{\\bf T}_0=\\sum_{n,\\alpha(\\ne 0,0)}\n{\\bf T}\\left({\\bf R}_{n,\\alpha}^{(0)}\\right)\n\\end{equation}\nIt is worth emphasizing that $H_{rel}$ contains a contribution from\nneighboring dots, originating from the interdot Coulomb interaction.\nA trivial angular dependent part can only be decoupled \nfrom $H_{rel}$, or, the 2-dimensional Schr\\\"odinger\nequation can be traced back to an ordinary radial Schr\\\"odinger equation,\n if the term ${\\bf r} \\cdot {\\bf D } \\cdot {\\bf r}$\nhas the same circular symmetry as the intra-dot Coulomb term\n$\\beta/(2 r)$. Therefore we confine ourselves to circular dots \non a cubic lattice,\nand we have\n\\begin{equation}\n{\\bf T}_0=\\frac{1}{a^3}\\sum_{N_1,N_2\\ne 0,0}\\;\n\\frac{1}{(N_1^2 +N_2^2 )^{3/2}}\\;{\\bf I}\\approx \\frac{4}{a^3}\\;{\\bf I}\n\\end{equation}\nwhere the simple result is in n.n. approximation.\nUsing the interaction parameter $p=2N\\beta/a^3$ (with $N=2$) defined above,\nwe obtain\n\\begin{equation}\n{\\bf D}=\\frac{1}{4}(\\omega_0^2+2p)\\;{\\bf I}\n\\end{equation}\nIn this way, dot interaction defines\nan effective confinement frequency \n$\\omega_{0,eff}^2 = \\omega_0^2+2 p$. This means\nthat {\\em the c.m. excitations have to be calculated (or interpreted) \nwith another confinement potential then the relative excitations}.\nIn our figures we present results for $\\omega_{0,eff}=0.2\\; a.u.^*$,\nwhich agrees with the bare confinement potential used in Sect.IV and\nthe mean value in Sect.III.\nBecause our results are presented in units of $\\omega_0$, they\ndepend on $\\omega_0$ only weakly \nthrough the differing influence of electron-electron\ninteraction. For the absolute values, however, the influence\nof the dot interaction can be tremendous.\\\\\nIn the relative motion there is a coupling between orbital and spin parts\nthrough the Pauli principle. For $N=2$ and a circular effective\nconfinement, Pauli principle demands that orbital states with \neven and odd relative angular momentum $m_i$\nmust be combined with singlet and triplet spin states, respectively\n( see e.g. Ref.\\onlinecite{Taut2e}).\nFor the c.m. motion there is no interrelation \nbetween orbital and spin part because the c.m. coordinate\nis fully symmetric with respect to particle exchange. \nConsequently, any c.m. wave function can be combined with\na given spin eigen function.\nThe only spin dependent term in the total energy \nconsidered here is the {\\em Zeeman term}, which reads in our units\n\\begin{equation}\n\\frac{E_B}{\\omega_0}= 0.9134\\cdot10^{-2}\\;g_s\\;\\frac{B[Tesla]}{\\omega_0[a.u.^*]}\n\\;\\frac{M_s}{2}\n\\end{equation}\nwhere we used $g_s=-0.44$ for the gyro-magnetic factor of GaAs \nfrom Ref.\\onlinecite{Merkt}. The \ntotal spin quantum number is $M_s=0$ for the\nsinglet state and $0,\\pm1$ for the triplet state.\nOne of the most interesting points in quantum dot physics is that\nthe total orbital angular momentum of the ground state depends on \nthe magnetic field\n(see e.g. Ref.s\\onlinecite{Merkt},\\onlinecite{Maksym}). \nThis feature is a consequence of electron-- electron interaction. \nFor our parameter values,\nthe relative orbital angular momentum \nof the ground state $m_i$ changes from \n0 to --1, from --1 to --2, and from --2 to --3 \nat $B=1.250$, $4.018$, and $5.005\\; Tesla$.\nThis corresponds to a sequence $M_s$=0,+1,0,+1 for the spin \nquantum number.\nFigures 7a-c show the excitation frequencies for three B-values\nlying within the first three regions. \n$m_f$ is the relative orbital\nangular momentum of the final state. \nAll excitations are\nincluded irrespective of selection rules. \nFor dipole transitions only two of them would remain (the lowest excitation\n with $m_f=m_i\\pm 1$). For $B=0$, the lowest excitation energy\n(in units $\\omega_0$) for noninteracting electrons would be 1.\nAs seen in in Fig.7a, electron-- electron interaction decreases this\nvalue by at least a factor of $1/2$. The same holds qualitatively \nfor finite $B$. This is connected to the fact, that the ground \nstate depends on $B$. Let's consider an example. For $B=1.250 \\;Tesla$\nthe ground state switches from $m_i=0$ to $m_i=-1$. This implies that\nfor $B$ approaching this transition field from below, the\nexcitation energy for\ndipole allowed transition from $m_i=0$ to $m_f=-1$ converges to $0$.\nIn other words, there is a level crossing at the the transition field.\nTherefore, very small transition energies and switching of the ground state\nare connected.\\\\[1cm]\n\n\\begin{figure}[!th]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=Fig.7a.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n{\\psfig{figure=Fig.7b.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n{\\psfig{figure=Fig.7c.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nIntradot excitation energies (in units of the {\\em effective}\nconfinement frequency) for B=0 (a), B=3 Tesla (b) and\nB=4.5 Tesla (c). The corresponding relative orbital angular momenta\nof the ground state are $m_i$=0 (a), --1(b), and --2(c) and\nthe spin angular momenta $M_s$=0 (a), +1(b), and 0(c).\n}\n\\label{Fig7c}\n\\end{center}\n\\end{figure}\n\n\n\nFor a qualitative understanding, Figures 7a-c\ncan be used together with Figures 1a, 1b, 4, and 5 to investigate\nthe relative position of collective and intra-dot excitations. \nThe conclusion is that for small dot interaction (for $p$ well below\n$p_{cr}$), the lowest intra-dot excitation energies\nlie well below the lowest c.m. excitations. \nApart from using a different terminology,\nthis conclusion agrees with the experimental findings in \nRef.\\onlinecite{Heitmann}.\\\\\nFig.s 7b and c, which belong to finite $B$, show the Zeeman splitting.\nAll transition energies to final states with odd $m_f$ are triplets because\nthe corresponding spin state is a triplet state. The thin lines of\na triplet belong to spin-- flip transitions.\\\\ [.5cm]\n\n\\begin{figure}[!th]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=Fig.8.ps,angle=-90,width=12.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nIntradot excitation energies (in units of the {\\em effective}\nconfinement frequency) as a function of the magnetic field.\nThe small Zeeman splitting is neglected.\nThe B-- values, where the angular momenta of the ground state\nchange, are indicated by vertical lines. The absolute value of the\nfinal state orbital momentum $m_f$ of the curves at B=0\ngrows from bottom to top by 1 starting with 0.\n}\n\\label{Fig8}\n\\end{center}\n\\end{figure}\n\nIn Fig.8 the $B$-- dependence of the lowest excitation energies is shown.\nIt is clearly seen that the curves exhibit a kink at those $B$-- values,\nwhere the ground state configuration changes. The size of the kink \ndecreases with increasing $B$. If this kink could be resolved experimentally\n(e.g. by electronic Raman spectroscopy), it would be a direct\nindication for the change of the ground state configuration, \nand thus an experimentally observable \nconsequence of electron-- electron interaction.\n\n\\newpage\n\n\\section{Summary}\nWe solved the Schr\\\"odinger equation for a lattice \nof {\\em identical} parabolic (but not necessarily circular) \nquantum dots with Coulomb\ninteraction (in dipole approximation) between the dots. \nWe provide an overview over the state of art\nof these systems which includes the results of former publications.\nReferences can be found in the text.\n \n\\begin{itemize}\n\\item\nSimilar to single dots, the center of mass coordinates of all \ndots can be separated from the relative coordinates.\nOnly the c.m. coordinates of different dots are coupled to each other. \nThe relative coordinates\nof different dots are neither coupled to each other nor to the c.m. coordinates.\n\\item\nThis gives rise to two types of excitations: two collective c.m. modes\nper dot \nand and a complex spectrum of intra-dot excitations.\nIn periodic arrays only the collective c.m. modes show dispersion.\nIntra-dot excitations are dispersion-less.\n\\item The c.m. system can be solved exactly and analytically\nproviding magneto-- phonon excitations characterized by a certain\nwave number $\\bf q$ within the Brillouin zone.\nFor ${\\bf q}=0$ and one dot per unit cell, interdot interaction does not \nhave any influence on the c.m. excitations.\n\\item All dipole allowed excitations (seen in FIR experiments) are not\ninfluenced by the dot interaction.\n\\item Interdot interaction between two dots influences the spectrum through\na single parameter $p=2 N \\beta/a^3$, where $a$ is the distance between\nthe dots, $N$ the number of electrons per dot and $\\beta$ the inverse\nbackground dielectric constant.\n\\item If $p$ exceeds a certain critical value $p_{cr}$, the lowest c.m. mode \nbecomes soft leading to an instability. This transition is independent of\nthe magnetic field.\n\\item For B=0 and and one circular dot per unit cell, the two\nc.m. modes are not only degenerate in the middle of the Brillouin zone, \nbut also at some points on the surface. If we use the n.n. approximation\n for the\nlattice sums in the dynamical matrix, degeneracy is maintained even on\nfull curves in the Brillouin zone.\n\\item Intra-dot excitations have to be calculated from an effective\nconfinement. \nIn circular dots with a cubic environment in\nnearest neighbor approximation the effective confinement frequency reads\n $\\omega_{0,eff}^2 = \\omega_0^2+2 p$. \nThis effective confinement differs from that for the\nc.m. motion.\n\\item For $p$ well below $p_{cr}$, the lowest intra-dot excitations\nare much smaller than the lowest collective excitations.\n\\item The intra-dot excitation energies versus magnetic field\nexhibit kinks at those fields, where the angular momentum of the \nground state changes.\n\\end{itemize}\nIn the Appendix we prove a Kohn Theorem for dot arrays with \nCoulomb interaction between the dots without the dipole approximation. \nThe individual confinement potentials can be arbitrarily arranged and can carry \ndifferent electron numbers, but have to be described by identical\nconfinement tensors. This means that for breaking Kohn's Theorem in\ndot arrays, we have to have at least two different confinement species.\n\n\\section*{Appendix}\nWe are going to prove that for an arbitrary\narray \\footnote{The dot centers can be arranged arbitrarily.}\n of identical parabolic quantum dot potentials\n\\footnote{The confinement tensors $\\bf \\Omega$ \nof all dots must be equal.}\n in an homogeneous\nmagnetic field: i) the total c.m. degree of freedom\ncan be separated from the rest, ii) the total c.m. Hamiltonian \nis not influenced \nby Coulomb interaction,\nand iii) the eigenvalues of the total c.m. Hamiltonian \nare independent of the electron number $N$ in each dot \\footnote{\nThe electron number in different dots can be different.} . \n\\\\\nThe Hamiltonian $H=H^{(0)}+V$ consists of an one--particle term $H=H^{(0)}$\nand the Coulomb interaction between all electrons $V$. The dot centers are\nlocated at ${\\bf R}_{\\alpha}^{(0)}$ and the electron coordinates are denoted by\n${\\bf r}_{i \\alpha}={\\bf R}_{\\alpha}^{(0)}+{\\bf u}_{i \\alpha}$. Then we have\n\\begin{equation}\nH^{(0)}=\\sum_{i \\alpha}\n\\left\\{\n\\frac{1}{2 m^*}\n\\left[\n{\\bf p}_{i \\alpha}+\n{1\\over c} {\\bf A}({\\bf R}_{\\alpha}^{(0)}+{\\bf u}_{i \\alpha})\n\\right]^2 +\n\\frac{1}{2} \\;{\\bf u}_{i \\alpha} \\cdot\n{\\bf C }\n\\cdot {\\bf u}_{i \\alpha}\n\\right\\}\n\\label{H-0}\n\\end{equation}\nFirst of all we shift the gauge center for each electron into the middle\nof the corresponding dot using an unitary transformation similar to\n(\\ref{gauge-shift}). \nThis transforms the shift ${\\bf R}_{\\alpha}^{(0)}$ in the argument \nof the vector potential away.\nNext we drop the index $\\alpha$ in (\\ref{H-0})\n so that the index\n'$i$' runs over all electrons in all dots. Now we perform a\ntransformation to new coordinates $ \\tilde {\\bf u}_i$\n\\begin{equation}\n{\\bf u}_i=\\sum_k Q_{ik}\\;(\\sqrt{N}\\; \\tilde {\\bf u}_k)~;~~~\n(\\sqrt{N} \\;\\tilde {\\bf u}_i)=\\sum_k Q_{ki}^* \\; {\\bf u}_k\n\\end{equation}\nwhere $Q_{ik}$ is an unitary matrix. This implies\n\\begin{equation}\n{\\bf p}_i=\\sum_k Q_{ik}^*\\;(\\frac{\\tilde {\\bf p}_k}{\\sqrt{N}} )~;~~~\n(\\frac{\\tilde {\\bf p}_i}{\\sqrt{N}} )=\\sum_k Q_{ki} \\; {\\bf p}_k\n\\end{equation}\nIt is possible to choose for the first column \n$Q_{k1}=\\frac{1}{\\sqrt{N}}$. The other\ncolumns need not be specified. \nThen $\\tilde {\\bf u}_1=(1/N)\\sum_i{\\bf u}_i={\\bf U}$ is the c.m.\nof all elongations, or, the c.m. of the electron coordinates\nwith respect to the weighted center of the dot locations\n${\\bf R}^{(0)}=(1/N)\\sum_\\alpha N_\\alpha\\;{\\bf R}_{\\alpha}^{(0)}$,\nwhere $N_\\alpha$ is the number of electrons in dot $\\alpha$.\nThe corresponding canonical momentum \n$\\tilde{\\bf p}_{1}=(1/i)\\nabla_{\\tilde {\\bf u}_1}={\\bf P}$\nis the c.m. momentum.\nInserting our transformation into (\\ref{H-0}) provides\n\\begin{equation}\nH^{(0)}=\\sum_i\n\\left\\{\n\\frac{1}{2 m^*}\n\\left[\n\\frac{1}{\\sqrt{N}} \\;\\tilde{\\bf p}_{i}+\n{\\sqrt{N}\\over c} {\\bf A}(\\tilde{\\bf u}_{i})\n\\right]^2 +\n\\frac{N}{2} \\;\\tilde{\\bf u}_{i} \\cdot\n{\\bf C }\n\\cdot \\tilde{\\bf u}_{i}\n\\right\\}\n\\label{H-0-trans}\n\\end{equation}\nThe term $i=1$ in (\\ref{H-0-trans}) is the (separated) c.m. \nHamiltonian \n\\begin{equation}\nH_{c.m.}=\\frac{1}{2 m^*}\n\\left[\n\\frac{1}{\\sqrt{N}}\\; \\tilde{\\bf P}+\n{\\sqrt{N}\\over c} {\\bf A}(\\tilde{\\bf U})\n\\right]^2 +\n\\frac{N}{2} \\;\\tilde{\\bf U} \\cdot\n{\\bf C }\n\\cdot \\tilde{\\bf U}\n\\end{equation}\nwhich agrees with (\\ref{H-cm}). Clearly, the Coulomb interaction\n$V$ in $H$ is independent of the c.m., and does not contribute to\n$H_{c.m.}$.\nFor the independence of the eigenvalues of\n$N$ see the discussion following (\\ref{H-cm}).\\\\\n\nThis prove, in particular\nthe step from (\\ref{H-0}) to (\\ref{H-0-trans}),\nis not correct if the dot confinement tensor $\\bf C$ depends\non $\\alpha$ (or '$i$' in the changed notation). Therefore, all dots\nmust have the same $\\bf C$, but can have different electron numbers\n$N_\\alpha$. In other words, the total c.m. excitations in dot arrays, \nwhich are seen in\nFIR spectra, are not affected by the e-- e-- interaction, if and only if\nall confinement tensors $\\bf C$ are equal. On the other hand, {\\em if we want\nto observe e-- e-- interaction in the FIR spectra and break\nKohn's Theorem, we have to use dot lattices with at least two different\nconfinement tensors}. The simplest way to implement this is using\na lattice with two non-circular dots per cell, which are \nequal in shape, but rotated\nrelative to each other by 90 degrees.\n\n\n\\section{Acknowledgment}\n\nI am indebted to D.Heitmann, H.Eschrig, and E.Zaremba and their groups\nfor very helpful discussion and the\nDeutsche Forschungs-- Gemeinschaft\nfor financial funding.\n\n\\newpage\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Hawrylak-Buch} L.Jacak, P.Hawrylak, and A.Wojs, {\\em Quantum Dots},\nSpringer 1998\n\n\\bibitem{Broido} K.Kempa, D.A.Broido, and P.Bakshi; Phys. Rev. \nB{\\bf 43},9343 (1991);\\\\\nA.O.Govorov and A.V.Chaplik; Sov.Phys.{\\bf 72}, 1037(1991)\\\\\nA.O.Govorov and A.V.Chaplik; J.Phys.: C {\\bf 6}, 6507(1994)\\\\\nA.V.Chaplik and A.O.Govorov ; J.Phys.: C {\\bf 8}, 4071(1996)\n\n\n\\bibitem{Halperin} J.Dempsey, N.F.Johnson, L.Brey, and B.I.Halperin;\nPhys.Rev. B {\\bf 42},11708 (1990)\n\n\\bibitem{Chakraborty} T.Chakraborty, V.Halonen and P.Pietil\\\"ainen;\nPhys.Rev. B {\\bf 43}, 14289 (1991)\n\n\\bibitem{other} P.A.Maksym; Physica B {\\bf 249}, 233 (1998); O.Dippel, P.Schmelcher,\nand L.S.Cederbaum; Phys.Rev.A {\\bf 49}, 4415 (1994); B.Schuh; J.Phys. A {\\bf 18}, 803 (1985)\n\n\\bibitem{Tsallis} C.Tsallis; J.Math.Phys. {\\bf 19}, 277 (1978)\\\\ \nY.Tikoshinsky; J.Math.Phys. {\\bf 20},406 (1979)\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{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{Heitmann} C.Sch\\\"uller, K.Keller, G.Biese, E.Ullrichs, L.Rolf,\nC.Steinebach, and D.Heitmann, Phys.Rev.Lett. {\\bf80},2673 (1998)\n\n\\bibitem{Kotthaus} C.Dahl, J.P.Kotthaus, H.Nickel and W.Schlapp;\\\\\nPhys.Rev. {\\bf 46}, 15590 (1992)\n\n\\end{thebibliography}\n\n\\end{document}\n"
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"name": "cond-mat0002155.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Hawrylak-Buch} L.Jacak, P.Hawrylak, and A.Wojs, {\\em Quantum Dots},\nSpringer 1998\n\n\\bibitem{Broido} K.Kempa, D.A.Broido, and P.Bakshi; Phys. Rev. \nB{\\bf 43},9343 (1991);\\\\\nA.O.Govorov and A.V.Chaplik; Sov.Phys.{\\bf 72}, 1037(1991)\\\\\nA.O.Govorov and A.V.Chaplik; J.Phys.: C {\\bf 6}, 6507(1994)\\\\\nA.V.Chaplik and A.O.Govorov ; J.Phys.: C {\\bf 8}, 4071(1996)\n\n\n\\bibitem{Halperin} J.Dempsey, N.F.Johnson, L.Brey, and B.I.Halperin;\nPhys.Rev. B {\\bf 42},11708 (1990)\n\n\\bibitem{Chakraborty} T.Chakraborty, V.Halonen and P.Pietil\\\"ainen;\nPhys.Rev. B {\\bf 43}, 14289 (1991)\n\n\\bibitem{other} P.A.Maksym; Physica B {\\bf 249}, 233 (1998); O.Dippel, P.Schmelcher,\nand L.S.Cederbaum; Phys.Rev.A {\\bf 49}, 4415 (1994); B.Schuh; J.Phys. A {\\bf 18}, 803 (1985)\n\n\\bibitem{Tsallis} C.Tsallis; J.Math.Phys. {\\bf 19}, 277 (1978)\\\\ \nY.Tikoshinsky; J.Math.Phys. {\\bf 20},406 (1979)\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{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{Heitmann} C.Sch\\\"uller, K.Keller, G.Biese, E.Ullrichs, L.Rolf,\nC.Steinebach, and D.Heitmann, Phys.Rev.Lett. {\\bf80},2673 (1998)\n\n\\bibitem{Kotthaus} C.Dahl, J.P.Kotthaus, H.Nickel and W.Schlapp;\\\\\nPhys.Rev. {\\bf 46}, 15590 (1992)\n\n\\end{thebibliography}"
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cond-mat0002157
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"author": "F. F. Haas"
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We present recent Monte Carlo results on surfaces of bcc-structured binary alloys which undergo an order-disorder phase transformation in the bulk. In particular, we discuss surface order and surface induced disorder at the bulk transition between the ordered (DO${}_3$) phase and the disordered (A2) phase. An intricate interplay between different ordering and segregation phenomena leads to a complex surface behavior, which depends on the orientation of the surface under consideration.
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"name": "paper.tex",
"string": "\\documentstyle{edbk}\n\n\\input epsf\n\n\\textwidth=13.cm\n\\textheight=21.cm\n\\oddsidemargin=1.3cm\n\\topmargin=0.cm\n%\n%\\newcommand{\\dir}{../../graphics/frank}\n\\newcommand{\\dir}{.}\n%\n\\newcommand{\\fig}[3]\n{\n%\\begin{center}\n \\noindent\n \\unitlength=1mm\n \\begin{picture}(#2,#3)\n \\put(0,0){\\leavevmode \\epsfxsize=#2mm \\epsffile{\\dir/#1}}\n \\end{picture}\n \\noindent\n%\\end{center}\n}\n\\normallatexbib\n%\n\\begin{document}\n\\articletitle{Order and Disorder Phenomena at Surfaces of Binary Alloys}\n\\author{F. F. Haas}\n\\affil{Institut f\\\"ur Physik, Johannes Gutenberg Universit\\\"at Mainz, D55099 Mainz}\n\\author {F. Schmid}\n\\affil{Max-Planck Institut f\\\"ur Polymerforschung, Ackermannweg 10, D55021 Mainz}\n\\email {schmid@mpip-mainz.mpg.de} \n\\author{K. Binder}\n\\affil{Institut f\\\"ur Physik, Johannes Gutenberg Universit\\\"at Mainz, D55099 Mainz}\n\\email {binder@chaplin.physik.uni-mainz.de}\n\n\\begin{abstract}\nWe present recent Monte Carlo results on surfaces of bcc-structured binary \nalloys which undergo an order-disorder phase transformation in the bulk. \nIn particular, we discuss surface order and surface induced disorder \nat the bulk transition between the ordered (DO${}_3$) phase and\nthe disordered (A2) phase. An intricate interplay between different ordering \nand segregation phenomena leads to a complex surface behavior, \nwhich depends on the orientation of the surface under consideration.\n\\end{abstract}\n\nThe structure and composition of alloys at external surfaces and internal \ninterfaces often differs significantly from that in the bulk. \nIn most cases, this refers only to very few top layers at the surface, over \na thickness of order 1 nm. In the vicinity of a bulk phase transition, however,\nthe thickness of the altered surface region can grow to reach mesoscopic \ndimensions, of order 10-100 nm. If the bulk transition is second order, for\nexample, the thickness of the surface region is controlled by the bulk\ncorrelation length, which diverges close to the critical point\\cite{surf}.\nClose to first order bulk transitions, mesoscopic wetting layers may \nform\\cite{wetting}.\n\nWhile these various surface phenomena are fairly well understood in simple\nsystems, such as surfaces of liquid mixtures against the wall of a \ncontainer, the situation in alloys is complicated due to the interplay \nbetween the local structure, the order and the composition profiles.\nIn alloys which undergo an order/disorder transition, for example, \nthe surface segregation of one alloy component can induce surface \norder\\cite{ich1,dosch2,diehl1} or partial surface order\\cite{reichert,ich2}\nat surfaces which are less symmetric than the bulk lattice with respect to \nthe ordered phase. Even more subtle effects can lead to surface order at\nfully symmetric surfaces\\cite{mailander,schweika1,schweika2,schweika3}. \nFurthermore, different types of order may be present in such \nalloys\\cite{defontaine}, which can interact in a way to affect the\nwetting behavior significantly\\cite{hauge,gerhard}. \n\nIn this contribution, we discuss a situation where such an interplay of \nsegregation and different types of ordering leads to a rather intriguing\nsurface behavior: Surface induced disorder in a binary (AB) alloy\non a body centered cubic (bcc) lattice close to the first order bulk transition \nbetween the ordered DO${}_3$ phase and the disordered bulk phase.\nSurface induced disorder is a wetting phenomenon, which can be observed when\nthe bulk is ordered and the surface reduces the degree of ordering \n-- usually due to the reduced number of interacting \nneighbors\\cite{lipowsky1,kroll,dosch1}. A disordered layer may then nucleate at \nthe surface, which grows logarithmically as the bulk transition is approached. \nAccording to the theoretical picture, the surface behavior is driven by\nthe depinning of the interface between the disordered surface layer and the \nordered bulk. \n\nIn order to study the validity of this picture, we consider a very idealized \nminimal model of a bcc alloy with a DO${}_3$ phase: The alloy is mapped on \nan Ising model on the bcc lattice with negative nearest and next nearest\nneighbor interactions. The Hamiltonian of the system then reads\n\\begin{equation}\n{\\cal H} = V \\sum_{\\langle ij \\rangle} S_i S_j\n+ \\alpha V \\sum_{\\langle \\langle ij \\rangle \\rangle } S_i S_j\n- H \\sum_i S_i,\n\\end{equation}\nwhere Ising variables $S=1$ represent A atoms, $S=-1$ B atoms, the\nsum $\\langle ij \\rangle$ runs over nearest neighbor pairs, \n$\\langle \\langle ij \\rangle \\rangle$ over next nearest neighbor pairs, and the \nfield $H$ is the appropriate combination of chemical potentials \n$\\mu_A$ and $\\mu_B$ driving the total concentration $c$ of A in the alloy,\n($c= (\\langle S \\rangle+1)/2$). \n\nThe phases exhibited by this model are shown in Figure 1\nIn the disordered (A2) phase, the A and B particles are distributed \nevenly among all lattice sites. In the ordered B2 and DO${}_3$ phases, they \narrange themselves as to form a superlattice on the bcc lattice. The\nparameter $\\alpha$ was chosen $\\alpha=0.457$, such that the highest \ntemperature at which a DO${}_3$ phase can still exist is roughly half the \nhighest temperature of the B2 phase, like in the experimental case\nof FeAl. The resulting phase diagram is shown in Figure 2.\n\nIn order to characterize the ordered phases, it is useful to divide the bcc \nlattice into four face centered cubic (fcc) sublattices as indicated on \nFigure 1, and to define the order parameters\n\\begin{eqnarray}\n\\psi_1 &=& \\big( \n \\langle S \\rangle_a + \\langle S \\rangle_b \n - \\langle S \\rangle_c - \\langle S \\rangle_d )/2 \\nonumber \\\\\n\\psi_2 &=& \\big( \\langle S \\rangle_a - \\langle S \\rangle_b \n + \\langle S \\rangle_c - \\langle S \\rangle_d )/2 \\\\\n\\psi_3 &=& \\big( \\langle S \\rangle_a - \\langle S \\rangle_b \n - \\langle S \\rangle_c + \\langle S \\rangle_d )/2, \\nonumber\n\\end{eqnarray}\nwhere $\\langle S \\rangle_{\\alpha}$ is the average spin on the sublattice\n$\\alpha$. In the disordered phase, all sublattice compositions are equal\nand all order parameters vanish as a consequence. The B2 phase is\ncharacterized by $\\psi_1 \\ne 0$ and the DO${}_3$ phase by\n$\\psi_1 \\ne 0$ and $\\psi_2 = \\pm \\psi_3 \\ne 0$. The two dimensional\nvector $(\\psi_2,\\psi_3)$ is thus an\n\n\\begin{minipage}[b]{6.cm}\n\\fig{phases.eps}{55}{30}\n\\vspace{1.cm}\n\\baselineskip=10pt\nFigure 1.\\\\\n{\\small \nOrdered phases on the bcc lattice: (a) disordered A2 structure, \n(b) ordered B2 and (c) DO${}_3$ structure.\nAlso shown is assignment of sublattices $a,b,c$ and $d$.\n}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[b]{6.cm}\n\\fig{phdiag.eps}{55}{60}\n\\baselineskip=10pt\nFigure 2.\\\\\n{\\small \nPhase diagram in the $T-H$ plane. First order transitions are solid lines,\nsecond order transitions dashed lines. Arrows indicate positions of a\ncritical end point (cep) and a tricritical point (tcp).\n}\n\\end{minipage}\\\\\n\\bigskip\n\\noindent\n\n\\noindent\norder parameter for DO${}_3$\nordering, and the latter can be characterized conveniently in terms of\nits absolute value\n\\begin{equation}\n\\psi_{23} = \\sqrt{(\\psi_2{}^2 + \\psi_3{}^2)/2}.\n\\end{equation}\n\nWe have studied free (110) and (100) surfaces of this model at the temperature \n$T = 1\\: k_B T/V$ in the DO${}_3$ phase close to the transition to the\ndisordered phase. To this end, we have first located the transition point\nvery accurately by thermodynamic integration\\cite{KB2}, $H_0/V=10.00771[1]$.\nWe have then performed extensive Monte Carlo simulations of slabs each\n100-200 layers thick, with free boundary conditions at the two \nconfining (110) or (100) planes, and periodic boundary conditions in the \nremaining directions. \n\nIn all of our simulations, the average value of the Ising variable in the\ntop layer was one, {\\em i.e.}, the top layer was completely filled with A \natoms. Having stated this, we shall disregard this layer in the following and\ndiscuss the structure starting from the next layer underneath the surface. \nThe layer order parameters $\\psi_i(n)$ and the layer compositions $c(n)$ \ncan be determined in a straightforward manner for (110) layers, since they\ncontain sites from all sublattices. \nIn the case of the (100) layers, it is useful to define $c$ and the $\\psi_i$\nbased on the sublattice occupancies on two subsequent layers.\n\n\\bigskip\n\nFig. 3 shows the calculated profiles for two choices of $H$ close to the \ntransition. One clearly observes the formation of a disordered film at the \nsurface, which increases in thickness as the transition point is approached. \nThe film is characterized by low order parameters $\\psi_1$ and $\\psi_{23}$,\nand by a slightly increased concentration $c$ of A sites. The structure very\nclose to the surface depends on its orientation: The composition profiles\ndisplay some characteristic oscillations at a (110) surface, and grow\nmonotonously at a (100) surface. The order $\\psi_{23}$ drops to zero. \nThe order $\\psi_1$ drops to zero at the (110) surface, and at\nthe (100) surface, it changes sign and increases again in the outmost two\nlayers. The latter is precisely an example of\nthe segregation induced ordering mentioned earlier\\cite{ich1,dosch2,diehl1}.\n\n\\bigskip\n\\bigskip\n\n\\noindent\n\\begin{minipage}[t]{13cm}\n(110) surface \\hfill (100) surface \\\\\n\\fig{prof_110.eps}{55}{85} \\hfill \n\\fig{prof_100.eps}{55}{85} \\\\\n\\baselineskip=10pt\nFigure 3.\\\\\n{\\small \nProfiles of the total concentration (top) and order parameters $\\psi_1$\n(bottom, circles) and $\\psi_{23}$ (bottom, squares) at the\n(110) surface (left) and at the (100) surface (right) for different\nfields $H$ in units of $V$ as indicated.\n}\n\\end{minipage}\\\\\n\\bigskip\n\nWe can thus distinguish between two interesting regions in these profiles:\nThe near-surface region, where the properties of the profiles still\nreflect the peculiarities of the surface,\nand the interfacial region, where the profiles are determined from the\nproperties of the interface separating the disordered surface film \nfrom the ordered bulk.\n\nThe structure of the profiles in the interfacial region is basically\ndetermined by the fluctuations of the interface, which are characterized\nby a transverse correlation length $\\xi_{\\parallel}$. The latter is in turn \ndriven by the thickness of the film and the interfacial tension $\\sigma$ \nor, more precisely, by a rescaled dimensionless interfacial tension\n\\begin{equation}\n1/\\omega = 4 \\pi \\xi_b{}^2 \\sigma/k_B T,\n\\end{equation}\nwith the bulk correlation length $\\xi_b$. The renormalization group theory\nof critical wetting\\cite{wetting}, which should apply here\\cite{kroll}, \npredicts that the transverse correlation length diverges according \nto a power law\n\\begin{equation}\n\\xi_{\\parallel} \\propto \\frac{1}{\\sqrt{\\omega}} (H_0 - H)^{-\\nu_{\\parallel}}\n\\end{equation}\nas $H_0$ is approached, with the exponent $\\nu_{\\parallel} = 1/2$.\nWith this knowledge, one can calculate the effective width\nof the order parameter profiles, \n$\\xi_{\\perp} \\propto \\sqrt{- \\omega \\ln(H_0-H)}$, the profiles of\nlayer susceptibilities etc. We have examined these carefully at\nthe (110) interface and the (100) interface, both for the order\nparameters $\\psi_1$ and $\\psi_{23}$, and we could fit everything nicely\ninto the theoretical picture. \nOur further discussion here shall focus on the near-surface region.\n\nAssuming that the order in the near-surface region is still determined by the \nfluctuations of the interface, the theory of critical wetting \npredicts a power law behavior\n\\begin{equation}\n\\label{b1}\n\\psi_{\\alpha,1} \\propto (H_0 - H)^{\\beta_1} \n\\end{equation}\nfor value $\\psi){\\alpha,1}$ of the order parameter $\\psi_{\\alpha}$ directly \nat the surface, regardless of the structure of the surface. Figure 4 shows \nthat, indeed, the surface order parameter $\\psi_{23,1}$ decays according to \na power law at both the (110) and the (100) surface, with the same exponent\nand the exponent is in both cases identical within the error,\n$\\beta_1 = 0.618$. Furthermore, we notice strong finite size effects \nclose to the $H_0$. Since these are asymptotically driven by \nthe ratio $(L/\\xi_{\\parallel})$, they can be exploited to determine\nthe behavior of $\\xi_{\\parallel}$ as the phase transition is approached, \n{\\em i.e.}, the exponent $\\nu_{\\parallel}$. The finite size scaling analysis \nyields $\\nu_{\\parallel}=1/2$\\cite{frank}, \nin agreement with the theory.\n\n\\bigskip\n\\bigskip\n\n\\noindent\n\\begin{minipage}[t]{13.cm}\n(110) surface \\hfill (100) surface \\\\\n\\fig{psi230.eps}{55}{50} \\hfill \\fig{psi230_100.eps}{55}{50}\\\\\n\\baselineskip=10pt\nFigure 4.\\\\\n{\\small \nOrder parameter $\\psi_{23}$ at the surface at the (110) surface (left)\nand the (100) surface (right) vs. $(H_0-H)/V$ for different\nsystem sizes $L\\times L \\times D$ as indicated. Solid line shows power\nlaw with exponent $\\beta_1 = 0.618$.\n}\n\\end{minipage}\\\\\n\\bigskip\n\nThe profiles of $\\psi_{23}$ thus seem entirely determined by the depinning\nof the interface, in agreement with the standard theory of critical wetting.\nThe situation is however different when one looks at the other order parameter,\n$\\psi_1$. This is not particularly surprising in the case of the (100) surface.\nWe have already noted that this surface breaks the symmetry with respect\nto the $\\psi_1$ ordering, hence the segregation of A particles to the\ntop layer induces additional $\\psi_1$ order at the surface (Figure 5). \nThe (110) surface, on the other hand, is not symmetry breaking. \nThe order $\\psi_1$ decays at the surface, yet with an exponent\n$\\beta_1 = 0.801$ which differs from that observed for $\\psi_{23}$ (Figure 6).\nEven more unexpected, the finite size effects cannot be analyzed consistently \nwith the assumption that the transverse correlation length diverges with the \nexponent $\\nu_{\\parallel}=1/2$, but rather suggest \n$\\nu_{\\parallel} = 0.7 \\pm 0.05$. The order parameter fluctuations of \n$\\psi_1$ at the surface seem to be driven \nby a length scale which diverges at $H_0$ with an exponent different from\nthat given by the capillary wave fluctuations of the depinning interface.\n\n\\bigskip\n\\bigskip\n\n\\noindent\n\\begin{minipage}[t]{6.cm}\n\\fig{psi10_100.eps}{55}{50}\n\\baselineskip=10pt\nFigure 5.\\\\\n{\\small \nOrder parameter $\\psi_{1}$ in the first layers underneath\nthe (100) surface vs. $(H_0-H)/V$ .\n}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{6.cm}\n\\fig{psi10.eps}{55}{50}\n\\baselineskip=10pt\nFigure 6.\\\\\n{\\small \nOrder parameter $\\psi_{1}$ at the (110) surface \nvs. $(H_0-H)/V$ for different\nsystem sizes $L\\times L \\times D$ as indicated. \nSolid line shows power\nlaw with exponent $\\beta_1 = 0.801$.\n}\n\\end{minipage}\n\n\\bigskip\n\nTo summarize, we have seen that the phenomenology of surface induced disorder \nin a relatively simple bcc alloy with just two coupled types of ordering \nis much more complex than predicted by the standard theory of surface induced\ndisorder and critical wetting.\nLooking at our profiles, we were able to distinguish between two regions, \nthe near-surface region and the interfacial regions. \nIn the situations studied in our simulations, it seemed that these regions \ncould be well separated from each other. \nSome rapid variations of the profiles in the near-surface region,\nare followed by smooth changes in the interfacial region. \nThe local surface structure affects the total composition profile relatively\nstrongly, and has practically no influence on the profile of the DO${}_3$\norder, $\\psi_{23}$. In the case of the B2 order, $\\psi_1$, the situation\nis more intriguing. The symmetry breaking (100) surface induces local order \nin the near-surface region which apparently does not couple to the interface.\nAt the non-symmetry breaking (110) surface, $\\psi_1$ was found to\nexhibit qualitatively new and unexpected power law behavior as the\nwetting transition is approached. \n\nThe last observation clearly requires further exploration in the future.\nThe picture will be even more complex in situations\nwhere the near-surface profiles and the interfacial profiles cannot be\nseparated any more. We expect that this could be the case, {\\em e.g.},\nat (111) surfaces, which break the symmetry with respect to both\nB2 and DO${}_3$ ordering.\n\n\\bigskip\n\n\\noindent\n\\small\n\\baselineskip=11pt\nF.F. Haas was supported by the Graduiertenf\\\"orderung of the Land \nRheinland-Pfalz. \n\n\\begin{chapthebibliography}{99}\n\\parskip=0pt\n\\baselineskip=11pt\n\\bibitem{surf} For reviews on surface critical phenomena see \nK. Binder in \n{\\it Phase Transitions and Critical Phenomena}, Vol. 8, p. 1 (1983),\nC. Domb and J.L. Lebowitz eds., Academic Press, London;\nS. Diehl, ibid, Vol. 10, p.1. (1986).\n\\bibitem{wetting} For reviews on wetting see, {\\em e.g.},\n P. G. de Gennes, Rev. Mod. Phys. {\\bf 57}, 827 (1985);\n S. Dietrich in {\\it Phase Transitions and Critical Phenomena},\n C. Domb and J.L. Lebowitz eds (Academic Press, New York, 1988), Vol. 12;\n M. Schick in {\\it Les Houches, Session XLVIII -- Liquids at Interfaces},\n J. Charvolin, J. F. Joanny, and J. Zinn-Justin eds\n (Elsevier Science Publishers B.V., 1990).\n\\bibitem{ich1} F. Schmid, Zeitschr. f. Phys. {\\bf B 91}, 77 (1993).\n\\bibitem{dosch2}\n S. Krimmel, W. Donner, B. Nickel, and H. Dosch,\n Phys. Rev. Lett. {\\bf 78}, 3880 (1997).\n\\bibitem{diehl1}\n A. Drewitz, R. Leidl, T. W. Burkhardt, and H. W. Diehl,\n Phys. Rev. Lett. {\\bf 78}, 1090 (1997);\n R. Leidl and H. W. Diehl,\n Phys. Rev. B {\\bf 57}, 1908 (1998);\n R. Leidl, A. Drewitz, and H. W. Diehl,\n Int. Journal of Thermophysics {\\bf 19}, 1219 (1998).\n\\bibitem{reichert} \nH. Reichert, P.J. Eng, H. Dosch, I.K. Robinson, Phys. Rev. Lett. {\\bf 74},\n2006 (1995).\n\\bibitem{ich2}\n F. Schmid, in {\\em Stability of Materials}, p 173, A. Gonis {\\em et al} eds.,\n (Plenum Press, New York, 1996).\n\\bibitem{mailander} L. Mail\\\"ander, H. Dosch, J. Peisl, R.L. Johnson,\n Phys. Rev. Lett. {\\bf 64}, 2527 (1990).\n\\bibitem{schweika1}\n W. Schweika, K. Binder, and D. P. Landau, \n Phys. Rev. Lett. {\\bf 65}, 3321 (1990).\n\\bibitem{schweika2}\n W. Schweika. D.P. Landau, and K. Binder, Phys. Rev. B {\\bf 53}, 8937 (1996).\n\\bibitem{schweika3}\n W. Schweika, D.P. Landau, in {\\em Computer Simulation Studies in \n Condensed-Matter Physics X}, P. 186 (1997).\n\\bibitem{defontaine} for reviews see\n D. De Fontaine in {\\em Solid State Physics \\bf 34}, p 73\n H. Ehrenreich, F. Seitz and D. Turnbell eds., Academic Press, \n New York 1979;\n K. Binder in {\\em Festk\\\"orperprobleme (Advances in\n Solid State Physics) \\bf 26}, p 133, P. Grosse ed., Vieweg, Braunschweig 1986.\n\\bibitem{lipowsky1}\n R. Lipowsky, J. Appl. Phys. {\\bf 55}, 2485 (1984).\n\\bibitem{hauge}\n E. H. Hauge, Phys. Rev. B {\\bf 33}, 3323 (1985).\n\\bibitem{gerhard}\n D. M. Kroll, G. Gompper, Phys. Rev. B {\\bf 36}, 7078 (1987);\n G. Gompper, D. M. Kroll, Phys. Rev. B {\\bf 38}, 459 (1988).\n\\bibitem{kroll}\n D. M. Kroll and R. Lipowsky, Phys. Rev. B {\\bf 28}, 6435 (1983).\n\\bibitem{dosch1}\n H. Dosch, {\\em Critical Phenomena at Surfaces and Interfaces\n (Evanescent X-ray and Neutron Scattering)},\n Springer Tracts in Modern Physics Vol 126 (Springer, Berlin, 1992).\n\\bibitem{KB2}\n K. Binder, Z. Phys. B {\\bf 45}, 61 (1981).\n\\bibitem{frank}\n F. F. Haas, Dissertation Universit\\\"at Mainz (1998);\n F. F. Haas, F. Schmid, and K. Binder, in preparation.\n\\end{chapthebibliography}\n\n\\end{document}\n\n\\end{document}\n"
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[
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"name": "cond-mat0002157.extracted_bib",
"string": "\\bibitem{surf} For reviews on surface critical phenomena see \nK. Binder in \n{\\it Phase Transitions and Critical Phenomena}, Vol. 8, p. 1 (1983),\nC. Domb and J.L. Lebowitz eds., Academic Press, London;\nS. Diehl, ibid, Vol. 10, p.1. (1986).\n\n\\bibitem{wetting} For reviews on wetting see, {\\em e.g.},\n P. G. de Gennes, Rev. Mod. Phys. {\\bf 57}, 827 (1985);\n S. Dietrich in {\\it Phase Transitions and Critical Phenomena},\n C. Domb and J.L. Lebowitz eds (Academic Press, New York, 1988), Vol. 12;\n M. Schick in {\\it Les Houches, Session XLVIII -- Liquids at Interfaces},\n J. Charvolin, J. F. Joanny, and J. Zinn-Justin eds\n (Elsevier Science Publishers B.V., 1990).\n\n\\bibitem{ich1} F. Schmid, Zeitschr. f. Phys. {\\bf B 91}, 77 (1993).\n\n\\bibitem{dosch2}\n S. Krimmel, W. Donner, B. Nickel, and H. Dosch,\n Phys. Rev. Lett. {\\bf 78}, 3880 (1997).\n\n\\bibitem{diehl1}\n A. Drewitz, R. Leidl, T. W. Burkhardt, and H. W. Diehl,\n Phys. Rev. Lett. {\\bf 78}, 1090 (1997);\n R. Leidl and H. W. Diehl,\n Phys. Rev. B {\\bf 57}, 1908 (1998);\n R. Leidl, A. Drewitz, and H. W. Diehl,\n Int. Journal of Thermophysics {\\bf 19}, 1219 (1998).\n\n\\bibitem{reichert} \nH. Reichert, P.J. Eng, H. Dosch, I.K. Robinson, Phys. Rev. Lett. {\\bf 74},\n2006 (1995).\n\n\\bibitem{ich2}\n F. Schmid, in {\\em Stability of Materials}, p 173, A. Gonis {\\em et al} eds.,\n (Plenum Press, New York, 1996).\n\n\\bibitem{mailander} L. Mail\\\"ander, H. Dosch, J. Peisl, R.L. Johnson,\n Phys. Rev. Lett. {\\bf 64}, 2527 (1990).\n\n\\bibitem{schweika1}\n W. Schweika, K. Binder, and D. P. Landau, \n Phys. Rev. Lett. {\\bf 65}, 3321 (1990).\n\n\\bibitem{schweika2}\n W. Schweika. D.P. Landau, and K. Binder, Phys. Rev. B {\\bf 53}, 8937 (1996).\n\n\\bibitem{schweika3}\n W. Schweika, D.P. Landau, in {\\em Computer Simulation Studies in \n Condensed-Matter Physics X}, P. 186 (1997).\n\n\\bibitem{defontaine} for reviews see\n D. De Fontaine in {\\em Solid State Physics \\bf 34}, p 73\n H. Ehrenreich, F. Seitz and D. Turnbell eds., Academic Press, \n New York 1979;\n K. Binder in {\\em Festk\\\"orperprobleme (Advances in\n Solid State Physics) \\bf 26}, p 133, P. Grosse ed., Vieweg, Braunschweig 1986.\n\n\\bibitem{lipowsky1}\n R. Lipowsky, J. Appl. Phys. {\\bf 55}, 2485 (1984).\n\n\\bibitem{hauge}\n E. H. Hauge, Phys. Rev. B {\\bf 33}, 3323 (1985).\n\n\\bibitem{gerhard}\n D. M. Kroll, G. Gompper, Phys. Rev. B {\\bf 36}, 7078 (1987);\n G. Gompper, D. M. Kroll, Phys. Rev. B {\\bf 38}, 459 (1988).\n\n\\bibitem{kroll}\n D. M. Kroll and R. Lipowsky, Phys. Rev. B {\\bf 28}, 6435 (1983).\n\n\\bibitem{dosch1}\n H. Dosch, {\\em Critical Phenomena at Surfaces and Interfaces\n (Evanescent X-ray and Neutron Scattering)},\n Springer Tracts in Modern Physics Vol 126 (Springer, Berlin, 1992).\n\n\\bibitem{KB2}\n K. Binder, Z. Phys. B {\\bf 45}, 61 (1981).\n\n\\bibitem{frank}\n F. F. Haas, Dissertation Universit\\\"at Mainz (1998);\n F. F. Haas, F. Schmid, and K. Binder, in preparation.\n"
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cond-mat0002158
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Crossover from classical to random-field critical exponents in As-doped TbVO$_{4}$
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[
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"author": "C.-H. Choo"
},
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"author": "H. P. Schriemer\\cite{currentaddress}"
},
{
"author": "and D.R. Taylor"
}
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Using birefringence techniques we have measured the critical exponents $\beta$, $\gamma$, and $\delta$ in As-doped TbVO$_{4}$, a structural realization of the random-field Ising model where random strain fields are introduced by V-As size mismatch. For pure TbVO$_{4}$ we observe the expected classical critical exponents, while for a mixed sample with 15\% As concentration our results are $\beta=0.31 \pm 0.03$, $\gamma=1.22 \pm 0.07$ and $\delta=4.2 \pm 0.7$. These values are consistent with the critical exponents for the short range pure Ising model in three dimensions in agreement with a prediction by Toh. The susceptibility data showed a crossover with temperature from classical to random field critical behaviour.
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"name": "crossoverms.tex",
"string": "\\documentstyle[aps,prb,multicol,graphics]{revtex}\n\n\\begin{document}\n\n\\title{Crossover from classical to random-field critical exponents in\nAs-doped TbVO$_{4}$} \n\\author{C.-H. Choo, H. P. Schriemer\\cite{currentaddress}, and D.R.\nTaylor} \n\\address{Department of Physics, Queen's University, Kingston,\nON, Canada K7L 3N6} \n\\date{Received \\today} \n\\maketitle \n\\begin{abstract}\nUsing birefringence techniques we have measured the critical exponents \n$\\beta$, $\\gamma$, and $\\delta$\nin As-doped TbVO$_{4}$, a structural realization of the random-field\nIsing model where random strain fields are introduced by V-As size\nmismatch. For pure TbVO$_{4}$ we observe\nthe expected classical critical exponents, while for a mixed sample with\n15\\% As concentration our results are $\\beta=0.31 \\pm 0.03$,\n$\\gamma=1.22 \\pm 0.07$ and $\\delta=4.2 \\pm 0.7$. These values are\nconsistent with the critical exponents for the short range pure Ising\nmodel in three dimensions in agreement with a prediction by Toh. The \nsusceptibility data showed a crossover with\ntemperature from classical to random field critical behaviour. \n\\end{abstract}\n%\\pacs{64.60.Fr, 75.40.Cx, 78.30.Ly}\n\n\\begin{multicols}{2}\n%\\section*{}\n\nDetermination of the critical properties of the random field Ising\nmodel\\cite{{Belanger},{Belanger2},{Nattermann}} (RFIM) has shown encouraging progress recently,\nboth experimentally and theoretically, after many years of\nuncertainties. Measurements on dilute antiferromagnets in a field, the\nmost frequently studied realization of the RFIM, have been difficult to\nobtain because dilution of the magnetic species inhibits\nequilibration close to the critical temperature. However recent\nexperiments by Slanic {\\it et al.}\\cite{Slanic} on samples with only 7\\%\ndilution have shown equilibrium behaviour through the transition\ntemperature, thus allowing confident determination of the critical\nexponents. In the last few years, theoretical investigations making use\nof a variety of analytical and computational techniques have led to\npredictions of the critical exponents of the RFIM that show reasonable\nconsistency with each other. A recent analysis by Fortin and\nHoldsworth\\cite{Fortin} supports earlier suggestions that the critical\nexponents for the RFIM in three dimensions ($d=3$) are those of the Ising\nmodel for reduced dimension $d'=1.5$. Nevertheless dimensional reduction\nhas not been rigorously proved, and even if the effective dimension\n$d'=1.5$ is correct the uncertainty in some of the calculated critical\nexponents is quite large. Likewise the experimental situation is still\nnot satisfactory, since the critical exponents measured by Slanic {\\it et\nal.}\\cite{Slanic} have substantial uncertainties and are only partially\nconsistent with theory. In addition, measurements of specific heat\ncritical exponents\\cite{{Belanger},{Slanic2}} in dilute antiferromagnets\nalso appear to disagree with theory.\n\nFor another realization of the RFIM, where random strain fields are\ngenerated by substitutional impurities in crystals undergoing structural\nIsing (Jahn-Teller) transitions, the results to date have also not been\nconclusive. Random fields due to As/V substitutions in DyVO$_{4}$, which\nhas $d=3$ Ising exponents, appeared to increase the susceptibility\nexponent $\\gamma$ as expected but had no effect on the order parameter\nexponent $\\beta$.\\cite{Reza} The interpretation of the effects\nof random fields in the As-doped DyVO$_{4}$ system is complicated by the\nfact that the true critical behaviour of pure DyVO$_{4}$ should be\nclassical due to the long range strain coupling. However because of the relative\nweakness of the long range to short range interactions, classical exponents\nare not observable at accessible temperatures $|t| \\geq 10^{-2}$, where $t=(T -\nT_{\\mathrm{D}})/T_{\\mathrm{D}}$ is the reduced temperature,\\cite{{GAGehring3},{Marques}}\nleading to \nuncertainty on what the effects of the random fields would be. We have\ntherefore extended these experiments to the related\nTbVO$_{4}$/TbAsO$_{4}$ system where the critical behaviour of the pure\ncompounds is unequivocally\nclassical\\cite{{Harley},{Elliot},{Wells},{Sandercock},{Berkhan}}, and\nsearched for changes in critical behaviour in mixed crystals due to\nrandom fields. Since this system starts from a different universality\nclass, the results cannot be compared directly with results from dilute\nantiferromagnets, but it is an important system that can\nindependently test theoretical models and predictions of random field\neffects. In contrast with the large number of theoretical investigations\nof the short-range RFIM, the literature on the random-strain\nversion of the RFIM is very limited, consisting primarily of a paper by\nToh.\\cite{Toh} In this paper, Toh compares the random-strain RFIM with\nlong range forces to that of the short-range RFIM under a\nrenormalization group analysis. The main result is that the critical\nexponents should change from classical values to values that are close\nto those of the $d=3$ pure short-range Ising model. \n\nThe lowest $4f$ electronic levels in TbVO$_{4}$ consists of 2 singlet\nstates $\\sim 18$ cm$^{-1}$ apart and a non-Kramers' doublet\napproximately halfway between. Coupling between the doublets and lattice\ndistortions leads to a Tb ion-ion interaction of the Ising\nform\\cite{Elliot} and a tetragonal-orthorhombic phase transition at\ntemperature T$_{\\mathrm{D}}$. Since the Tb ions are coupled\npredominantly to $k\\sim 0$ acoustic phonons and to bulk strains, the\nion-ion interaction is very long range. The order parameter is the\nmacroscopic strain $a-b$ where $a$ and $b$ are basal plane unit cell\nparameters in the orthorhombic phase. The orthorhombic distortion gives\nrise to birefringence, $\\Delta n$, which is proportional to $a-b$ (at\nleast to a good approximation\\cite{Harley}).\n\nThe full Hamiltonian for the coupled electron-phonon TbVO$_{4}$ system\nin an external magnetic field can be written as\n\\begin{equation}\nH=- \\case1/2 \\sum_{ij} J_{\\it ij}\\sigma^{\\it z}_{\\it i} \\sigma^{\\it z}_{\\it j} -\n\\case1/2 \\epsilon \\sum_{\\it i} (1 + \\tau^{\\it z}_{\\it i})\\sigma^{\\it x}_{\\it i} -\n\\text{B}\\sum_{\\it i} {\\it m^{x}_{i}} \n\\label{meanfield} \\end{equation} \nwhere $m^{x}_{i} = \\case1/4{\\text{g}}\\mu_{\\text{B}}(1 + \\sigma^z_{i})\\tau^x_{i}$ \nand $\\sigma^{z}, \\sigma^{x}, \\tau^{z}\n\\text{and } \\tau^{x}$ are Pauli type\noperators\\cite{{Elliot},{Gehring2}}. $J_{ij}$ describes the ion-ion\ninteractions, $2\\epsilon$ is the high temperature splitting between the\nouter singlets and B is the magnetic field applied along the $x$ (or\n$a$) axis (i.e. along the 110 direction). The field B is able to induce an\northorhombic distortion because of the strongly anisotropic Tb magnetic\nmoment in the orthorhombic phase. In the mean field approximation, a\nLandau expansion of the free energy, with $\\epsilon=0$, shows that\nB$^{2}$/T is effectively an ordering field,\\cite{Page} and this \nis supported by the experimental data\nthat follows. For B and $\\epsilon$ small, TbVO$_{4}$ is well described\nby an Ising model Hamiltonian. As the mode softening at the transition\nin this type of system is anisotropic\\cite{Cowley}, classical critical\nexponents are expected, and observed, rather than $d=3$ Ising exponents. \n\nIn the mixed compound, Tb(As$_{x}$V$_{1-x}$)O$_{4}$, a fraction $x$ of\nthe V atoms are replaced by As atoms, generating random, static\nstrain fields, one component of which has the right symmetry to couple\nto the order parameter.\\cite{GAGehring2} For $\\epsilon=\\text{B}=0$ the\nHamiltonian has the form of the RFIM,\n\\begin{equation} H =-\\case1/2\n\\sum_{ij} J_{\\it ij} \\sigma^{\\it z}_{\\it i} \\sigma^{\\it z}_{\\it j} -\n\\sum_{\\it i}h_{\\it i}\\sigma^z_{\\it i} \\label{randomfield} \n\\end{equation} \nwhere $h_{i}$ is\nthe random local strain field which is expected to have a Gaussian distribution about\n$h=0$. In his analysis of this type of random field system with\nanistropic mode softening, Toh predicts changes to the upper critical\ndimension, thus modifying the values of critical exponents at $d=3$.\n\nCrystals of Tb(As$_{x}$V$_{1-x}$)O$_{4}$ with impurity concentrations of\n$x= 0$ and 0.15 were prepared using the flux growth method at the\nUniversity of Oxford. The crystals were cut and\npolished perpendicular to the $c$ axis, with thickness of about 1 mm. They\nwere mounted in a strain-free manner with the $c$ axis horizontal in a\nhelium optical cryostat. The crystal could be rotated about a vertical\naxis, allowing alignment of the $c$ axis parallel to the laser beam and\none of the $a$ axes parallel to a horizontal magnetic field. A circular \naperture in the sample holder of about 1 mm in diameter limited the\nsampled area to a small region of the crystal, and thus \nreduced effects due to any inhomogeneous composition, temperature and\nordering field\nthat may be present in the crystal.\n\nThe light source for birefringence experiments was a HeNe laser \noperating at 543.5\nnm, a wavelength that should give reasonable birefringence in this\ncrystal.\\cite{Hikel} Photoelastic modulation and lock-in detection \nallowed sensitive measurement\\cite{Ferre} of the phase shift $\\phi$ due \nto the orthorhombic distortion. Adjustment of a Babinet compensator \nensured that the detector output was proportional \nto $\\phi$. In a typical experiment, the data acquisition system brought the sample \nto each desired temperature, waited up to 15 minutes for equilibration, and then \nramped the magnetic field up and down while recording field, light intensity, \nand temperature. \n\n\nBelow T$_{\\mathrm{D}}$, \nthe orthorhombic distortions are equally likely\nto be in the 110 or 1$\\overline{1}$0 direction, leading to the formation\nof twinned orthorhombic domains separated by domain walls. To avoid\nthe cancellation of birefringence due to \nmultiple domains, the crystal is forced into a\nsingle domain by applying a magnetic field that favours the distortion\naxis parallel to the magnetic field.\\cite{KAGehring,GAGehring1} Thus to\ndetermine the birefringence, and hence $\\beta$, below T$_{\\mathrm{D}}$,\nwe recorded data over a range of magnetic fields and extrapolated the\nhigh magnetic field data back to zero field. With the rather small\nmagnetic fields available ($< 0.35$ T) it was sometimes difficult to\nachieve a single domain at temperatures close to T$_{\\mathrm{D}}$ , at\n$|$T - T$_{\\mathrm{D}}| < 0.2 $ K for $x = 0.15$, but this became\nprogressively easier further away from T$_{\\mathrm{D}}$. For the $x=0$\n sample where pinning is presumably weaker, single domain structure at\n$|$T - T$_{\\mathrm{D}}| < 0.75 $ K was relatively easy to achieve. For\nthis reason, the critical isotherm exponent, $\\delta$, obtained from the\ndependence of the induced birefringence on ordering field at the\ntransition temperature, is more reliable for pure TbVO$_{4}$ than that\nfor the mixed sample. Above T$_{\\mathrm{D}}$, the change in induced\nbirefringence resulting from a change in the ordering field gives the\nsusceptibility and hence $\\gamma$. Although the exponents for pure\nTbVO$_{4}$ are known to be classical,\\cite{Harley}, their measurement \nprovides a comparison with the\nmixed sample where modified critical exponents are expected.\nIn experiments, T$_{\\mathrm{D}}$ appeared to vary slightly from run to run \nbecause of effects such as mounting strains and temperature gradients. \nTo minimize the effects of a variable T$_{\\mathrm{D}}$ on the results, \nefforts were made to measure\nall exponents of one particular sample on the same run. In the power-law \nfits to the data, T$_{\\mathrm{D}}$ was chosen to optimize both the $\\beta$ and\n$\\gamma$ fits simultaneously. The consistency in the results of repeated\nexperiments and the quality of the fits give confidence in the\nresults.\n\nThe birefringence with increasing and decreasing ordering fields for\nvarious temperatures below T$_{\\mathrm{D}}$ in the $x=0.15$ sample is\nshown in Fig.\\ \\ref{figure1}. Hysteresis observed in the low field\nregion is attributed to pinning of the multidomain structure.\nThe data from the higher field region where hysteresis is absent were\nextrapolated to determine the zero field birefringence. A log-log plot\nof both the $x=0.15$ and $x=0$ data, fitted to a power law of the form\n$\\Delta n \\propto |t|^{\\beta}$ is shown in Fig.\\ \\ref{figure2}. Using\nT$_{\\mathrm{D}} = 29.26 \\pm 0.03 $ K and data in the range $0.005 < |t|\n<0.027$, we obtained a value of $\\beta = 0.31 \\pm 0.03$ for the mixed\nsample. At larger $|t|$ we found no convincing evidence for crossover \nbehaviour towards the classical exponent. For the pure sample, \nwe obtained $\\beta = 0.46 \\pm 0.06$ for T$_{\\mathrm{D}} = 32.32 \\pm 0.04 $ K \nfrom data in the range $0.004 < |t| <0.037$.\n\nFigure\\ \\ref{figure3} presents susceptibility data for selected \ntemperatures above T$_{\\mathrm{D}}$. At temperatures close to\nT$_{\\mathrm{D}}$, these slopes deviate from linearity, attributable \nin part to the presence of non-zero birefringence at\nzero field, probably caused by internal strains\\cite{Harley}. The\nsusceptibility exponent $\\gamma$ was found by fitting the various\nvalues of $\\chi(T)$ to the relation $\\chi^{-1} \\propto |t|^{\\gamma}$.\nFigure\\ \\ref{figure4} shows a log-log plot of the susceptibility $\\chi$ {\\it versus} \nthe reduced temperature $|t|$ for $x =0$ and $0.15$ samples. As can be\nseen from Fig.\\ \\ref{figure4}, the log-log plot for $x=0.15$ shows two\nlinear fits with the data closer to T$_{\\mathrm{D}}$ having a larger\nslope than that for data further away. Data in the range $0.027 > |t| > 0.01$ \n(closest to T$_{\\mathrm{D}}$) were optimized first\nto a linear fit giving T$_{\\mathrm{D}} = 29.26 \\pm 0.03 $ K and \n$\\gamma = 1.22 \\pm 0.07$. A linear\nfit to data in the range $0.100 > |t| > 0.021$ (further from T$_{\\mathrm{D}}$) \nwith T$_{\\mathrm{D}}$ unchanged yields a value of $\\gamma =\n0.89 \\pm 0.03$. We did not attempt to fit these data to a crossover \nfunction, but they suggest a crossover temperature near $|t|=0.024$ \n($29.96$ K). The log-log plot of the susceptibility {\\it versus} reduced temperature for pure\nTbVO$_{4}$ in Fig.\\ \\ref{figure4} does not show a crossover effect. A power \nlaw fit in the range $0.058 > |t| > 0.005$ gives T$_{\\mathrm{D}} =\n32.32 \\pm 0.04 $ K and $\\gamma = 0.92 \\pm 0.07$.\n\nNonlinearity in the dependence of the order parameter on the ordering\nfield at temperatures below T$_{\\mathrm{D}}$ can be attributed to the\nprogressive detwinning of the sample with increasing field\\cite{Reza2}\n(i. e. changing from a multidomain to a single domain structure). The\ncritical isotherm exponents $\\delta$ were extracted from the field-induced \nbirefringence data at temperatures closest to that of the\npreviously determined transition temperatures for $x=0.15$ and $x=0$\nsamples. Only the higher field $\\Delta$n data were fitted to the power\nlaw (B$^{2}$/T$_{\\mathrm{D}})^{1/\\delta}$. For the $x=0.15$ sample,\nwe obtained a value of $\\delta = 4.2 \\pm 0.7$ at $T=29.24 \\pm 0.03$ K\nand for the pure sample $\\delta = 2.6 \\pm 0.4$ at $T=32.33 \\pm 0.03$ K. \nThe uncertainty in locating T$_{\\mathrm{D}}$ is \nincorporated in the uncertainty in $\\delta$.\n\nOur values of the critical exponents for pure TbVO$_{4}$ agree\nsatisfactorily with the values \n$\\beta=0.5, \\gamma=1$, and $\\delta=3$ expected for a mean field system. \nFor the random-field sample our values $\\beta=0.31\n\\pm 0.03$, $\\gamma=1.22 \\pm 0.07$, and $\\delta = 4.2 \\pm 0.7$ are in \ngood agreement with the exponents for the $d=3$\nIsing model,\\cite{Guillou} $\\beta=0.33$, $\\gamma=1.24$, and $\\delta=4.8$. \n\nToh's predictions\\cite{Toh} for the effects of random fields on the exponents \nin this type of system are therefore well supported by our results. Toh noted \nthat while the\nanisotropic strain interactions in TbVO$_{4}$ reduce the upper critical\ndimension $d^{*}$ from 4 for the standard Ising model to 2, resulting in\nclassical exponents\\cite{Cowley}, the introduction of\nrandom strain fields raises $d^{*}$ from 2 to 4 again, leaving the\ncritical exponents the same as for the standard Ising model in $d=3$. \nIn view of the focus in recent theoretical analysis on dimensional\nreduction it is of interest to examine Toh's analysis and our results in\nterms of changes in effective dimension $d'$ instead of in $d^{*}$. Thus\nfor the Ising model we regard the upper critical dimension to be fixed\nat 4; in pure TbVO$_{4}$, the anisotropic strain interactions presumably\nraise the effective dimension from 3 to 5, giving classical exponents as\nbefore. If changes in critical properties due to random fields can be \ndescribed by dimensional reduction, the consensus prediction\\cite{{Nattermann},{Fortin}} \nis that random fields reduce the effective dimensionality by $2-\\eta$, \nwhere $\\eta$ is the exponent for the decay of magnetization correlations. \nIn the present case, since $\\eta=0.032$ for $d=3$,\\cite{Guillou} the\ndimensionality should be reduced by 1.97 or essentially 2, resulting in $d'=3$. \nHence our results, within their accuracy, are consistent\nwith dimensionality reduction by random fields of $2-\\eta \\sim 2$ for this system.\n\n\\begin{references}\n\\bibitem[*]{currentaddress} Current address: Van der Waals-Zeeman Instituut, Universiteit\nvan Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands.\n\\bibitem{Belanger} D. P. Belanger and A. P. Young, J. Magn. Magn. Mater. {\\bf\n100}, 272 (1991).\n\\bibitem{Belanger2} D. P. Belanger, in {\\it Spin Glasses and Random Fields},\nedited by A. P. Young (World Scientific, Singapore, 1998), p. 251.\n\\bibitem{Nattermann} T. Nattermann, {\\it ibid.}, p. 277.\n\\bibitem{Slanic} Z. Slanic, D. P. Belanger and J. A. Fernandez-Baca, Phys.\nRev. Lett. {\\bf 82}, 426 (1999).\n\\bibitem{Fortin} J. Y. Fortin and P. C. Holdsworth, J. Phys. A: Math. Gen.\n{\\bf 31}, 85 (1998).\n\\bibitem{Slanic2}Z. Slanic and D. P. Belanger, J. Magn. Magn. Mater. {\\bf\n186}, 65 (1998).\n\\bibitem{Reza} K. A. Reza and D. R. Taylor, Phys. Rev. B {\\bf 46}, 11425\n(1992).\n\\bibitem{GAGehring3} G. A. Gehring and M. C. Marques, J. Phys. C: Solid \nState Phys. {\\bf 13}, 3135 (1980).\n\\bibitem{Marques} M. C. Marques, J. Phys. C: Solid State Phys. {\\bf 13}, 3149 (1980).\n\\bibitem{Harley} R. T. Harley and R. M. Macfarlane, J. Phys. C: Solid State Phys.\n{\\bf 8}, L451 (1975).\n\\bibitem{Elliot} R. J. Elliott, R. T. Harley, W. Hayes and S. R. P. Smith,\nProc. Roy. Soc. Lond. A {\\bf 328}, 217 (1972).\n\\bibitem{Wells} M. R. Wells and R. D. Worswick, Phys. Lett. {\\bf A42}, 269 (1972).\n\\bibitem{Sandercock} J. R. Sandercock, S. B. Palmer, R. J. Elliott, W. Hayes,\nS. R. P. Smith and A. P. Young, J. Phys. C: Solid State Phys. {\\bf 5}, 3126 (1972).\n\\bibitem{Berkhan} W. Berkhahn, H. G. Kahle, L. Klein and H. C. Schopper,\nPhys. Stat. Sol.(b) {\\bf 55}, 265 (1973).\n\\bibitem{Toh} H. S. Toh, J. Phys. A: Math. Gen. {\\bf 25}, 4767 (1992).\n\\bibitem{Gehring2} G. A. Gehring and K. A. Gehring, Rep. Prog. Phys. {\\bf 38},\n1 (1975).\n\\bibitem{Page} J. H. Page, S. R. P. Smith, D. R. Taylor and R. T. Harley, J.\nPhys. C: Solid State Phys. {\\bf 12}, L875 (1979).\n\\bibitem{Cowley} R. Cowley, Phys. Rev. B {\\bf 13}, 4877 (1976).\n\\bibitem{GAGehring2} G. A. Gehring, S. J. Swithenby and M. R. Wells, Solid\nState Comm. {\\bf 18}, 31 (1976).\n\\bibitem{Hikel} W. Hikel, H. Hess and H. G. Kahle, J. Phys:\nCondens. Matter {\\bf 1}, 2137 (1989).\n\\bibitem{Ferre}J. Ferr\\'e and G.A.\nGehring, Rep. Prog. Phys. {\\bf 47}, 513 (1984). See arrangement (i), p. 520.\n\\bibitem{KAGehring} K. A. Gehring, A. P. Malozemoff, W. Staude and R. N. Tyte,\nSolid State Commun. {\\bf 9}, 511 (1971).\n\\bibitem{GAGehring1} G. A. Gehring, A. P. Malozemoff, W. Staude and R. N. Tyte,\nJ. Phys. Chem. Solids {\\bf 33}, 1487 (1972).\n\\bibitem{Reza2} K. A. Reza and D. R. Taylor (unpublished).\n\\bibitem{Guillou} J. C. Le Guillou, J. Zinn-Justin, Phys. Rev. Lett.\n{\\bf 39}, 95 (1977).\n\\end{references}\n\n\\end{multicols}\n\n\\clearpage\n\\begin{figure}\n\\vspace*{4.0cm}\n\\begin{center}\n\\scalebox{0.75}{\\includegraphics[0,0][7.9in,8.7in]{Fig1.eps}}\n\\vspace{0.2cm}\n\\caption{Changes in phase $\\phi$ (birefringence) in $x=0.15$ sample at selected \ntemperatures below T$_{\\mathrm{D}}$ for increasing (lower points) and decreasing \n(upper points) ordering fields. The lines join data points.} \n\\label{figure1}\n\\end{center}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\vspace*{4.0cm}\n\\begin{center}\n\\scalebox{0.75}{\\includegraphics[0,0][7.5in,9.0in]{Fig2.eps}}\n\\vspace{0.2cm}\n\\caption{Log-log plots of birefringence {\\it versus} reduced temperature\nfor $x=0$ and $0.15$. The slopes give the order parameter critical exponent $\\beta$.} \n\\label{figure2}\n\\end{center}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\vspace*{4.0cm}\n\\begin{center}\n\\scalebox{0.75}{\\includegraphics[0,0][7.6in,9.1in]{Fig3.eps}}\n\\vspace{0.2cm}\n\\caption{Changes in birefringence {\\it versus} ordering field in $x=0.15$ sample\nat selected temperatures above T$_{\\mathrm{D}}$. Slopes in the small-field limit give \nsusceptibilities.} \n\\label{figure3}\n\\end{center}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\vspace*{4.0cm}\n\\begin{center}\n\\scalebox{0.75}{\\includegraphics[0,0][8.1in,9.0in]{Fig4.eps}}\n\\vspace{0.2cm}\n\\caption{Log-log plots of susceptibility {\\it versus} reduced temperature for\n$x=0$ and $0.15$. The data for the mixed sample show a crossover from classical to \nrandom-field critical behaviour.} \n\\label{figure4}\n\\end{center}\n\\end{figure}\n\n\\end{document}\n"
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{
"name": "cond-mat0002158.extracted_bib",
"string": "\\bibitem[*]{currentaddress} Current address: Van der Waals-Zeeman Instituut, Universiteit\nvan Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands.\n\n\\bibitem{Belanger} D. P. Belanger and A. P. Young, J. Magn. Magn. Mater. {\\bf\n100}, 272 (1991).\n\n\\bibitem{Belanger2} D. P. Belanger, in {\\it Spin Glasses and Random Fields},\nedited by A. P. Young (World Scientific, Singapore, 1998), p. 251.\n\n\\bibitem{Nattermann} T. Nattermann, {\\it ibid.}, p. 277.\n\n\\bibitem{Slanic} Z. Slanic, D. P. Belanger and J. A. Fernandez-Baca, Phys.\nRev. Lett. {\\bf 82}, 426 (1999).\n\n\\bibitem{Fortin} J. Y. Fortin and P. C. Holdsworth, J. Phys. A: Math. Gen.\n{\\bf 31}, 85 (1998).\n\n\\bibitem{Slanic2}Z. Slanic and D. P. Belanger, J. Magn. Magn. Mater. {\\bf\n186}, 65 (1998).\n\n\\bibitem{Reza} K. A. Reza and D. R. Taylor, Phys. Rev. B {\\bf 46}, 11425\n(1992).\n\n\\bibitem{GAGehring3} G. A. Gehring and M. C. Marques, J. Phys. C: Solid \nState Phys. {\\bf 13}, 3135 (1980).\n\n\\bibitem{Marques} M. C. Marques, J. Phys. C: Solid State Phys. {\\bf 13}, 3149 (1980).\n\n\\bibitem{Harley} R. T. Harley and R. M. Macfarlane, J. Phys. C: Solid State Phys.\n{\\bf 8}, L451 (1975).\n\n\\bibitem{Elliot} R. J. Elliott, R. T. Harley, W. Hayes and S. R. P. Smith,\nProc. Roy. Soc. Lond. A {\\bf 328}, 217 (1972).\n\n\\bibitem{Wells} M. R. Wells and R. D. Worswick, Phys. Lett. {\\bf A42}, 269 (1972).\n\n\\bibitem{Sandercock} J. R. Sandercock, S. B. Palmer, R. J. Elliott, W. Hayes,\nS. R. P. Smith and A. P. Young, J. Phys. C: Solid State Phys. {\\bf 5}, 3126 (1972).\n\n\\bibitem{Berkhan} W. Berkhahn, H. G. Kahle, L. Klein and H. C. Schopper,\nPhys. Stat. Sol.(b) {\\bf 55}, 265 (1973).\n\n\\bibitem{Toh} H. S. Toh, J. Phys. A: Math. Gen. {\\bf 25}, 4767 (1992).\n\n\\bibitem{Gehring2} G. A. Gehring and K. A. Gehring, Rep. Prog. Phys. {\\bf 38},\n1 (1975).\n\n\\bibitem{Page} J. H. Page, S. R. P. Smith, D. R. Taylor and R. T. Harley, J.\nPhys. C: Solid State Phys. {\\bf 12}, L875 (1979).\n\n\\bibitem{Cowley} R. Cowley, Phys. Rev. B {\\bf 13}, 4877 (1976).\n\n\\bibitem{GAGehring2} G. A. Gehring, S. J. Swithenby and M. R. Wells, Solid\nState Comm. {\\bf 18}, 31 (1976).\n\n\\bibitem{Hikel} W. Hikel, H. Hess and H. G. Kahle, J. Phys:\nCondens. Matter {\\bf 1}, 2137 (1989).\n\n\\bibitem{Ferre}J. Ferr\\'e and G.A.\nGehring, Rep. Prog. Phys. {\\bf 47}, 513 (1984). See arrangement (i), p. 520.\n\n\\bibitem{KAGehring} K. A. Gehring, A. P. Malozemoff, W. Staude and R. N. Tyte,\nSolid State Commun. {\\bf 9}, 511 (1971).\n\n\\bibitem{GAGehring1} G. A. Gehring, A. P. Malozemoff, W. Staude and R. N. Tyte,\nJ. Phys. Chem. Solids {\\bf 33}, 1487 (1972).\n\n\\bibitem{Reza2} K. A. Reza and D. R. Taylor (unpublished).\n\n\\bibitem{Guillou} J. C. Le Guillou, J. Zinn-Justin, Phys. Rev. Lett.\n{\\bf 39}, 95 (1977).\n"
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cond-mat0002160
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Correlation Effects in Multi-Band Hubbard Model and \\Anomalous Properties of FeSi
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{
"name": "fesi00p.tex",
"string": "%%% injpsj.tex for JPSJ.sty <ver.1.0>\n\n%\\documentstyle[seceq]{jpsj}\n%\\documentstyle[twocolumn,seceq]{jpsj}\n%\\documentstyle[seceq,short,epsf]{jpsj}\n%\\documentstyle[seceq,preprint,epsf]{jpsj}\n%\\documentstyle[twocolumn,epsf]{jpsj}\n\\documentstyle[preprint,epsf]{jpsj}\n\n%%%\\def\\sf{\\rm}\n%%%\\renewcommand\\figureheight[1]{\\vspace{24pt}\\mbox{\\rule{0cm}{#1}}}\n\\def\\runtitle{Correlation Effects in Multi-Band Hubbard Model and Anomalous\nProperties of FeSi}\n\\def\\runauthor{Kentaro {\\sc Urasaki} and Tetsuro {\\sc Saso}}\n\n\\title\n{\nCorrelation Effects in Multi-Band Hubbard Model and \\\\Anomalous\nProperties of FeSi\n}\n\n\\author\n{\nKentaro {\\sc Urasaki}\\footnote{e-mail address: \nkentaro@krishna.th.phy.saitama-u.ac.jp}\nand Tetsuro {\\sc Saso}\\footnote{e-mail address: \nsaso@phy.saitama-u.ac.jp}\n}\n\n\\inst\n{\nDepartment of Physics, Faculty of Science, Saitama University, Urawa, 338-8570\n}\n\n\\recdate\n{February 5, 2000}\n\n\\abst\n{\nThe two-band Hubbard model with the density of states \nobtained from the band calculation\nis applied for FeSi, which is suggested to be\na Kondo insulator or a correlated band insulator.\nUsing this model, the correlation effects on FeSi \nare investigated in terms of \nthe self-consistent second-order perturbation theory combined with the\nlocal approximation. \nThe calculated optical conductivity spectrum \nreproduces the experiments by Damascelli {\\it et al.} \nsemiquantitatively and the specific heat explains the anomalous contribution \nat about 250 K observed in FeSi. Inclusion of the spin fluctuation and the\nextension to the case of strong correlation are also discussed.\n}\n\n\\kword\n{two-band Hubbard model, FeSi, optical conductivity, \nspecific heat\n}\n\n\\begin{document}\n\\sloppy\n\\maketitle\n\nIn the study of a specific material among the strongly correlated electron\nsystems, the effect of the band structures often plays a crucial role when one\ncompares a theoretical calculation to the experiments. Use of a simple\ntheoretical model might not capture the salient features of the material.\nDevelopment of a theoretical method that is capable of taking proper account\nof the realistic features of the material is necessary. We report our recent\napproach to the study of the anomalous properties of FeSi in such direction.\n\nFeSi is well known for more than thirty years and a number of studies from\nvarious aspects have been done,\nstimulated by the fascinating physical properties.\nThe early study by Jaccarino {\\it et al.}\\cite{Jaccarino67}\nshowed that\nthe susceptibility is much enhanced over the value expected from the band\nparamagnetism at finite temperatures and has a broad peak at about \n500 K.\nIt was also reported that the specific heat seems \nto have an anomalous enhancement at about 250 K.\nThese behaviors were explained by a band model with an energy gap, but\nunphysically narrow bands were necessary, so that\nthis difficulty has attracted interests of many researchers.\nFrom the conductivity measurements, \nFeSi is an insulator at low temperatures but shows metallic \nbehavior at room temperature. \nTo explain these unusual properties of FeSi, \nseveral theoretical approaches \nhave been proposed, \nbut the most successful one is the spin fluctuation scenario \nby Takahashi and Moriya.\\cite{Takahashi79}\nIt explains the anomalous magnetic property of FeSi \nand their idea of the thermally induced magnetic moment\nwas confirmed by the neutron scattering experiment.\\cite{Shirane87} \n\nThe recent optical studies\\cite{Schlesinger93,Ohta94,Damascelli97,Paschen97},\nhowever, revealed \nthe unusual properties of FeSi again.\nSchlesinger {\\it et al.} reported \nthat the gap of about 60 meV ($\\sim$700 K) opened \nat low temperatures \nis filled and almost closed at room temperature (about 250$\\sim$300 K), \nwhich they attributed to the correlation effect. \nThe following experiments also reported the evidence of \nthe correlation effects at low\ntemperatures.\\cite{Saitoh95,Chernikov97,DiTusa98,Fath98} \nIn these contexts, Aeppli and Fisk\\cite{Aeppli92} suggested that FeSi can be viewed as a \nKondo insulator or a strongly correlated insulator.\n\nKondo insulators have been found in the f-electron systems \nand typical examples are YbB$_{12}$\\cite{Kasaya85} \nand Ce$_3$Bi$_4$Pt$_3$\\cite{Hundley90} and so on.\nThey have correlated f-bands and small energy gaps at low temperatures. \nAlthough there are many similarities among FeSi and these materials, \nthe correlation in FeSi may not be so strong. \nHowever, the same physics can be recognized both in \nFeSi and Kondo insulators, \nif one reexamines the experimental data carefully.\n%Unfortunately, there is no typical model \n%that can describe the correlation effects on insulators \n%in d electron systems, like the symmetric case of \n%the periodic Anderson model for f electron systems. \nFrom this aspect, Fu and Doniach\\cite{Fu95} proposed an \nextended Hubbard model with \ntwo mixed conduction bands, which is based on their band calculation\\cite{Fu94} \nfor FeSi, \nand confirmed the importance of the \ncorrelation effects in physical quantities.\nTheir calculation, however, seems to include some errors about the \ntreatment of the self-energies. \nTherefore, we reinvestigated this model carefully and \ncalculated the correlation effects in more correct way,\\cite{Urasaki98} \nand confirmed that the correlation effects do play important roles, \nbut the shape of the spectrum in the optical conductivity \ndid not coincide with the experimental data, \nbecause of the use of the too simple model Hamiltonian. \n\nTherefore in the present report, we use an extended two-band Hubbard model \nwith the density of states obtained from the band calculation, \nand attempt to explain the low temperature anomalies of FeSi \nobserved in the optical conductivity\\cite{Urasaki99} and the specific heat \nconsistently.\n\nThe band calculations\\cite{Mattheiss93,Jarlborg95,Galakhov95,Kulatov97,Yamada99} \nfor FeSi predict that the ground state \nis a band insulator and a recent calculation\\cite{Yamada99} reproduces the \n gap size close to the observed one. \nTherefore, we start from the band insulator model, \nwhich consists of two Hubbard bands for d-electrons \nas follows. \n\\begin{eqnarray}\nH=\n& &\\sum_{ij\\sigma} \n(t^1_{ij}c_{i1\\sigma}^\\dagger c_{j1\\sigma} +t^2_{ij}c_{2i\\sigma}^\\dagger \nc_{2j\\sigma})\\cr \n&+&U\\sum_{i}(n_{i1\\uparrow}n_{i1\\downarrow}+ \nn_{i2\\uparrow}n_{i2\\downarrow})\\cr \n&+&U_2\\sum_{i}(n_{i1\\uparrow}n_{i2\\downarrow}+ \nn_{i2\\uparrow}n_{i1\\downarrow})\\cr \n&+&U_3\\sum_{i}(n_{i1\\uparrow}n_{i2\\uparrow}+ \nn_{i2\\downarrow}n_{i1\\downarrow})\\cr \n&-&J\\sum_{i}(c_{i1\\uparrow}^\\dagger c_{i1\\downarrow} \nc_{i2\\downarrow}^\\dagger c_{i2\\uparrow} + \nc_{i2\\uparrow}^\\dagger c_{i2\\downarrow} \nc_{i1\\downarrow}^\\dagger c_{i1\\uparrow} ), \n\\end{eqnarray} \nwhere the $c^\\dagger_{ia\\sigma}(c_{ia\\sigma})$ creates (destroys) \nan electron on site $i$ in band $a=$1, 2 with spin $\\sigma$. \nThe tight binding parameters $t^a_{ij}$ should be fitted \nto the band calculation and $U$, $U_2$, $U_3$ and $J$ \ndenote the Coulomb and exchange interactions. \n\nSince one can expect that the optical \nconductivity spectrum reflects the structure of \nthe quasi-particle density of states (DOS) of a system, \nwe use the DOS obtained from the \nband calculation for FeSi by Yamada {\\it et al.}\\cite{Yamada99} \nfor the initial DOS \nso as to enable detailed comparison with the experiment. \n\n\nFurthermore, we start from the following general \nexpression of the current operator, \n\\begin{eqnarray}\nj=e\\sum_{\\sigma,{\\bf k}}\\sum_{mm^\\prime}\nv^{mm^\\prime}_{\\bf k}c^\\dagger_{m{\\bf k}}\nc_{m^\\prime{\\bf k}},\n\\end{eqnarray}\nwhere $m$ denotes the band indices and \nderive the convenient expression for the optical conductivity. \nFor simplicity, \nwe set the intra- and interband contributions to be equal \n($v^{mm^\\prime}_{\\bf k}=v_{\\bf k}$). \nMoreover, we assume that the momentum \nconservation is violated in real systems by some \ndefects and phonon-assisted transitions. \nTherefore, using the linear response theory, we consider \nthe current-current correlation function as below, \n\\begin{eqnarray}\\label{eq:cfunc2}\nK(i\\omega_n)&=&\n\\int^\\beta_0d\\tau e^{i\\omega_n \\tau}\\sum_{mm^\\prime}\n\\sum_{{\\bf kk^\\prime}\\sigma\\sigma^\\prime}\nv_{\\bf k}v_{\\bf k^\\prime}\\cr\n&&\\!\\!\\!\\times\n<T_\\tau c^\\dagger_{m{\\bf k}\\sigma}(\\tau)c_{m{\\bf k}\\sigma}(\\tau)\nc^\\dagger_{m^\\prime{\\bf k^\\prime}\\sigma^\\prime}(0)\nc_{m^\\prime{\\bf k^\\prime}\\sigma^\\prime}(0)>\\cr\n&&\\cr\n&&\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\simeq-\\frac{1}{\\beta}\\sum_{mm^\\prime}\n\\sum_{l}\\sum_{{\\bf kk^\\prime}\\sigma}v_{\\bf k}v_{\\bf k^\\prime}\n{\\cal G}^m_{{\\bf k}\\sigma}(i\\nu_l)\n{\\cal G}^{m^\\prime}_{{\\bf k}^\\prime\\sigma}(i\\nu_l+i\\omega_n)\\cr\n&&\\cr&&\n\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\!\\!\\!\n\\times[\\delta_{\\bf kk^\\prime}\n+\\Gamma_{\\bf kk^\\prime}^{mm^\\prime\\sigma}(i\\nu_l;i\\omega_n)\n{\\cal G}^m_{{\\bf k^\\prime}\\sigma}(i\\nu_l)\n{\\cal G}^{m^\\prime}_{{\\bf k}\\sigma}(i\\nu_l+i\\omega_n)\n],\n\\end{eqnarray}\nwhere $\\Gamma_{\\bf kk^\\prime}^{mm^\\prime\\sigma}(i\\nu_l;i\\omega_n)$\ndenotes the vertex function, \nand set $[\\dots]$ constant. \nFor the present case, this leads to the following expression for the \noptical conductivity,\n\\begin{eqnarray}\\label{eq:j-dos}\n\\sigma(\\omega,T)&=&\\frac{\\pi(ev)^2}{\\hbar}\\sum_\\sigma\n\\int^\\infty_{-\\infty}d\\nu \n\\frac{f(\\nu)-f(\\nu+\\omega)}{\\omega}\\cr\n&& \\hspace{-1cm}\\times \n[\\rho^\\sigma_{1}(\\nu)+\\rho^\\sigma_{2}(\\nu)]\n[\\rho^\\sigma_{1}(\\nu+\\omega)+\\rho^\\sigma_{2}(\\nu+\\omega)\n], \n\\end{eqnarray}\nwhere $\\rho^\\sigma_{a}(\\nu)$ denotes the DOS for the band $a$. \nThis joint-DOS-like form for the optical conductivity is \nsimple but convenient for the present case. \nWe set $(ev)^2/\\hbar=1$ for simplicity. \n\n\\begin{figure}\\vspace{0.5cm} \n\\epsfxsize=8cm\n\\centerline{\\epsfbox{fig1.eps}}\n\\caption{Fig. 1 Calculated optical conductivity \nwithin the Hartree-Fock approximation or a \nrigid band model.}\n\\label{fig:1}\n\\end{figure}\nFirstly, we show the optical conductivity obtained \nfrom the Hartree-Fock approximation (HFA) or a rigid band model in Fig. 1. \nThe used DOS is displayed in Fig. 2 by the solid line for $T=0$.\nThe DOS is independent of the temperature within HFA. \nAt 0 K, \nonly the interband contribution survives and reproduces \nthe shape of the spectrum of the experiment at 4 K in Fig. 3. \nTherefore, \nthe band calculation by Yamada {\\it et al.}\\cite{Yamada99} \nseems to \ngive a good result about the whole structure of the DOS at $T=0$ \nbut with a slightly smaller gap size\n(see the comparison with the experiment below). \nWithin the rigid band model, however, since the gap is filled only with \nthe intraband (Drude) contribution, \nthe temperature variation is monotonous and the spectrum does not become flat \nat a temperature of the order of the gap size.\nThis disagreement was shown by Fu {\\it et al.} first. \nOhta {\\it et al.}\\cite{Ohta94} also calculated the optical conductivity \nin the joint-DOS form from their band calculation, but \nthe flat part of the optical conductivity spectrum \nwithin the gap could not be reproduced. \nTherefore, the rigid band model is not sufficient to explain the experiments. \n\nNext, we investigate the correlation effect in \nthe low energy and low temperature region of this model.\nTherefore we calculate the correlation effect \nby the self-consistent second-order perturbation theory (SCSOPT)\ncombined with the local approximation.\nThe second-order self-energies are given by \n\\begin{eqnarray}\\label{eq:sigma}\n\\Sigma_1^{(2)\\sigma}(\\omega)&=&\n\\int\\!\\!\\!\\int\\!\\!\\!\\int^\\infty_{-\\infty}\nd\\varepsilon_1 d\\varepsilon_2 d\\varepsilon_3 \\cr\\cr\n&&[U^2\\rho_1^{-\\sigma}(\\varepsilon_1)\n\\rho_1^{\\sigma}(\\varepsilon_2)\n\\rho_1^{-\\sigma}(\\varepsilon_3) \\cr\\cr\n&&+U_2^2\\rho_2^{-\\sigma}(\\varepsilon_1)\n\\rho_1^{\\sigma}(\\varepsilon_2)\n\\rho_2^{-\\sigma}(\\varepsilon_3) \\cr\\cr\n&&+U_3^2\\rho_2^{\\sigma}(\\varepsilon_1)\n\\rho_1^{\\sigma}(\\varepsilon_2)\n\\rho_2^{\\sigma}(\\varepsilon_3) \\cr\\cr\n&&+J^2\\rho_2^{-\\sigma}(\\varepsilon_1)\n\\rho_2^{\\sigma}(\\varepsilon_2)\n\\rho_1^{-\\sigma}(\\varepsilon_3) \n]\\cr\\cr\n\\times&&\n\\hspace{-5mm}\\frac{f(-\\varepsilon_1)f(\\varepsilon_2)f(\\varepsilon_3)\n+f(\\varepsilon_1)f(-\\varepsilon_2)f(-\\varepsilon_3)}\n{\\omega+\\varepsilon_1-\\varepsilon_2-\\varepsilon_3+{\\rm i}\\delta},\\cr\n\\Sigma_2^{(2)\\sigma}(\\omega)&=&(1\\leftrightarrow 2),\n\\end{eqnarray}\nwhere $\\rho_a^\\sigma(\\omega)=\n-(1/\\pi){\\rm Im}G_a^{\\sigma}(\\omega+{\\rm i}\\delta)$ and\n\\begin{eqnarray}\\label{eq:g}\nG_a^\\sigma(\\omega)\n&&=\\frac{1}{N}\\sum_{\\bf k}G_a^{\\sigma}({\\bf k},\\omega)\\cr\n&&=\\int^{\\infty}_{-\\infty}d\\varepsilon\\rho^{0\\sigma}_a(\\varepsilon)\n\\frac{1}{\\omega-\\varepsilon-\\Sigma^{(2)\\sigma}_a(\\omega)}.\n\\end{eqnarray}\nHere, $N$ is the number of sites, $f(\\varepsilon)$ the Fermi function \nand $\\rho^{0\\sigma}_a(\\varepsilon)$ \nthe DOS of band $a$ for the non-interacting case. \nTo make numerical calculation easy, \nwe take $\\delta$ finite ($\\delta=10^{-7}$) in eq. (\\ref{eq:g}) and \nconvert these equations with the transformations\\cite{MullerHartmann89}\n\\begin{eqnarray}\nA_a^\\sigma(\\tau)&=&\\int^\\infty_{-\\infty}d\\epsilon\ne^{-{\\rm i}\\tau\\varepsilon}\\rho_a^\\sigma(\\varepsilon)f(\\varepsilon),\\cr\nB_a^\\sigma(\\tau)&=&\\int^\\infty_{-\\infty}d\\epsilon\ne^{-{\\rm i}\\tau\\varepsilon}\\rho_a^\\sigma(\\varepsilon)f(-\\varepsilon).\n\\end{eqnarray}\nThese equations have to be solved \nself-consistently. \nIn this paper, we set $U_2=U-J$ and $U_3=U-2J$ \nin order to reduce the number of parameters.\nIn this case, the Hamiltonian is rotationally invariant \nin spin and real spaces if the two bands are degenerate.\\cite{Parmenter73} \n\nIn the following results, $U=0.5$ eV and $J=0.35U$ are chosen \nso as to reproduce the shape and the \ntemperature dependence of the optical conductivity spectrum. \nThe solid line for $T=0$ in Fig. 2 indicates the initial DOS at 0 K, \nand the correlation effect is absent except the Hartree-Fock contribution \nsince the band 1 is filled and the band 2 is empty. \n\nNote that the gap in the DOS is widened by 16 $\\%$ so as to \nreproduce the shape of the spectrum of \nthe optical conductivity at 4 K in the experiment, which does not \nchange the essence of the following result. \nThen, the gap size ($E_g$) of $75$ meV is obtained \nif the steepest parts of the DOS at \nthe both sides of the gap are extrapolated and the tails are neglected. \n(If we regard the gap as the region inside the tails of the gap edge, \nwe obtain 60 meV.) \nThe band 1 and 2 in our Hamiltonian correspond to \nthe upper and lower part of the DOS \nwith respect to the Fermi level ($E_F=0$) as is seen in Fig. 2, \nwhere we introduce a cut off for each band \nso as to include one state per spin in each band. \nThen the band width for the band 1 and 2 are about 0.56 eV and about \n0.85 eV, respectively. \nAlthough the DOS is asymmetric, \nthe chemical potential is fixed at $\\omega=0$ \nand assumed to be temperature independent. \nOne can see in Fig. 2 that the correlation is introduced at finite $T$ \nthrough the thermally excited electrons and holes \nand the gap existing at 0 K is almost filled up \nat the temperature of the order of its size, \nwhich results in the temperature variation of the interband contribution \nof the optical conductivity (see below). \n\\begin{figure}\\vspace{0.5cm} \n\\epsfxsize=8cm\n\\centerline{\\epsfbox{fig2.eps}}\n\\caption{The temperature dependence of the quasi-particle DOS \nand the initial DOS obtained from the band calculation(Ref. 26)\nat $T=0$. At finite $T$, the DOS is strongly temperature dependent \ndue to the correlation effects. \n}\n\\label{fig:2}\n\\end{figure}\n\n\\begin{figure}\n\\epsfxsize=8cm\n\\centerline{\\epsfbox{fig3.eps}}\n\\caption{(a)The temperature dependence of the optical conductivity \ncalculated with the eq. (\\ref{eq:j-dos}). \n(b)The experimental data from Ref. 11. \nThe peaks due to phonons observed in the gap are omitted. }\n\\label{fig:3}\n\\end{figure}\nIn Fig. 3(a), the temperature variation of the optical conductivity \ncalculated from the temperature-dependent DOS in Fig. 2 is shown. \nIn our calculation (Fig. 3(a)), \nthe gap is almost filled up at 300 K as well as \nthe rapid increase in the gap region from 150 to 300 K is seen. \nThis is consistent with the experiment (Fig. 3(b)), \nwhere the gap is filled rapidly from 100 K to 300 K. \nReflecting the correlation effects, the peak at the gap edge shifts \nto lower frequency \nregion, as is seen in the experiment. \nIn our calculation, however, \nthere are dips between the Drude and the interband contributions in contrast\nto the experiment. \nThis may be caused by the simplification \nin deriving eq. (\\ref{eq:j-dos}). \nHowever, the almost flat spectrum is obtained at 300 K, \nwhich comes from the temperature dependence of the \ninterband contribution. \n\nWe also calculate the temperature variation of the specific heat \nwith the same parameters as in the optical conductivity. \nStarting from the equation of motion,\\cite{Fetter71}\nwe obtain the following expression for the total energy \nper site:\n\\begin{eqnarray}\\label{eq:sh}\nE&=&\\frac{1}{2}\\sum_\\sigma\\int^\\infty_{-\\infty}d\\omega\nf(\\omega)\\left[ \\omega\\{\\rho^\\sigma_{1}(\\omega)+\\rho^\\sigma_{2}(\\omega)\\} \n\\right.\\cr\n&& \\hspace{-5mm} +\\frac{1}{N}\\sum_{\\bf k}\\left. \\{\\varepsilon^1_{\\bf \nk}\\rho^\\sigma_{1}({\\bf k},\\omega)\n+\\varepsilon^2_{\\bf k}\\rho^\\sigma_{2}({\\bf k},\\omega) \\} \\right],\n\\end{eqnarray}\n\\begin{figure}\\vspace{0.5cm} \n\\epsfxsize=8cm \n\\centerline{\\epsfbox{fig4.eps}}\n\\caption{Calculated specific heat using the same parameter as in Fig. 3(a).}\n\\label{fig:4}\n\\end{figure}\nwhere $\\varepsilon^1_{\\bf k}$ ($\\varepsilon^2_{\\bf k}$) \nis the Fourier transformation of $t^1_{ij}$ ($t^2_{ij}$). \nThe specific heat can be calculated from the numerical differentiation \nof the energy as $C_V=(\\partial E/\\partial T)_V$. \nThe difference between the cases with $U=0$ and $0.5$ eV in Fig. 4 \nindicates the contribution from the correlation effect, \nwhich results in a peak of about 4 J/K mol at about 250 K, \nand explains the ``anomalous\" \ncontribution ($\\sim$6 J/K mol) in the specific heat at about 250 K \nreported by Jaccarino {\\it et al.}\\cite{Jaccarino67}\nNote that \nthey evaluated the anomaly by subtracting \nthe specific heat of CoSi after the \nnormal electronic contributions $\\gamma_{\\rm FeSi}$ \nand $\\gamma_{\\rm CoSi}$ are removed, respectively. \nIn the above calculations, we confirmed that the correlation effect is\nessential \nto explain the temperature dependence of the optical conductivity\nand the specific heat in FeSi.\nAt higher temperatures or for magnetic properties, however,\nit is also important to take the spin fluctuations\n\\cite{Takahashi79,Saso99} \ninto account. \n\nThe self-consistent renormalization (SCR) theory of spin fluctuations \nhas succeeded in describing the itinerant magnetism and \nthe quantum critical phenomena \nwith a small number of parameters.\\cite{Moriya85} \nOn the other hand, the dynamical mean field theory (DMFT) \nis one of the most powerful schemes to take account of the \nstrong local correlation. \nOne of the authors has proposed a new and practical scheme \nthat unifies DMFT and SCR in a microscopic way.\\cite{Saso99} \nApplication of this theory to FeSi may improve the present calculation towards \nthe inclusion of the effects of spin fluctuations at finite temperatures and \nthe intermediate coupling. \n\n\\section*{Acknowledgements}\nThe authors would like to thank Professor H. Yamada for providing them \nthe details of the band calculation (LMTO-ASA) for FeSi and \nfor his useful comments.\nThis work is supported by Grant-in-Aid for Scientific Research No.11640367\nfrom the Ministry of Education, Science, Sports and Culture.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Jaccarino67}V. Jaccarino, G. K. Wertheim, \nJ. H. Wernick, L. R. Walker and S. Arajs: \nPhy. Rev. {\\bf 160} (1967) 476.\n\n\\bibitem{Takahashi79}\nY. Takahashi and T. Moriya:\nJ. Phys. Soc. Jpn. {\\bf 46} (1979) 1451;Y. Takahashi, J. Phys.: Cond. Matter\n{\\bf 9} (1997) 2593.\n\n\\bibitem{Shirane87} G. Shirane, J. E. Fisher, Y. Endoh and K. Tajima: \nPhys. Rev. Lett. {\\bf 59} (1987) 351;\nK. Tajima, Y. Endoh, J. E. Fisher and G. Shirane: Phys. Rev. B {\\bf 38} (1988)\n6954.\n\n\\bibitem{Schlesinger93}\nZ. Schlesinger, Z. Fisk, Hai-Tao Zhang, M. B. Maple,\nJ. F. DiTusa and G. Aeppli: \nPhys. Rev. Lett. {\\bf 71} (1993) 1748.\n\n\\bibitem{Ohta94} H. Ohta, S. Kimura, E. Kulatov, S. V. 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Shamin, A. V. Postnikov, St Uhlenbrock, M. Neumann, Z. W. Lu, \nB. M. Klein and Zhu-Pei Shi: J. Phys.: Condens. Matter {\\bf 7} (1995) 5529. \n\n\\bibitem{Kulatov97}\nE. Kulatov and H. Ohta: \nJ. Phys. Soc. Jpn. {\\bf 66} (1997) 2386.\n\n\\bibitem{Yamada99}\nH. Yamada, K. Terao, H. Ohta, T.Arioka and E. Kulatov: \nJ. Phys.: Condens. Matter {\\bf 11} (1999) L309. \n\n\\bibitem{Parmenter73}\nR. H. Parmenter: \nPhys. Rev. B {\\bf 8} (1973) 1273.\n\n\\bibitem{MullerHartmann89} E. M\\\"uller-Hartmann: Z. Phys. {\\bf 76} 211\n(1989).\n\n\\bibitem{Fetter71} A. L. Fetter and J. D. Walecka, {\\it Quantum Theory of \nMany-Particle Systems} (McGraw Hill, 1971).\n\n\\bibitem{Saso99}\nT. Saso: J. Phys. Soc. Jpn {\\bf 68} (1999) 3941. \n\n\\bibitem{Moriya85}\nT. Moriya, \"Spin fluctuations in Itinerant Electron Magnetism\" \n(Springer, 1985).\n\n\\end{thebibliography}\n%\\newpage\n%\\baselineskip=6ex\n%\\noindent\n%{\\bf Figure Captions}\n\n%\\noindent\n%Figure 1: The initial DOS obtained from the band calculation(Ref. 26) at $T=0$. \n%At finite $T$, the DOS is strongly temperature dependent \n%due to the correlation effects.\n\n%\\noindent\n%Figure 2: (a)The temperature dependence of the optical conductivity \n%calculated with the eq. (\\ref{eq:j-dos}). \n%(b)The experimental data from Ref. 11. \n%The peaks due to phonons observed in the gap are omitted. \n\n%\\noindent\n%Figure 3: Calculated optical conductivity within the Hartree-Fock\n%approximation.\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002160.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Jaccarino67}V. Jaccarino, G. K. Wertheim, \nJ. H. Wernick, L. R. Walker and S. Arajs: \nPhy. Rev. {\\bf 160} (1967) 476.\n\n\\bibitem{Takahashi79}\nY. Takahashi and T. Moriya:\nJ. Phys. Soc. Jpn. {\\bf 46} (1979) 1451;Y. Takahashi, J. Phys.: Cond. Matter\n{\\bf 9} (1997) 2593.\n\n\\bibitem{Shirane87} G. Shirane, J. E. Fisher, Y. Endoh and K. Tajima: \nPhys. Rev. Lett. {\\bf 59} (1987) 351;\nK. Tajima, Y. Endoh, J. E. Fisher and G. Shirane: Phys. Rev. B {\\bf 38} (1988)\n6954.\n\n\\bibitem{Schlesinger93}\nZ. Schlesinger, Z. Fisk, Hai-Tao Zhang, M. B. Maple,\nJ. F. DiTusa and G. Aeppli: \nPhys. Rev. Lett. {\\bf 71} (1993) 1748.\n\n\\bibitem{Ohta94} H. Ohta, S. Kimura, E. Kulatov, S. V. Halilov, T. Nanba, M.\nMotokawa, M. Sato and K. Nagasawa: J. Phys. Soc. Jpn. {\\bf 63} (1994) 4206.\n\n\\bibitem{Paschen97} S. Paschen, E. Felder, M. A. Chernikov, L. Degiorgi, \nH. Schwer, H. R. Ott, D. P. Young, J. L. Sarrao and Z. Fisk: Phys. Rev. B \n{\\bf 56} (1997) 12916.\n\n\\bibitem{Damascelli97} A. Damascelli, K. Schulte, D. van der Marel, M. \nF\\\"{a}th and A. A. Menovsky: Physica B {\\bf 230-232} (1997) 787.\n\n\\bibitem{Saitoh95}\nT. Saitoh, A. Sekiyama, T. Mizokawa, A. Fujimori, K. Ito, H. Nakayama \nand M. Shiga: Solid State Commun. {\\bf 95} (1995) 307.\n\n\\bibitem{Chernikov97}\nM. A. Chernikov, L. Degiorgi, E. Felder, S. Paschen, A. D. Bianchi, H. R. Ott,\nJ. L. Sarrao, Z. Fisk and D. Mandrus: \nPhys. Rev. B {\\bf 56} (1997) 1366. \n\n\\bibitem{Fath98}\nM. F\\\"{a}th, J. Aarts, A. A. Menovsky, G. J. Nieuwenhuys and J. A. Mydosh: \nPhys. Rev. B {\\bf 58} (1995) 15483. \n\n\\bibitem{DiTusa98} J. F. DiTusa, K. Friemelt, E. Bucher, G. Aeppli \nand A. P. Ramirez: \nPhys. Rev. B {\\bf 58} (1998) 10288.\n\n\\bibitem{Aeppli92}\nG. Aeppli and Z. Fisk: \nComments Condens. Matter Phys. {\\bf 16} (1992) 155.\n\n\\bibitem{Kasaya85} M. Kasaya: J. Mag. Magn. Mater. {\\bf 47 \\& 48} (1985) 429. \n\n\\bibitem{Hundley90}M. F. Hundley, P. C. Canfield, J. D. Thompson,\nZ. Fisk and J. M. Laurence: Phys. Rev. B {\\bf 42} (1990) 4862.\n\n\\bibitem{Fu95}\nC. Fu and S. Doniach: \nPhys. Rev. B {\\bf 51} (1995) 17439.\n\n\\bibitem{Fu94}\nC. Fu, M. P. C. M. Krijn and S. Doniach: \nPhys. Rev. B {\\bf 49} (1994) 2219.\n\n\\bibitem{Urasaki98}\nK. Urasaki and T. Saso: \nPhys. Rev. B {\\bf 58} (1998) 15528.\n\n\\bibitem{Urasaki99}\nK. Urasaki and T. Saso: J. Phys. Soc. Jpn. {\\bf 68} (1999) 3477. \n\n\\bibitem{Mattheiss93}\nL. F. Mattheiss and D. R. Hamann: \nPhys. Rev. B {\\bf 47} (1993) 13114.\n\n\\bibitem{Jarlborg95} T. Jarlborg: Phys. Rev. B {\\bf 51} (1995) 11106.\n\n\\bibitem{Galakhov95}\nV. R. Galakhov, E. Z. Kurmaev, V. M. Cherkashenko, Yu M. Yarmoshenko, \nS. N. Shamin, A. V. Postnikov, St Uhlenbrock, M. Neumann, Z. W. Lu, \nB. M. Klein and Zhu-Pei Shi: J. Phys.: Condens. Matter {\\bf 7} (1995) 5529. \n\n\\bibitem{Kulatov97}\nE. Kulatov and H. Ohta: \nJ. Phys. Soc. Jpn. {\\bf 66} (1997) 2386.\n\n\\bibitem{Yamada99}\nH. Yamada, K. Terao, H. Ohta, T.Arioka and E. Kulatov: \nJ. Phys.: Condens. Matter {\\bf 11} (1999) L309. \n\n\\bibitem{Parmenter73}\nR. H. Parmenter: \nPhys. Rev. B {\\bf 8} (1973) 1273.\n\n\\bibitem{MullerHartmann89} E. M\\\"uller-Hartmann: Z. Phys. {\\bf 76} 211\n(1989).\n\n\\bibitem{Fetter71} A. L. Fetter and J. D. Walecka, {\\it Quantum Theory of \nMany-Particle Systems} (McGraw Hill, 1971).\n\n\\bibitem{Saso99}\nT. Saso: J. Phys. Soc. Jpn {\\bf 68} (1999) 3941. \n\n\\bibitem{Moriya85}\nT. Moriya, \"Spin fluctuations in Itinerant Electron Magnetism\" \n(Springer, 1985).\n\n\\end{thebibliography}"
}
] |
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cond-mat0002161
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Surface induced disorder in body-centered-cubic alloys
|
[
{
"author": "F. F. Haas"
},
{
"author": "F. Schmid${}^{\\ddag}$"
},
{
"author": "K. Binder"
}
] |
We present Monte Carlo simulations of surface induced disordering in a model of a binary alloy on a bcc lattice which undergoes a first order bulk transition from the ordered DO3 phase to the disordered A2 phase. The data are analyzed in terms of an effective interface Hamiltonian for a system with several order parameters in the framework of the linear renormalization approach due to Br\'ezin, Halperin and Leibler. We show that the model provides a good description of the system in the vicinity of the interface. In particular, we recover the logarithmic divergence of the thickness of the disordered layer as the bulk transition is approached, we calculate the critical behavior of the maxima of the layer susceptibilities, and demonstrate that it is in reasonable agreement with the simulation data. Directly at the (110) surface, the theory predicts that all order parameters vanish continuously at the surface with a nonuniversal, but common critical exponent $\beta_1$. However, we find different exponents $\beta_1$ for the order parameter $(\psi_2,\psi_3)$ of the DO3 phase and the order parameter $\psi_1$ of the B2 phase. Using the effective interface model, we derive the finite size scaling function for the surface order parameter and show that the theory accounts well for the finite size behavior of $(\psi_2,\psi_3)$, but not for that of $\psi_1$. The situation is even more complicated in the neighborhood of the (100) surface, due to the presence of an ordering field which couples to $\psi_1$.
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[
{
"name": "paper.tex",
"string": "\\documentstyle[aps,multicol,pre]{revtex}\n\n\\input epsf\n%\\documentstyle[aps,preprint,epsf]{revtex} \\newenvironment{multicols}[1]{}{}\n%\\textwidth=14.cm\n%\\textheight=20.cm\n%\\oddsidemargin=1.3cm\n%\\topmargin=0.cm\n%\\pagestyle{empty}\n%\n%\\newcommand{\\lettersize}{\\baselineskip=0.8cm}\n%\n\\newcommand{\\name}{frank}\n%\n%\\newcommand{\\dir}{/home/plato/schmid/graphics/\\name}\n\\newcommand{\\dir}{.}\n%\n\\newcommand{\\fig}[4]\n{\n%\\begin{center}\n \\noindent\n \\unitlength=1mm\n \\begin{picture}(#2,#3)\n \\put(0,0){\\leavevmode \\epsfxsize=#2mm \\epsffile{\\dir/#1}}\n \\end{picture}\n \\noindent\n%\\end{center}\n#4\n}\n%\n%\n\\begin{document}\n\n\\title{Surface induced disorder in body-centered-cubic alloys}\n\\author{F. F. Haas, F. Schmid${}^{\\ddag}$, K. Binder}\n\\address{\nInstitut f\\\"ur Physik, Universit\\\"at Mainz, D-55099 Mainz, Germany \\\\\n${\\ddag}$ Max-Planck-Institut f\\\"ur Polymerforschung, Ackermannweg 10, \nD-55021 Mainz, Germany}\n\n\\maketitle\n\\tighten\n\n\\begin{abstract}\nWe present Monte Carlo simulations of surface induced disordering in a model\nof a binary alloy on a bcc lattice which undergoes a first order bulk \ntransition from the ordered DO3 phase to the disordered A2 phase.\nThe data are analyzed in terms of an effective interface Hamiltonian \nfor a system with several order parameters in the framework of the linear \nrenormalization approach due to Br\\'ezin, Halperin and \nLeibler. We show that the model provides a good description of the system in \nthe vicinity of the interface. In particular, we recover the logarithmic\ndivergence of the thickness of the disordered layer as the bulk transition\nis approached, we calculate the critical behavior of the maxima of the layer \nsusceptibilities, and demonstrate that it is in reasonable agreement with \nthe simulation data. Directly at the (110) surface, the theory predicts that \nall order parameters vanish continuously at the surface\nwith a nonuniversal, but common critical exponent $\\beta_1$.\nHowever, we find different exponents $\\beta_1$ for the order parameter\n$(\\psi_2,\\psi_3)$ of the DO3 phase and the order parameter $\\psi_1$\nof the B2 phase. Using the effective interface model, we derive the \nfinite size scaling function for the surface order parameter and show\nthat the theory accounts well for the finite size behavior of \n$(\\psi_2,\\psi_3)$, but not for that of $\\psi_1$. \nThe situation is even more complicated in the neighborhood of the\n(100) surface, due to the presence of an ordering field which couples\nto $\\psi_1$.\n\\end{abstract}\n\n%\\clearpage\n\n\\begin{multicols}{2}\n\n\\section{Introduction}\n\nFirst order phase transitions in the bulk of systems can drive a variety of \ninteresting wetting phenomena at their surfaces and interfaces. They\nhave attracted much attention over many years\\cite{wetting}, and are still \nvery actively investigated\\cite{wetting2}. \nProminent examples are the wetting of a liquid on a \nsolid substrate at liquid-vapour coexistence, or the wetting of one component \nof a binary fluid below the demixing temperature on the walls of a container.\nThese systems are representatives of a generic situation, which has been \nstudied in particular detail: Three phases coexist, substrate, liquid and \nvapour. The substrate acts as inert ``spectator'' which basically provides the\n``boundary conditions'' for the liquid-vapour system. The liquid-vapour \ntransition can be described by a single order parameter ({\\em e.g.}, the \ndensity), which can take two equilibrium bulk values at coexistence \n(the liquid density or the gas density). \nObviously, the liquid phase will only wet the substrate \nif it is preferentially adsorbed by the latter.\nAs one approaches the liquid-vapour coexistence from the vapour side,\ndifferent scenarios are possible, depending on the substrate interactions\nand on the temperature: Either the liquid film covering the substrate \nremains microscopic at coexistence (``partial wetting''), or it grows \nmacroscopically thick (``complete wetting'').\nThe transition from partial to complete wetting can be first order or\ncontinuous (``critical wetting'').\n%In the first case, the first order wetting transition is continued by a \n%line of first order ``prewetting'' transitions at off-coexistence, \n%where the thickness of the liquid film makes a finite jump.\nSince critical wetting is only expected\non certain substrates at a specific temperature, it is rather difficult\nto observe experimentally (An experimental observation of critical wetting \nwith long range forces has been reported in \\cite{ragil}, and with short \nrange forces in \\cite{ross}).\n\nWetting phenomena are also present in alloys which undergo a discontinuous \norder-disorder transition in the bulk\\cite{lipowsky1,kroll}. \nIn many cases, surfaces are neutral with respect to the symmetry of the \nordered phase, but reduce the degree of ordering due to the reduced number of \ninteracting neighbors. \nThe surfaces can thus be wetted by a layer of disordered alloy,\n{\\em i.e.}, ``surface induced disorder'' (SID) occurs\\cite{lipowsky2,dosch1}. \nThe situation is reminiscent of liquid-vapour wetting; \nhowever, the underlying symmetry in the system restricts the possible \nwetting scenarios significantly.\n\nWe shall illustrate this for systems with purely short range interactions:\nWe consider a Landau free energy functional of the form\n\\begin{eqnarray}\n\\label{landau}\n{\\cal F}\\{ {\\bf m} \\} & = & \n\\int \\! d \\vec{r} \\! \\int_0^{\\infty} \\!\\!\\!\\! dz \\; \\Big\\{ \\;\n\\frac{g}{2} (\\nabla {\\bf m})^2 + f_b\\Big({\\bf m}(\\vec{r},z)\\Big) \\; \\Big\\} \n\\nonumber \\\\\n&& \\qquad \\qquad\n+ \\int \\! d \\vec{r} \\; f_s \\Big({\\bf m}(\\vec{r},z=0)\\Big).\n\\end{eqnarray}\nHere the vector {\\bf m} subsumes the relevant order parameters, \nthe $z$-axis is taken to be perpendicular to the surface, and $d\\vec{r}$ \nintegrates over the remaining spatial dimensions. \nThe offset of the bulk free energy\ndensity $f_b({\\bf m})$ is chosen such that $f_b({\\bf m}_b)=0$ in the bulk.\nThe surface contribution $f_s({\\bf m})$ accounts for the influence of the \nsurface on the order parameter, {\\em i.e.}, the preferential adsorption of \none phase or in the case of SID the disordering effect.\nIn mean field approximation, the functional (\\ref{landau}) is minimized\nby the bulk equation\n\\begin{equation}\ng \\; \\frac{d^2 m_i}{d z^2} = \\partial_i f_b({\\bf m}),\n\\end{equation}\nwhich describes the motion of a classical particle of mass $g$ in the\nexternal potential $(-f_b({\\bf m}))$, subject to the boundary condition \nat $z=0$\n\\begin{equation}\ng \\; \\frac{d m_i}{d z} = - \\partial_i f_s({\\bf m})\n%\\qquad \\mbox{at} \\qquad z=0\n%\\end{equation} \n%\\begin{equation}\n\\quad \\mbox{with} \\quad\n\\Big| g \\frac{d{\\bf {\\bf m}}}{dz} \\Big| = \\sqrt{2 g f_b({\\bf m})}\n\\end{equation}\n\\end{multicols} \\twocolumn\n\n\\begin{figure}[t]\n\\noindent\n\\fig{cahn_old.eps}{75}{75}{\n\\vspace*{-0.4cm}\n\\caption{ Cahn construction (schematic) for a second order wetting transition:\n(a) Critical wetting,\n(b) Partial and complete wetting,\n(c) Off bulk coexistence, approaching critical wetting.\nInsets show the corresponding order parameter profiles.\nSee text for more explanation.}\n\\label{cahn_wet}\n}\n\\end{figure}\n\nIf the order parameter has just one component, this equation can be \nsolved graphically by the Cahn construction\\cite{cahn}. \nThis is illustrated in Fig. 1 for \nthe case of a continuous wetting transition. The corresponding order parameter \nprofiles are shown as insets. Complete wetting is encountered if $f_s'(m)$ \ncrosses $\\sqrt{2 g f_b}$ at the outer side of the minimum corresponding to the \nadsorbed phase. Partial wetting is found if the crossing point is located \nbetween the two minima (Fig. \\ref{cahn_wet}b). Critical wetting connects the \ntwo regimes, {\\em i.e.}, $f_s'(m)$ crosses $\\sqrt{2 g f_b}$ right at the \nadsorbed phase minimum of $f_b$ (Fig. \\ref{cahn_wet}a). Fig. \\ref{cahn_wet}c) \nshows a case where the system is off bulk coexistence. \n\nNow, let us consider the case of surface induced disorder. Here, several\nequivalent ordered phases exist, and the ordered state breaks a symmetry.\nFor neutral surfaces which do not discriminate between the ordered phases, \n$f_b$ and $f_s$ have the same symmetry. This implies that $f_s$ is extremal \nin the disordered phase (${\\bf m}=0$), {\\em i.e.}, $|\\partial f_s|$ is zero \nat ${\\bf m}=0$ and thus crosses $\\sqrt{2 g f_b}$ at ${\\bf m}=0$ \n\\begin{figure}[b]\n\\noindent\n\\hspace*{-0.3cm}\\fig{cahn_sid_old.eps}{80}{37}{\n\\vspace*{-0.3cm}\n\\caption{Cahn construction (schematic) for surface induced disorder in a \nsystem with a one-component order parameter $m$ (a) at bulk coexistence and \n(b) off bulk coexistence. \nDashed line shows surface term $f_s'(m)$ for critical wetting, \ndotted line for partial wetting.}\n\\label{cahn_sid}\n}\n\\end{figure}\\noindent\n(Fig. \\ref{cahn_sid}). \nComparing that with the scenario sketched above (Fig. \\ref{cahn_wet}), \nwe find that surface induced disordering corresponds to \neither partial or critical wetting\\cite{kroll} \n-- the symmetry of the surface interactions\nexcludes the possibility of complete wetting\\cite{fn1}.\nThe off-coexistence situation (Fig. \\ref{cahn_sid}b)) resembles\nthat in Fig. \\ref{cahn_wet}c).\n\nAlloys which exhibit surface induced disorder thus seem particularly \nsuited to study critical wetting. Unfortunately, the simplification\ndue to the symmetry of the system often goes along with severe\ncomplications in other respect: Usually, one has to deal with a number\nof order parameters and other coupled fields, which interact in a way that \nmay not always be transparent. If the surface under consideration does\nnot have the symmetry of the bulk lattice with respect to\nthe ordered phases, the interplay of order parameters and surface segregation \ncreates effective ordering surface fields\\cite{ich1,ich3,dosch2,diehl1},\nwhich may affect the critical behavior at the surface\\cite{diehl1,upton}. \nIn the case of a one component order parameter, such a field drives the\nsystem from critical wetting to partial wetting.\nWhen several order parameters are involved, this is not necessarily the\ncase\\cite{helbing,gerhard1,gerhard2}. More subtle effects can lead to surface\norder even at fully symmetric surfaces\\cite{dosch4,schweika1,schweika3}.\n\nExperimentally, surface induced disorder has been investigated at the (100) \nsurface of Cu${}_3$Au\\cite{sundaram,rae,alvarado,dosch3}. A number of studies \nhave provided evidence that the order parameter right at the surface vanishes\ncontinuously as the bulk transition is approached\\cite{sundaram,rae,alvarado}, \nand established the relation with the existence of a disordered surface \nlayer of growing thickness\\cite{dosch3}.\nThe related case of ``interface induced disorder'' has been studied among\nother in Cu-Pd, where the width of anti-phase boundaries was shown to diverge\nlogarithmically as the temperature of the transition to the disordered\nphase was approached from below\\cite{ricolleau}.\n\nThe first simulation studies of surface induced disorder in different \nsystems have reproduced the continuous decrease of the surface order \nparameter at the bulk first order transition\\cite{sundaram2,gerhard2,helbing}, \nand the logarithmic growth of a disordered layer near the surface\\cite{helbing}.\nA detailed study of surface induced disorder at the (111) surface of\nCuAu has been published recently by Schweika {\\em et al}\\cite{schweika}. \nThe critical behavior of various quantities has been analyzed, and critical \nexponents were found which agree well with the theory of critical wetting. \nMost notably, Schweika {\\em et al} observe nonuniversal exponents, as\npredicted by renormalization group theories of wetting phenomena\\cite{bhl,fh}.\nIn contrast, Monte Carlo simulations of critical wetting in a simple Ising \nmodel have given results which were more consistent with mean field \nexponents\\cite{KB1}. This latter finding has intrigued theorists for some \ntime, and a number of theories have been put forward to account for the \nunexpected lack of fluctuation effects\\cite{fjin,boulter,swain1}. \nThe nonuniversality of the exponents observed by Schweika {\\em et al} seems \nto indicate that the fluctuations are restored in the case of SID. \nAlternatively, it may also stem from a competition of different length \nscales associated with the local order parameter and the local \ncomposition\\cite{gerhard1,gerhard2,hauge}.\n\nIn the present work, we study surfaces of a bcc-based alloy close to the\nfirst order transition from the ordered DO${}_3$ phase to the disordered\nphase. Our work is thus closely related to that of Schweika {\\em et al}. \nIt differs in that the order parameter structure in the bcc case\nis much more complex than in the fcc-alloy:\nWhereas only one (three dimensional) order parameter drives the transition\nconsidered by Schweika {\\em et al}, we have to deal with two qualitatively \ndifferent order parameters, which are entangled with each other in a rather \nintriguing way. In fact, we shall see that one of them behaves as expected \nfrom the theory of critical wetting, whereas the other exhibits different \ncritical exponents, which do not fit into the current picture. \n\nA similar system has been investigated some time ago by \nHelbing {\\em et al}\\cite{helbing}. The systems studied there were \nrather small, and a detailed analysis of the critical behavior was \nnot possible. Helbing {\\em et al} report evidence for the presence of a \nlogarithmically growing disordered layer at the (100)-surface as phase \ncoexistence was approached. In retrospect, this result seems surprising,\nsince the (100) surface breaks the symmetry with respect to one of \nthe order parameters, and we know nowadays that this nucleates an\nordering surface field. In order to elucidate the influence of this\nordering field in more detail, we have thus considered both\nthe (110) surface, which has the full symmetry of the bulk lattice,\nand the (100) surface.\n\nOur paper is organized as follows: In the next section, we provide some\ntheoretical background on the theory of wetting in systems with several\norder parameters. Section \\ref{model} is devoted to some general remarks on \norder-disorder transitions in bcc-alloys, and to the presentation of the model \nand the simulation method. Our results are discussed in section \\ref{results}. \nWe summarize and conclude in section \\ref{summary}.\n\n\\section{Effective interface theory of surface induced disorder}\n\\label{wetting}\n\n\\subsection{General considerations}\n\\label{general}\n\nWe have already sketched one of the popular mean field approaches to wetting \nproblems in the introduction. Since the bulk of the system is not critical, \none can expect fluctuations to be negligeable for the most part. Only the\nfluctuations of the local position $l(\\vec{r})$ of the interface between the \ngrowing surface layer and the bulk phase remain important\\cite{lkz}. \nAs the interface moves into the bulk, capillary wave excursions of larger \nand larger wavelengths become possible. These introduce long-range \ncorrelations parallel to the surface, characterized by a correlation length \n$\\xi_{\\parallel}$ which diverges at wetting. \n\nIn light of these considerations, fluctuation analyses often replace the \nLandau free energy functional (\\ref{landau}) by an effective interface \nHamiltonian\\cite{bhl,lkz,fisher}\n\\begin{equation}\n\\label{hinter}\n{\\cal H} \\{ l \\} / k_B T \n= \\int \\!\\! d \\vec{r} \\; \\Big\\{ \\frac{1}{8 \\pi \\omega} \n(\\nabla {l})^2 + V_0(l) \\Big\\}.\n\\end{equation}\nHere all lengths are given in units of the bulk correlation length $\\xi_b$\nin the phase adsorbed at the surface,\nthe parameter $\\omega$ is the dimensionless inverse of the \ninterfacial tension $\\sigma$\n\\begin{equation}\n\\omega = k_B T/4 \\pi \\sigma \\xi_b{}^2,\n\\end{equation}\nand the potential $V_0(l)$ describes effective interactions between the\ninterface and the surface. The wetting transition is thus identified with a \ndepinning transition of the interface from the surface. \n\nIn the linearized theory, the partition function of the Hamiltonian\n(\\ref{hinter}) is approximated by\n\\begin{equation}\n{\\cal Z} \\approx\n%\\int {\\cal D} \\{l\\} e^{-{\\cal H}\\{l\\}/k_BT} \\approx\n\\int \\! {\\cal D} \\{l\\} \\; e^{-\\int \\!\\! d\\vec{r}\n\\; (\\nabla l)^2/8 \\pi \\omega } \\;\\;\n\\big[ \\, 1 + \\int \\!\\! d \\vec{r} \\; V_0(l)\\, \\big].\n\\end{equation}\nIt is convenient to switch from the real space $\\vec{r}$ to the Fourier space\n$\\vec{q}$. The integration over short wavelength fluctuations with wavevector \n$|\\vec{q}| > \\lambda^{-1}$, where $\\lambda$ is arbitrary, \nis then straightforward: One separates $l$ into a \nshort wavelength part $\\hat{l}(\\vec{q}) = l(\\vec{q})\\;\\theta (q-\\lambda^{-1})$ \nand a long wavelength part $\\overline{l} = l - \\hat{l}$, and exploits\nthe relation $V_0(\\overline{l}+x) = \\exp[x \\; d/dl] V_0(\\overline{l})$, \nto obtain the unrescaled coarse grained potential \\cite{lf}\n\\begin{eqnarray}\n\\overline{V}_{\\lambda}(\\overline{l}) & = &\n\\int \\! {\\cal D} \\{\\hat{l}\\} \\; \\exp \\Big[{-\\frac{1}{4 \\pi^2}\\int \\!\\! d\\vec{q}\n\\; \\big\\{ \\frac{|\\vec{q}\\; \\hat{l}|^2}{8 \\pi \\omega } \n+ \\hat{l}\\frac{d}{d l}} \\big\\} \\Big] \\;\nV_0(\\overline{l}) \\nonumber\\\\\n&=& \\exp \\Big[{\\frac{\\xi_{\\perp,\\lambda}^2}{2} \\; (\\frac{d}{dl})^2} \\Big] \nV_{\\lambda}(\\overline{l}) \n\\\\\n%&=& \\frac{1}{\\sqrt{2 \\pi \\xi_{\\perp,\\lambda}^2}} \\int \\!\\! dh \\; \n%e^{-h^2/2 \\xi_{\\perp,\\lambda}^2} \\; V(\\overline{l}+h), \\nonumber\n\\mbox{with} && \\xi_{\\perp,\\lambda}^2\n=\\frac{\\omega}{\\pi}\\int_{q >1/\\lambda}^{1/\\Lambda} \\!\\!\\!\\ d \\vec{q} \\;\n\\frac{1}{q^2}\n= 2 \\omega \\ln(\\lambda/\\Lambda),\n\\label{xip1}\n\\end{eqnarray}\nwhere $\\Lambda$ is a microscopic cutoff length. After rescaling \n$\\vec{r}\\to\\vec{r}/\\lambda$, $\\overline{V}_{\\lambda}(l)\\to V_{\\lambda}(l) \n=\\lambda^{d-1}\\overline{V}_{\\lambda}(\\overline{l}\\lambda^{\\zeta})$, and \nnoting that the roughness exponent $\\zeta$ is zero for capillary waves \nin $d=3$ dimensions, this can be rewritten as\n\\begin{equation}\n\\label{gauss}\nV_{\\lambda}(l) = \n\\frac{\\lambda^{2}}{\\sqrt{2 \\pi \\xi_{\\perp,\\lambda}^2}} \\int \\!\\! dh \\; \ne^{-h^2/2 \\xi_{\\perp,\\lambda}^2} \\; V_0(l+h). \n\\end{equation}\nRenormalizing the potential $V_0(l)$ thus amounts to convoluting it\nwith a Gaussian of width $\\xi_{\\perp,\\lambda}^2$\\cite{bhl}, which is\nthe width of a free interface on the length scale $\\lambda$ parallel to the \ninterface. In the case of a bound interface, a natural choice \nfor $\\lambda$ is $\\xi_{\\parallel}$, the parallel correlation length of the \ninterface. Since the remaining fluctuations after the renormalization should\nbe small on this length scale, the procedure can be made self consistent\nby equating $\\xi_{\\parallel}$ with its mean field value \n\\begin{equation}\n\\label{ddv}\n4 \\pi \\omega \\frac{d^2}{d l^2} V_{\\lambda}(l)\\Big|_{l=\\langle l \\rangle} =\n(\\xi_{\\parallel}/\\lambda)^{-2} = 1 \\quad \\mbox{at}\n\\quad \\lambda = \\xi_{\\parallel}\n\\end{equation}\nwhere the average position of the interface $\\langle l \\rangle$ is the \nposition of the minimum of $V_{\\lambda}(l)$. Note that the renormalized\nfree energy density per area $\\xi_{\\parallel}{}^2$ is of order unity. \nThe singular part $F_s$ of the total interface free energy thus scales like \n$ F_s \\propto \\xi_{\\parallel}^{-2}$.\nFrom the renormalized Hamiltonian (\\ref{hinter}),\n\\begin{equation}\n{\\cal H}_{\\xi_{\\parallel}} \\{ l \\} / k_B T = \\frac{1}{4 \\pi^2}\n\\int_0^{\\xi_{\\parallel}/\\Lambda} \\!\\! d \\vec{q} \\; \n\\frac{1}{8 \\pi \\omega} (q^2 + 1)\\; \\big|\\, l(\\vec{q})\\, \\big|^2\n\\end{equation}\nwe can now calculate the distribution probability to find the\ninterface at a position $h$,\n\\begin{equation}\n\\label{p1}\nP(h) = \\big\\langle \\delta[h - l(0)] \n\\big\\rangle_{{\\cal H}_{{\\xi}_{\\parallel}^2}}\n= \\frac{1}{\\sqrt{2 \\pi \\xi_{\\perp}}} \\; e^{-h^2/2 \\xi_{\\perp}^2}, \n\\end{equation}\nand the joint probability distribution that the interface is found at \n$h$ and $h'$ at two points separated by $\\vec{r}$ from each other.\n\\begin{eqnarray}\n\\label{p2}\nP^{(2)}(h,&&h',\\vec{r}) = \n \\big\\langle \\delta[h - l(0)]\\; \\delta[h' - l(\\vec{r})]\n\\big\\rangle_{{\\cal H}_{{\\xi}_{\\parallel}}} \\nonumber\\\\\n=&& \\frac{1}{2 \\pi \\sqrt{g(0)^2-g(r)^2}} \\nonumber\\\\\n&& \\times \\;\n\\exp\\big[- \\frac{(h-h')^2}{4(g(0)-g(r))}\n-\\frac{(h+h')^2}{4(g(0)+g(r))}\\big],\n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\label{gr}\ng(r) = \\big\\langle l(0) \\; l(\\vec{r}) \n\\big\\rangle_{{\\cal H}_{{\\xi}_{\\parallel}}} \n = \\frac{\\omega}{\\pi} \\int_0^{\\xi_{\\parallel}/\\Lambda} \\!\\! \n\\frac{d \\vec{q}}{q^2+1} \\; e^{i \\vec{q}\\vec{r}/\\xi_{\\parallel}}\n\\end{equation}\nis the height-height correlation function of the interface and\n\\begin{equation}\n\\label{g0}\n\\xi_{\\perp}^2 = g(0) \\approx 2 \\omega \\ln (\\xi_{\\parallel}/\\Lambda).\n\\end{equation}\nAn analogous expression has been derived by Bedeaux and Weeks for a free \nliquid-gas interface in a gravitational field\\cite{bedeaux}. In three \ndimensions, the height-height correlation function for $r \\gg \\Lambda$ \nand $\\xi_{\\parallel} \\gg \\Lambda$ is a Bessel function $K$,\n\\begin{equation}\ng(r) = 2 \\omega \\; K_0(r/\\xi_{\\parallel}).\n\\end{equation}\n\nWe assume that the average order parameter profile \n$\\langle m(z) \\rangle $ is given by the average over mean field order \nparameter profiles $m_{\\mbox{\\small bare}}(z-l)$, \ncentered around the local interface positions \n$l$, which are distributed according to the distribution function $P(l)$. \n\\begin{equation}\n\\label{maver}\n\\langle m(z) \\rangle = \\int \\! dl \\; P(l) \\; m_{\\mbox{\\small bare}}(z-l)\n\\end{equation}\nThe distribution functions $P(h)$ and $P^{(2)}(h,h',r)$ can then be used\nto calculate various characteristics of the profiles: \n\nFor example, the effective width of the order parameter profile,\n$W = 1/\\big( 2 \\; {\\partial \\langle m \\rangle }/\n{\\partial z} \\big)_{\\langle l \\rangle}$,\nis broadened by $P(h)$ and diverges according to \\cite{ich2,andreas}\n\\begin{equation}\n\\label{ew2}\nW^2 \\approx W_0{}^2 + \\frac{\\pi}{2} \\; \\xi_{\\perp}{}^2\n\\end{equation}\nwhere $W_0$ denotes the ``intrinsic width'' of the mean field profile,\n$W_0 = 1/(2 \\; dm_{\\mbox{\\small bare}}/dz)|_{z=0}$. \n\nAnother quantity of interest is the layer-layer susceptibility, which\ndescribes the order parameter fluctuations at a given distance from\nthe surface,\n\\begin{equation}\n\\label{chinn0}\n\\chi_{zz} = \\int \\! d \\vec{r} \\; \\Big\\{\\big\\langle m(0)m(\\vec{r})\\big\\rangle_z\n- \\big\\langle m \\big\\rangle^2_z\n\\Big\\}.\n\\end{equation}\nSince it has the dimension of a square length, one deduces immediately that\n$\\chi_{zz}$ scales like $\\xi_{\\parallel}{}^2$ in the interfacial region.\nFor a more detailed analysis, we rewrite $\\chi_{zz}$ as \n\\begin{eqnarray}\n\\chi_{zz} &=& \n\\int \\! d \\vec{r} \\; \n\\int \\! dh \\; dh' \\; m_{\\mbox{\\small bare}}(z-h) \n\\; m_{\\mbox{\\small bare}}(z-h') \\nonumber \\\\\n& &\\times\n\\int \\! d \\vec{r} \\; \n\\big\\{{P^{(2)}(h,h',r)} - {P(h)\\; P(h')} \\big\\},\n\\end{eqnarray}\nexpand the joint probability $P^{(2)}(h,h',r)$ in powers of\n$\\Delta(r) = g(r)/\\xi_{\\perp}^2 \n= K_0(r/\\xi_{\\parallel})/\\ln(\\xi_{\\parallel}/\\Lambda)$, \n\\begin{eqnarray}\nP^{(2)}&&(h,h',r) = P(h)\\; P(h')\\; \\Big\\{ \\;\n 1 + \\frac{h \\; h'}{\\xi_{\\perp}^2} \\Delta(r) \\nonumber\\\\\n && + \\; \\frac{1}{2} \\: \\big[ 1 - \\frac{h^2 + h'^2}{\\xi_{\\perp}^2} + \n\\frac{h^2 h'^2}{\\xi_{\\perp}^4} \\big] \\Delta(r)^2 + \\cdots \\Big\\}, \n\\end{eqnarray}\nand recall $\\int \\! dr\\,r\\,K_0(r)=1$ and $\\int \\! dr\\,r\\,K_0(r)^2 = 1/2$.\nIf the intrinsic width of the profile $m_{\\mbox{\\small bare}}(z)$ is small \ncompared to $\\xi_{\\perp}$, the intrinsic profile can be approximated by a simple\nstep profile in the interfacial region, \n$m_{\\mbox{\\small bare}}(z) = m_b \\theta(z)$, \nwhere $m_b$ is the bulk order parameter. One then obtains\n\\begin{eqnarray}\n\\chi_{zz} &=& m_b{}^2 \\xi_{\\parallel}{}^2\ne^{- (z-\\langle l \\rangle) ^2/\\xi_{\\perp}^2} \\nonumber\\\\\n\\label{chizz}\n&&\\times \\; \\Big ( \\, \\frac{2 \\omega}{\\xi_{\\perp}{}^2} \n+ \\big(\\frac{2 \\omega}{\\xi_{\\perp}{}^2}\\big)^2 \\;\n\\frac{(z-\\langle l \\rangle)^2}{4 \\xi_{\\perp}{}^4} \n+ \\cdots \\Big).\n\\end{eqnarray}\n\nSo far, these results are valid for infinite systems.\nThe restriction to finite lateral dimension $L$ affects the interface \ndistribution $P(h)$ (\\ref{p1}) in two ways: It introduces a lower cutoff \n$\\xi_{\\parallel}/L$ in the integrals over $\\vec{q}$ ({\\em e.g.}, (\\ref{gr})), \nand the mean position of the interface (the zeroth mode) is no longer fixed at \nthe minimum of the renormalized potential, but distributed according to\n$\\exp[-L^2 V_{\\xi_{\\parallel}}(h)]$. The width of the distribution\nfunction $P(h)$ is now given by\n\\begin{eqnarray}\n\\label{xipfss}\n\\xi_{\\perp}^2 &=& \\frac{\\omega}{\\pi} \n\\int_{\\xi_{\\parallel}/L}^{\\xi_{\\parallel}/\\Lambda} \\!\\! \\frac{d \\vec{q}}{q^2+1} \n+ L^2 \\; \\frac{d^2 V_{\\xi_{\\parallel}}(h)}{d h^2} \\Big|_{h = \\langle l \\rangle}\n\\nonumber\\\\\n&=& 2 \\omega \\ln(\\frac{\\xi_{\\parallel}}{\\Lambda}) \n- \\omega \\ln(1+(\\frac{\\xi_{\\parallel}}{L})^2) + \n4 \\pi \\omega \\; (\\frac{\\xi_{\\parallel}}{L})^2.\n\\end{eqnarray}\n\n\\subsection{Bare and renormalized effective interface potential}\n\\label{special}\n\nWe shall now apply these general considerations to a specific potential \n$V_0(l)$, designed to describe systems with short range interactions and \nseveral order parameters and nonordering densities. Effective interface \npotentials for systems with two order parameters have been derived by \nHauge\\cite{hauge} and Kroll and Gompper\\cite{gerhard1}. Their approach can \nreadily be generalized to the case of arbitrary many order parameters and \nnonordering densities. We choose the coordinate system in the order parameter \nand density space $\\{ \\bf m \\}$ such that ${\\bf m} = 0$ in the phase which \nwets the surface, and that the coordinate axes $m_i$ point in the \ndirections of the principal curvatures of the bulk free energy function\n$f_b({\\bf m})$. Close to this phase, $f_b$ can then be approximated by the\nquadratic form\n\\begin{equation}\n\\label{fb}\nf_b({\\bf m}) = \\frac{g}{2} \\sum_i \\frac{1}{\\lambda_i{}^2} m_i{}^2 + \\mu,\n\\end{equation}\nwhere $\\mu$ is the field which drives the system from coexistence, and the\n$\\lambda_i$ have the dimension of a length. We number the coordinate axes $i$ \n($i \\ge 0$) such that the $\\lambda_i$ are arranged in descending order. The \nlargest of these dominates the correlations at large distances and is thus\nthe correlation length $\\xi_b$, {\\em i.e.}, $\\lambda_0 = \\xi_b \\equiv 1$ in \nour units. The surface contribution has the form\n\\begin{equation}\n\\label{fs}\nf_s({\\bf m}) = \\sum_i h_{i,1} m_i + \\frac{1}{2}\\sum_{ij} c_{ij} m_i m_j.\n\\end{equation}\nFollowing Hauge and Kroll/Gompper, we now assume that the actual profile \nfrom the adsorbed phase to the bulk phase is close to the profile of a free \ninterface between these two phases. Close to the surface region, we \nthus approximate the former by the test function \n\\begin{equation}\nm_i(z) = v_i \\exp{(z-l)/\\lambda_i}\n\\end{equation}\n(at $z \\ll l$), where $l$ denotes the position of the effective interface. \nInserting this into eqn. (\\ref{landau}) with (\\ref{fb}) and (\\ref{fs}),\nwe obtain the effective interface potential\n\\begin{equation}\n\\label{vbare}\nV_0(l) = \\sum_i a_i \\: e^{-l/\\lambda_i} \n+ \\sum_{ij} b_{ij} \\: e^{-l (1/\\lambda_i + 1/\\lambda_j)} + \\mu l\n\\end{equation}\nfor $l \\gg 0$, with $a_i = h_{i,1} v_i$ and \n$b_{ij} = \\frac{1}{2}(c_{ij} - g \\delta_{ij}/\\lambda_i) v_i v_j$. \nThis expression is of course only valid for large $l$. Notably, it fails\nat $l=0$, since the true potential $V_0(l)$ must diverge there.\nWe shall suppose that the leading term $b_{00} \\equiv b$ in the second sum is \npositive and dominates over the more rapidly decaying terms, and disregard the\nlatter in the following.\n\nAt $\\omega = 0$ (or in mean field approximation), the interface ist flat, \nand its position is given by the minimum of $V_0(l)$. At nonzero $\\omega$,\nthe potential has to be renormalized as described in the previous section.\nNow the renormalization is straightforward if the fluctuations are \nsufficiently small that the interface position $\\langle l \\rangle$ at\nwetting is well in the asymptotic tail of the potential (weak fluctuation\nlimit). According to a criterion introduced by \nBr\\'ezin, Halperin, and Leibler\\cite{bhl}, this is true as long as \n$\n\\int_0^{\\infty} dl\\: e^{-(l-\\langle l \\rangle)^2/2 \\xi_{\\perp}^2} V_0(l)\n\\approx\n\\int_{-\\infty}^{\\infty} dl \\: \ne^{-(l-\\langle l \\rangle)^2/2 \\xi_{\\perp}^2} V_0(l),\n$ {\\em i.e.}, \n\\begin{equation}\n\\label{wfl}\n2 \\xi_{\\perp}^2 - \\langle l \\rangle < 0 \n\\qquad \\mbox{and} \\qquad\n\\xi_{\\perp}^2/\\lambda_i - \\langle l \\rangle < 0\n\\end{equation}\nfor all $\\lambda_i$. For $\\lambda_i > 1/2$, the first inequality enforces\nthe second one. In a system with one order parameter, it leads to the \nwell-known inequality $\\omega < 1/2$\\cite{bhl,fh}. As we shall see shortly, this\ncondition is also sufficient to ensure the validity of the weak fluctuation \nlimit in a system with several order parameters. Since $\\omega$ in our \nsimulations turns out to be much smaller than 1/2, we shall not discuss the \nother regimes in the present paper.\n\nIn the weak fluctuation limit, the renormalized potential takes the form\n\\begin{equation}\n\\label{vxi}\n\\frac{V_{\\xi_{\\parallel}}(l)}{\\xi_{\\parallel}^2}\n= \\sum_{i: \\lambda_i < 1/2} a_i \\; e^{-l/\\lambda_i} \\; \n(\\frac{\\xi_{\\parallel}}{\\Lambda})^{\\omega/\\lambda_i^2} + \nb \\; (\\frac{\\xi_{\\parallel}}{\\Lambda})^{4 \\omega} e^{-2 l} + \n\\mu l.\n\\end{equation}\nThe cutoff parameter $\\Lambda$ is of the order of the correlation length,\n$\\Lambda \\approx \\xi_b = 1$, and will be dropped hereafter.\n\n\\subsection{Free energy scaling}\n\\label{scaling}\n\nNow our task is to determine $\\xi_{\\parallel}$ self consistently by use \nof eqn. (\\ref{ddv}), which will yield the scaling behavior of the singular\npart of the surface free energy, $F_s \\propto \\xi_{\\parallel}^{-2}$.\nBefore generalizing to several order parameters, we shall briefly discuss\nthe situation in a system with only one length scale $\\lambda_0$. \nThe formal alikeness of the more general theory with this often discussed \nspecial case can thus be highlighted. Moreover, many of the results \nderived for one order parameter carry over directly to the case of several \norder parameters.\n\nIn a system with one order parameter, the singular free energy has the scaling \nform\n\\begin{equation}\n\\label{fs0}\nF_s \\propto \\xi_{\\parallel}{}^{-2} = {8 \\pi \\omega} \\; {\\mu} \\; \nf(\\Phi_0),\n\\end{equation}\nwhere the scaling function $f(\\Phi_0)$ depends on the dimensionless parameter\n\\begin{equation}\n\\Phi_0 = C_0 \\; \\mu^{(\\omega-1)/2} \\; a_0 \n\\qquad \\mbox{with} \\qquad\nC_0 = \\sqrt{(8\\pi \\omega)^{\\omega}/2 b}.\n\\end{equation}\nDepending on the value of $\\Phi_0$, one can distinguish between different\nregimes:\n\\begin{eqnarray}\n\\lefteqn{\\Phi_0 \\gg 1:} \\hspace{1.7cm}\n&& f(\\Phi_0) = 1/2 \\; g_1 \\! \\big( 2^{\\omega}\\Phi_0^{-2} \\big) \n\\nonumber \\\\\n \\mbox{with} \\quad && \ng_1(x) \\approx 1 + x - (2+\\omega) x^2 + \\cdots \\nonumber\\\\\n \\lefteqn{\\mbox{(complete wetting)}} \\\\\n\\lefteqn{|\\Phi_0| \\ll 1:} \\hspace{1.7cm}\n&& f(\\Phi_0) \\approx 1 - \\frac{1}{2}\\Phi_0\n+ \\frac{2+\\omega}{8} \\Phi_0^2 + \\cdots \\nonumber \\\\\n \\lefteqn{\\mbox{(critical wetting, field like)}} \\\\\n\\lefteqn{\\Phi_0 \\ll -1:} \\hspace{1.7cm} \n&& f(\\Phi_0) = (\\Phi_0^2/2)^{1/(1-\\omega)} \\; \ng_2 \\big((\\Phi_0^2/2)^{-1/(1-\\omega)}\\big) \\nonumber \\\\\n\\mbox{with} \\quad && \ng_2(x) \\approx \\; 1 + \\frac{3 x}{2(1-\\omega)} - \n\\frac{(2 + 7 \\omega) x^2}{8(1-\\omega)^2} + \\cdots \\nonumber\\\\\n \\lefteqn{\\mbox{(partial wetting)}} \n\\end{eqnarray}\nThe point $\\Phi_0=0$ is the critical wetting point. If one approaches this\npoint from the partial wetting side $a_0 \\to 0^-$ on the coexistence line \n$\\mu = 0$, the parallel correlation length $\\xi_{\\parallel}$ diverges with\nthe well-known nonuniversal exponent\n\\begin{equation}\n\\xi_{\\parallel} = (2 \\pi \\omega/b)^{-1/2(1-\\omega)} \\; (-a_0)^{-1/(1-\\omega)},\n\\end{equation}\nand the distance between the average position of the interface and the surface\ndiverges asymptotically like\n\\begin{equation}\n\\label{ell0}\n\\langle l \\rangle \\to - (1 + 2 \\omega)/(1+\\omega) \\; \\ln (-a_0).\n\\end{equation}\n\nThe relevant regime for most cases of surface induced disorder is however the \ncritical wetting regime, where the critical wetting point is approached under\na finite angle to the coexistence line in $(a_0, \\mu)$ space.\nHere the parallel correlation length $\\xi_{\\parallel}$ scales like\n\\begin{equation}\n\\label{xipar}\n\\xi_{\\parallel} = \\frac{1}{\\sqrt{8 \\pi \\omega}} \\; \\mu^{-\\nu_{\\parallel}} \n\\qquad \\mbox{with} \\qquad \\nu_{\\parallel} = 1/2. \n\\end{equation}\nas $\\mu$ approaches zero, the width of the interface diverges with\n\\begin{equation}\n\\label{xiperp}\nW^2 \\to \\frac{\\pi}{2} \\xi_{\\perp}^2 = - \\frac{\\pi}{2} \\omega \\ln (\\mu),\n\\end{equation}\nand its average position with\n\\begin{equation}\n\\label{ell1}\n\\langle l \\rangle \\approx - (\\omega + 1/2) \\; \\ln (\\mu).\n\\end{equation}\n\nThese results can be used to derive the\nlayer-bulk susceptibility of the order parameter in the interfacial region\n\\begin{equation}\n\\label{chil}\n\\chi_{0,\\langle l \\rangle} \n= \\frac{\\partial \\langle m_0 \\rangle }{\\partial \\mu} \\Big|_{\\langle l \\rangle}\n%&=& \\frac{\\partial \\langle m \\rangle }{\\partial \\langle l \\rangle} \\;\n% \\frac{\\partial \\langle l \\rangle} {\\partial \\mu}\n%+ \\frac{\\partial \\langle m \\rangle }{\\partial \\xi_{\\perp}{}^2} \\;\n% \\frac{\\xi_{\\perp}{}^2} {\\partial \\mu} \\nonumber \\\\\n\\propto -\\frac{\\partial \\langle m_0 \\rangle}{\\partial z} \n\\Big|_{\\langle l \\rangle}\\;\n \\frac{\\partial \\langle l \\rangle} {\\partial \\mu}\n\\propto \\frac{1}{\\mu \\sqrt{ln (\\mu)}}.\n\\end{equation}\nIn the step approximation $m_{0,\\mbox{\\small bare}}(z) = m_{0,b} \\theta(z)$,\nthe layer-bulk susceptibility in the interfacial region can be calculated in \nmore detail:\n\\begin{equation}\n\\label{chiz}\n\\chi_{0,z} = \\frac{m_{0,b}}{\\sqrt{2 \\pi} \\, \\xi_{\\perp} \\, \\mu} \\;\ne^{-(z-\\langle l \\rangle)^2/2 \\xi_{\\perp}{}^2}\n\\big( \\omega +\\frac{1}{2} - \\frac{z-\\langle l \\rangle}{2 \\ln \\mu}\\big).\n\\end{equation}\nIt has a slightly asymmetric peak of width $\\xi_{\\perp}$ at \n$z = \\langle l \\rangle$, the height of which scales like $1/\\mu$. \n\nThe layer-layer susceptibility could already be derived in the previous\nsection. It also has a peak at the interface, which is however a factor of\n$\\sqrt{2}$ narrower. Its height scales like\n\\begin{equation}\n\\label{chill}\n\\chi_{\\langle l \\rangle\\langle l \\rangle} \n\\propto \\xi_{\\parallel}{}^2/\\xi_{\\perp}{}^2\n\\propto -1/(\\mu \\; \\ln(\\mu)).\n\\end{equation}\n\nNext we determine the critical behavior of the order parameter at the \nsurface, $m_{0,1}$, \n\\begin{equation}\n\\label{m1_1}\nm_{0,1} \\propto - \\frac{\\partial F_s}{\\partial h_{0,1}}\n\\propto - \\frac{\\partial\\xi_{\\parallel}^{-2}}{\\partial a_0}\n\\propto \\mu^{\\beta_{0,1}},\n\\qquad \\beta_{0,1} = \\frac{1+\\omega}{2} \n\\end{equation}\nIt will prove useful to rederive the exponent $\\beta_{0,1}$ \nin an alternative way: The surface order parameter in mean\nfield theory is given by $m_{\\mbox{\\small bare}}(0)=m_b\\exp(-l/\\lambda_0)$ \nAveraging the profile according to eqn. (\\ref{maver}) yields\n\\begin{equation}\n\\label{m1_2}\nm_{0,1} = m_b \\langle e^{-l/\\lambda_0} \\rangle_{P(l)} \n= m_b e^{- \\langle l \\rangle/\\lambda_0 + \\xi_{\\perp}^2/2 \\lambda_0^2}.\n\\end{equation}\nAfter inserting $\\lambda_0 = 1$ and using eqns. (\\ref{ell1}) and \n(\\ref{xiperp}), one recovers the power law of eqn. (\\ref{m1_1}) \nwith the same exponent $\\beta_{0,1}$.\nThe approach has the advantage that it allows for a straightforward \ncalculation of finite size effects on surface critical behavior:\nWe simply replace the expression (\\ref{xiperp}) for $\\xi_{\\perp}$ in the \ninfinite system by eqn. (\\ref{xipfss}) to obtain\n\\begin{equation}\n\\label{m1s1}\nm_{0,1} \\propto m_b \\; \\mu^{\\beta_{0,1}} \\; \n\\hat{M}_0( 8 \\pi \\omega \\; \\mu L^{1/\\nu_{\\parallel}} )\n\\end{equation}\nwith the scaling function\n\\begin{equation}\n\\label{m1s2}\n\\hat{M}_0(x) = \\big( \\frac{x}{x+1} \\big)^{\\omega/2} \\;\ne^{2 \\pi \\omega /x}.\n\\end{equation}\n\nWe are now ready to generalize these results to the case of several order \nparameters and nonordering densities. Formally, the theory turns out to remain \nvery similar. The self consistent determination of $\\xi_{\\parallel}$ leads to a \ngeneralized version of the scaling form for the singular part of the surface \nfree energy (\\ref{fs0}),\n\\begin{equation}\n\\label{fs1}\nF_s \\propto \\xi_{\\parallel}{}^{-2} = {8 \\pi \\omega} \\; {\\mu} \\; \nf(\\{\\Phi_i\\}),\n\\end{equation}\nwhere the scaling variables are\n\\begin{equation}\n\\Phi_i = C_i \\; \\mu^{(1-2 \\lambda_i)(1-\\omega/\\lambda_i)/2 \\lambda_i} \\; a_i \n\\end{equation}\n\\begin{displaymath}\n\\qquad \\mbox{with} \\qquad\nC_i = (8\\pi \\omega)^{\\omega/2 \\lambda_i^2 \\cdot (2 \\lambda_i - 1)}\n(2 b)^{-1/2 \\lambda_i}.\n\\end{displaymath}\n\nAs in the one-order parameter case, we have to distinguish between different\nregimes depending on the values of the scaling variables. \n\n\\subsection{Symmetry preserving and symmetry breaking surfaces}\n\\label{surface}\n\nLet us first assume that the effect of nonordering densities can be\ndisregarded ({\\em e.g.}, because the associated length scales are small,\n$\\lambda_i < 1/2$), and consider the case of a symmetry preserving surface.\nNo ordering surface fields are then present, {\\em i.e.}, \n$a_i \\propto h_i = 0$ for all contributions $i$. The system is thus in a\n``multicritical wetting regime'', where $|\\Phi_i | \\ll 1$ for all $i$, \nand the scaling function can be expanded as\n\\begin{equation}\nf(\\{ \\Phi_i \\}) = 1 - \\sum_i \\Phi_i \\: \\frac{2 \\lambda_i - 1}{2 \\lambda_i^2}\n + \\cdots\n\\end{equation}\nThe effective interface position $\\langle l \\rangle$, and the correlation\nlength $\\xi_{\\parallel}$ are given by eqns. (\\ref{ell1}) and (\\ref{xipar})\nas in the case of normal critical wetting. Hence all the results related to\ninterfacial properties, such as the interfacial width, the interfacial\nlayer susceptibilities etc., remain unchanged. In particular, the\ncriterion for the validity of the weak fluctuation limit is still\n$\\omega < 1/2$ (from eqns. (\\ref{wfl}), (\\ref{xiperp}) and (\\ref{ell1})).\nThe surface order parameters obey the power law\n\\begin{equation}\nm_{i,1} \\propto - \\frac{\\partial \\xi_{\\parallel}^{-2}}{\\partial a_i}\n\\propto \\mu^{\\beta_{i,1}}, \\quad\n\\beta_{i,1} = \\frac{1}{2 \\lambda_i} \n+ \\frac{\\omega}{2 \\lambda_i^2} (2 \\lambda_i - 1).\n\\end{equation}\nFollowing the lines of (\\ref{m1_2}), one also obtains the finite size scaling \nfunction\n\\begin{equation}\n\\label{mscal}\n\\hat{M}_i(x) = \\big( \\frac{x}{x+1} \\big)^{\\omega/2 \\lambda_i^2} \\;\ne^{2 \\pi \\omega /x \\lambda_i^2}.\n\\end{equation}\n\nA whole sequence of surface exponents is thus predicted, one for each order \nparameter. In practice, however, one hardly ever measures only one \"pure\" order \nparameter $m_i$. Instead, one expects to observe some combination of \ncontributions with different exponents $\\beta_{i,1}$, which will be dominated \nby the leading exponent $\\beta_{0,1} = (\\omega + 1)/2$ in the asymptotic \nlimit $\\mu \\to 0$.\n\nThe situation changes when at least one of the $a_i$ becomes nonzero at\ncoexistence. This is the case, {\\em e.g.}, at a symmetry breaking surface, \nwhere one or several surface fields become nonzero, or even at a symmetry\npreserving surface if the length scale associated with a nonordering\ndensity exceeds half the bulk correlation length, $\\lambda_i > 1/2$.\n\nLet $a_J e^{-l/\\lambda_J}$ be the leading nonvanishing term in the potential \n(\\ref{vbare}). As one approaches coexistence, $\\mu \\to 0$, the scaling variable \n$\\Phi_J$ increases and one eventually enters a regime $|\\Phi_J| \\gg 1$. \nFor negative $a_J$, ($\\Phi_J \\ll -1$), the wetting becomes partial, {\\em i.e.}, \nno surface induced disordering takes place.\nFor positive $a_J$, ($\\Phi_J \\gg 1$), different scenarios are possible,\ndepending on the sign and the amplitude of the higher order terms $a_i$,\n($i>J$) in eqn. (\\ref{vxi}). If they are positive or sufficiently small, \nsuch that\n\\begin{equation}\n\\label{small}\n|a_i a_J^{-\\lambda_J/\\lambda_i}| \\ll 1,\n\\end{equation}\nthe disordered phase wets the surface. The effective interface \nposition $\\langle l \\rangle$ diverges asymptotically like\n\\begin{equation}\n\\label{ell2}\n\\langle l \\rangle \\approx - \\lambda_J (1 + \\omega/2 \\lambda_J^2) \\; \\ln (\\mu),\n\\end{equation}\nthe parallel correlation length scales like\n\\begin{equation}\n\\xi_{\\parallel} = \\sqrt{\\lambda_J/4 \\pi \\omega \\mu},\n\\end{equation}\nand the scaling function in eqn. (\\ref{fs1}) takes the form\n\\begin{displaymath}\nf(\\{\\Phi_i\\}) =\n\\frac{1}{2 \\lambda_J}\n\\Big( 1 + \\sum_i \\Phi_i \\Phi_J^{-\\lambda_J/\\lambda_i} K_J(\\lambda_i) \\Big.\n\\end{displaymath}\n\\begin{equation}\n\\label{fs2}\n\\qquad \\Big. + \\; \\frac{1}{2} \\Phi_J^{-2 \\lambda_J} K_J(\\frac{1}{2}) \\; \\Big).\n\\end{equation}\nwith\n\\begin{displaymath}\n K_J(\\lambda_i) = \\frac{\\lambda_J^{\\lambda_J/\\lambda_i}}{\\lambda_i} \\;\n(\\frac{\\lambda_J}{\\lambda_i} - 1) \\;\n(2 \\lambda_J)^{\\omega/2 \\lambda_i^2 \\cdot (1-\\lambda_i/\\lambda_J)}.\n\\end{displaymath}\nAccording to eqn. (\\ref{wfl}), the weak fluctuation regime here is bounded by \n$\\omega < 2 \\lambda_J^2$, thus encompassing the regime $\\omega < 1/2$.\n\nThe criterion (\\ref{small}) is motivated as follows: If one of the higher\norder $a_i$ is negative and large, the interface potential \n$V_{\\xi_{\\parallel}}(l)$ may exhibit a second minimum closer to the\nsurface, which competes with the minimum at large $l$ and may prevent the\nformation of an asymptotically diverging wetting layer. \nThe inspection of the free energy scaling function (\\ref{fs2}) reveals \nthat the transition to such a partial wetting regime is appropriately described\nin terms of the combined scaling variable\n\\begin{displaymath}\n\\tilde{\\Phi}_{i,J} = \\Phi_i \\Phi_J^{-\\lambda_J/\\lambda_i}\n\\propto\na_i a_J^{-\\lambda_J/\\lambda_i} \n\\mu^{(\\lambda_J/\\lambda_i - 1)(1- \\omega/(2\\lambda_i\\lambda_J))}.\n\\end{displaymath}\nThis quantity has to be large at the point $\\mu_0$ where the one minimum\nof $V_{\\xi_{\\parallel}}(l)$ splits up in two.\nThe condition (\\ref{small}) ensures that $\\tilde{\\Phi}_{i,J}$ is small\nfor all $\\mu$.\n\nThe wetting is critical with respect to all order parameters $m_i$ with length \nscales $\\lambda_i$ larger than $\\lambda_J$. As coexistence is approached,\nthey vanish at the surface according to the power law\n\\begin{equation}\n\\label{b1_2}\nm_{i,1} \\propto - \\frac{\\partial \\xi_{\\parallel}^{-2}}{\\partial a_i}\n\\propto \\mu^{\\beta_{i,1}}, \\quad\n\\beta_{i,1} = \\frac{\\lambda_J}{\\lambda_i}\n+ \\frac{\\omega}{2 \\lambda_i^2} (\\frac{\\lambda_i}{\\lambda_J} - 1).\n\\end{equation}\nThe finite size scaling function $\\hat{M}_i(x)$ is again given by \n(\\ref{mscal}), with the scaling variable $x = 4 \\pi \\omega \\mu L^2 /\\lambda_J$.\nNote that the exponents $\\beta_{i,1}$ are nonuniversal even in the mean field \nlimit ($\\omega = 0$). This remarkable effect has first been discovered by \nHauge\\cite{hauge} and later studied by Kroll/Gompper in an fcc Ising \nantiferromagnet using a mean field approximation\\cite{gerhard1}, \nMonte Carlo simulations, and a linear renormalization group study similar to \nthe one presented here\\cite{gerhard2}. However, $\\langle l \\rangle$ in this\nwork is taken from eqn. (\\ref{ell0}) rather than determined self consistently,\nhence the resulting critical exponents differ somewhat from those calculated\nhere. As in the case of the symmetry preserving surface, a whole set of\nexponents is predicted by eqn. (\\ref{b1_2}). In the asymptotic limit\n$\\mu \\to 0$, however, the surface behavior is expected to be governed\nby the leading exponent\n\\begin{equation}\n\\label{b1_3}\n\\beta_{0,1} = \\frac{\\xi_b}{\\lambda_J}\n+ \\frac{\\omega}{2} (\\frac{\\xi_b}{\\lambda_J} - 1).\n\\end{equation}\nWe have reinserted the bulk correlation length $\\xi_b \\equiv 1$ here.\n\nFinally, we discuss the critical behavior of the surface susceptibilities.\nThe corresponding critical exponents can be shown to obey simple scaling laws. \nIn the case of the surface-bulk susceptibility, the relation follows trivially:\n\\begin{equation}\n\\label{chi1}\n\\chi_{i,1} \\propto \\frac{\\partial m_{i,1}}{\\partial \\mu} \n\\propto \\mu^{-\\gamma_{i,1}}, \n\\qquad \\gamma_{i,1} = 1 - \\beta_{i,1}.\n\\end{equation}\nIn the case of the surface-surface susceptibility, it depends on the regime\nunder consideration. In the ''critical wetting regimes'' discussed here,\nthe free energy scaling function $f$ can be expressed as a Taylor series\nin powers of the scaling variables $\\Phi_i$ or $\\tilde{\\Phi}_{i,J}$, \nrespectively, and one\nobtains\n\\begin{equation}\n\\label{chi11}\n\\chi_{i,11} \\propto \\frac{\\partial m_{i,1}}{\\partial h_{i,1}} \n\\propto \\mu \\frac{\\partial^2 f}{\\partial a_i{}^2}\n\\propto \\mu^{-\\gamma_{i,11}}, \n\\qquad \\gamma_{i,11} = 1 - 2 \\beta_{i,1}.\n\\end{equation}\nThe dominating exponents in the asymptotic limit are \n$\\gamma_{0,1}$ and $\\gamma_{0,11}$.\n\n\\section{Modeling order-disorder transition in bcc-alloys}\n\\label{model}\n\nFig. \\ref{phases} shows some typical structures of binary (AB) bcc-alloys \n({\\em e.g.}, FeAl \\cite{kubaschewski}). It is useful to divide the bcc lattice \ninto four fcc-sublattices a--d as indicated in the Figure. \nThe phase transitions are then conveniently \ndescribed in terms of a set of order parameters\\cite{duenweg}\n\n\\begin{eqnarray}\n\\psi_1 &=& \\big( c_a + c_b - c_c - c_d) \\nonumber \\\\\n\\psi_2 &=& \\big( c_a - c_b + c_c - c_d) \\\\\n\\psi_3 &=& \\big( c_a - c_b - c_c + c_d), \\nonumber\n\\end{eqnarray}\nwhere $c_{\\alpha}$ denotes the composition on the sublattice $\\alpha$,\n{\\em i.e.}, the average concentration of one component $A$ there. \nIn the disordered phase, all sublattice compositions are equal and\nthese order parameters vanish. The B2 phase is characterized by \nnonzero $\\psi_1$, and the DO${}_3$ phase in addition by nonzero\n$\\psi_2 = \\pm \\psi_3$. By symmetry, physical quantities have to be \ninvariant under sublattice exchanges $(a \\leftrightarrow b)$, \n$(c \\leftrightarrow d)$, and $(a,b) \\leftrightarrow (c,d)$. \nThe leading terms in a Landau expansion of the free energy $F$ thus read\n\\begin{eqnarray}\n\\label{flandau}\n F &=& F_0 + A_1 \\psi_1^2 + A_2 (\\psi_2^2 + \\psi_3^2) \n + B \\; \\psi_1 \\psi_2 \\psi_3 \\\\\n&& + C_1 \\psi_1^4 + C_2 (\\psi_2^4 + \\psi_3^4) \n + C_3 \\psi_2^2 \\psi_3^2 + C_4 \\psi_1^2 (\\psi_2^2+\\psi_3^2),\\nonumber\n\\end{eqnarray}\nWe point out in particular the cubic term $B \\psi_1 \\psi_2 \\psi_3$. \nIt can be read in two ways. On the one hand, it describes how the B2-order \ninfluences the DO${}_3$ order: The order parameter $\\psi_1 $ breaks the \nsymmetry with respect to indi-\n\\begin{figure}[b]\n\\noindent\n\\hspace*{0.5cm}\\fig{phases.eps}{75}{35}{\n\\vspace*{0.5cm}\n\\caption{bcc lattice with (a) disordered A2 structure (b) B2 structure, \nand (c) DO${}_3$ structure. Also shown is the assignment of \nsublattices a,b,c,d.}\n\\label{phases}\n}\n\\end{figure}\n\\noindent\nvidual sign reversal of $\\psi_2$ or $\\psi_3$ \nand orients $(\\psi_2,\\psi_3)$ such that \n$\\psi_2 = - \\mbox{sign}(B \\psi_1)\\; \\psi_3$. Conversely, one can interpret\nthe product $\\psi_2 \\psi_3$ as an effective ordering field acting on $\\psi_1$. \nWe shall come back to this point later.\n\nAt the presence of surfaces, the situation is even more complicated.\nFirst, we can always expect that one component enriches at the surface, \nsince there are no symmetry arguments to prevent that.\nEven if no explicit surface field coupling to the total concentration $c$ is \napplied, the component which is in excess with respect to the ideal \nstoechiometry of the bulk phase ((3:1) in the DO${}_3$ phase) \nwill segregate to the surface. Second, we have already mentioned \nthat the Landau expansion of the surface free energy $f_s$ depends \non the orientation of the surface\\cite{ich1,diehl1}. \nThe (110) surface has the same symmetry with respect to sublattice exchanges \nas the bulk, hence the Landau expansion of the surface free energy must have \nthe form (\\ref{flandau}). In case the order is sufficiently suppressed at the \nsurface, one can thus hope to find classical surface induced disordering here.\nIn the case of the (100) surface, the symmetry with respect to the exchange \n$(a,b)\\leftrightarrow (c,d)$ is broken. The surface enrichment\nof one component then induces an effective ordering surface field, which \ncouples to the order parameter \n$\\psi_1$\\cite{ich3}. \nOther ordering fields \ncoupling to $\\psi_2$ and $\\psi_3$ are still forbidden by symmetry. The full \nspectrum of possible ordering surface fields is allowed in the case of\nthe (111) surface.\n\nIn order to model these phase transitions, we consider an Ising model\nof spins $S_i = \\pm 1$ on the bcc-lattice with antiferromagnetic interactions \nbetween up to next nearest neighbors,\n\\begin{equation}\n{\\cal H} = V \\sum_{\\langle ij \\rangle} S_i S_j\n+ \\alpha V \\sum_{\\langle \\langle ij \\rangle \\rangle } S_i S_j \n- H \\sum_i S_i.\n\\end{equation}\nwhere the sum $\\langle ij \\rangle$ runs over nearest neighbour \nand $\\langle \\langle ij \\rangle \\rangle$ over next nearest neighbour pairs.\nSpins $S=+1$ represent A-atoms and $S=-1$ B-atoms, hence the concentration\n$c$ of A is related to the average spin $\\langle S \\rangle$ via\n\\begin{equation}\n\\label{cc}\nc = (\\langle S \\rangle + 1)/2,\n\\end{equation}\nand the field $H$ represents a chemical potential.\nThe parameter $\\alpha = 0.457$ is chosen such that the highest temperature\nwhich can still support a B2 phase is about twice as high as the highest \ntemperature of the DO${}_3$ phase, like in the experimental phase diagram\nof FeAl. The phase diagram of our model is shown in Figure \\ref{phdiag}.\n\nThe surface simulations were performed in a \n$L \\times L \\times D$ geometry with periodic boundary conditions in the \n$L$ direction and free boundary conditions in the $D$ \ndirection, varying $D$ from 100 to 200 and $L$ from 20 to 100. \nIn order to handle systems of that size efficiently, we have \ndeveloped\\cite{frank} a multispin code\\cite{bhanot}, which allowed to store \nthe configurations bitwise instead of bytewise\\cite{fn2}. \nOur Monte Carlo runs had total lengths of up to $2\\cdot 10^6$ \nMonte Carlo sweeps.\n\n\\begin{figure}\n\\noindent\n\\hspace*{1.5cm}\\fig{phdiag.eps}{65}{68}{\n\\vspace*{0.cm}\n\\caption{Phase diagram of our model in the $T-H$ plane. Solid lines mark first\norder phase transitions, dashed lines second order phase transitions.\nArrows indicate the positions of a critical end point (cep) and a\ntricritical point (tcp).\n}\n\\label{phdiag}\n}\n\\end{figure}\n\\noindent \n\n\\section{Simulation results}\n\\label{results}\n\nWe have studied (110) and (100) oriented surfaces at $T = 1 \\: V/k_B$\nclose to the first order bulk transition between the ordered DO${}_3$ phase \nand the disordered A2 phase. The exact bulk transition point was\ndetermined previously from bulk simulations by thermodynamic \nintegration\\cite{KB2}, $H_0/V = 10.00771(1)$\\cite{frank}.\nIn the presence of such a high \nbulk field, the very top layer of a free (110) or (100) surface is completely \nfilled with $A$ particles, {\\em i.e.}, Ising spins $S=1$. Consequently, the \norder parameters $\\psi_{\\alpha}$ and the layer susceptibilities vanish there.\nIn the following, we shall generally disregard this top layer and \nanalyze the profiles starting from the second layer. \n\n\\subsection{(110) Surfaces -- DO${}_3$ order}\n\nWe begin with a detailed discussion of surface induced disordering\nat (110) surfaces, {\\em i.e.}, surfaces with the full symmetry of the bulk. \nFigure \\ref{psi23} shows profiles of the order parameter of DO${}_3$ ordering\nper site\n\\begin{equation}\n\\psi_{23} = \\sqrt{(\\psi_2{}^2 + \\psi_3{}^2)/2}.\n\\end{equation}\nOne clearly sees how a disordered layer forms and grows in thickness as the \nbulk transition is approached. In order to extract an interface position \n$\\langle l \\rangle$ and an effective interfacial width $W$, \nwe have fitted the profiles to a shifted tanh function\n\\begin{equation}\n\\label{fit}\n\\psi_{23}(n) = \\psi_{23}^{\\mbox{\\tiny bulk}}\n\\Big( 1 + \\exp \\big[- 2 \\; (z-\\langle l \\rangle)/W \\: \\big] \\; \\Big)^{-1}.\n\\end{equation}\nThe results are shown in Figs. \\ref{ll} and \\ref{w2}. Sufficiently close to \nthe bulk transition, at $(H_0-H)/V < 0.005$, the data are consistent with the \nlogarithmic divergence predicted by eqns. (\\ref{ell1}) and (\\ref{ew2}).\nIntuitively, one would expect that an effective interface theory is only\napplicable if $l > W$, \n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{psi23.eps}{75}{64}{\n\\vspace*{0.cm}\n\\caption{Profiles of $\\psi_{23}$ near a (110) surface at temperature \n$T=1 \\: V/k_B $ for different fields $H$ in units of $V$ as indicated. The bulk\ntransition is at $H_0/V = 10.00771(1)$.\nZeroth (top) layer is not shown ($\\psi_{23}(0) \\equiv 0$, see text).\n}\n\\label{psi23}\n}\n\\end{figure}\n\n\\begin{figure}\n\\noindent\n\\hspace*{1.cm}\\fig{ll.eps}{70}{60}{\n\\vspace*{0.cm}\n\\caption{Position of the interface as estimated from the fit (65)\nin units of (110) layers vs. $(H-H_0)/V$.}\n\\label{ll}\n}\n\\end{figure}\n\\begin{figure}\n\\noindent\n\\hspace*{0.8cm}\\fig{w2.eps}{74}{60}{\n\\vspace*{0.cm}\n\\caption{Squared interfacial width as estimated from the fit (65)\nin units of (110) layers vs. $(H-H_0)/V$. Long dashed\nline shows squared interface position $\\langle l \\rangle^2$ for\ncomparison.}\n\\label{w2}\n}\n\\end{figure}\n\\noindent\n{\\em i.e.}, the width of the interface is smaller than\nthe distance of the interface from the surface. Indeed, Fig. \\ref{w2} shows\nthat the logarithmic behavior sets in approximately at the value of $H$ where \n$l$ begins to exceed $W$. The prefactors of the logarithms in Figs \\ref{ll} \nand \\ref{w2} are predicted to be $(r/2+\\omega/r) \\sqrt{2} \\xi_b/a_0$ \nin the case of $\\langle l \\rangle$ (Fig. \\ref{ll}), and \n$\\pi \\omega \\xi_b^2/a_0^2$ in the case of $W^2$ (Fig. \\ref{w2}), \nwhere $\\xi_b$ is the bulk correlation length, $a_0$ the lattice constant, \na factor $\\sqrt{2}$ or $2$ accounts for the distance of (110) layers from \neach other in units of $a_0$, and the parameter \n$r = \\mbox{max}(1,2 \\lambda_J/\\xi_b)$ depends on the length scale\n$\\lambda_J$ of composition fluctuations (see the discussion in section \n\\ref{surface}). We shall see below that the surface data suggest \n$\\beta_1 = r/2 + \\omega (1/r - 1/2) = 0.618$. \nInserting this result, one derives \n$ 4.5[7] < \\xi_b/a_0 < 5.4[8]$ from Fig. \\ref{ll}, and $\\xi_b/a_0 > 7.8[8]$\nfrom Fig. \\ref{w2}. These values do not agree with each other within\nin the statistical error; the interfacial width seems to decrease too \nfast as one moves away from coexistence. \nYet the difference seems still acceptable, especially considering\nhow small the region of apparent logarithmic behavior is. It has been\nobserved in other systems\\cite{andreas2}, that the vicinity of surfaces\nalso affects the intrinsic width $W_0$ of an interface.\nMoreover, many non-diverging terms have been neglected in eqns. (\\ref{ell1}) \nand (\\ref{ew2}) which lead to systematic errors if one is not close enough \nto $H_0$. We note that $\\xi_b$ seems rather large for a system which is not \ncritical in the bulk. On the other hand, Fig. \\ref{psi23} shows that the bulk \norder parameter $\\psi_{23}$ decreases considerably as one approaches the \nphase transition point. This observations suggests that a critical point\nis at least nearby, although preempted by the first order transition from \nthe DO${}_3$ phase to the disordered phase. \n\nNext we consider the profiles of the layer susceptibilities of the\norder parameter $\\psi_{23}$. They can be determined from the simulation \ndata by use of the fluctuation relations\\cite{schweika}\n\\begin{eqnarray}\n\\chi_{z} &=& \\frac{N_{\\mbox{\\tiny total}}}{k_B T}\n \\Big( \\langle \\psi(z) \\psi_{\\mbox{\\tiny total}} \\rangle - \n\\langle \\psi(z) \\rangle \\: \\langle \\psi_{\\mbox{\\tiny total}} \\rangle \\Big) \\\\\n\\chi_{zz} &=& \\frac{N_{\\mbox{\\tiny layer}}}{k_B T}\n \\Big( \\langle \\psi(z)^2 \\rangle - \\langle \\psi(z) \\rangle^2 \\Big),\n\\end{eqnarray}\nwhere $\\psi$ is the order parameter under consideration, \n$N_{\\mbox{\\tiny layer}}$ denotes the number of sites in a layer, and\n$N_{\\mbox{\\tiny total}}$ the total number of sites.\nFig. \\ref{chiprof} shows that both the layer-bulk susceptibility $\\chi_z$ and \nthe layer-layer suszeptibility $\\chi_{zz}$ exhibit the expected peak in\nthe vicinity of the interface (eqns. (\\ref{chiz}) and (\\ref{chizz})).\nThe centers of the peaks \ncan be fitted nicely by Gaussians of width $\\xi_{\\perp}$ and \n$\\xi_{\\perp}/\\sqrt{2}$, respectively, where \n$\\xi_{\\perp}$ is calculated from the width $W$ of the order parameter \nprofile using $\\xi_{\\perp} = \\sqrt{2/\\pi} \\, W$. The wings of the\npeaks are not Gaussian any more, but asymmetric -- the layer susceptibilities\nare enhanced at the bulk side of the interface, and suppressed at the\nsurface side. Such an asymmetry has been predicted qualitatively for $\\chi_z$\nin eqn. (\\ref{chiz}), but not for $\\chi_{zz}$ (cf. eqn. (\\ref{chizz})). \nEven in the case of $\\chi_z$, the observed asymmetry is so strong that it\ncannot be brought into quantitative agree-\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{chin23.eps}{70}{65}{ \n\\vspace*{-60mm} \n\n(a) \n\n\\vspace*{60mm}\n}\n\n\\noindent\n\\hspace*{0.6cm}\\fig{chinn23.eps}{68}{55}{\n\\vspace*{-60mm} \n\n(b) \n\n\\vspace*{58mm}\n\\caption{\nProfiles of the layer-bulk susceptibility $\\chi_z$ (a) and\nthe layer-layer susceptibility $\\chi_{zz}$ (b) per site of\nthe order parameter $\\psi_{23}$ in units of $k_B T$, \nfor different fields $H$ in units of $V$ as indicated.\nSolid line shows the fit of a Gaussian of width \n(a) $\\xi_{\\perp}= (2/\\pi)^{1/2} W$ and (b) $\\xi_{\\perp}/2^{1/2}$\nto the profile corresponding to $H=10.004$. \nZeroth (top) layer is not shown ($\\chi(0) \\equiv 0$, see text).\n}\n\\label{chiprof}\n}\n\\end{figure}\n\\noindent\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{chin23m.eps}{75}{62}{ \n\\vspace*{0.2cm}\n\\caption{\nMaximum of the layer-bulk susceptibility $\\chi_z$ per site of\nthe order parameter $\\psi_{23}$ in units of $k_B T$ vs. $(H_0-H)/V$ \nfor different system sizes as indicated.\nSolid line shows a fit to a $(H_0-H)^{-1}$ behavior, \nand dashed line the same with logarithmic correction (see text).\n}\n\\label{chin23m}\n}\n\\end{figure}\n\\noindent\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{chinn23m.eps}{75}{62}{\n\\vspace*{0.2cm}\n\\caption{\nMaxima of the layer-layer susceptibility $\\chi_{zz}$ per site of\nthe order parameter $\\psi_{23}$ in units of $k_B T$ vs. $(H_0-H)/V$, \nfor different system sizes as indicated.\nInset shows bare data, with a fit to a power law behavior with \nunknown exponent (dotted line). In the main plot, the bulk contribution \nto $\\chi_{zz}$ has been subtracted. Solid line indicates the slope of \n$(H_0-H)^{-1}$, and dashed line the whole theoretical prediction including\nthe logarithmic correction.\n}\n\\label{chinn23m}\n}\n\\end{figure}\n\\noindent\nment with the theory. We recall that\nthe linear theory approximates the capillary waves of the interface by those \nof a free interface with some suitable long-wavelength cutoff, {\\em i.e.}, \nthey are taken to be distributed symmetrically about the mean interface \nposition. The failure of the theory to describe the details of the\nprofiles of $\\chi_z$ and $\\chi_{zz}$ presumably reflects the fact that\nthe capillary waves are in fact asymmetric. Nevertheless, the main\nfeatures of the profiles are captured by the theory.\n\nThe centers of the peaks are slightly more distant from the surface than \n$\\langle l \\rangle$ in Fig. \\ref{ll}, but the difference is not significant \n(up to three layers at $(H_0-H)/V=0.0007$). According to the theoretical\nprediction (\\ref{chil}) and (\\ref{chill}), the heights of the peaks should \ndiverge with $1/(H_0-H)$ with different logarithmic corrections.\nOur data are shown in Figs. \\ref{chin23m} and \\ref{chinn23m}.\nThe maxima of the layer-bulk susceptibility are best fitted by the\nsimple $1/(H-H_0)$ behavior, which the theory predicts as long as the \ninterfacial width is dominated by the intrinsic width $W_0$. \nIn the regime $(H_0-H)/V<0.005$, where the capillary wave broadening\nof the interface becomes significant, the data are also consistent \nwith the logarithmically corrected version \n$\\chi_z^{max} \\propto 1/(H_0-H)\\sqrt{|\\ln (H_0-H)|}$\n(see Fig. \\ref{chin23m}). \n\nThe analysis of the layer-layer susceptibility is more subtle. \nFrom a double logarithmic plot of the raw data, one is tempted to conclude\nthat the predicted $1/(H_0-H)$ \nbehavior is not valid; the data rather\nsuggest a divergence with a critical exponent $0.63$ \n(Fig. \\ref{chinn23m}, inset). However, since we are not aware of any \ntheoretical explanation which could motivate such an exponent, we believe \nthat the apparent power law behavior over roughly two decades of $(H_0-H)$ \nis most likely accidental. Looking at the values of $\\chi_{zz}$ close to the \ncenter of the slab (Fig. \\ref{chiprof}b)), one recognizes that the contribution\nof bulk fluctuations to $\\chi_{zz}$ is significant even close to $H_0$. \nThe situation is complicated by the fact that the bulk fluctuations increase \nconsiderably in the vicinity of $H_0$, although their amplitude does not \ndiverge. Within the crude approximation that the capillary waves of the \ninterface and the bulk fluctuations are uncorrelated, one can subtract \nthe latter as ``background''. The thereby corrected data agree reasonably well \nwith the theory, especially when taking into account the logarithmic \ncorrection $\\chi_{zz}^{max} \\propto 1/(H_0-H)|\\ln(H_0-H)|$ \n(Fig. \\ref{chinn23m}). \n\nWe proceed to study the properties of the system directly at the surface.\nFigure \\ref{psi230} shows the order parameter $\\psi_{23,1}$ in the first layer \n(recalling that the top (zeroth) \nlayer is discarded) as a function of $(H_0-H)$ for various \n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{psi230.eps}{75}{63}{ \n\\vspace*{0.cm} \n\\caption{\nOrder parameter $\\psi_{23,1}$ at the surface (first layer) vs. $(H_0-H)/V$ for \ndifferent system sizes $L\\times L \\times D$ as indicated. Solid line indicates\npower law with the exponent $\\beta_1 = 0.618$.\n}\n\\label{psi230}\n}\n\\end{figure}\n\\noindent\n\n\\begin{figure}\n\\hspace*{0.cm}\\fig{psi230s.eps}{80}{68}{\n\\vspace*{-0.cm} \n\\caption{\nFinite-size scaled plot of the surface order parameter $\\psi_{23,1}$ \nvs. $(H_0-H)/V$ for system sizes $L\\times L \\times D$ as indicated. Data were\nscaled with exponents $\\nu_{\\parallel}=1/2$ and $\\beta_1=0.618$. Dashed\nline shows the finite size scaling function predicted by eqn. (51).\n}\n\\label{psi230s}\n}\n\\end{figure}\n\\noindent\nsystem sizes. One notices finite size effects if the dimension $L$ \nparallel to the interface is small. As long as $L$ is large enough, the data \nexhibit a power law behavior with \nthe exponent $\\beta_1=0.618[4]$. \nWe emphasize that $\\beta_1$ clearly differs from 1/2 here. It is close to the\nvalue $\\beta_1=0.64$ found by Schweika et al in their simulations of \nsurface induced disorder in fcc-alloys\\cite{schweika}. As discussed in\nsection \\ref{surface}, several factors may lead to such a nonuniversal\nexponent -- capillary wave fluctuations, and/or the presence of a\nlength scale $\\lambda_J > \\xi_b/2$, which competes with the correlation\nlength $\\xi_b$ and would have to be associated with the nonordering \ncomposition fluctuations in the case of the symmetry preserving (110) surface.\nUsing eqn. (\\ref{b1_3}), we can derive upper bounds for the capillary\nparameter, $\\omega < 0.236$, and for $\\lambda_J$, $\\lambda_J/\\xi_b < 0.618$. \n\nAfter applying finite size scaling with the exponents $\\beta_1$ and\n$\\nu_{\\parallel} = 1/2$ (cf. eqn. (\\ref{m1s1}), the data collapse onto a \nsingle master curve. The form of the latter can be calculated from \neqn. (\\ref{m1s1}),\n\\begin{equation}\n\\psi_{23,1} \\: L^{\\beta_1/\\nu_{\\parallel}}\n\\propto \\frac{x^{r/2+\\omega/r}}{(x+1)^{\\omega/2}} \n\\; e^{2 \\pi \\omega /x}\n\\end{equation}\nwith $x \\propto (H_0-H) L^{1/\\nu_{\\parallel}}$ and\n$r = \\mbox{max}(1, 2 \\lambda_J/\\xi_b)$, where the two unknown \nproportionality constants are fit parameters and $\\omega=0.236$ was\nused (the result is only very barely sensitive to the choice of $\\omega$).\nFig. \\ref{psi230s} shows that the data agree nicely with the theoretical \nprediction.\n\nFigure \\ref{chin230} shows the layer-bulk susceptibility at the surface\nfor the order parameter $\\psi_{23}$. According to eqn. (\\ref{chi1}),\nit should diverge with the exponent $\\gamma_1 = 1-\\beta_1 = 0.382$.\nIndeed, the fit to our data in the region $(H_0-H)/V < 0.02$ yields \n$\\gamma_1 =0.37[5]$. \nIn the case of the layer-layer susceptibility,\nthe theory (\\ref{chi11}) predicts $\\gamma_{11} = 1-2\\beta_1 =-0.236$, \n{\\em i.e.}, $\\chi_{11}$ does not diverge at the phase\ntransition. In fact, it first increases as $H_0$ is approached, \n\n\\begin{figure}\n\\noindent\n\\hspace*{0.cm}\\fig{chin230.eps}{78}{68}{\n\\vspace*{-0.cm} \n\\caption{\nSurface layer-bulk susceptibility per site of the order parameter $\\psi_{23}$ \nvs. $(H_0-H)/V$ for different system sizes as indicated.\nSolid line marks a power law with the exponent $\\gamma_1=0.37$.\n}\n\\label{chin230}\n}\n\\end{figure}\n\\noindent\nbut then decreases for $(H_0-H)/V < 0.02$ (not shown). The layer-layer \nsusceptibility at the surface here behaves in a \nsimilar way as observed by Schweika {\\em et al} in their studies of surface \ninduced disorder at the (111) surface of an fcc-based alloy\\cite{schweika}.\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{profiles.eps}{75}{104}{\n\\vspace*{-0.1cm} \n\\caption{\nProfiles of the total composition $c = (\\langle S \\rangle+1)/2$ (top) and \nof the order parameter $\\psi_1$ (bottom) for different fields $H$ in units\nof $V$ as indicated. \nTop (zeroth) layer is not shown ($c(0)\\equiv 1, \\psi_1(0)\\equiv(0)$).\nThin dashed lines with squares show for comparison the profiles of \n$\\psi_{23}$ from Fig. 5.\n}\n\\label{profiles}\n}\n\\end{figure}\n\\noindent\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.cm}\\fig{chin1m.eps}{80}{65}{ \n\\vspace*{-0.cm}\n\\caption{\nMaximum of the layer-bulk susceptibility $\\chi_z$ per site of\nthe order parameter $\\psi_{1}$ in units of $k_B T$ vs. $(H_0-H)/V$ \nfor different system sizes as indicated.\nSolid line shows a fit to a $(H_0-H)^{-1}$ behavior, \nand dashed line the same with the appropriate logarithmic correction.\n}\n\\label{chin1m}\n}\n\\end{figure}\n\\noindent\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.cm}\\fig{chinn1m.eps}{80}{70}{\n\\vspace*{-0.2cm}\n\\caption{\nMaxima of the layer-layer susceptibility $\\chi_{zz}$ per site of\nthe order parameter $\\psi_{1}$ minus bulk contribution\nin units of $k_B T$ vs. $(H_0-H)/V$, for different system sizes as indicated.\nSolid line indicates the slope of \n$(H_0-H)^{-1}$, and dashed line the whole theoretical prediction including\nthe logarithmic correction.\nInset shows bare data, with a fit to a power law behavior (dotted line).\n}\n\\label{chinn1m}\n}\n\\end{figure}\n\\noindent\n\n\\subsection{(110) Surfaces -- B2 order}\n\nFrom the results discussed so far, we conclude that the behavior of the \norder parameter $\\psi_{23}$ can be understood nicely within the effective \ninterface theory of critical wetting. However, we shall see that this holds\nonly in part for the second order parameter, $\\psi_1$. \n\nFig. \\ref{profiles} shows profiles of $\\psi_1$ for different fields $H$. \nThey resemble those of $\\psi_{23}$, in particular the inflection point of the \nprofiles is located approximately at the same distance from the surface. \nThe upper part of Fig. \\ref{profiles} displays profiles of the total \nconcentration $c$ of $A$ particles (eqn. (\\ref{cc}). They exhibit some \ncharacteristic, $H$-independent oscillations in the first four layers, \nand the $A$ concentration is slightly enhanced in the disordered region.\nHowever, the overall variation is rather small.\n\nThe layer susceptibility profiles of the order parameter $\\psi_1$\nare qualitatively similar to those of $\\psi_{23}$ and not shown here. \nFig. \\ref{chin1m} demonstrates that the maximum of the layer-bulk\nsusceptibility evolves with the field $H$ as theoretically predicted,\n$\\chi_z^{max} \\propto 1/(H_0-H)\\sqrt{|\\ln (H_0-H)|}$. \nIn the case of the layer-layer susceptibility, the agreement with the \ntheoretically expected behavior \n$(\\chi_{zz}^{max} - \\chi_{zz}^{bulk}) \\propto 1/(H_0-H)|\\ln(H_0-H)|$ \nis not quite as convincing, but \nthe data are still consistent with the theory \nfor $(H-H_0)/V < 0.01$ (Fig. \\ref{chinn1m}). \nNote that the bare values of $\\chi_{zz}^{max}$ would again rather suggest \na power law, $\\chi_{zz}^{max} \\propto (H_0-H)^{-0.53}$ (Fig. \\ref{chinn1m},\ninset), which is however most likely accidental.\n\nHence the behavior of the order parameter $\\psi_1$ in the vicinity of the \ninterface is similar to that of the order parameter $\\psi_{23}$ and consistent \nwith the theory of critical wetting. The agreement however does not persist\nwhen looking right at the surface. Figs. \\ref{psi10} and \\ref{psi10s} \nshow how the value of $\\psi_1$ in the first surface layer depends on \n$(H_0-H)/V$. A power law behavior is found over one and a half decades of \n$(H_0-H)/V$, yet the exponent $\\beta_1(\\psi_{1})=0.801$ differs from that of \n$\\psi_{23,1}$, $\\beta_1(\\psi_{23})=0.618$ (Fig. \\ref{psi10}).\nMoreover, the data for different system sizes do not collapse if\none performs finite size scaling with the exponent $\\nu_{\\parallel}=1/2$\n(Fig. \\ref{psi10s}(a)). The collapse is significantly better\nif one assumes that the parallel correlation length diverges with the\nexponent $\\nu_{\\parallel} = 0.7 \\pm 0.05$ (Fig. \\ref{psi10s} (b)). \n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{psi10.eps}{77}{63}{ \n\\vspace*{-0.1cm} \n\\caption{\nOrder parameter $\\psi_{1}$ at the surface vs. $(H_0-H)/V$ for different\nsystem sizes $L\\times L \\times D$ as indicated. Solid line shows\npower law with the exponent $\\beta_1 = 0.801$.\n}\n\\label{psi10}\n}\n\\end{figure}\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.cm}\\fig{psi10s1.eps}{80}{65}{}\\\\\n\\noindent\n\\hspace*{0.2cm}\\fig{psi10s2.eps}{75}{65}{\n\\vspace*{0.1cm} \n\\caption{\nFinite-size scaled plots of the order parameter $\\psi_{1}$ at the surface \nvs. $(H_0-H)/V$ for system sizes $L\\times L \\times D$ as indicated. \nExponents are $\\beta_1=0.801$, $\\nu_{\\parallel}=0.5$ in (a), and\n$\\nu_{\\parallel}=0.7$ in (b). \n}\n\\label{psi10s}\n}\n\\end{figure}\n\nWe have no explanation for these unexpected findings. The discussion in\nsection \\ref{wetting} has shown that several surface exponents $\\beta_{i,1}$ \nmay be present in a system with several order parameters. Even though we\nhave argued that only the smallest exponent should survive in the asymptotic\nlimit $\\mu \\to 0$, the other power law contributions may conceivably \nstill dominate the behavior of certain quantities over a wide range of $\\mu$.\nHowever, the critical exponent $\\nu_{\\parallel}$ should in all cases \nremain invariably $\\nu_{\\parallel}=1/2$. Our results seem to indicate that \nthe behavior of the order parameter $\\psi_1$ at the surface is governed \nby a length scale, which differs from that given of the interfacial \nfluctuations, but which nonetheless diverges as $H_0$ is approached. \nNote that $\\nu_{\\parallel} \\approx 0.7$ is close to the exponent \n$\\nu = 0.63$ with which the bulk correlation length diverges at an Ising type \ntransition in three dimensions. Likewise, \nthe exponent $\\beta_1 = 0.801$\nfound here resembles the surface critical exponent of the ordinary transition,\n$\\beta_1\\sim 0.8$ \\cite{diehl2,KB3}. One might thus suspect that $\\psi_1$ in \nthe disordered surface layer becomes critical at $H_0$. However, such a co-\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{chin10.eps}{75}{63}{\n\\vspace*{-0.cm} \n\\caption{\nSurface layer-bulk susceptibility per site of the order parameter $\\psi_{1}$ \nvs. $(H_0-H)/V$ for different system sizes as indicated.\n}\n\\label{chin0}\n}\n\\end{figure}\n\\noindent\nincidence would seem rather surprising.\nFurthermore, we have noted earlier that \nthe combination $\\psi_2 \\psi_3$ acts as an ordering field on $\\psi_1$, \nhence $\\psi_1$ cannot become critical as long as $\\psi_{23}$ is not \nstrictly zero.\n\nFigure \\ref{chin0} shows the layer-bulk susceptibility at the surface as\na function of $(H_0-H)/V$. It decreases as $H_0$ is approached, hence the \nscaling relation $\\beta_1 + \\gamma_1 = 1$ is obviously not met for the order \nparameter $\\psi_{1,1}$. \n\n\\subsection{(100) Surfaces}\n\\label{100}\n\nFinally, we turn to the discussion of (100) surfaces. As already mentioned\nearlier, (100) surfaces break the symmetry with respect to the order\nparameter $\\psi_1$, an ordering surface field coupling to this order \nparameter is allowed and thus usually present\\cite{ich1,diehl1}. This field \nis often closely related to surface segregation\\cite{ich1,ich2}. \nIn our case, the excess component $A$ of the $DO{}_3$ segregates in the\nsurface layer and induces a staggered concentration field in the layers \nunderneath, which is equivalent to $\\psi_1$ ordering.\n\nThis is demonstrated in Fig. \\ref{prof_100}. The order parameters and the\ncomposition $c$ are defined based on the sublattice occupancies on two \nsubsequent layers of distance $a_0/2$, starting from the first layer underneath\nthe surface. The top layer is again disregarded, since it is entirely\nfilled with $A$ or $S\\equiv 1$. The profiles of $\\psi_1$ clearly\ndisplay the signature of an additional ordering tendency at the surface,\nwhich in fact reverses the sign of $\\psi_1$ in the top layers.\nHowever, the effect is rather weak and does not influence the system\nsignificantly deeper in the bulk. The profiles can be analyzed like those\nat the (110) surface, and mean interface positions and mean interfacial widths \ncan be extracted to yield figures very similar to Figs. \\ref{ll} and \\ref{w2}.\nThe amplitudes of the logarithmic divergences can again be used\nto estimate the bulk correlation length $\\xi_b$. From the mean \ninterface position, one \n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{prof_100.eps}{75}{113}{\n\\vspace*{-0.cm} \n\\caption{\nProfiles of the total concentration of $A$ (top, diamonds), of the order \nparameters $\\psi_1$ (bottom, circles) and $\\psi_{23}$ (bottom, squares) \nat $H=10.003$ (filled symbols) and at $H=10.007$ (open symbols).\nZeroth (top) layer is not shown.\n}\n\\label{prof_100}\n}\n\\end{figure}\n\\noindent\n\n\\begin{figure}\n\\noindent\n\\hspace*{0.5cm}\\fig{psi230_100.eps}{75}{65}{ \n\\vspace*{-0.cm} \n\\caption{\nOrder parameter $\\psi_{23}$ at the surface vs. $(H_0-H)/V$ for different\nsystem sizes $L\\times L \\times D$ as indicated. Solid line shows\npower law with the exponent $\\beta_1 = 0.61$.\n}\n\\label{psi230_100}\n}\n\\end{figure}\\noindent\ncalculates $4.9[7] < \\xi_b/a_0 < 5.8[8]$, and from the interfacial width,\n$\\xi_b/a_0 > 7.5[9]$, in agreement with the values obtained for the (110) \nsurface. Likewise, the study of the layer susceptibilities at the interface \ndoes not offer new surprises. The maxima of the layer-bulk susceptibilities\nfor both $\\psi_1$ and $\\psi_{23}$ grow according to a power law \n$\\chi_z \\propto (H_0-H)^{-1}$. The layer-layer susceptibility in the\ninterfacial region seems to grow with a different exponent ($\\sim 0.6$ like\nin the case of the (110) surface), yet after subtracting the ``background'' \nthe data are also consistent with the theoretically expected behavior.\nLast, we study how the surface value of the order parameter $\\psi_{23}$ \nevolves as the transition $H_0$ is approached. Fig. \\ref{psi230_100}\nshows that it vanishes according to a power law with the exponent \n$\\beta_1=0.61[2]$, which is within the error the same exponent as in the case\nof the (110) surface. As far as the surface behavior of $\\psi_{23}$ is \nconcerned, the (100) and the (110) surface are thus basically equivalent. \nThe weak ordering tendency of $\\psi_1$ has an at most slightly perturbing\neffect on the profiles of $\\psi_{23}$.\n\n\\section{Summary and Outlook}\n\\label{summary}\n\nWe have presented an extensive Monte Carlo study of\nsurface induced disorder in a simple spin lattice model for bcc-based \nbinary alloys. Our work complements earlier Monte Carlo simulations of\nSchweika {\\em et al}\\cite{schweika}, who have studied surface induced disorder \nin fcc-based alloys within a similar model.\nLike these authors, we observe critical wetting behavior\nwith nonuniversal exponents. We have discussed our results in terms of\nan effective interface model designed to describe a system with several\norder parameters. In such a complex material, nonuniversal exponents may\nresult both from fluctuation effects and from a competition of length scales.\n\nDue to the complicated order parameter structure in our system, however,\nour data could not fully be explained within a theory which traces \neverything back to the properties of a single interface between a\ndisordered and an ordered phase. The theory provides a satisfactory\npicture for the behavior of the order parameter describing \nthe DO${}_3$ ordering, $\\psi_{23}$, and in general for the structure\nin the interfacial region. However, it fails to predict the behavior of the \norder parameter of B2 ordering, $\\psi_1$, directly at the surface. \nOur data thus indicate that the fluctuations of $\\psi_1$ at the surface \nrequire special treatment. Parry and coworkers\\cite{swain1,parry} have \nrecently suggested an approach to a theory of wetting based on an effective \ninterface Hamiltonian with two ``interfaces'', the usual one separating\nthe phase adsorbed at the surface and the bulk phase, and a second one which \naccounts in an effective way for the fluctuations directly at the surface.\nOur problem seems to call for such an approach.\nUnfortunately, we are far from understanding even the constituting elements, \nthe fluctuations of $\\psi_1$ at the wall. We seem to observe a coupling \nbetween critical wetting and some kind of surface critical behavior of \n$\\psi_1$, the origin of which is unclear.\n\nHence already our simple, highly idealized model exhibits a complex and rather\nintriguing wetting behavior. In real alloys, numerous additional complications \nare present which will lead to an even richer and more interesting \nphenomenology. For example, long range \ninteractions are known to influence wetting transitions significantly. \nThe effect of van-der-Waals forces on wetting has been investigated in\ndetail\\cite{wetting}. Van-der-Waals forces are important in liquid-vapour \nsystems or binary fluids, but presumably irrelevant in alloys.\nInstead, elastic interactions caused by lattice distortions presumably \nplay an important role. \n\nFurthermore, real surfaces are never ideally smooth,\nbut have steps and islands. We have seen that the orientation of\nthe surface affects the surface ordering. In our study, we did not\nobserve dramatic differences between the (110) surface and the (100) surface. \nNevertheless, we expect that the influence of the surface orientation on the \nwetting behavior can be quite substantial, {\\em e.g.}, in situations with \nstrong surface segregation, or if surface orientations are involved \nwhich also break the symmetry with respect to the DO${}_3$ order \n({\\em e.g.}, the (111) surface).\nLikewise, we can expect that steps and islands will affect the ordering\nand the wetting properties of the alloy. It is well known in general that \nthe wetting behavior on corrugated or rough surfaces differs from\nthat on smooth surfaces\\cite{borgs,netz,swain2}. In addition, even a few steps or\nislands on an otherwise smooth, but symmetry breaking surface of an alloy \ncan have a dramatic effect on the ordering behavior, since every step\nchanges the sign of the ordering surface field.\n\n\\section*{Acknowledgments}\n\nWe wish to thank M. M\\\"uller and A. Werner for helpful discussions.\nF.F.H. acknowledges financial support from the Graduiertenf\\\"orderung of \nthe Land Rheinland Pfalz, and F.S. has been supported from the Deutsche \nForschungsgemeinschaft through the Heisenberg program.\n\n\\begin{thebibliography}{99}\n\\bibitem{wetting} for reviews on wetting see, {\\em e.g.},\n P. G. de Gennes, Rev. Mod. Phys. {\\bf 57}, 827 (1985);\n S. Dietrich in {\\it Phase Transitions and Critical Phenomena},\n C. Domb and J.L. Lebowitz eds (Academic Press, New York, 1988), Vol. 12;\n M. Schick in {\\it Les Houches, Session XLVIII -- Liquids at Interfaces},\n J. Charvolin, J. F. Joanny, and J. Zinn-Justin eds\n (Elsevier Science Publishers B.V., 1990).\n\\bibitem{wetting2} for short overviews on recent progress see, {\\em e.g.}, \n E. M. Blokhuis and B. Widom, \n Curr. Opns in Coll. Interf. Science {\\bf 1}, 424 (1996);\n G. H. Findenegg and S. Herminghaus,\n Curr. Opns in Coll. Interf. 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Phys. {\\bf 66}, 3667 (1977).\n\\bibitem{fn1} Under certain, rather unusual circumstances, the\n critical wetting point could be approached on a complete wetting path,\n however (see section \\ref{scaling}).\n\\bibitem{ich1} F. Schmid, Zeitschr. f. Phys. {\\bf B 91}, 77 (1993).\n\\bibitem{ich3}\n F. Schmid, in {\\em Stability of Materials}, p 173, A. Gonis {\\em et al} eds.,\n (Plenum Press, New York, 1996).\n\\bibitem{dosch2}\n S. Krimmel, W. Donner, B. Nickel, and H. Dosch, \n Phys. Rev. Lett. {\\bf 78}, 3880 (1997).\n\\bibitem{diehl1} \n A. Drewitz, R. Leidl, T. W. Burkhardt, and H. W. Diehl,\n Phys. Rev. Lett. {\\bf 78}, 1090 (1997);\n R. Leidl and H. W. Diehl,\n Phys. Rev. B {\\bf 57}, 1908 (1998); \n R. Leidl, A. Drewitz, and H. W. Diehl,\n Intnl. Journal of Thermophysics {\\bf 19}, 1219 (1998).\n\\bibitem{upton}\n P. J. Upton, D. Abraham, to be published.\n\\bibitem{helbing}\n W. Helbing, B. D\\\"unweg, K. Binder, and D. P. Landau,\n Z. Phys. B {\\bf 80}, 401 (1990).\n\\bibitem{gerhard1}\n D. M. Kroll, G. Gompper, Phys. Rev. B {\\bf 36}, 7078 (1987);\n\\bibitem{gerhard2}\n G. Gompper, D. M. Kroll, Phys. Rev. B {\\bf 38}, 459 (1988).\n\\bibitem{dosch4}\n L. Mail\\\"ander, H. Dosch, J. Peisl, R.L. Johnson,\n Phys. Rev. Lett. {\\bf 64}, 2527 (1990).\n\\bibitem{schweika1}\n W. Schweika, K. Binder, and D. P. Landau,\n Phys. Rev. Lett. {\\bf 65}, 3321 (1990).\n\\bibitem{schweika3}\n W. Schweika, D.P. Landau, in {\\em Computer Simulation Studies in\n Condensed-Matter Physics X}, P. 186 (1997).\n\\bibitem{sundaram}\n V. S. Sundaram, B. Farrell, R. S. Alben, and W. D. Robertson, \n Phys. Rev. Lett. {\\bf 31}, 1136 (1973).\n\\bibitem{rae}\n E. G. McRae and R. A. Malic, Surf. Sci. {\\bf 148}, 551 (1984).\n\\bibitem{alvarado} \n S. F. Alvarado, M. Campagna, A. Fattah, and W. Uelhoff, Z. Phys. B {\\bf 66}, 103 (1987).\n\\bibitem{dosch3}\n H. Dosch, L. Mail\\\"ander, A. Lied, J. Peisl, F. Grey, R. L. Johnson, and\n S. Krummacher, Phys. Rev. Lett. {\\bf 60}, 2382 (1988);\n H. Dosch, L. Mail\\\"ander, H. Reichert, J. Peisl, and R. L. Johnson, \n Phys. Rev. B {\\bf 43}, 13172 (1991).\n\\bibitem{ricolleau}\n Ch. Ricolleau, A. Loiseau, F. Ducastelle, and R. Caudron,\n Phys. Rev. Lett. {\\bf 68}, 3591 (1992).\n\\bibitem{sundaram2}\n V. S. Sundaram, R. S. Alben, \n and W. D. Robertson, Surf. Sci. {\\bf 46}, 653 (1974).\n\\bibitem{schweika} \n W. Schweika. D.P. Landau, and K. Binder, Phys. Rev. B {\\bf 53}, 8937 (1996).\n\\bibitem{KB1} \n K. Binder, D. P. Landau, and D. M. Kroll, \n Phys. Rev. Lett. {\\bf 56}, 2272 (1986);\n K. Binder and D. P. Landau, Phys. Rev. B {\\bf 37}, 1745 (1988);\n K. Binder, D. P. Landau, and S. Wansleben, \n Phys. Rev. B {\\bf 40}, 6971 (1989).\n\\bibitem{bhl}\n E. Br\\'ezin, B. I. Halperin, and S. Leibler, \n Phys. Rev. Lett. {\\bf 50}, 1387 (1983).\n\\bibitem{fh}\n D. S. Fisher and D. A. Huse, Phys. Rev. B {\\bf 32}, 247 (1985).\n\\bibitem{fjin}\n M. E. Fisher and A. J. Jin, Phys. Rev. Lett. {\\bf 69}, 792 (1992).\n\\bibitem{boulter}\n C. J. Boulter, Phys. Rev. Lett. {\\bf 79}, 1897 (1997).\n\\bibitem{swain1}\n P. S. Swain and A. O. Parry, Europhys. Lett. {\\bf 37}, 207 (1997).\n\\bibitem{hauge}\n E.H. Hauge, Phys. Rev. B {\\bf 33}, 3322 (1986).\n\\bibitem{lkz}\n R. Lipowsky, D. M. Kroll, and R. K. P. Zia, \n Phys. Rev. B {\\bf 27}, 4499 (1983).\n\\bibitem{fisher}\n For a review see \n M. E. Fisher in {\\em Statistical Mechanics of Membranes and Surfaces},\n D. R. Nelson, T. Piran, R. B. Weinberg eds. \n (World Scientific, Singapore, 1989).\n\\bibitem{lf}\n R. Lipowsky and M. E. Fisher, Phys. Rev. B {\\bf 36}, 2126 (1987).\n\\bibitem{bedeaux}\n D. Bedeaux and J. D. Weeks, J. Chem. Phys. {\\bf 82}, 972 (1985).\n\\bibitem{ich2}\n F. Schmid and K. Binder, Phys. Rev. B {\\bf46}, 13553 (1992);\n F. Schmid, K. Binder, Phys. Rev. {\\bf B 46}, 13565 (1992).\n\\bibitem{andreas}\n A. Werner, F. Schmid, M. M\\\"uller, and K. Binder, \n J. Chem. Phys. {\\bf 107}, 8175 (1997).\n\\bibitem{kubaschewski} O. Kubaschewski in \n {\\em Iron -- Binary Phase Diagrams}, p 5, (Springer, Berlin 1982).\n\\bibitem{duenweg}\n B. D\\\"unweg and K. Binder, Phys. Rev. B {\\bf 36}, 6935 (1987).\n\\bibitem{frank}\n For details see F. F. Haas, Dissertation \n (Johannes Gutenberg Universit\\\"at Mainz 1998).\n\\bibitem{bhanot}\n G. Bhanot, D. Duke, and R. Salvator, J. Stat. Phys. {\\bf 44}, 985 (1986).\n\\bibitem{fn2} Our multispin code differs from the one described in \\cite{bhanot}\n in that we could not make use of bit operations, unfortunately, due to the \n complicated nature of the interactions in our model.\n\\bibitem{KB2}\n K. Binder, Z. Phys. B {\\bf 45}, 61 (1981).\n\\bibitem{andreas2}\n K. Binder, M. M\\\"uller, F. Schmid, A. Werner,\n J. Stat. Phys. {\\bf 95}, 1045 (1999).\n\\bibitem{diehl2}\n H. W. Diehl, in {\\em Phase Transitions and Critical Phenomena}, Vol. 8,\n C. Domb and J. Lebowitz eds. (Academic, London, 1983).\n\\bibitem{KB3}\n D. P. Landau and K. Binder, Phys. Rev. B {\\bf 41}, 4633 (1990).\n\\bibitem{parry}\n C. J. Boulter and A. O. Parry, Phys. Rev. Lett. {\\bf 74}, 3403 (1995);\n A. O. Parry and C. J. Boulter, Physica A {\\bf 218}, 77 (1995);\n C. J. Boulter and A. O. Parry, Physica A {\\bf 218}, 109 (1995);\n A. O. Parry, J. Phys.: Cond. Matt. {\\bf 8}, 10761 (1996).\n\\bibitem{borgs}\n C. Borgs, J. De Coninck, R. Koteck\\'y, and M. Zinque,\n Phys. Rev. Lett. {\\bf 74}, 2292 (1995);\n\\bibitem{netz}\n R. R. Netz and D. Andelman, Phys. Rev. E {\\bf 55}, 687 (1997).\n\\bibitem{swain2}\n A. O. Parry, P. S. Swain, and J. A. Fox,\n J. Phys.: Cond. Matt. {\\bf 8}, L659 (1996);\n P. S. Swain and A. O. Parry, \n J. Phys. A: Math. Gen. {\\bf 30}, 4597 (1997);\n P. S. Swain and A. O. Parry,\n Eur. Phys. J. B {\\bf 4}, 459 (1998).\n\\end{thebibliography}\n%\\end{multicols}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002161.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{wetting} for reviews on wetting see, {\\em e.g.},\n P. G. de Gennes, Rev. Mod. Phys. {\\bf 57}, 827 (1985);\n S. Dietrich in {\\it Phase Transitions and Critical Phenomena},\n C. Domb and J.L. Lebowitz eds (Academic Press, New York, 1988), Vol. 12;\n M. Schick in {\\it Les Houches, Session XLVIII -- Liquids at Interfaces},\n J. Charvolin, J. F. Joanny, and J. Zinn-Justin eds\n (Elsevier Science Publishers B.V., 1990).\n\\bibitem{wetting2} for short overviews on recent progress see, {\\em e.g.}, \n E. M. Blokhuis and B. Widom, \n Curr. Opns in Coll. Interf. Science {\\bf 1}, 424 (1996);\n G. H. Findenegg and S. Herminghaus,\n Curr. Opns in Coll. Interf. Science {\\bf 2}, 301 (1997).\n\\bibitem{ragil} K. Ragil, J. Meunier, D. Broseta, J. O. Indekeu, and D. Bonn,\n Phys. Rev. Lett. {\\bf 77}, 1532 (1996).\n\\bibitem{ross} D. Ross, D. Bonn, and J. Meunier, Nature {\\bf 400}, 737 (1999);\n D. Ross, D. Bonn, and J. Meunier, preprint 1999.\n\\bibitem{lipowsky1} \n R. Lipowsky, Phys. Rev. Lett. {\\bf 49}, 1575 (1982); \n R. Lipowsky and W. Speth, Phys. Rev. B {\\bf 28}, 3983 (1983).\n\\bibitem{kroll}\n D. M. Kroll and R. Lipowsky, Phys. Rev. B {\\bf 28}, 6435 (1983).\n\\bibitem{lipowsky2}\n R. Lipowsky, J. Appl. Phys. {\\bf 55}, 2485 (1984).\n\\bibitem{dosch1}\n H. Dosch, {\\em Critical Phenomena at Surfaces and Interfaces \n (Evanescent X-ray and Neutron Scattering)}, \n Springer Tracts in Modern Physics Vol 126 (Springer, Berlin, 1992).\n\\bibitem{cahn} J. W. Cahn, J. Chem. Phys. {\\bf 66}, 3667 (1977).\n\\bibitem{fn1} Under certain, rather unusual circumstances, the\n critical wetting point could be approached on a complete wetting path,\n however (see section \\ref{scaling}).\n\\bibitem{ich1} F. Schmid, Zeitschr. f. Phys. {\\bf B 91}, 77 (1993).\n\\bibitem{ich3}\n F. Schmid, in {\\em Stability of Materials}, p 173, A. Gonis {\\em et al} eds.,\n (Plenum Press, New York, 1996).\n\\bibitem{dosch2}\n S. Krimmel, W. Donner, B. Nickel, and H. Dosch, \n Phys. Rev. Lett. {\\bf 78}, 3880 (1997).\n\\bibitem{diehl1} \n A. Drewitz, R. Leidl, T. W. Burkhardt, and H. W. Diehl,\n Phys. Rev. Lett. {\\bf 78}, 1090 (1997);\n R. Leidl and H. W. Diehl,\n Phys. Rev. B {\\bf 57}, 1908 (1998); \n R. Leidl, A. Drewitz, and H. W. Diehl,\n Intnl. Journal of Thermophysics {\\bf 19}, 1219 (1998).\n\\bibitem{upton}\n P. J. Upton, D. Abraham, to be published.\n\\bibitem{helbing}\n W. Helbing, B. D\\\"unweg, K. Binder, and D. P. Landau,\n Z. Phys. B {\\bf 80}, 401 (1990).\n\\bibitem{gerhard1}\n D. M. Kroll, G. Gompper, Phys. Rev. B {\\bf 36}, 7078 (1987);\n\\bibitem{gerhard2}\n G. Gompper, D. M. Kroll, Phys. Rev. B {\\bf 38}, 459 (1988).\n\\bibitem{dosch4}\n L. Mail\\\"ander, H. Dosch, J. Peisl, R.L. Johnson,\n Phys. Rev. Lett. {\\bf 64}, 2527 (1990).\n\\bibitem{schweika1}\n W. Schweika, K. Binder, and D. P. Landau,\n Phys. Rev. Lett. {\\bf 65}, 3321 (1990).\n\\bibitem{schweika3}\n W. Schweika, D.P. Landau, in {\\em Computer Simulation Studies in\n Condensed-Matter Physics X}, P. 186 (1997).\n\\bibitem{sundaram}\n V. S. Sundaram, B. Farrell, R. S. Alben, and W. D. Robertson, \n Phys. Rev. Lett. {\\bf 31}, 1136 (1973).\n\\bibitem{rae}\n E. G. McRae and R. A. Malic, Surf. Sci. {\\bf 148}, 551 (1984).\n\\bibitem{alvarado} \n S. F. Alvarado, M. Campagna, A. Fattah, and W. Uelhoff, Z. Phys. B {\\bf 66}, 103 (1987).\n\\bibitem{dosch3}\n H. Dosch, L. Mail\\\"ander, A. Lied, J. Peisl, F. Grey, R. L. Johnson, and\n S. Krummacher, Phys. Rev. Lett. {\\bf 60}, 2382 (1988);\n H. Dosch, L. Mail\\\"ander, H. Reichert, J. Peisl, and R. L. Johnson, \n Phys. Rev. B {\\bf 43}, 13172 (1991).\n\\bibitem{ricolleau}\n Ch. Ricolleau, A. Loiseau, F. Ducastelle, and R. Caudron,\n Phys. Rev. Lett. {\\bf 68}, 3591 (1992).\n\\bibitem{sundaram2}\n V. S. Sundaram, R. S. Alben, \n and W. D. Robertson, Surf. Sci. {\\bf 46}, 653 (1974).\n\\bibitem{schweika} \n W. Schweika. D.P. Landau, and K. Binder, Phys. Rev. B {\\bf 53}, 8937 (1996).\n\\bibitem{KB1} \n K. Binder, D. P. Landau, and D. M. Kroll, \n Phys. Rev. Lett. {\\bf 56}, 2272 (1986);\n K. Binder and D. P. Landau, Phys. Rev. B {\\bf 37}, 1745 (1988);\n K. Binder, D. P. Landau, and S. Wansleben, \n Phys. Rev. B {\\bf 40}, 6971 (1989).\n\\bibitem{bhl}\n E. Br\\'ezin, B. I. Halperin, and S. Leibler, \n Phys. Rev. Lett. {\\bf 50}, 1387 (1983).\n\\bibitem{fh}\n D. S. Fisher and D. A. Huse, Phys. Rev. B {\\bf 32}, 247 (1985).\n\\bibitem{fjin}\n M. E. Fisher and A. J. Jin, Phys. Rev. Lett. {\\bf 69}, 792 (1992).\n\\bibitem{boulter}\n C. J. Boulter, Phys. Rev. Lett. {\\bf 79}, 1897 (1997).\n\\bibitem{swain1}\n P. S. Swain and A. O. Parry, Europhys. Lett. {\\bf 37}, 207 (1997).\n\\bibitem{hauge}\n E.H. Hauge, Phys. Rev. B {\\bf 33}, 3322 (1986).\n\\bibitem{lkz}\n R. Lipowsky, D. M. Kroll, and R. K. P. Zia, \n Phys. Rev. B {\\bf 27}, 4499 (1983).\n\\bibitem{fisher}\n For a review see \n M. E. Fisher in {\\em Statistical Mechanics of Membranes and Surfaces},\n D. R. Nelson, T. Piran, R. B. Weinberg eds. \n (World Scientific, Singapore, 1989).\n\\bibitem{lf}\n R. Lipowsky and M. E. Fisher, Phys. Rev. B {\\bf 36}, 2126 (1987).\n\\bibitem{bedeaux}\n D. Bedeaux and J. D. Weeks, J. Chem. Phys. {\\bf 82}, 972 (1985).\n\\bibitem{ich2}\n F. Schmid and K. Binder, Phys. Rev. B {\\bf46}, 13553 (1992);\n F. Schmid, K. Binder, Phys. Rev. {\\bf B 46}, 13565 (1992).\n\\bibitem{andreas}\n A. Werner, F. Schmid, M. M\\\"uller, and K. Binder, \n J. Chem. Phys. {\\bf 107}, 8175 (1997).\n\\bibitem{kubaschewski} O. Kubaschewski in \n {\\em Iron -- Binary Phase Diagrams}, p 5, (Springer, Berlin 1982).\n\\bibitem{duenweg}\n B. D\\\"unweg and K. Binder, Phys. Rev. B {\\bf 36}, 6935 (1987).\n\\bibitem{frank}\n For details see F. F. Haas, Dissertation \n (Johannes Gutenberg Universit\\\"at Mainz 1998).\n\\bibitem{bhanot}\n G. Bhanot, D. Duke, and R. Salvator, J. Stat. Phys. {\\bf 44}, 985 (1986).\n\\bibitem{fn2} Our multispin code differs from the one described in \\cite{bhanot}\n in that we could not make use of bit operations, unfortunately, due to the \n complicated nature of the interactions in our model.\n\\bibitem{KB2}\n K. Binder, Z. Phys. B {\\bf 45}, 61 (1981).\n\\bibitem{andreas2}\n K. Binder, M. M\\\"uller, F. Schmid, A. Werner,\n J. Stat. Phys. {\\bf 95}, 1045 (1999).\n\\bibitem{diehl2}\n H. W. Diehl, in {\\em Phase Transitions and Critical Phenomena}, Vol. 8,\n C. Domb and J. Lebowitz eds. (Academic, London, 1983).\n\\bibitem{KB3}\n D. P. Landau and K. Binder, Phys. Rev. B {\\bf 41}, 4633 (1990).\n\\bibitem{parry}\n C. J. Boulter and A. O. Parry, Phys. Rev. Lett. {\\bf 74}, 3403 (1995);\n A. O. Parry and C. J. Boulter, Physica A {\\bf 218}, 77 (1995);\n C. J. Boulter and A. O. Parry, Physica A {\\bf 218}, 109 (1995);\n A. O. Parry, J. Phys.: Cond. Matt. {\\bf 8}, 10761 (1996).\n\\bibitem{borgs}\n C. Borgs, J. De Coninck, R. Koteck\\'y, and M. Zinque,\n Phys. Rev. Lett. {\\bf 74}, 2292 (1995);\n\\bibitem{netz}\n R. R. Netz and D. Andelman, Phys. Rev. E {\\bf 55}, 687 (1997).\n\\bibitem{swain2}\n A. O. Parry, P. S. Swain, and J. A. Fox,\n J. Phys.: Cond. Matt. {\\bf 8}, L659 (1996);\n P. S. Swain and A. O. Parry, \n J. Phys. A: Math. Gen. {\\bf 30}, 4597 (1997);\n P. S. Swain and A. O. Parry,\n Eur. Phys. J. B {\\bf 4}, 459 (1998).\n\\end{thebibliography}"
}
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cond-mat0002162
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[
{
"name": "asym.tex",
"string": "\\section{The asymmetric XX-chain}\nWe also considered\nthe asymmetric XX-chain which is defined by the Hamiltonian \n\\begin{eqnarray}\n\\label{Ha}\n\\fl\nH_{\\rm a}=\\sum_{j=1}^{L-1}\\left[ p \\sigma_j^+\\sigma_{j+1}^- + q \\sigma_j^-\\sigma_{j+1}^+ \\right]\n\\nonumber \\\\\n\\lo+\\frac{1}{\\sqrt{8}}\\left[\n\\alpha_-'\\sigma_1^-+\\alpha_+'\\sigma_1^+ +\\alpha_z\\sigma_1^z \n+\\beta_-'\\sigma_L^-+\\beta_+'\\sigma_L^+ +\\beta_z \\sigma_L^z\\right] .\n\\end{eqnarray}\nWithout loss of generality we restrict ourselves to $\\sqrt{pq}=\\frac{1}{2}$.\nUnder this condition $H_{\\rm a}$ can be mapped on the symmetric chain \\eref{HXX}\nwith boundary parameters\n\\begin{equation}\n\\label{boundaryDM}\n\\alpha_-=Q^{\\frac{1-L}{2}}\\alpha_-' \\quad \\alpha_+=Q^{\\frac{L-1}{2}}\\alpha_+'\n\\quad \\beta_-=Q^{\\frac{L-1}{2}}\\beta_-' \\quad \\beta_+=Q^{\\frac{1-L}{2}}\\beta_+'\n\\end{equation}\nby a similarity transformation,\nwhere $Q=\\sqrt{\\frac{q}{p}}$.\nThe diagonal terms remain unchanged.\nWe will separate two cases in the following. First we will consider\n hermitian bulk terms, i.e. \n\\begin{equation}\n\\label{kk}\np=\\rme^{\\rmi\\pi \\kappa}/2, q=\\rme^{-\\rmi\\pi \\kappa}/2. \n\\end{equation}\nThis corresponds to Dzyaloshinsky-Moriya type interactions in the bulk.\nThereafter we will turn to non-hermitian bulk terms with $p,q$ in $\\mathbb{R}$.\n\\subsection{Dzyaloshinsky-Moriya interactions}\nWe restricted ourselves to hermitian boundaries.\nIn this case the value of $\\kappa$ has only an effect onto the spectra if\nnon-diagonal boundary terms are present at both ends of the chain.\nWe introduce the parametrization\n\\begin{equation}\n\\alpha_+'=R_{\\alpha}'\\rme^{\\rmi\\pi\\varphi'}\n\\quad \\alpha_-'=R_{\\alpha}'\\rme^{-\\rmi\\pi \\varphi'}\\quad\n \\beta_+'=R_{\\beta}'\\rme^{\\rmi\\pi(\\chi'+\\varphi')}\n\\quad \\beta_-'=R_{\\beta}'\\rme^{-\\rmi\\pi(\\chi'+\\varphi')}.\n\\label{paradm}\n\\end{equation}\nAccording to \\eref{boundaryDM} and \\eref{parann} the mapping onto symmetric bulk terms yields\n\\begin{equation}\n\\label{prim}\nR_{\\alpha}'=R_{\\alpha} \\quad R_{\\beta}'=R_{\\beta} \\quad\n\\varphi=\\varphi'+\\kappa\\frac{1-L}{2} \\quad\n\\chi=\\chi'+\\kappa(L-1) \\mbox{mod}\\, 2 .\n\\end{equation}\nRemember that in the case of length independent boundary terms, the\nvalue of $\\chi$ contains the full information about the partition\nfunction.\nObserve now, that it is possible to choose values of $\\kappa$ such\nthat the value of $\\chi$ in \\eref{prim} becomes independent of $L$ as long\nas one considers only certain sequences of lattice lengths.\nThis is exactly the case\nif $\\kappa/2$ is rational, i.e. $\\kappa/2=m/n, m \\in \\mathbb{Z}, n\\in \\mathbb{N}^+$.\nIn this case $\\chi$ is independent of $L$ for $L=ln+r, 0\\leq r < n, l\\in \\mathbb{N}$,\ni.e.\n\\begin{equation}\n\\label{chip}\n\\chi=\\chi'+ \\frac{2 m(r-1)}{n}.\n\\end{equation}\nThe results obtained for the symmetric case can be adopted immediately.\nThis is not possible for irrational values of $\\kappa/2$.\nNote that this kind of commensurability and incommensurability has also\n been observed for the\nperiodic chain with this type of interaction \\cite{AlcWre}.\n\\subsection{$p,q \\in \\mathbb{R}$}\nWithout loss of generality, we are going to restrict ourselves to $p>q$ which\nimplies $Q<1$.\nWe have to distinguish two different situations.\nIf $\\alpha_-'\\beta_+'$ equals zero we can adopt the results\nfor the symmetric case with $F=C=0$ (see appendix A for details).\nOne has just to exchange $\\alpha_{\\pm}$ and $\\beta_{\\pm}$ by \n$\\alpha_{\\pm}'$ and $\\beta_{\\pm}'$.\nIf otherwise\n$\\alpha_-'\\beta_+'\\neq 0$ the situation changes.\nThe energy gaps suggest the partition function for the long\nchain \\eref{Hlong} with boundary terms given by \\eref{boundaryDM}, i.e.\n\\begin{eqnarray}\n{\\cal Z}_{\\rm long}=\\tr z^{ \\frac{2L}{(Q+1/Q)\\pi}(H - e_{\\infty}L-f_{\\infty}) }\n=\\frac{2}{\\eta(z)}\n\\sum_{m \\in \\mathbb{Z}/2} z^{2(m+\\frac{\\Delta_x}{2})^2+ 2mL\\Delta_y}\n\\label{puq}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{Deltax}\n\\Delta_x=\\frac{\\rmi}{2\\pi}\\ln\\Gamma \\qquad \\Delta_y= \\frac{\\rmi}{\\pi}\\frac{Q-Q^{-1}}{Q+Q^{-1}}\n\\end{equation}\nand\n\\begin{eqnarray}\n\\label{Gamma}\n\\Gamma&=\n&(1-Q^2(1-2\\alpha_z^2-2\\alpha_-'\\alpha_+')-2\\alpha_z^2Q^4) \\nonumber \\\\\n&&\\times (1-Q^2(1-2\\beta_z^2-2\\beta_-'\\beta_+')-2\\beta_z^2Q^4)/(\\alpha_-'^2\\beta_+'^2Q^4(1+Q^2)^2) .\n\\end{eqnarray}\nNote that the value of $\\Delta_y$ is purely imaginary.\nHence the length dependent term given in the partition function is a phase.\nSuch a term already appeared in the toroidal partition function\nfor the asymmetric model with\nperiodic boundary conditions \\cite{NohKim}. \nNote also that the result \\eref{puq}\n simplifies to the partition function we obtained in the Neumann-Neumann case for $H_{\\rm long}$\n if one sets $Q=1$ (see \\eref{zlongnn}).\n\nIn \\cite{paper1} we computed the exact ground-state energy on the finite chain for boundaries defined by\n\\begin{equation}\n\\label{dddd}\n\\alpha_z=\\beta_z=0 \\qquad \\alpha_+'\\alpha_-'=\\beta_+'\\beta_-'=1.\n\\end{equation}\nIf we introduce the parameter\n$\\chi'$ via $\\beta_+'=\\rme^{\\rmi\\pi\\chi'} \\alpha_+'$,\nthen \n\\eref{Gamma} simplifies to\n\\begin{equation}\n\\Gamma=\\left(\\rme^{\\rmi\\pi\\chi'}Q^2\\right)^{-2} .\n\\end{equation}\nExpanding the exact expression obtained in \\cite{paper1} leads to\n\\begin{eqnarray}\n\\label{puqe0}\n\\fl\nE_0\\sim -\\frac{Q+Q^{-1}}{2\\pi}L - \\frac{Q+Q^{-1}+(Q-Q^{-1})\\frac{1}{2}\\ln\\Gamma}{2\\pi}\n - \\frac{(Q+Q^{-1})\\pi}{2 L}\\left(\\frac{1}{24} +\\frac{(\\ln \\Gamma)^2}{8\\pi}\\right) .\n\\end{eqnarray}\nIt is only for this case that we can perform the projection \nonto the $(+,+)$-sector.\n We obtain \n\\begin{equation}\n\\label{puqodd}\n{\\cal Z}=\\frac{1}{\\eta(z)}\n\\sum_{m \\in \\mathbb{Z}}z^{2(m+\\chi'/2-\\rmi \\ln Q/\\pi)^2+2mL\\Delta_y}\n\\quad \\mbox{for odd $L$}\n\\end{equation}\n\\begin{equation}\n\\label{puqeven}\n{\\cal Z}=\\frac{1}{\\eta(z)}\n\\sum_{m \\in \\frac{2\\mathbb{Z}+1}{2}}z^{2(m+\\chi'/2-\\rmi \\ln Q/\\pi)^2+2mL\\Delta_y}\n\\quad \\mbox{for even $L$}.\n\\end{equation}\nNote that our results for $H_{\\rm long}$ only apply if \\eref{Gamma} is different from zero.\nIf the nominator of $\\Gamma$ vanishes we obtain logarithmic terms in the asymptotic \n behaviour of the energy gaps\nsimilar the ones obtained for the symmetric chain.\nWe obtained the fermion energies\n\\begin{eqnarray}\n\\fl\n2\\Lambda \\sim \\frac{Q+Q^{-1}}{2L} \\Biggl\\{ k\\pi \\pm \\frac{1}{2}\\Biggl[ \\arg \\Delta \n-\\rmi\\ln L \n\\Biggr]\\Biggr\\}\\pm \\rmi \\frac{Q-Q^{-1}}{2} \n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\Delta=\\frac{2\\alpha_-'^2\\beta_+'^2 Q^4 (1+Q^2)^2}{2AQ^2+4Q^4(B+E^2)-6Q^6(D+2E^2)+8E^2Q^8}.\n\\label{deltapuq}\n\\end{equation}\nThe values of $A,B,D,E$ are given in \\eref{koeff}.\nWe are not going to consider the case, where the denominator in \\eref{deltapuq}\nvanishes.\n\n"
},
{
"name": "bib.tex",
"string": "\\newpage\n\\section*{References}\n\\begin{thebibliography}{99}\n\\bibitem{paper1} {Bilstein U, Wehefritz B 1999 \\JPA {\\bf 32} 191}\n\\bibitem{ARev}{Affleck I 1989 Field Theory Methods and Quantum Critical\nPhenomena \\\\ {\\it Fields, Strings and Critical Phenomena}\nLes Houches 1988 (North-Holland: Amsterdam)}\n\\bibitem{Sal}{Saleur H 1998 {\\it Lectures on Non Perturbative Field Theory and\nQuantum Impurity Problems} \\\\ cond-mat/9812110}\n\\bibitem{AlcBaaGriRit2} {Alcaraz F C, Baake M, Grimm U and Rittenberg V\n1988 \\JPA {\\bf 21} L117}\n\\bibitem{AlcBarBat}{Alcaraz F C, Barber M N, Batchelor M T\n1987 \\PRL {\\bf 58} 771}\n\\bibitem{AlcBarBat2}{Alcaraz F C, Barber M N, Batchelor M T\n1988 \\APNY {\\bf 182} 280}\n\\bibitem{OshAff}{Oshikawa M, Affleck I 1997 \\NP B {\\bf 495} 533}\n\\bibitem{AlcBaaGriRit1} {Alcaraz F C, Baake M, Grimm U and Rittenberg V\n1989 \\JPA {\\bf 22} L5}\n\\bibitem{Aff2} {Affleck I 1998 \\JPA {\\bf 31} 2761}\n\\bibitem{AlcWre}{Alcaraz F C, Wreszinski W F 1990\n{\\it J. Stat. Phys.} {\\bf 58} 45}\n\\bibitem{paper0}{Bilstein U, Wehefritz B 1997 \\JPA {\\bf 30} 4925}\n\\bibitem{Car}{Cardy J L 1986 \\NP B {\\bf 275} 200}\n\\bibitem{NohKim}{Noh J D, Kim D 1996 \\PR E {\\bf 53} 3225}\n\\bibitem{LSM}{Lieb E, Schultz T and Mattis D 1961 {\\em Ann. Phys.}\n{\\bf 16} 407}\n\\bibitem{Gui}{Guinea F 7518 \\PR B {\\bf 32} 7518}\n\\bibitem{BugSha}{Bugrij A I, Shadura V N 1990 \\PL {\\bf 150} 171}\n\\bibitem{LambertW}{Corless R M, Gonnet G H, Hare D E G, Jeffrey D J, Knuth D E \n{\\it On the Lambert W Function} Maple Share Library}\n\\end{thebibliography}\n"
},
{
"name": "con.tex",
"string": "\\section{Conclusions}\nIn this paper we considered the spectra of the XX--model with boundary fields \ngiven by the Hamiltonian $H$ in \\eref{HXX}. \nIn order to obtain our results we also studied the Hamiltonian $H_{\\rm long}$ \ndefined by \\eref{Hlong}. Furthermore we considered the asymmetric XX--chain \\eref{Ha}.\nHere we separated two cases. First we considered the case where the values of $p$ and $q$\nin \\eref{Ha} are given by $p=\\rme^{\\rmi\\pi \\kappa}/2$ and $q=\\rme^{-\\rmi \\pi \\kappa}/2$\n(this corresponds to Dzyaloshinsky-Moriya interactions). Second we considered \nthe case where $p$ and $q$ are real numbers.\n\nFor periodic boundary conditions the partition function for $H$ is given by the\npartition function of the free boson with periodic boundary conditions \\eref{bosonp}\n\\cite{AlcBaaGriRit2,AlcBarBat,AlcBarBat2}.\nThe partition function for $H_{\\rm long}$ with periodic boundary conditions \nis just the expression \\eref{bosonp} multiplied by 4. The spectra of the asymmetric chain\nwith Dzyaloshinsky-Moriya interactions and periodic boundary conditions have been studied\nin \\cite{AlcWre}, whereas the partition function for\n real values of $p$ and $q$ and periodic boundary conditions \nwas given in \\cite{NohKim}.\n\nThe results we obtained for $H$ and $H_{\\rm long}$ can be encoded in terms of the parameters\ngiven in \\eref{newparam}. For reasons discussed in the text we obtained the energy gaps\nonly for the cases given in \\tref{structure}.\nNote that as long as we restrict ourselves to hermitian boundary terms\nwe studied the most general case. This is not true for non-hermitian boundaries.\n\nFor $H$ with hermitian boundaries we obtained the partition functions corresponding \nto one of the three boson partition functions \\eref{bosondd},\\eref{bosonnn} and\n\\eref{bosondn}. \nHowever, we found these partition functions also for certain non-hermitian boundary terms.\n\nWe obtained the Neumann-Neumann partition function \\eref{bosonnn}\nif all non-diagonal terms\nare present, where the value of $\\Delta$ in \\eref{bosonnn} is given by \\eref{deltann}.\nThe Dirichlet-Neumann\npartition function \\eref{bosondn} is obtained for boundary terms which satisfy \\eref{dnnonhermparam}.\nWe found the Dirichlet-Dirichlet partition function \\eref{bosondd} for two types of boundary terms \ngiven by \\eref{dd2} respectively \\eref{ddnonherm}.\nThe value of $\\Delta$ in \\eref{bosondd} is given by \\eref{deltadd} in both cases.\nFurthermore, for the case of non-hermitian boundaries we found a case which is special.\nFor boundary terms given by \\eref{pa} the partition function is the Dirichlet-Dirichlet partition\nfunction \\eref{bosondd} multiplied by 2, where the value of $\\Delta$ is $0$ or $1/2$ for\nodd respectively even values of the lattice length.\n\nThe partition functions for $H_{\\rm long}$ have also been considered. If all non-diagonal\nterms are present it is given by \\eref{zlongnn}. For the other cases it is just the\npartition function of $H$ multiplied by 4. We also computed the values of the free surface \nenergies of $H_{\\rm long}$ and $H$ for certain boundary terms. They are given in \\tref{tabfac}.\nIn this table we have also \ngiven the values of the lowest highest weights which appear in the spectra of $H_{\\rm long}$ for\nthese boundary terms.\n\nFor the last two cases in \\tref{structure} we obtained logarithmic corrections to\nthe free surface energy (equation \\eref{loge0}). Here we found also logarithmic terms \nin the asymptotic behaviour of the energy gaps (equation \\eref{aslog}).\n\nIn the case of Dzyaloshinsky-Moriya interactions, we restricted ourselves \nto hermitian chains. The value of the phase $\\kappa$ (see \\eref{kk}) has only an\neffect on the spectra if non-diagonal boundary terms are present at both ends of the chain.\nIn this case we obtain the Neumann-Neumann partition function \\eref{bosonnn} if\n$\\kappa/2$ is a rational number.\nThe value of $\\chi$ in the definition of $\\Delta$ in \\eref{deltann} has just to be \nexchanged by the expression in \\eref{chip}.\n\nThe partition function for the asymmetric chain \\eref{Ha} with real values of $p$ and $q$\nhas only been obtained for one special type of boundaries (see equation \\eref{dddd}).\nIt is given by \\eref{puqodd} for $L$ odd respectively by \\eref{puqeven} for $L$ even.\nThe asymptotic behaviour \nof the ground-state energy for this case is given in \\eref{puqe0}.\n\n"
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"name": "intro.tex",
"string": "\\section{Introduction}\nIn the first paper of this series \\cite{paper1} we have shown \nhow to diagonalize the Hamiltonian of the XX--model with boundaries\n\\begin{equation}\n\\fl\nH=\\frac{1}{2}\\sum_{j=1}^{L-1}\n\\left[\\sigma_j^+\\sigma_{j+1}^-+\\sigma_j^-\\sigma_{j+1}^+\\right]\n+\\frac{1}{\\sqrt{8}}\\left[\\alpha_-\\sigma_1^-+\\alpha_+\\sigma_1^+\n+\\alpha_z\\sigma_1^z+\\beta_+\\sigma_L^++\\beta_-\\sigma_L^-+\\beta_z\\sigma_L^z\\right]\n\\label{HXX}\n\\end{equation}\nby introducing an auxiliary Hamiltonian \n\\begin{eqnarray}\n\\fl\nH_{\\rm long}=\\frac{1}{2}\\sum_{j=1}^{L-1}\n\\left[\\sigma_j^+\\sigma_{j+1}^-+\\sigma_j^-\\sigma_{j+1}^+\\right] \\nonumber \\\\\n+\\frac{1}{\\sqrt{8}}\\left[\\alpha_-\\sigma_0^x\\sigma_1^-+\\alpha_+\\sigma_0^x\\sigma_1^+\n+\\alpha_z\\sigma_1^z+\\beta_+\\sigma_L^+\\sigma_{L+1}^x+\\beta_-\\sigma_L^-\\sigma_{L+1}^x\n+\\beta_z\\sigma_L^z\\right]\n\\label{Hlong}\n\\end{eqnarray}\nwhich in turn may be diagonalized\nin terms of free fermions. The parameters $\\alpha_{\\pm},\\beta_{\\pm},\\alpha_z$ and $\\beta_z$\nare arbitrary complex numbers.\nNote that $H_{\\rm long}$ commutes with $\\sigma_0^x$ and $\\sigma_{L+1}^x$.\nHence the spectrum of $H_{\\rm long}$ decomposes into four sectors \n$(+,+),(+,-),(-,-),(-,+)$ corresponding to the eigenvalues $\\pm 1$ of $\\sigma_0^x$\nand $\\sigma_{L+1}^x$. \nThe spectrum of $H$ is obtained by projecting onto the $(+,+)$--sector.\n\nWhile in \\cite{paper1} our focus was on the diagonalization\nof the finite chain, here\n we are going to examine the asymptotic behaviour of \n the energy levels for large values of $L$.\nIn the case of conformal invariance \nthis information is usually encoded in the partition function,\n which we define by\n\\begin{equation}\n{\\cal Z}= \\lim_{L\\to \\infty}\\tr z^{ \\frac{L}{\\xi}(H-e_{\\infty}L-f_{\\infty})}\n\\label{partf}\n\\end{equation}\nwhere $z<1$ and $e_{\\infty}$ and $f_{\\infty}$ denote the free bulk and the free\nsurface energy respectively. \n$\\xi$ may be considered as a normalization constant here. We will also consider the partition\nfunction for $H_{\\rm long}$.\n\nIn the case of periodic boundary conditions the continuum limit of the XXZ--chain can\nbe described by a free boson field $\\Phi$ (see e.g. \\cite{ARev}), with action\n\\begin{equation}\nS=\\frac{1}{2}\\int dx_1 dx_2 \\left[ \\left(\\partial_1 \\Phi\\right)^2+\n\t\t\t \\left( \\partial_2 \\Phi\\right)^2 \\right]\n\\end{equation}\nbeing compactified on a circle of radius $r$, i.e. we identify \n$\\Phi$ with $\\Phi+2\\pi r$ (we follow the notation of \\cite{Sal}).\nThe radius of compactification $r$ is related to the value of the\nanisotropy in the XXZ--model. At the free fermion-point the anisotropy \nis zero and we have $r=1/\\sqrt{4\\pi}$.\n\nConsider the boson field on a cylinder of length $L$ and\ncircumference $N$ imposing periodic boundary conditions in the direction\nof the cylinders length.\nQuantizing the field with time in $N$-direction \n yields the partition\nfunction \\cite{Sal}\n\\begin{equation}\n\\label{bosonp}\n{\\cal Z}_{\\rm periodic}=\\frac{1}{\\eta(q)^2}\\sum_{m,n \\in \\mathbb{Z}} \n\tq^{\\frac{1}{2}\\left( \\frac{m^2}{2\\pi r^2} + 2\\pi r^2 n^2\\right)}\n\\end{equation}\nwhere $q=\\rme^{-2\\pi N/L}$ and \n\\begin{equation}\n\\eta(q)=q^{\\frac{1}{24}}\\prod_{n=1}^{\\infty}(1-q^n).\n\\end{equation}\nThis expression coincides with the partition function\n for the periodic XXZ-chain, which has been studied \nextensively in \\cite{AlcBaaGriRit2,AlcBarBat,AlcBarBat2}.\nThe partition function for $H_{\\rm long}$ in the periodic case \nis just $4 {\\cal Z}_{\\rm periodic}$.\n\nSuppose now the case of open boundaries at both ends of the cylinder.\n There are two different \n boundary conditions preserving conformal invariance,\n i.e. the Dirichlet and the von Neumann (see e.g. \\cite{Sal,OshAff} ).\nThis yields three different partition functions corresponding \nto the various possible combinations of the two boundary conditions.\nThe respective partition functions can be found in \\cite{Sal,OshAff}.\nImposing Dirichlet boundary conditions at both ends of the\ncylinder results in\n\\begin{equation}\n\\label{bosondd}\n{\\cal Z}_{\\rm DD}(\\Delta) =\\frac{1}{\\eta(q)}\\sum_{n \\in \\mathbb{Z}}\n\tq^{\\frac{1}{2} (2 \\sqrt{\\pi} r n+\\Delta)^2\t}\n\\end{equation}\nwhere $q=\\rme^{-\\pi N/L}$ and $\\Delta=(\\Phi_0-\\Phi_L)/\\sqrt{\\pi}$.\nBy $\\Phi_0$ and $\\Phi_L$ we denote the values of\nthe boson field at the boundaries.\nThis type of partition function \n has also been obtained for the open XXZ-chain \nwith diagonal, hermitian boundary fields \\cite{AlcBaaGriRit1}.\n\nThe Neumann-Neumann partition function \nis given by\n\\begin{equation}\n\\label{bosonnn}\n{\\cal Z}_{\\rm NN}(\\Delta) =\\frac{1}{\\eta(q)}\\sum_{m \\in \\mathbb{Z}}\n q^{2(m/(2\\sqrt{\\pi}r) +\\Delta/2)^2 } \n\\end{equation}\nwhere $\\Delta=(\\tilde{\\Phi}_0-\\tilde{\\Phi}_L)/\\sqrt{\\pi}$ and $\\tilde{\\Phi}_0,\\tilde{\\Phi}_L$ denote the values\nof the dual field $\\tilde{\\Phi}$ at the boundaries \\cite{Sal,OshAff}.\nThe Dirichlet-Neumann boundary condition yields \\cite{Sal,OshAff}\n\\begin{equation}\n\\label{bosondn}\n{\\cal Z}_{\\rm DN} =\\frac{1}{2\\eta(q)}\\sum_{k \\in \\mathbb{Z}}\n q^{\\frac{1}{4} (k-1/2)^2 } .\n\\end{equation}\nIn this case no free parameter appears. \n\nWhich boundary conditions of the Hamiltonian $H$ lead to the partition functions\n\\eref{bosonnn} and \\eref{bosondn} is yet not known.\nWe will close this gap. \nWe are going to see that, as long as the Hamiltonian $H$ is hermitian, the partition\nfunction of the chain corresponds to one of the three boson partition\nfunctions for open boundaries. However, this may also be the case for certain non-hermitian\nboundary terms.\nSee section 2 for a survey over the boundary conditions which \nwe will consider.\n\nIn \\cite{Aff2} it has been argued that non-diagonal boundary terms\ncorrespond to von Neumann boundary conditions\nfor the boson field. Our results will show that this assumption is correct\nin the hermitian case, but not necessarily for non-hermitian boundaries.\n\nIn this paper we also study the asymmetric XX--chain with boundaries (see equation\n\\eref{Ha}). This includes the quantum chain with Dzyaloshinsky-Moriya interactions,\nwhich was already studied in \\cite{AlcWre} for periodic boundary conditions. \nThe partition function for real values of $p$ and $q$ in \\eref{Ha} and periodic boundary conditions was \nobtained in \\cite{NohKim}.\nThe special case $p=1$ and $q=0$ with boundaries has already been studied in \\cite{paper0}.\n\nThis paper is organized as follows: We will start with a summary of our results for\nthe symmetric chain with hermitian and non-hermitian boundaries in section 2. The\nasymmetric XX-chain will be discussed in section 3. We will give our conclusions in section 4.\nThe appendix is dedicated to the computation of the fermion energies which yield \nthe energy gaps of the chain.\n\n\n"
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"string": "\\documentclass[12pt]{iopart}\n%\\textwidth18.5cm\n%\\hoffset=-.3truein\n%\\renewcommand{\\rmdefault}{ppl}\n%\\renewcommand{\\rmdefault}{ptm}\n%\\SetSymbolFont{letters}{normal}{OML}{ptmcm}{m}{it}\n%\\SetSymbolFont{operators}{normal}{OT1}{ptmcm}{m}{n}\n\n\\usepackage{amsfonts}\n\\usepackage{lscape}\n\n\\begin{document}\n\\input{epsf.tex}\n%\\renewcommand{\\baselinestretch}{1.5}\n\\begin{center}\\Large The XX--model with boundaries.\\\\ Part II: \nFinite size scaling and partition functions\n\\\\[1cm]\n\\normalsize\nUlrich Bilstein\\footnote{E-mail address: \\tt bilstein@theoa1.physik.uni-bonn.de}\\\\[0.5cm]\n Universit\\\"{a}t Bonn \\\\\n Physikalisches Institut, Nu\\ss allee 12,\n D-53115 Bonn, Germany\\\\[1.3cm]\n\\end{center} \n{\\bf \\small Abstract.}\n\\small\nWe compute the continuum limit of the spectra \n for the XX--model with arbitrary complex\nboundary fields. \nIn the case of hermitian boundary terms one obtains the partition functions of the free \ncompactified boson field on a cylinder with \nNeumann-Neumann, Dirichlet-Neumann or Dirichlet-Dirichlet \nboundary conditions. \nThis applies also for certain non-hermitian boundaries.\nFor special cases we also compute the free surface energy.\nFor certain non-hermitian boundary terms the results are more complex.\nHere one obtains logarithmic corrections to the free surface energy.\nThe asymmetric version of the XX--model with boundaries (this includes the Dzyaloshinsky-Moriya\ninteraction)\nis also discussed.\\\\[0.2cm]\n\\normalsize\n%\\pacs{007, 08.15}\n\\input{intro.tex}\n\\input{outline.tex}\n\\input{resnonherm.tex}\n\\input{asym.tex}\n\\input{con.tex}\n\\ack \nI would like to thank Birgit Wehefritz for her contribution to the early stage of this work.\nI would also like to thank Vladimir Rittenberg for many discussions and constant encouragement.\nI would like to thank Paul Pearce for helpful comments.\nI am grateful to Klaus Krebs for carefully reading the manuscript and many fruitful \ndiscussions. This work was supported by the TMR Network Contract FMRX-CT96-0012 of the European \nCommission. \n\\input{resherm.tex}\n\\input{bib.tex}\n\\end{document}\n"
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{
"name": "outline.tex",
"string": "\\section{Summary of results}\nThe structure of our results\nis most simply encoded in terms of new parameters $F,C,G,K,J$, where\n\\begin{eqnarray}\n\\label{newparam}\n\\fl\nF=4 \\alpha_-\\alpha_+\\beta_+\\beta_- \\nonumber \\\\\n\\fl\nC=\\alpha_-^2\\beta_+^2+\\alpha_+^2\\beta_-^2 \\nonumber \\\\\n\\fl\nG =2\\alpha_-\\alpha_+(1+2\\beta_z^2)+2\\beta_-\\beta_+(1+2\\alpha_z^2)\n\\nonumber \\\\\n\\fl\nK =2(\\alpha_-\\alpha_++\\beta_-\\beta_+)+(1+2\\alpha_z^2)(1+2\\beta_z^2)\\nonumber \\\\\n\\fl\nJ =8\\alpha_z\\beta_z-4\\alpha_z^2\\beta_z^2+2(\\alpha_-\\alpha_++\\beta_-\\beta_+) +2\\alpha_z^2+2\\beta_z^2\n-1 .\n\\end{eqnarray}\nThe six different situations we considered are given in \n\\tref{structure}. \nThis table also indicates which kind of partition function we obtained for $H$.\nFor hermitian boundaries the cases in this table cover the whole parameter \nspace. This is not the case for non-hermitian boundary terms.\n\n\\begin{table}\n\\caption{Conditions on the parameters $F,C,G,K,J$ being considered in this paper.\n An asterisk indicates that the corresponding value may be arbitrary.\nThe partition functions ${\\cal Z}_{DD},{\\cal Z}_{NN}$ and ${\\cal Z}_{DN}$ \nare defined in \\eref{bosondd},\\eref{bosonnn} and \\eref{bosondn}.\nFor the last two cases we obtained anomalous behaviour of the energies (see section 2.5).\nWe use h. and n-h. as a shortcut for hermitian and non-hermitian.\n}\n\\begin{indented}\n\\item[]\n\\begin{tabular}{@{}lccccrr}\n\\br\nF & C & G & K & J & & \\\\\n\\mr\n$\\neq 0$ & * & * & * & * & ${\\cal Z}_{NN}$ & h. and n-h. \\\\\n0 & 0 & $\\neq 0$ & * & * & ${\\cal Z}_{DN}$ & h. and n-h. \\\\\n0 & 0 & 0 & $\\neq 0$ & * & ${\\cal Z}_{DD}$ & h. and n-h. \\\\\n0 & 0 & 0 & 0 & 0 & $2\\times{\\cal Z}_{DD}$ & n-h. \\\\ \\mr\n0 & $\\neq0$ & $\\neq 0$ & * & * & anom. & n-h. \\\\\n0 & 0 & 0 & 0 & $\\neq 0$ & anom. & n-h. \\\\\n\\br\n\\end{tabular}\n\\end{indented}\n\\label{structure}\n\\end{table}\nFor the cases in \\tref{structure} we computed analytically the energy gaps in the leading\norder as shown in appendix A. The exact ground-state energies for certain cases being\nlisted in \\tref{tabfac} have already been obtained in \\cite{paper1}.\n\nHowever, the information of the energy gaps is already sufficient to obtain the\npartition functions for $H_{\\rm long}$ upto a factor $z^{-\\frac{1}{24}+h_{\\rm min}}$, where\n$h_{\\rm min}$ denotes the lowest highest weight appearing in the representation\nof the Virasoro algebra.\nThis information has to be extracted from the ground-state energies \\cite{Car}\n\\begin{equation}\n\\label{gstate}\nE_0=e_{\\infty}L+f_{\\infty}-\\frac{1}{L}\\left(\\frac{1}{24}-h_{\\rm min}\\right)+ \\ldots\n\\end{equation}\nThe value of the free bulk-energy is already known from the periodic chain, i.e. $e_{\\infty}=-\\frac{1}{\\pi}$ \n\\cite{LSM}. The values of the free surface energies $f_{\\infty}$ and of $h_{\\rm min}$ have been obtained \nfor the cases given in \\tref{tabfac} expanding the exact expressions for the finite chain given in \\cite{paper1}.\n\nThe results confirm the partition functions we will give for $H_{\\rm long}$ in the following.\nFor the cases not being listed in \\tref{tabfac} we checked our expressions numerically. \nThe partition functions for $H$ have been obtained afterwards by projecting onto the $(+,+)$-sector\n(see \\cite{paper1} for details on the projection mechanism). We are now going \nto discuss the cases given in \\tref{structure} in detail.\n\\subsection{Neumann-Neumann boundary conditions}\nThe condition $F\\neq 0$ simply implies that all non-diagonal \nboundary terms are present.\n The diagonal terms \nmay be arbitrary. It is quite instructive to introduce a new parametrisation\nof the boundary parameters $\\alpha_{\\pm}$ and $\\beta_{\\pm}$ as follows:\n\\begin{equation}\n\\alpha_+=R_{\\alpha}\\rme^{\\rmi\\pi\\varphi}\\quad \\alpha_-=R_{\\alpha}\\rme^{-\\rmi\n\\pi\\varphi}\\quad\n \\beta_+=R_{\\beta}\\rme^{\\rmi\\pi(\\chi+\\varphi)}\\quad \\beta_-=R_{\\beta}\\rme^{-\\rmi\\pi(\\chi+\\varphi)}\n\\label{parann}\n\\end{equation}\nwhere $R_{\\alpha},R_{\\beta} \\in \\mathbb{R}^+,\\chi,\\varphi \\in \\mathbb{R},-1<\\chi\\leq 1$.\nThe value of $\\Delta$ which enters the partition function \\eref{bosonnn} is then\ngiven by \n\\begin{equation}\n\\label{deltann}\n\\Delta=\\left\\{ \\begin{array}{ll} \\chi & \\mbox{for $L$ odd} \\\\\n\t\t\t \\chi+1 & \\mbox{for $L$ even} \\end{array} \\right. .\n\\end{equation}\n\\input{tabfu.tex}\nThe partition function for $H_{\\rm long}$ is given by \n\\begin{equation}\n\\label{zlongnn}\n{\\cal Z}_{\\rm long}=\n2 {\\cal Z}_{NN}(\\chi)+2{\\cal Z}_{NN}(\\chi+1)\n\\end{equation}\n for even and odd values of $L$.\nNote that the parametrization \\eref{parann} does not work for non-hermitian boundaries.\nIn this case we may define \nthe parameter $\\chi$ via\n\\begin{equation}\n\\chi=\\frac{\\rmi}{2\\pi} \\ln\\left( \\frac{\\alpha_-\\beta_+}{\\alpha_+\\beta_-}\\right)\n\\end{equation}\nwhere the logarithm is taken in a way such that\n $0\\leq|\\mbox{Re}(\\chi)|\\leq\\frac{1}{2}$\nfor Re$(\\alpha_+\\beta_-+\\alpha_-\\beta_+) > 0$ and $1\\geq |\\mbox{Re}(\\chi)| \\geq \\frac{1}{2}$\nfor Re$(\\alpha_+\\beta_-+\\alpha_-\\beta_+) < 0$. This rule is consistent with\nthe parametrisation \\eref{parann},\nwhich implies Re$(\\alpha_+\\beta_-+\\alpha_-\\beta_+)=2R_{\\alpha}R_{\\beta}\\cos( \\chi\\pi)$.\nHowever, we cannot perform the projection onto the $(+,+)$-sector in general if\nthe boundaries are non-hermitian. In this case \nour result for the partition function of $H$ is restricted to the cases listed in \\tref{tabfac}\nwhich satisfy\n the conditions $\\alpha_z=\\beta_z=0$ or $\\alpha_+=\\alpha_-,\\beta_+=\\beta_-$ \\cite{paper1}.\nThe result for $H_{\\rm long}$ is valid in general.\n\\subsection{Dirichlet-Neumann boundary conditions}\nThe conditions for the Dirichlet-Neumann case yield\n\\begin{equation}\n\\beta_{\\pm}=0 \\quad \\alpha_{\\pm}\\neq 0 \\quad \\beta_z\\neq \\pm\\frac{\\rmi}{\\sqrt{2}} \\quad\n\\mbox{or}\\quad\n\\beta_{\\pm}\\neq0 \\quad \\alpha_{\\pm}= 0 \\quad \\alpha_z\\neq \\pm\\frac{\\rmi}{\\sqrt{2}} .\n\\label{dnnonhermparam}\n\\end{equation}\nThe value of $\\alpha_z$ respectively of $\\beta_z$ may be arbitrary.\nThe partition function for $H_{\\rm long}$ is just ${\\cal Z}_{\\rm long}=4 {\\cal Z}_{DN}$\nfor $L$ even and odd.\nThe case of one non-diagonal boundary at the end of a semi-infinite XX--chain\nhas already been considered in \\cite{Gui}.\nThe author has shown, that the model can be decoupled into\ntwo Ising models\nwith different boundary conditions. One of them being subject to a free boundary\ncondition at the end, the other one being subject to\nthe fixed boundary condition.\n The partition functions for the Ising model with\nfree and mixed boundary conditions are well known \\cite{Car,BugSha}.\nTaking the product of them also results in \\eref{bosondn}.\n\\subsection{Dirichlet-Dirichlet boundary conditions}\nThe Dirichlet-Dirichlet partition function is obtained for \ntwo different types of boundary terms.\nThe first type is given by \n\\begin{equation}\n\\label{ddnonherm}\n\\beta_z\\neq \\pm \\frac{\\rmi}{\\sqrt{2}} \\quad \\alpha_z\\neq \\pm \\frac{\\rmi}{\\sqrt{2}}\n\\quad \\beta_-=\\alpha_-=0.\n\\end{equation}\nwhere instead of the last equation we might have also chosen $\\beta_+=\\alpha_+=0$.\nHowever, for this type of boundaries the non-diagonal boundary terms \nhave no influence on the energies even \non the finite chain.\nThe second type is given by\n\\begin{equation}\n\\label{dd2}\n\\beta_{\\pm}=0 \\quad \\alpha_{\\pm}\\neq 0 \\quad \\beta_z=\\pm\\frac{\\rmi}{\\sqrt{2}}\n\\quad \\mbox{or} \\quad\n\\beta_{\\pm}\\neq 0 \\quad \\alpha_{\\pm}= 0 \\quad \\alpha_z=\\pm\\frac{\\rmi}{\\sqrt{2}}\n\\end{equation}\nwhere the values of $\\alpha_z$ respectively $\\beta_z$ may be chosen arbitrarily.\nNote that for this second type the boundary terms are non-hermitian.\nFor both types of boundaries the value of $\\Delta$ which enters \\eref{bosondd} is given by\n\\begin{equation}\n\\label{deltadd}\n\\Delta=\\frac{1}{2\\pi}\\mbox{arccos}\\left( (-1)^{L+1}\\frac{J}{K}\\right) .\n\\end{equation}\nThe partition function for $H_{\\rm long}$ is ${\\cal Z}_{\\rm long}=4 {\\cal Z}_{DD}(\\Delta)$ \nfor even and odd $L$.\n\n\\subsection{$F=C=G=K=J=0$}\nFor the case $F=C=G=K=J=0$ the boundary\nparameters are nearly fixed,\ni.e.\n\\begin{equation}\n\\alpha_z=\\pm\\frac{\\rmi}{\\sqrt{2}} \\quad \\beta_z=\\mp\\frac{\\rmi}{\\sqrt{2}}\n\\label{pa}\n\\end{equation}\nand $\\alpha_+=\\beta_+=0$ or $\\alpha_-=\\beta_-=0$.\nThis case is special.\nThe partition function we obtained does not fit\ninto the picture suggested by the analogy to the free boson.\nHere an additional zero mode appears. \nThe value of $\\Delta$ is given by $\\Delta=0$ for odd $L$ and by $\\Delta=1/2$ for even \n$L$. The partition function for $H_{\\rm long}$ is ${\\cal Z}_{\\rm long}=8{\\cal Z}_{DD}(\\Delta)$\nfor $L$ even and $L$ odd.\n\n\n\n\n\n\n\n\n\n\n\n\n"
},
{
"name": "resherm.tex",
"string": "\\appendix\n\\section{Determination of the energy gaps}\nIn \\cite{paper1} we have seen that the spectrum of\n$H_{\\rm long}$ is given in terms of free fermions.\nThe fermionic energies $2\\Lambda_n$ are given by\n\\begin{equation}\n2\\Lambda_n=\\frac{1}{2}(x_n+x_n^{-1}).\n\\label{egap}\n\\end{equation}\nwhere the $x_n$ are the roots of the polynomial\n\\begin{eqnarray}\n\\label{pol}\n\\fl\np(x^2) = \\Bigl[ x^{4L+8}+1 - A (x^{4L+6}+x^2)\n+(B+E^2)(x^{4L+4} +x^4) \\nonumber \\\\\n\\lo\n+ (D+2E^2) (x^{4L+2}+x^6) +E^2 (x^{4L} +x^8) -2E (x^{2L+8}+x^{2L})\n\\nonumber\\\\ \\lo{+}\n\\left((A-B-D-1)/2-(-1)^{L} C - 2 E^2\\right) (x^{2L+6}+x^{2L+2})\n\\nonumber \\\\\n \\lo + \\left(A-B-D-1+ 2 (-1)^L C+ 4 E -4 E^2\\right) x^{2L+4} \\left. \\Bigr]\n\\right/(x^2-1)^2\n\\end{eqnarray}\nwhere the coefficients appearing in \\eref{pol} are functions of the boundary parameters:\n\\begin{eqnarray}\n\\label{koeff}\nA = 2 (\\alpha_-\\alpha_+ + \\beta_-\\beta_+ + \\alpha_z^2 + \\beta_z^2-1)\\qquad\nC = \\alpha_-^2 \\beta_+^2+ \\alpha_+^2 \\beta_-^2 \\nonumber \\\\\nB = ( 2 \\alpha_-\\alpha_+-1)( 2 \\beta_-\\beta_+-1) +\n 4 \\beta_z^2(\\alpha_-\\alpha_+-1) +4\\alpha_z^2 ( \\beta_+\\beta_- -1)\n\\nonumber\\\\\n D = \\beta_z^2 (4 \\alpha_-\\alpha_+ -2) + \\alpha_z^2( 4 \\beta_+\\beta_- -2) \\qquad\n E = 2 \\alpha_z \\beta_z .\n\\end{eqnarray}\nSince there appear\n4 zeros for each fermion, namely $x_n,x_n^{-1},-x_n$ and $-x_n^{-1}$,\nthe polynomial yields $L+1$ fermionic energies (see \\cite{paper1} for details).\nIn addition to these fermions there always exists a fermion with energy $2\\Lambda_0=0$,\nwhich we named 'spurious' zero mode.\nAll possible combinations of these $L+2$ fermions build up the spectrum\nof $H_{\\rm long}$. The spectrum of $H$ is then obtained by excluding the 'spurious'\nzero mode from the set of fermion energies and then taking the sector with an even or with an odd number\nof fermions being excited with respect to the vacuum. How to decide \nwhether one has to pick up the even or the odd sector has been discussed in detail in \\cite{paper1}.\n\nSince in this paper we are interested into the large $L$ behaviour\nof the low lying energy levels, we look for the\nasymptotics of the zeros of $p(x^2)$ in the vicinity of the point $x=\\pm \\rmi$\nusing the ansatz $x=\\rme^{\\rmi \\frac{\\pi}{2}-\\rmi\\frac{\\phi}{L}}$,\nwhere we assume $\\phi$ to be a constant in leading order.\nFor the first four cases given in \\tref{structure} we obtained the following \nequations:\n\\begin{itemize}\n\\item[$\\circ$]\n$F\\neq 0$\n\\begin{equation}\n\\cos(2\\phi)+\\frac{2C}{F}=0\n\\end{equation}\n\\item[$\\circ$]\n$F=C=0,G\\neq 0$\n\\begin{equation}\n\\phi\\sin(2\\phi)=0\n\\end{equation}\n\\item[$\\circ$]\n$F=C=G=0,K\\neq 0$\n\\begin{equation}\n\\phi^2\\left[\\cos(2\\phi)+(-1)^{L+1}\\frac{J}{K}\\right] =0\n\\end{equation}\n\\item[$\\circ$]\n$F=C=G=K=J=0$\n\\begin{equation}\n\\phi^4\\exp(2\\rmi\\phi)=(-1)^L\n\\end{equation}\n\\end{itemize}\nSolving these equations for $\\phi$ yields the energy gaps (cf. \\eref{egap}) in leading order.\nWe obtained the following expressions:\n\\begin{itemize}\n\\item[$\\circ$]\n$F\\neq 0$\n\\begin{equation}\n2\\Lambda\\sim\\frac{1}{L}\\left[ \\frac{2n-1}{2}\\pi \\pm \\frac{1}{2}\\mbox{arccos}\n\\left(\\frac{2C}{F}\\right) \\right] \\qquad 1\\leq n\n\\end{equation}\n\\item[$\\circ$]\n$F=C=0,G\\neq 0$\n\\begin{equation}\n2\\Lambda\\sim \\frac{1}{L}\\frac{n}{2}\\pi \\qquad 0\\leq n\n\\end{equation}\n\\item[$\\circ$]\n$F=C=G=0,K\\neq 0$\n\\begin{equation}\n2\\Lambda \\sim \\frac{1}{L}\\left[ \\frac{2n-1}{2}\\pi \\pm \\frac{1}{2}\\mbox{arccos}\n\\left((-1)^{L+1}\\frac{J}{K}\\right) \\right] \\qquad 1\\leq n\n\\end{equation}\nIn addition to these modes there appears an additional zero mode.\n\\item[$\\circ$]\n$F=C=G=K=J=0$\n\\begin{equation}\n2\\Lambda\\sim\\frac{1}{L}n\\pi \\qquad 1\\leq n \\qquad \\mbox{for $L$ even}\n\\end{equation}\n\\begin{equation}\n2\\Lambda\\sim\\frac{1}{L}\\frac{2n-1}{2}\\pi \\qquad 1\\leq n \\qquad \\mbox{for $L$ odd}\n\\end{equation}\nFurthermore we found 3 additional zero modes for $L$ even and 2 additional zero \nmodes for $L$ odd.\n\\end{itemize}\nFor the last two cases in \\tref{structure}\nthe asymptotic zeros of the polynomial can only be found\nusing the ansatz $x=\\rme^{\\rmi\\frac{\\pi}{2}-\\rmi\\frac{\\phi}{L}}$ if\nwe assume the imaginary part of $\\phi$ to diverge as $L$ goes to infinity.\nHence we generalize our ansatz to $x=\\rme^{\\rmi \\frac{\\pi}{2}-\\rmi\\frac{\\phi(L)}{L}}$,\nwhere $\\phi(L)$ is a complex function.\nFurthermore we assume\n$\\lim_{L\\to \\infty}\\frac{\\phi(L)}{L}=0$ and $\\lim_{L \\to \\infty} \\rme^{-\\rmi\\phi(L)}=0$.\nThe second assumption will be explained shortly.\nIn both cases our ansatz leads to the solution of \nan equation of the form \n\\begin{equation}\n\\label{lam}\na\\rme^{-2\\rmi\\phi(L)}\\left[1+\\Or\\left(\\frac{\\phi(L)}{L}\\right)\\right]+\\frac{2\\rmi\\phi(L)}{L}\n\\left[b+\\Or\\left(\\frac{\\phi(L)}{L}\\right)\\right]=0 .\n\\end{equation}\nFrom this expression one can see that our second assumption is indeed necessary to\nsolve this equation, since otherwise the first term would be finite \nfor all values of $L$, whereas the second term vanishes as $L$ goes to infinity.\nNeglecting the terms of order \n $\\phi(L)/L$ this equation is solved\nin terms of the so called Lambert W function ${\\cal L}$ which\nis defined by the property ${\\cal L}(x) \\rme^{{\\cal L}(x)}=x$.\nWe obtain\n\\begin{equation}\n\\phi(L)=-\\rmi{\\cal L}(\\Delta L)/2\n\\quad\n\\mbox{where}\n\\quad\n\\Delta=-\\frac{a}{b} .\n\\end{equation}\nThe asymptotic behaviour of ${\\cal L}$ is well known \\cite{LambertW}, i.e.\n\\begin{equation}\n{\\cal L}(L)\\sim 2\\rmi k\\pi +\\ln L -\\ln(\\ln L+2\\rmi \\pi k) + ...\n\\end{equation}\nNote that this expression is in accordance with our assumptions we made concerning\nthe asymptotic behaviour of $\\phi(L)$.\n\nIn this paper we also considered the asymmetric chain, which is similar\nto the symmetric chain with length dependent boundary parameters (cf. \\eref{boundaryDM}).\nThis length dependence enters the polynomial \\eref{pol} only\nvia the coefficient $C$, which becomes\n\\begin{equation}\nC= \\alpha_-'^2\\beta_+'^2 Q^{2-2L}+\\alpha_+'^2\\beta_-'^2 Q^{2L-2} .\n\\label{CQ}\n\\end{equation}\nFor $Q<1$ (the case considered in section 3.2) the first term \n on the RHS of \\eref{CQ} diverges exponentially whereas\nthe second term vanishes exponentially as a function of the lattice length $L$.\nHence we have to distinguish two different situations.\nIf $\\alpha_-'\\beta_+'$ equals zero we may find the asymptotic zeros of the polynomial\nas for the symmetric case with $F=C=0$.\nIf otherwise\n$\\alpha_-'\\beta_+'\\neq 0$ then $C$ diverges as $L$ is increased.\nThis has to be compensated by modifying our ansatz to \n$x=\\rme^{\\rmi \\frac{\\pi}{2}-\\rmi\\frac{\\phi}{L}+\\ln Q}$ .\nThis works as long as the nominator of $\\Gamma$ in \\eref{Gamma} is \ndifferent from zero.\nOtherwise we have to modify our ansatz another time to\n$x=\\rme^{\\rmi \\frac{\\pi}{2}-\\rmi\\frac{\\phi(L)}{L}+\\ln Q}$, where\n$\\lim_{L\\to \\infty}\\frac{\\phi(L)}{L}=0$ and $\\lim_{L \\to \\infty} \\rme^{-\\rmi\\phi(L)}=0$.\nThis leads again to an equation of the form \\eref{lam} which can be solved as described above.\n\n\n\n\n\n \n\n\n"
},
{
"name": "resnonherm.tex",
"string": "\\subsection{Anomalous behaviour}\nFor the last two cases in \\tref{structure} we obtained logarithmic \ncorrections to the free surface energy.\nNote that there is no case in \\tref{tabfac} which corresponds \nto the boundaries in question.\nHowever, we computed the ground-state energy $E_0$ of $H_{\\rm long}$ \nnumerically for one example of boundaries which lead to the anomalous \nbehaviour. Our computations suggest \n\\begin{equation}\n\\label{loge0}\nE_0\\sim -\\frac{L}{\\pi} +f_{\\infty} +\\frac{1}{L}\\left( \\rho_1 \\ln L + \\rho_2 \\right)\n\\end{equation}\nwhere $\\rho_1,\\rho_2$ are complex numbers. \\Fref{re} and \\fref{im} show the\nreal respectively the imaginary part of the Casimir amplitude.\nThe computation has been done for \n$\\alpha_z=1/\\sqrt{2},\\beta_z=\\rmi/\\sqrt{2},\\alpha_{\\pm}=\\beta_{\\pm}=0$ which corresponds \nto the last case in \\tref{structure}.\nIn both cases the fermionic energies scale as \n\\begin{equation}\n\\label{aslog}\n2\\Lambda \\sim \\frac{1}{L} \\left[ k\\pi \\pm \\frac{1}{2}\\left( \\arg \\Delta -\n \\rmi\\ln L \\right)\\right] \\qquad k=0,1,2,....\n\\end{equation}\nwhere the value of $\\Delta$ is given by\n\\begin{equation}\n\\Delta=-\\frac{4C}{G} \n\\end{equation}\nfor $F=0,C,G\\neq 0$ respectively by\n\\begin{equation}\n\\Delta=(-1)^L\\frac{2J}{1-4\\alpha_z^2\\beta_z^2}\n\\end{equation}\nfor $F=C=G=K=0,J\\neq 0$.\nOur result for the last case makes only sense for $(\\alpha_z\\beta_z)^2\\neq 1/4$.\nThe factor 2 in \\eref{aslog} is just a remnant from our notation in \\cite{paper1}.\nNote the integer spacing of the energy gaps \\eref{aslog}.\n\\begin{figure}[tbp]\n\\setlength{\\unitlength}{1mm}\n\\def\\setl{ \\setlength\\epsfxsize{8.0cm}}\n\\begin{picture}(155,90)\n\\put(28,80){\n \\makebox{\n \\setl\n \\epsfbox{eo.ps}}\n }\n\\put(10,80){\\footnotesize Re$[ L (E_0+L/\\pi+0.25)]$}\n\\put(90,0){$L$}\n\\end{picture}\n\\caption{ Real part of the Casimir amplitude\n\t$(\\alpha_z=1/\\sqrt{2},\\beta_z=\\rmi/\\sqrt{2},\\alpha_{\\pm}=\\beta_{\\pm}=0)$.}\n\\label{re}\n\\end{figure}\n\\begin{figure}[tbp]\n\\setlength{\\unitlength}{1mm}\n\\def\\setl{ \\setlength\\epsfxsize{8.0cm}}\n\\begin{picture}(155,90)\n\\put(28,80){\n \\makebox{\n \\setl\n \\epsfbox{eoi.ps}}\n }\n\\put(25,80){\\footnotesize Im$( L E_0)$}\n\\put(90,0){$L$}\n\\end{picture}\n\\caption{Imaginary part of the Casimir amplitude\n\t $(\\alpha_z=1/\\sqrt{2},\\beta_z=\\rmi/\\sqrt{2},\\alpha_{\\pm}=\\beta_{\\pm}=0)$.}\n\\label{im}\n\\end{figure}\n\n\n\n\n\n\n\n"
},
{
"name": "tabfu.tex",
"string": "\\begin{landscape}\n\\fulltable{ The values of the parameters $F,C,G,K,J$ for\nthe cases where the ground-state energy of $H_{\\rm long}$ has been obtained in \\cite{paper1}. \\\\\nThe table shows the free surface energy $f_{\\infty}$ and the lowest highest weight $h_{\\rm min}$ (cf. \\eref{gstate}). The \nexpressions for $f_{\\infty}$ are also valid for $H$.\\\\\n$\\zeta$ denotes a free parameter, which has to be chosen such that $0\\leq \\mbox{Re}\\zeta \\leq 2\\pi$\nfor case 12\n and $-\\pi<\\mbox{Re} \\zeta<\\pi$ otherwise .\n }\n\\footnotesize\n\\label{tabfac}\n\\begin{tabular}{@{}lcccccccc}\n\\br\n&&&&&&& \\centre{2}{$h_{\\rm min}$} \\\\\n&&&&&&& \\crule{2} \\\\\ncase & $F$ & $(-1)^L C$ & $G$ & $H$ & $J$ & $f_{\\infty}$ & $L$ even & $L$ odd \\\\\n\\mr\n1 & 2 & 0 & 5 & 4 & 2 & $1/4-3/(2\\pi)$ & \\multicolumn{2}{c}{$1/32$} \\\\\n2 & 0 & 0 & 2 & 3 & 1 & $1/4-4/\\pi$ & \\multicolumn{2}{c}{$1/16$} \\\\\n3 & 0 & 0 & 1 & 2 & 0 & $1/2-3/{2\\pi}$ & \\multicolumn{2}{c}{$1/16$} \\\\\n4 & 0 & 0 & 4 & 4 & 2 & $-1/(2\\pi)$ & \\multicolumn{2}{c}{$1/16$} \\\\\n5 & 1 & $\\frac{1}{2}$ & 3 & 3 & 1 & $1/2-2/\\pi$ & $0$ & $1/8$ \\\\\n6 & 1 & $-\\frac{1}{2}$ & 3 & 3 & 1 & $1/2-2/\\pi$ & $1/8$ & $0$ \\\\\n7 & 2 & $1$ & 5 & 4 & 2 & $1/4-3/(2\\pi)$ & $0$ & $1/8$ \\\\\n8 & 2 & $-1$ & 5 & 4 & 2 & $1/4-3/(2\\pi)$ & $1/8$ & $0$ \\\\\n9 & 4 & $ (-1)^L2 \\cos\\zeta $ & 8 & 5 & 3 & $-1/\\pi$ & \\multicolumn{2}{c}{$\\zeta^2/(8\\pi^2)$}\\\\\n10 & $ 2+2\\cos \\zeta $ & $1+\\cos \\zeta $ & $4 +4 \\cos \\zeta$ &\n$3+2\\cos \\zeta$ & $1+\\cos \\zeta$ & $1/2-1/\\pi-\\cos(\\zeta/2)/2$ & $0$ & $1/8$ \\\\\n11 & $4 +4\\cos\\zeta$ & 0 & $6+6\\cos\\zeta$ & $4+2\\cos\\zeta$ &\n$2+2\\cos\\zeta$ & $1/4-1/(2\\pi)-\\cos(\\zeta/2)/2$ & \\multicolumn{2}{c}{$1/32$} \\\\\n12 & $2-2\\cos\\zeta$ & $\\cos\\zeta-1$ & $2-2\\cos\\zeta$ &\n4 & 4 & $1/2-(\\cos(\\zeta/4)+\\sin(\\zeta/4))/2$ & $1/8$ & $0$ \\\\\n13 & $4+4\\cos\\zeta$ & $-2-2\\cos\\zeta$ & $4+4\\cos\\zeta$ &\n $4+2\\cos\\zeta$ & $4+2\\cos\\zeta$ & $1/2-1/(2\\sqrt{2})-\\cos(\\zeta/2)/2$ & $1/8$ & $0$ \\\\\n14 & $8+8\\cos\\zeta$ & $-2-2\\cos\\zeta$ & $8+8\\cos\\zeta$ &\n $6+2\\cos\\zeta$ & $2\\cos\\zeta-2$ & $-\\cos(\\zeta/2)/2$ & $0$ & $1/8$ \\\\\n15 & $8+8\\cos\\zeta$ & $-4-4\\cos\\zeta$ & $6+6\\cos\\zeta$ &\n$6+2\\cos\\zeta$ & $6+2\\cos\\zeta$ & $-\\cos(\\zeta/2)/2$ & $1/8$ & $0$ \\\\\n16 & 0 & 0 & 0 &\n$2+2\\cos\\zeta$ & $2+2\\cos\\zeta$ & $1/2-\\cos(\\zeta/2)/2$ & $1/8$ & $0$ \\\\\n\\br\n\\end{tabular}\n\\label{tabfuck}\n\\endfulltable\n\\end{landscape}\n"
}
] |
[
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"name": "cond-mat0002162.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{paper1} {Bilstein U, Wehefritz B 1999 \\JPA {\\bf 32} 191}\n\\bibitem{ARev}{Affleck I 1989 Field Theory Methods and Quantum Critical\nPhenomena \\\\ {\\it Fields, Strings and Critical Phenomena}\nLes Houches 1988 (North-Holland: Amsterdam)}\n\\bibitem{Sal}{Saleur H 1998 {\\it Lectures on Non Perturbative Field Theory and\nQuantum Impurity Problems} \\\\ cond-mat/9812110}\n\\bibitem{AlcBaaGriRit2} {Alcaraz F C, Baake M, Grimm U and Rittenberg V\n1988 \\JPA {\\bf 21} L117}\n\\bibitem{AlcBarBat}{Alcaraz F C, Barber M N, Batchelor M T\n1987 \\PRL {\\bf 58} 771}\n\\bibitem{AlcBarBat2}{Alcaraz F C, Barber M N, Batchelor M T\n1988 \\APNY {\\bf 182} 280}\n\\bibitem{OshAff}{Oshikawa M, Affleck I 1997 \\NP B {\\bf 495} 533}\n\\bibitem{AlcBaaGriRit1} {Alcaraz F C, Baake M, Grimm U and Rittenberg V\n1989 \\JPA {\\bf 22} L5}\n\\bibitem{Aff2} {Affleck I 1998 \\JPA {\\bf 31} 2761}\n\\bibitem{AlcWre}{Alcaraz F C, Wreszinski W F 1990\n{\\it J. Stat. Phys.} {\\bf 58} 45}\n\\bibitem{paper0}{Bilstein U, Wehefritz B 1997 \\JPA {\\bf 30} 4925}\n\\bibitem{Car}{Cardy J L 1986 \\NP B {\\bf 275} 200}\n\\bibitem{NohKim}{Noh J D, Kim D 1996 \\PR E {\\bf 53} 3225}\n\\bibitem{LSM}{Lieb E, Schultz T and Mattis D 1961 {\\em Ann. Phys.}\n{\\bf 16} 407}\n\\bibitem{Gui}{Guinea F 7518 \\PR B {\\bf 32} 7518}\n\\bibitem{BugSha}{Bugrij A I, Shadura V N 1990 \\PL {\\bf 150} 171}\n\\bibitem{LambertW}{Corless R M, Gonnet G H, Hare D E G, Jeffrey D J, Knuth D E \n{\\it On the Lambert W Function} Maple Share Library}\n\\end{thebibliography}"
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cond-mat0002163
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Convergence to the critical attractor of dissipative maps: Log-periodic oscillations, fractality and nonextensivity
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"author": "F.A.B.F. de Moura"
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For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble $W(t)$ depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of $W(t)$ for the circle map whose critical attractor is dense. In this case, we found $W(t)$ to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of non-extensive Tsallis entropies. Pacs Numbers: 05.45.Ac, 05.20.-y, 05.70.Ce
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"name": "lanl.tex",
"string": "\\documentstyle[aps,epsfig]{revtex}\n\n\\textwidth 17.4cm\n\n\\begin{document}\n \n%\\draft\n\\title{Convergence to the critical attractor of dissipative maps: \nLog-periodic oscillations, fractality and nonextensivity} \n\\vspace{2.5cm}\n\n\\author{F.A.B.F. de Moura}\\address{Departamento de F\\'{\\i}sica,\nUniversidade Federal de Pernambuco, 50670-901 Recife - PE, Brazil}\n\\vspace{2cm}\n\\author{U. Tirnakli\\thanks{e-mail: tirnakli@sci.ege.edu.tr}}\n\\address{Department of Physics, Faculty of Science,\nEge University 35100 Izmir, Turkey \\\\\nand \\\\\nCentro Brasileiro de Pesquisas F\\'{\\i}sicas, \nRua Xavier Sigaud 150, 22290-180 Rio de Janeiro - RJ, Brazil}\n\\vspace{2cm}\n\\author{M. L. Lyra}\\address{Departamento de F\\'{\\i}sica, Universidade\nFederal de Alagoas, 57072-970 Macei\\'o - AL, Brazil}\n\n\n\\maketitle\n\\vspace{1cm}\n\n\\begin{abstract}\nFor a family of logistic-like maps, we investigate the rate of \nconvergence to the critical attractor when an ensemble of initial \nconditions is uniformly spread over the entire phase space. \nWe found that the phase space volume occupied by the ensemble $W(t)$ \ndepicts a power-law decay with log-periodic oscillations reflecting \nthe multifractal character of the critical attractor. \nWe explore the parametric dependence of the power-law exponent and the \namplitude of the log-periodic oscillations with the attractor's \nfractal dimension governed by the inflexion of the map near its \nextremal point. Further, we investigate the temporal evolution of \n$W(t)$ for the circle map whose critical attractor is dense. \nIn this case, we found $W(t)$ to exhibit a rich pattern with a slow \nlogarithmic decay of the lower bounds. These results are discussed \nin the context of non-extensive Tsallis entropies. \n\nPacs Numbers: 05.45.Ac, 05.20.-y, 05.70.Ce\n\n\\end{abstract}\n\n\n\n%\\newpage\n\n\\vspace{1cm}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{introduction}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nNonlinear low-dimensional dissipative maps can describe a great variety \nof systems with few degrees of freedom. The underlying nonlinearity can \ninduce the system to exhibit a complex behavior with quite structured \npaths in the phase space. The sensitivity to initial conditions is a \nrelevant aspect associated to the structure of the dynamical attractor.\n In general, the sensitivity is measured as the \neffect of any uncertainty on the system's variables. For systems \nexhibiting periodic or chaotic orbits, the effect of any uncertainty \non initial conditions depicts an exponential temporal evolution with \n$\\xi (t) \\equiv\\lim_{\\Delta x(0)\\rightarrow 0} \\Delta x(t)/\\Delta x(0) \n\\sim e^{\\lambda t}$, where $\\lambda$ is the Lyapunov exponent, $\\Delta\nx(0)$ and $\\Delta x(t)$ are the uncertainties at times $0$ \nand $t$. When the Lyapunov exponent $\\lambda < 0$, $\\xi (t)$ \ncharacterizes the rate of contraction towards periodic orbits. On the \nother hand, for $\\lambda >0$, it characterizes the rate of divergence \nof chaotic orbits. \nAt bifurcation and critical points (i.e., onset to chaos) the \nLyapunov exponent $\\lambda$ vanishes. Recently, it was shown that this \nfeature is related to a power-law sensitivity to initial conditions on \nthe form\\cite{Zheng,Lyra1,Lyra2}\n\n\\begin{equation}\n\\xi (t) = [1+(1-q)\\lambda_qt]^{1/(1-q)}~~~,\n\\end{equation}\nwith $\\lambda_q$ defining a characteristic time scale after which the\npower-law behavior sets up. \n\n\nA quantitative way to measure the sensitivity to initial conditions is \nto follow, from a particular partition of the phase space, the temporal \nevolution of the number of cells $W(t)$ occupied by an ensemble of\nidentical copies of the system. \nFor periodic and chaotic orbits $W(t) = W(0)e^{\\lambda t}$.\nIn the particular case of equiprobability, the well-known Pesin equality \nreads $K=\\lambda$ if $\\lambda\\ge 0$\\cite{pesin} \nwith $K$ being the\nKolmogorov-Sinai entropy \\cite{KS} defined as the \nvariation per unit time of the standard Boltzmann-Gibbs entropy\n$S=-\\sum p_i\\ln{p_i}$. \nThis equality provides a link between the sensitivity to initial\nconditions and the dynamic evolution of the relevant entropy. \n\nAt bifurcation and critical points and for an ensemble of initial\nconditions concentrated in a single cell, i.e. $W(0)=1$, it has been\nshown that \n\\begin{equation}\nW(t) = [1+(1-q)K_qt]^{1/(1-q)}~~~,\n\\end{equation}\nwith $K_q$ being the generalized Kolmogorov-Sinai entropy \ndefined as the rate of variation of the non-extensive Tsallis \nentropy $S_q=(1-\\sum p_i^q)/(q-1)$\\cite{Tsallis}.\nThe Pesin equality can be generalized\nas $K_q=\\lambda_q$ if $\\lambda_q \\ge 0$\\cite{Zheng}. \nTsallis entropies have been successfully applied to recent studies of a \nseries of non-extensive systems and provided a theoretical \nbackground to the understanding of some of their unusual physical \nproperties\\cite{bjp,html}. \n\n\nThe expansion towards the critical attractor\nof an ensemble of initial conditions concentrated around the inflexion \npoint of the map can be characterized by a proper $S_q$ evolving at a\nconstant rate. Scaling arguments have shown that the appropriate \nentropic index $q$ is related to the multifractal\nstructure of the critical dynamical attractor by\\cite{Lyra2}\n\\begin{equation}\n\\frac{1}{1-q} = \\frac{1}{\\alpha_{min}}-\\frac{1}{\\alpha_{max}}\n\\end{equation}\nwhere $\\alpha_{min}$ and $\\alpha_{max}$ are the extremal singularity\nstrengths of the multifractal spectrum of the critical\nattractor\\cite{Halsey}. The above scaling relation has been shown to \nhold for the families of generalized logistic and circle\nmaps\\cite{Lyra2,Lyra3,Lyra4,Lyra5}.\n\n\nHowever, the temporal evolution of critical dynamical systems can be \nstrongly dependent on the particular initial ensemble. Although some\nscaling laws can be found for an ensemble of initial conditions\nconcentrated around the map inflexion point, these are usually not\nuniversal with respect to a general ensemble. In this work, we are going\nto numerically investigate the critical temporal evolution of the volume\nof the phase space occupied by an ensemble of initial conditions spread\nover the entire phase space. This ensemble is expected to contract towards\nthe critical attractor. Using a family of \none-dimensional generalized logistic maps having $d_f<1$, we will perform\na detailed study of the parametric dependence of \n$W(t)$ on the fractal dimension of the critical attractor. \nDue to the discrete scale invariance of the critical\nattractor, the convergence displays log-periodic oscillations\\cite{Sornette}.\nWe are also going to explore the dependence of the amplitude of these\noscillations with respect to the attractor's fractal dimension. Further,\nthe behavior of $W(t)$ will be investigated for the one-dimensional\ncritical circle map having $d_f=1$. For this map, the temporal evolution\nis expected to display distinct trends since the critical \nattractor is dense. \n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{The convergence to the critical attractor of generalized\nlogistic maps}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nLogistic-like maps are the simplest one-dimensional nonlinear dynamical\nsystems which allow a close investigation of a series of critical\nexponents related to the onset of chaotic orbits. This family reads\n\\begin{equation}\nx_{t+1}=1-a|x_t|^z~~~;~~~(z>1~~;~~0<a<2~~; t=0,1,2,...~~; \nx_t\\in [-1,1])~~.\n\\end{equation}\nHere $z$ is the inflexion of the map in the neighborhood of the extremal\npoint $\\bar{x}=0$. These maps are well known to have topological\nproperties not dependent of $z$. However, the metrical properties, such as\nFeigenbaum exponents\\cite{Feigenbaum,Coullet} and the multifractal\nspectrum of the critical attractor do depend on $z$. In particular the\nfractal dimension of the critical attractor $d_f(z) <1$\\cite{Beck} and\ntherefore it does not fill a finite fraction of the phase space. For a set\nof initial conditions spread in the vicinity of the inflexion point, it\nwas found that the volume in phase space occupied by the ensemble grows\nfollowing a rich pattern with the upper bounds $W_{max}(t)$ governed by a\npower-law $W_{max}(t)\\propto t^{1/(1-q)}$, where $q$ is the entropic index\ncharacterizing the relevant Tsallis entropy that grows at a constant \nrate. It has been shown that the dynamic exponent $1/(1-q)$ is directly\nrelated to geometric scaling exponents related to the extremal sets of\nthe dynamic attractor\\cite{Lyra2}.\n\n\nDue to the presence of long-range spatial and temporal correlations at\ncriticality, one expects the critical exponent governing the temporal\nevolution to be sensitive to the particular initial ensemble. Indeed,\nthe multifractal spectrum characterizing the critical dynamical attractor\nindicates that an infinite set of exponents are needed to fully\ncharacterize the scaling behavior. In particular, an ensemble consisting\nof a set of identical systems whose initial conditions is spread over the\nentire phase-space is a common one when studying non-linear as well as \nthermodynamical systems. \n\n\nHere, we will follow the dynamic evolution, in phase space, of an\nensemble of initial conditions uniformly distributed over the phase-space\nand explore its relation with the generalized fractal dimensions of the\ncritical attractor. In practice, a partition of the phase space on\n$N_{box}$ cells of equal size is performed and a set of $N_c$ identical\ncopies of the system is followed whose initial conditions are uniformly\nspread over the phase-space. The ratio $r=N_c/N_{box}$ is a control \nparameter giving the degree of sampling of the phase-space.\n\nWithin the non-extensive Tsallis statistics, there is a proper entropy $S_q$\n evolving at a constant rate such that\n\\begin{equation}\nK_q = \\lim_{N_{box}\\rightarrow\\infty} [S_q(t) - S_q(0)]/t\n\\end{equation}\ngoes to a constant value as $t\\rightarrow\\infty$. Notice that $K_q<0$ for\nthe process of convergence towards the critical attractor. Assuming that\nall cells of the partition are occupied with equal probability, the\nentropy $S_q(t)$ can be written as\n\\begin{equation}\nS_q(t) = \\frac{1-\\sum_{i=1}^{W(t)} p_i^q}{q-1} = \\frac{W(t)^{1-q}-1}{1-q}\n\\end{equation}\nThe last two equations imply that the number of occupied cells evolves in\ntime as\n\\begin{equation}\nW(t) = [W(0)^{1-q}+(1-q)K_qt]^{1/(1-q)}\n\\end{equation}\nwith the exponent $\\mu = -1/(q-1) >0 $ governing the asymptotic\npower-law decay. \n\n\nIn figure~1, we show our results for $W(t)/N_{box}$ in the standard\nlogistic map with inflexion $z=2$ and from distinct\npartitions of the phase space with sampling ratio $r=0.1$ . We observe\nthat, after a short transient period when $W(t)$ is nearly constant, a\npower-law contraction of the volume occupied by the ensemble sets up.\n$W(t)$ saturates at a finite fraction corresponding to the phase space\nvolume occupied by the critical attractor on a given finite partition. The\nsaturation is postponed when a finer partition is used once the fraction\noccupied by the critical attractor vanishes in the limit\n$N_{box}\\rightarrow\\infty$. \n\n\nIn figure~2, we show $W(t)/N_{box}$ for a given fine partition of the\nphase-space and distinct sampling ratios $r$. We notice that the crossover\nregime to the power-law scaling is quite short for large values of $r$ so\nthat a clear power-law scaling regime sets up even at early times. This \nfeature is consistent with eq.(7) which states that the crossover time\n$\\tau$ scales as $\\tau\\sim 1/W(0)^{q-1}$. \nFurther, the scaling regime exhibits log-periodic oscillations once the\nmultifractal nature of the critical attractor is closely probed by such\ndense ensemble. A general form for $W(t)$ reflecting the discrete scale \ninvariance of the attractor can be written as\n\n\\begin{equation}\nW(t) = t^{-\\mu}P\\left( \\frac{\\ln{t}}{\\ln{\\lambda}}\\right)\n\\end{equation}\nwhere $P$ is a function of period unity and $\\lambda$ is the \ncharacteristic scaling factor between the periods of two consecutive\noscillations. These log-periodic\noscillations have been observed in a large number of systems exhibiting\ndiscrete scale invariance\\cite{Sornette}. In general the amplitude of\nthese oscillations ranges form $10^{-4}$ up to $10^{-1}$.\nKeeping only the first term in a Fourier series of\n$P(\\ln{t}/\\ln{\\lambda})$, one can write $W(t)$ in the form\n\\begin{equation}\nW(t) = c_0t^{-\\mu}\\left[ 1 +\n2\\frac{c_1}{c_0}\n\\cos{\\left(2\\pi\\frac{\\ln{t}}{\\ln{\\lambda}} +\\phi\\right)}\\right]\\;\\; .\n\\end{equation}\n\n\nLog-periodic modulations correcting a pure power-law have been found\nin several systems, as, for example, \ndiffusion-limited-aggregation\\cite{Sornette2}, crack growth\\cite{Ball},\nearthquakes\\cite{Newman} and financial markets\\cite{Drozdz}. \nIt has also been observed in thermodynamic systems with fractal-like\nenergy spectrum\\cite{Mendes,Vallejos}. The factors\ncontrolling the log-periodic relative amplitude $2c_1/c_0$ are not well\nknown for most of the systems where it has been observed. In the present\nstudy, we can closely investigate the factors which may control these\namplitudes by measuring it as a function of the map inflexion $z$ for a\nfixed partition and sampling ratio (see figure 3). We found that these\noscillations have amplitudes decaying exponentially with $z$ as shown in\nfigure~4. It is interesting to point out that the fractal dimension of\nthe attractor is a monotonically decreasing function of $z$. Therefore,\nthe above trend indicates a possible correlation between the\namplitude of the log-periodic oscillations and the fractal dimension of\nthe dynamical attractor. \n\n\nWe also measured the critical exponent $\\mu$ as a function of the map\ninflexion $z$. Our results are summarized in the Table. It is a decreasing\nfunction of $z$ as can be seen in figure~5. The volume occupied by the\nensemble depicts a fast contraction for $z\\sim 1$ where the fractal\ndimension is small. On the other side, a very slow contraction is\nobserved for large values of $z$, pointing towards a saturation or at most\nto a logarithmic decrease of $W(t)$ in the limit of dense attractors. We\nwould like to point out here that the exponent governing the expansion\nof the volume occupied by an ensemble of initial conditions concentrated\naround the inflexion point exhibits a reversed trend. Although scaling\narguments have shown that this exponent can be written in terms of\nscaling exponents characterizing the extremal sets in the attractor, we\ncould not devise a simple scaling relation between $\\mu$ and the\nmultifractal singularity spectrum. However, we observed\nthat, when plotted against the fractal dimension of the attractor as\nshown in figure~6, the dynamic exponent $\\mu$ is very well fitted by\n$\\mu\\propto (1-d_f)^2$, which indicates $d_f$ as the relevant geometric\nexponent coupled to the dynamics of the uniform ensemble. \nWe would like to mention here that the \nsame dynamic exponents were obtained for the generalized periodic maps\nwhich belong to the same universality class of logistic-like\nmaps\\cite{Lyra3}.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{The convergence to the critical attractor of the circle map}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe results from the previous section indicate that a slow convergence to\nthe critical attractor shall be expected for dense critical attractors.\nHowever, it is not clear in what fashion this convergence will take place\nwhen the dynamical attractor fills the phase space with a\nmultifractal probability density as occurs for the one-dimensional\ncritical circle map\n\n\\begin{equation}\n\\theta_{t+1} = \\theta_t +\\Omega -\\frac{1}{2\\pi}\\sin{(2\\pi\\theta_t)}\n~~~ mod(1)\n\\end{equation}\nwhere $0\\leq\\theta_t<1$ is a point on a circle. The circle map describes\ndynamical systems possessing a natural frequency $\\omega_1$ which are\ndriven by an external force of frequency $\\omega_2$ ($\\Omega =\n\\omega_1/\\omega_2$ is the bare winding number) and belongs to the same\nuniversality class of the forced Rayleigh-B\\'enard \nconvection\\cite{Jensen}. For $\\Omega = 0.606661...$ the circle map has a\ncubic inflexion ($z=3$) in the vicinity of the point $\\bar{\\theta}=0$.\nStarting from a given point on the circle, it generates a quasi-periodic\norbit which fills the phase-space and the dynamical attractor is\na multifractal with fractal dimension $d_f=1$\\cite{Halsey}. \n\n\nIn figure~7 we show our results for the temporal evolution of the\nphase-space volume occupied by an ensemble of initial conditions uniformly\nspread over the circle. $W(t)$ exhibits a rich pattern which resembles the\none observed for the sensitivity function associated to the expansion of\nthe phase-space from initial conditions concentrated around the inflexion\npoint. However, $W(t)$ does not present any power-law regime. Instead, the\nlower bounds display a slow logarithmic decrease with time, saturating at\na finite volume fraction. The saturation is a feature related to the\nfinite partition used in the numerical calculation. This minimum\ndecreases logarithmically with the number of cells in the phase-space as \nshown in figure~8. We also observed the same behavior for generalized\ncircle maps with an arbitrary inflexion $z$\\cite{Lyra5}. The critical\nattractors within this family have all $d_f=1$ although \nthey exhibit a $z$-dependent multifractal singularity spectra. \nThe $z$-independent scenario for $W(t)$ corroborates the \nconjecture \nthat $d_f$ is the relevant geometric exponent coupled to the dynamics \nof the uniform ensemble.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Summary and Conclusions}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn this work, we studied the temporal evolution in phase space of an\nensemble of identical copies of one-dimensional nonlinear dissipative\nmaps. We found that the phase-space volume occupied by an initially\nuniform ensemble displays a power-law decay with log-periodic oscillations\nwhenever the dynamical attractor has a fractal dimension $d_f<1$. The\namplitude of the oscillations was found to depict a monotonic parametric\ndependence on $d_f$. For dense multifractal attractors, $W(t)$ presents \nonly a slow logarithmic contraction of its lower bounds followed by a\nrich pattern.\n\n\nThe critical exponent characterizing the contraction of the uniform\nensemble was found to have no direct relation to the one governing the\nexpansion from a set of initial conditions concentrated around the\ninflexion point. In particular, no power-law was found for the contraction\nin the standard and generalized circle maps, in contrast to the\n$z$-dependent power-law expansion. These results indicate that the\nrelevant Tsallis entropy evolving at a constant rate (modulated by\nlog-periodic oscillations) is characterized by an entropic index $q$\nthat depends on the initial ensemble. It would be valuable to\ninvestigate the possible existence of classes of ensembles with a common\ndynamics in phase-space and, therefore, characterized by the same\nentropic index $q$. The non-universality of $q$ with respect to the\ninitial ensemble is related to the multifractal character of the dynamical\nattractor. However, as for the ensemble concentrated at the vicinity of\nthe inflexion point, the exponent governing the dynamics of the uniform\nensemble is coupled to a geometric scaling exponent, in particular to the\nproper fractal dimension of the attractor. This result comes in favor of\nthe concept that the degree of nonextensivity of the entropy measure\nevolving at a constant rate is related to the fractal nature of the\ndynamical attractor. \n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{acknowledgments}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nUT acknowledges the partial support of BAYG-C program of TUBITAK \n(Turkish agency) as well as CNPq and PRONEX (Brazilian agencies).\nThis work was partially supported by CNPq and CAPES (Brazilian research\nagencies). \nMLL would like to thank the hospitality of the Physics Department at\nUniversidade Federal de Pernambuco during the Summer School 2000 where\nthis work was partially developed.\n\n\n\\newpage\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{thebibliography}{99}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Zheng}C. Tsallis, A.R. Plastino and W.-M. Zheng, Chaos Solitons\nFractals {\\bf 8}, 885 (1997).\n\n\\bibitem{Lyra1}U.M.S. Costa, M.L. Lyra, A.R. Plastino and C.Tsallis,\nPhys. Rev. E {\\bf 56}, 245 (1997).\n\n\\bibitem{Lyra2}M.L. Lyra and C. Tsallis, Phys. Rev. Lett. {\\bf 80}, 53\n(1998).\n\n\\bibitem{KS} A.N. Kolmogorov, Dok. Acad. Nauk SSSR {\\bf 119}, \n861 (1958); Ya. G. Sinai, Dok. Acad. Nauk SSSR {\\bf 124}, 768 (1959).\n\n\\bibitem{pesin} Ya. Pesin, Russ. Math. Surveys {\\bf 32}, 55 (1977).\n\n\\bibitem{Tsallis}C. Tsallis, J. Stat. Phys. {\\bf 52}, 479 (1988).\n\n\\bibitem{bjp}For a recent review see: C. Tsallis, Braz. J. Phys. {\\bf 29}, \n1 (1999) [http://www.sbf.if.usp.br/WWW\\_pages/Journals/BJP/index.htm].\n\n\\bibitem{html}A complete list of references on the subject of nonextensive \nTsallis statistics can be found at http://tsallis.cat.cbpf.br/biblio.htm\n\n\\bibitem{Halsey} T.A. Halsey {\\em et al}, Phys. Rev. A {\\bf 33}, 1141\n(1986).\n\n\\bibitem{Lyra3}M.L. Lyra, Ann. Rev. Comp. Phys. {\\bf 6}, 31 (1999).\n\n\\bibitem{Lyra4}C.R. da Silva, H.R. da Cruz and M.L. Lyra, Braz. J. Phys.\n{\\bf 29}, 144 (1999) \n[http://www.sbf.if.usp.br/WWW\\_pages/Journals/BJP/index.htm].\n\n\\bibitem{Lyra5}U. Tirnakli, C. Tsallis and M.L. Lyra, Eur. Phys. J. B\n{\\bf 11}, 309 (1999).\n\n\\bibitem{Sornette}D. Sornette, Phys. Rep. {\\bf 297}, 239 (1998).\n\n\\bibitem{Feigenbaum}M.J. Feigenbaum, J. Stat. Phys. {\\bf 19}, 25 (1978);\n{\\bf 21}, 669 (1979).\n\n\\bibitem{Coullet}P. Coullet and C. Tresser, J. Phys. (Paris) Colloq. {\\bf\n5} C25 (1978).\n\n\\bibitem{Beck} C. Beck and F. Schlogl, in {\\em Thermodynamics of Chaotic\nSystems} (Cambridge University Press, Cambridge, 1993)\n\n\\bibitem{Jensen}M.H. Jensen {\\em et al}, Phys. Rev. Lett. {\\bf 55}, 2798\n(1985).\n\n\\bibitem{Sornette2}D. Sornette {\\em et al}, Phys. Rev. Lett. {\\bf 76},\n251 (1996).\n\n\\bibitem{Ball}R.C. Ball and R. Blumenfeld, Phys. Rev. Lett. {\\bf 65},\n1784 (1990).\n\n\\bibitem{Newman}W.I. Newman, D.L. Turcotte and A.M. Gabrielov, Phys. Rev.\nE {\\bf 52}, 4827 (1995).\n\n\\bibitem{Drozdz}S. Drozdz, F. Ruf, J. Speth and M. Wojcik, \nEur. Phys. J. B {\\bf 10}, 589 (1999).\n\n\\bibitem{Mendes} C. Tsallis {\\em et al}, Phys. Rev. E {\\bf 56}, R4922\n(1997).\n\n\\bibitem{Vallejos} R.O. Vallejos {\\em et al}, Phys. Rev. E {\\bf 58}, 1346\n(1998).\n\n\\end{thebibliography}\n\n\\newpage\n\n\n\\section*{FIGURE AND TABLE CAPTIONS}\n\n\\noindent\n{\\bf Figure 1 -} The volume occupied by the ensemble $W(t)$ as a function\nof time in the standard logistic map ($z=2$) and with sampling ratio\n$r=0.1$. From top to bottom $N_{box} = 2000, 8000, 32000, 128000$.\n\n~\n\n\\noindent\n{\\bf Figure 2 -} The volume occupied by the ensemble $W(t)$ as a function\nof time in the standard logistic map ($z=2$) and for a partition\ncontaining $N_{box} = 128000$ cells. Notice the emergence of log-periodic\noscillations for large sampling ratios.\n\n~\n\n\\noindent\n{\\bf Figure 3 -} The periodic function $W(t)/(c_0t^{-\\mu})$ versus time\nwithin the scaling regime and for $r=10$. Data from map inflexions \n$z=1.1, 1.25, 1.5, 2.0$ are shown. The amplitude of the oscillations\ndecreases monotonically as $z$ increases, but the characteristic scaling\nfactor between the periods of two consecutive oscillations\nis roughly $z$-independent.\n\n~\n\n\\noindent\n{\\bf Figure 4 -} The amplitude of the log-periodic oscillations \n$2c_1/c_0$ as a function of the map inflexion $z$ for sampling ratio \n$r=10$. The monotonic decrease of the oscillations indicates a close \nrelation between these and the fractal dimension of the underlying \ndynamical attractor.\n\n~\n\n\\noindent\n{\\bf Figure 5 -} The dynamic exponent $\\mu$ governing the contraction of\nthe occupied phase space volume [$W(t)\\propto t^{-\\mu}$] as a function of \nthe map inflexion $z$. \n\n~\n\n\\noindent\n{\\bf Figure 6} - The parametric dependence of the dynamic exponent $\\mu$\nwith the fractal dimension $d_f$ of the critical attractor. It is very\nwell fitted to the form $\\mu\\propto (1-d_f)^2$, indicating that $d_f$\nseems to be the relevant geometric exponent coupled to the dynamics of the\nuniform ensemble.\n\n~\n\n\\noindent\n{\\bf Figure 7 -} The volume occupied by the ensemble $W(t)$ as a \nfunction of time in the standard critical circle map. The lower bounds \ndisplay a slow logarithmic decay with time saturating at a finite volume\nfraction due to the finite partition of the phase space.\n\n~\n\n\\noindent\n{\\bf Figure 8 -} The asymptotic lower bounds for the occupied volume in\nthe phase space versus the number of cells $N_{box}$. The logarithmic\ndecay agrees with the prediction that $\\mu (d_f\\rightarrow 1)\\rightarrow\n0$. The same behavior was observed for the family of generalized circle \nmaps and corroborates the conjecture that $d_f$ is the relevant\ngeometric exponent coupled to the dynamics of the uniform ensemble.\n\n~\n\n\\noindent\n{\\bf Table } - Numerical values, within the $z$-generalized family of \nlogistic maps, of: {\\em i}) the dynamic exponent $\\mu$ governing the \ncontraction towards the critical attractor of the uniform ensemble; \n{\\em ii}) the entropic index $q$ of the proper Tsallis entropy \ndecreasing at a constant rate; {\\em iii}) the fractal dimension $d_f$ \nof the critical attractor. These values also hold for the generalized \nperiodic maps. The last line represents our results for the \n$z$-generalized circle maps. \n\n\n\\newpage\n\\begin{center}\n{\\bf Table}\n\n\\vspace{1.52cm}\n\n\\begin{tabular}{||c|c|c|c||} \\hline\n$z$ & $~\\mu = -1/(1-q)~$ & $q$ & $d_f$ \\\\ \\hline\n$1.10$ & $1.62 \\pm 0.02$ & $1.62\\pm 0.01$ & $~0.32 \\pm 0.02~$ \\\\ \\hline\n$1.25$ & $1.23 \\pm 0.01 $ & $1.81\\pm 0.01$ & $0.40\\pm 0.01$ \\\\ \\hline\n$1.5$ & $0.95 \\pm 0.01$ & $2.05\\pm 0.01$ & $0.47\\pm 0.01$ \\\\ \\hline\n$1.75$ & $0.80 \\pm 0.01$ & $~2.25\\pm 0.015~$ & $0.51\\pm 0.01$ \\\\ \\hline\n$2.0$ & $ 0.71 \\pm 0.01$ & $2.41\\pm 0.02$ & $0.54\\pm 0.01$ \\\\ \\hline\n$2.5$ & $0.59 \\pm 0.01$ & $2.70\\pm 0.02$ & $ 0.58\\pm 0.01$ \\\\ \\hline\n$3.0$ & $0.515 \\pm 0.005$ & $2.94\\pm 0.02$ &$ 0.60\\pm 0.01$ \\\\ \\hline\n$5.0$ & $0.395 \\pm 0.005$ & $3.53\\pm 0.03$ &$ 0.66\\pm 0.01$ \\\\ \\hline\n$z$-circular & & & \\\\\nmaps & $ 0.0 $ & $\\infty$ &$ 1.0 $ \\\\ \\hline \n\\end{tabular}\n\\end{center}\n\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002163.extracted_bib",
"string": "\\begin{thebibliography}{99}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Zheng}C. Tsallis, A.R. Plastino and W.-M. Zheng, Chaos Solitons\nFractals {\\bf 8}, 885 (1997).\n\n\\bibitem{Lyra1}U.M.S. Costa, M.L. Lyra, A.R. Plastino and C.Tsallis,\nPhys. Rev. E {\\bf 56}, 245 (1997).\n\n\\bibitem{Lyra2}M.L. Lyra and C. Tsallis, Phys. Rev. Lett. {\\bf 80}, 53\n(1998).\n\n\\bibitem{KS} A.N. Kolmogorov, Dok. Acad. Nauk SSSR {\\bf 119}, \n861 (1958); Ya. G. Sinai, Dok. Acad. Nauk SSSR {\\bf 124}, 768 (1959).\n\n\\bibitem{pesin} Ya. Pesin, Russ. Math. Surveys {\\bf 32}, 55 (1977).\n\n\\bibitem{Tsallis}C. Tsallis, J. Stat. Phys. {\\bf 52}, 479 (1988).\n\n\\bibitem{bjp}For a recent review see: C. Tsallis, Braz. J. Phys. {\\bf 29}, \n1 (1999) [http://www.sbf.if.usp.br/WWW\\_pages/Journals/BJP/index.htm].\n\n\\bibitem{html}A complete list of references on the subject of nonextensive \nTsallis statistics can be found at http://tsallis.cat.cbpf.br/biblio.htm\n\n\\bibitem{Halsey} T.A. Halsey {\\em et al}, Phys. Rev. A {\\bf 33}, 1141\n(1986).\n\n\\bibitem{Lyra3}M.L. Lyra, Ann. Rev. Comp. Phys. {\\bf 6}, 31 (1999).\n\n\\bibitem{Lyra4}C.R. da Silva, H.R. da Cruz and M.L. Lyra, Braz. J. Phys.\n{\\bf 29}, 144 (1999) \n[http://www.sbf.if.usp.br/WWW\\_pages/Journals/BJP/index.htm].\n\n\\bibitem{Lyra5}U. Tirnakli, C. Tsallis and M.L. Lyra, Eur. Phys. J. B\n{\\bf 11}, 309 (1999).\n\n\\bibitem{Sornette}D. Sornette, Phys. Rep. {\\bf 297}, 239 (1998).\n\n\\bibitem{Feigenbaum}M.J. Feigenbaum, J. Stat. Phys. {\\bf 19}, 25 (1978);\n{\\bf 21}, 669 (1979).\n\n\\bibitem{Coullet}P. Coullet and C. Tresser, J. Phys. (Paris) Colloq. {\\bf\n5} C25 (1978).\n\n\\bibitem{Beck} C. Beck and F. Schlogl, in {\\em Thermodynamics of Chaotic\nSystems} (Cambridge University Press, Cambridge, 1993)\n\n\\bibitem{Jensen}M.H. Jensen {\\em et al}, Phys. Rev. Lett. {\\bf 55}, 2798\n(1985).\n\n\\bibitem{Sornette2}D. Sornette {\\em et al}, Phys. Rev. Lett. {\\bf 76},\n251 (1996).\n\n\\bibitem{Ball}R.C. Ball and R. Blumenfeld, Phys. Rev. Lett. {\\bf 65},\n1784 (1990).\n\n\\bibitem{Newman}W.I. Newman, D.L. Turcotte and A.M. Gabrielov, Phys. Rev.\nE {\\bf 52}, 4827 (1995).\n\n\\bibitem{Drozdz}S. Drozdz, F. Ruf, J. Speth and M. Wojcik, \nEur. Phys. J. B {\\bf 10}, 589 (1999).\n\n\\bibitem{Mendes} C. Tsallis {\\em et al}, Phys. Rev. E {\\bf 56}, R4922\n(1997).\n\n\\bibitem{Vallejos} R.O. Vallejos {\\em et al}, Phys. Rev. E {\\bf 58}, 1346\n(1998).\n\n\\end{thebibliography}"
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cond-mat0002164
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Spin and charge inhomogeneities in high-$T_c$ cuprates: \\ Evidence from NMR and neutron scattering experiments
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[
{
"author": "Dirk K. Morr $^{1}$"
},
{
"author": "J\\\"{o}rg Schmalian $^{2}$"
},
{
"author": "and David Pines $^{1,3}$"
}
] |
In this communication we consider the doping dependence of the strong antiferromagnetic spin fluctuations in the cuprate superconductors. We investigate the effect of an incommensurate magnetic response, as recently observed in inelastic neutron scattering (INS) experiments on several YBa$_2$Cu$_3$O$_{6+x}$ compounds, on the spin-lattice and spin-echo relaxation rates measured in nuclear magnetic resonance (NMR) experiments. We conclude that a consistent theoretical description of INS and NMR can be reached if one assumes spatially inhomogeneous but locally commensurate spin correlations and that NMR and INS experiments can be described within a single theoretical scenario. We discuss a simple scenario of spin and charge inhomogeneities which includes the main physical ingredients required for consistency with experiments.
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[
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"name": "NMR.tex",
"string": "\\documentstyle[twocolumn,aps,epsfig,floats]{revtex}\n\\begin{document}\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname %\n@twocolumnfalse\\endcsname\n\\title{Spin and charge inhomogeneities in \n high-$T_c$ cuprates: \\\\\n Evidence from NMR and neutron scattering experiments\n}\n\\author{Dirk K. Morr $^{1}$, J\\\"{o}rg Schmalian $^{2}$, \nand David Pines $^{1,3}$}\n\\address{ $^{1}$ University of Illinois at Urbana-Champaign, \nLoomis Laboratory of Physics, 1110 W. Green St., Urbana, IL 61801\\\\\n$^{2}$ISIS Facility, Rutherford Appleton Laboratory, \n Chilton, Didcot, Oxfordshire, OX11 0QX United Kingdom \\\\\n$^{3}$ Institute for Complex Adaptive Matter, University of California, and \nLANSCE-Division, Los Alamos National Laboratory, Los Alamos, NM 87545}\n\\date{\\today}\n\\draft\n\\maketitle\n\\begin{abstract}\nIn this communication we consider the doping dependence of the strong \nantiferromagnetic spin fluctuations in the cuprate superconductors. \n We investigate the effect of an \nincommensurate magnetic response, as recently observed in inelastic\n neutron scattering (INS) experiments on \nseveral \nYBa$_2$Cu$_3$O$_{6+x}$ compounds, on the spin-lattice and spin-echo \nrelaxation rates measured in nuclear magnetic resonance (NMR) experiments. We conclude that a consistent\ntheoretical description of INS and NMR can be reached if one assumes \nspatially \n inhomogeneous but locally commensurate spin correlations and that NMR and INS experiments can be described \nwithin a single theoretical scenario. We discuss a \nsimple scenario of spin and charge inhomogeneities which includes the main physical \ningredients \nrequired for consistency with experiments. \n\\end{abstract}\n\n\\pacs{PACS numbers: 74.25.Ha,74.25.Nf} \n]\n\n\\narrowtext\n\n\\section{Introduction}\n\\label{intro}\nUnderstanding the doping, frequency and temperature dependence of the magnetic\n response in the high-$T_c$ cuprates is one of the most challenging \nproblems for the high-$T_c$ community.\n Insight into the nature of the strong antiferromagnetic fluctuations \nvery likely holds the key to the unusual normal state properties of the \ncuprates \\cite{general,DJS}, \nas seen in a variety of experimental techniques. \nBoth nuclear magnetic resonance (NMR) and inelastic neutron \nscattering (INS) experiments are important tools to probe spin \nexcitations in cuprates \\cite{NMRreview,Mason1,Bourges1}. \nWhile INS experiments provide insight into the momentum \nand frequency resolved imaginary part of the spin susceptibility, $\\chi({\\bf \nq},\\omega)$, \nNMR experiments yield information on the momentum averaged \nreal and imaginary part of $\\chi({\\bf q},\\omega)$ in the zero frequency limit.\n However, in contrast to NMR experiments, INS measurements rely on the \nexistence \nof large single crystals and often\n suffer from a rather limited experimental resolution. \nTherefore, these experimental techniques are complementary \nin the information provided on spin excitations in the cuprates. \n\nOne of the central questions in the cuprate superconductors is whether \nINS and NMR experiments can be simultaneously understood \nwithin a single theoretical scenario. \nThus far it is not even been clear whether one can reach agreement\nbetween INS and NMR data as far as the order of magnitude of the \nspin susceptibility is concerned. This problem is caused, in part, by the fact\nthat NMR is sensitive to all magnetic fluctuations, \nregardless of whether they are\nrelated to a pronounced momentum dependence of the spin \nsusceptibility.\n In contrast, INS typically probes only those\nparts of the susceptibility which are strongly momentum dependent;\nthe rest are typically attributed to the ``background\" of the signal,\nas might be caused by the scattering of neutrons on nuclei and phonons.\n\nThe situation has been further complicated by recent INS experiments on \nYBa$_2$Cu$_3$O$_{6+x}$ which observe a crossover from a substantial \nincommensuration in the magnetic response at low frequencies, to a commensurate \nstructure in $\\chi$ at higher frequencies \\cite{Tra92,Dai97,Dai98}. The \nquestion \nthus arises: does this incommensurate order \noriginate from spatially inhomogeneous correlations \nbetween the doped holes \\cite{FN1}, which would preserve a locally commensurate \nmagnetic response, or does it reflect homogeneous incommensuration,\n most likely due to Fermi surface effects. \n\nIn this communication we argue that since NMR experiments probe the local \nspin environment around a nucleus, they are able to distinguish between a \nlocally commensurate and incommensurate magnetic response. \nEarlier theoretical studies of NMR experiments on \nYBa$_2$Cu$_3$O$_{6+x}$ and YBa$_2$Cu$_4$O$_8$ used a phenomenological form of \n$\\chi$ \\cite{MMP90}:\n\\begin{equation}\n\\chi({\\bf q},\\omega) = { \\alpha \\xi^2 \\over 1+\\xi^2 ( {\\bf q}-{\\bf Q})^2 \n-i \\omega/\\omega_{sf} } \\, , \n\\label{chi}\n\\end{equation}\nwhere $\\xi$ is the magnetic correlation length in units of the lattice \nconstant, \n$a_0$, $\\omega_{\\rm sf}$ an energy\nscale characterizing the diffusive spin excitations, $\\alpha$ an overall \ntemperature independent constant, and ${\\bf Q}$ is the position of the\npeak in momentum space which was assumed to be commensurate, i.e. ${\\bf \nQ}=(\\pi,\\pi)$.\nUsing this expression for $\\chi({\\bf q},\\omega)$ a rather detailed quantitative\nunderstanding of various NMR data has been reached \\cite{Zha96,Sok93,BP95}.\nIn particular, from the analysis of the longitudinal spin lattice relaxation \nrate\nof the $^{63}$Cu nuclei, $1/^{63}T_1$, and the spin spin relaxation time, \n$1/T_{\\rm 2G}$, \nscaling laws like $\\omega_{\\rm sf} \\propto \\xi^{-z}$ with dynamical critical \nexponent, $z$,\n have been deduced. It was found that $z\\approx 1$ between a lower crossover \ntemperature, $T_*$, \nand a higher one, $T_{cr}$, whereas $z\\approx 2$ above $T_{cr}$. In the \ntemperature range\n where $z=1$ scaling applies, it follows that $T_1T/T_{2G}$ is independent of\n temperature; a conjecture which was experimentally verified \nby Curro {\\it et al.} for YBa$_2$Cu$_4$O$_8$ \\cite{Cur97}.\nBelow the pseudogap temperature, $T_*$, $\\omega_{\\rm sf}$ and \n$\\xi$ decouple and a quasiparticle and spin pseudogap emerges. \nMoreover, it was recently shown that the temperature dependence of \n the incommensurate peak width, as determined in \n INS experiments on La$_{2-x}$Sr$_x$CuO$_4$ \\cite{AMH97}, \n showed $z=1$ scaling over a wide range of temperatures, in agreement with the \nNMR findings for YBa$_2$Cu$_4$O$_8$.\nBecause of its appearance in both NMR and INS results, we will therefore assume in\nthe following analysis that $z=1$ is the\nproper scaling behavior for magnetically underdoped cuprates between $T_*$ and \n$T_{\\rm cr}$.\n\nOne of our central results is that the attempt to understand the NMR data for \nYBa$_2$Cu$_4$O$_8$ with homogeneous incommensuration is inconsistent with $z=1$ \nscaling. This implies that the local magnetic response, which is probed in NMR \nexperiments, is commensurate. Should INS experiments show that this compound \nexhibits a (globally) incommensurate magnetic response similar to \nthat seen in \nLa$_{2-x}$Sr$_x$CuO$_4$, this would strongly support a dynamic charge and spin inhomogeneity (stripe) origin of \nthe incommensuration. In Sec.~V we discuss a {\\it spin and charge inhomogeneity} \nscenario which reconciles the global incommensuration with the local \ncommensuration in $\\chi$.\n\nIn a second important result we carry out a quantitative comparison of the strength of the \nantiferromagnetic spin fluctuations, as measured by NMR and INS experiments.\nWe find that agreement between the results obtained from these quite different experimental techniques, which explore not only a different frequency range, but different wavevector regimes, can be obtained within a factor of 2. Given the large uncertainties in \ndetermining the absolute value of $\\chi''$, we believe that this result \ndemonstrates that a consistent description of INS and NMR data can be achieved \nwithin the framework of Eq.(1).\n\n\n\nAlthough the form of $\\chi$ in Eq.(\\ref{chi}) was originally invented to \nunderstand the spin response at very low frequencies\nand above the superconducting transition temperature \\cite{MMP90}, it has proved \ninteresting to investigate to what extent one can understand INS data at higher \nfrequencies within the same framework. \nThe results by Aeppli {\\em et al.} support the picture of a unique \nincommensurate spin \nresponse from zero energy up to $15\\, {\\rm meV}$, the highest energy used \nin Ref.\\cite{AMH97}.\nWithin the error bars of the experiment, only for momentum values away from the \npeak maximum do systematic deviations from Eq.(\\ref{chi}) occur.\nSuch deviations indicate the presence of lattice corrections to the continuum \nlimit\nused in Eq.(\\ref{chi}). Lattice corrections are expected to be extremely\n important for the local, momentum averaged, susceptibility:\n\\begin{equation}\n\\chi_{loc}''(\\omega)= {1 \\over 4 \\pi^2} \\int_{BZ} d^2 {\\bf q} \\, \\chi''({\\bf \nq},\\omega) \\, ,\n\\label{chiloc}\n\\end{equation}\nsince the phase space of momenta away from the peak maxima is considerable in \ntwo dimensions. For higher frequencies spin excitations away from the \nantiferromagnetic peak and energetically large compared to $\\omega_{\\rm sf}$ \ncome into play and we therefore expect a poor description of \n$\\chi''_{loc}(\\omega)$ in terms of Eq.(\\ref{chi}). This conclusion is \nindependent of whether the peak position commensurate or incommensurate \npeaks. \n\nDeviations from the universal continuum limit of $\\chi$ are expected to be also \nof \n relevance for the relaxation rates measured in NMR \nexperiments since these are weighted momentum averages of \n$\\chi({\\bf q},\\omega)$.\nThus, it is worthwhile to study\nwhether one can find indications for lattice corrections from an \nanalysis of NMR and INS data.\nSuch corrections to $\\chi$ are also of relevance\nfor our understanding of the lifetime of so-called {\\em cold} quasiparticles if \none assumes that the lifetime of these quasiparticles is also dominated by \nscattering off spin fluctuations \\cite{HR,SP96,IM98,Sto_pc}.\n\nOur paper is organized as follows. \nIn Sec.~\\ref{theory} we give a brief overview of the \ntheoretical framework in which we analyze the INS and NMR data. \nIn Sec.~\\ref{1248} we analyze NMR data on YBa$_2$Cu$_4$O$_8$ for both \ncommensurate and incommensurate magnetic response and discuss the \nrole of lattice corrections. In Secs.~\\ref{1237} and \\ref{1236} we analyze NMR \ndata \non two YBa$_2$Cu$_3$O$_{6+x}$ compounds. In Sec.~\\ref{INS} we \ndiscuss the consistency between NMR and INS data, as well as the role \nof lattice corrections for the local susceptibility. \nIn Sec.~\\ref{mag_stripes} we propose a {\\em spin and charge inhomogeneity} scenario as \na \npossible way\nto reconcile NMR and INS data. Finally, in Sec.~\\ref{concl} we draw our \nconclusions. \n\n\n\n\\section{Theoretical Overview}\n\\label{theory}\n\nWe briefly discuss the theoretical framework \nin which we discuss NMR and INS experiments. \nIn order to analyze the low frequency NMR data, we use the \ndynamical spin susceptibility of Eq.(\\ref{chi}), where for the commensurate \ncase ${\\bf Q}=(\\pi,\\pi)$. To allow for an incommensurate structure of $\\chi$, \nwe use Eq.(\\ref{chi}) with ${\\bf Q}=(1,1 \\pm \\delta)\\pi$ and ${\\bf Q}=(1 \\pm \n\\delta,1)\\pi$ and sum over all four peaks \\cite{Zha96}.\nThe inclusion of the correct form of lattice corrections is rather difficult \nsince it requires a microscopic model which is \n beyond the scope of this paper. \nIn general we expect lattice corrections to appear in the form of an\nupper momentum cutoff, $\\Lambda$.\nIn the following we choose a soft cutoff procedure for $\\delta {\\bf q} \\equiv \n{\\bf q}-{\\bf Q}$ in the denominator of Eq.(\\ref{chi}):\n\\begin{equation}\n\\delta {\\bf q}^2 \\rightarrow \n\\delta {\\bf q}^2 \\left(1+ { \\delta {\\bf q}^2 \\over \\Lambda^2 } \\right) \\, .\n\\label{lattice}\n\\end{equation}\nWe also expect a weak momentum dependence of\n$\\alpha$ and $\\omega_{\\rm sf}$ once the continuum description\nbreaks down; however, to keep the number of tunable parameters small we \nignore these effects.\n \n\n \nIn NMR experiments, one measures the spin-lattice relaxation rate $1/T_{1x}$, \nwith applied magnetic \nfield in $x$-direction,\n and the spin-echo rate $1/ T_{2G}$, which can be expressed in terms of\nthe dynamical susceptibility as:\n\\begin{eqnarray}\n{1 \\over T_{1x} T} &=& { k_B \\over 2 \\hbar} (\\hbar^2 \\gamma_n \\gamma_e)^2 \n{ 1 \\over N} \\sum_q F_x(q) \\lim_{\\omega \\rightarrow 0} { \\chi''(q, \\omega) \n\\over \n\\omega} \\ , \\label{T1T} \\\\\n\\Big({1 \\over T_{2G} }\\Big)^2 &=& { 0.69 \\over 128 \\hbar^2} (\\hbar^2 \\gamma_n \n\\gamma_e)^4 \\Bigg\\{ { 1 \\over N} \\sum_q \\Big[F_{ab}^{eff}(q) \\chi'(q, \n\\omega)\\Big]^2 \\nonumber \\\\\n& & - \\Big[\\sum_q F_{ab}^{eff}(q) \\chi'(q, \\omega)\\Big]^2 \\Bigg\\}\n\\label{T2G} \\ ,\n\\end{eqnarray}\nwhere $x=ab,c$ describes the direction of the external magnetic field. \nThe $^{63}Cu$ form factors are given by \n\\begin{eqnarray}\n^{63}F_c(q) &=&\\Bigg[A_{ab}+2B \\Big( cos(q_x)+cos(q_y) \\Big) \\Bigg]^2 \\ , \\\\\n^{63}F_{ab}^{eff}(q)&=&\\Bigg[A_c+2B \\Big( cos(q_x)+cos(q_y) \\Big) \\Bigg]^2 \\ , \n\\\\\n^{63}F_{ab}(q)&=&{1 \\over 2} \\Big[ ^{63}F_{ab}^{eff}(q) + ^{63}F_c(q) \\Big] \\ .\n\\end{eqnarray}\nwhere $A_{ab}, A_c$ and $B$ are the on-site and transferred hyperfine coupling \nconstants, respectively \\cite{Zha96}.\n\nIt follows from Eq.(\\ref{chi}) that the spin excitations in the normal state \nare \ncompletely described by three parameters, $\\alpha, \\xi$, and $\\omega_{sf}$. In \norder to extract these parameters from the experimental NMR data, we also need \nto obtain the three hyperfine coupling constants, \n$A_{ab}, A_c$, and $B$; hence we have six unknown parameters in the above \nequations, and \n require six equations to determine them. So far we have two, Eqs.(\\ref{T1T}) \nand (\\ref{T2G}). \nAn additional constraint arises from the temperature independence of the \n$^{63}$Cu Knight shift\nin a magnetic field parallel to the $c$ axis in YBa$_2$Cu$_3$O$_7$ \n\\cite{Zha96} which yields\n\\begin{equation}\nA_c+4B \\approx 0 \\ .\n\\label{constr1}\n\\end{equation}\nA fourth constraint comes from the anisotropy of the $^{63}$Cu spin-lattice \nrelaxation \nrates \\cite{Barrett}, which\nfor YBa$_2$Cu$_3$O$_{7}$ was measured to be \n\\begin{equation}\n^{63}R ={T_{1c} \\over T_{1ab} } \\approx 3.7 \\pm 0.1 \\ .\n\\label{constr2}\n\\end{equation}\nA fifth constraint involving the hyperfine coupling constants can be obtained \nby \n plotting the Knight shift $^{63}K_{ab}$ versus $\\chi_0(T)$ \\cite{Zha96}, \nwhich \nyields\n\\begin{equation}\n4B+A_{ab} \\approx 200 { kOe \\over \\mu_B} \\ .\n\\label{constr3}\n\\end{equation}\nA final constraint is obtained from the earlier conjecture that the\nantiferromagnetic correlation length at the crossover temperature \n$T_{\\rm cr}$ is approximately two lattice constants \\cite{BP95}. \nRecent microscopic calculations have confirmed this conjecture, \nshowing that indeed $\\xi(T_{cr}) $ is of the order of a few lattice \nconstants\\cite{SPS97}.\n \n\nTo the extent that experimental data for $T_1$ and $T_{2G}$ are available \nup to $T_{cr}$, one can fit the above set of six equations \nself-consistently to the data, and thus obtain not only the temperature \nindependent \nparameters $\\alpha, A_{ab}, A_c$, and $B$, \nbut also the temperature dependence of $\\xi$ and $\\omega_{sf}$. It turns out, \nhowever, that this analysis is only possible for \nYBa$_2$Cu$_4$O$_8$, since this is the single compound for which data up to \nsufficiently high temperatures have been obtained \\cite{Cur97}. \nOur corresponding results will be presented in section~\\ref{1248}.\nIn order to extract the relevant parameters from NMR data on the related, \nwidely \nstudied, YBa$_2$Cu$_3$O$_{6+x}$ compounds, and thus to make a comparison \nbetween \nNMR and \nINS experiments possible, we need to make two assumptions, which, at least \npartly, will be supported by the experimental data for YBa$_2$Cu$_4$O$_8$. \n First, we assume a relation between $\\omega_{sf}$ and $\\xi$, given by \n\\cite{Sok93,BP95}\n\\begin{equation}\n\\omega_{sf} = { \\hat{c} \\, \\xi^{-z} } \\ ,\n\\label{scale1}\n\\end{equation}\nwhere $\\hat{c}$ is a temperature independent constant. \nFor YBa$_2$Cu$_4$O$_8$, where we can independently extract $\\omega_{sf}$ and \n$\\xi$, we will show that there is indeed a crossover from \n$z=1$ behavior at low temperature to $z=2$ behavior at higher temperatures.\nThe second assumption concerns the temperature dependence of the magnetic \ncorrelation length, $\\xi(T)$, which we take to be that obtained by one of us \nusing a renormalization group (RG) approach \\cite{Sch98}: \nfor $z=1$ the temperature dependence of the magnetic correlation length \nis given by\n\\begin{equation}\n\\xi^{-2} = \\xi_0^{-2} + b T^2 \\ ,\n\\label{xi}\n\\end{equation}\na result which is in good agreement with INS experiments on \nLa$_{2-x}$Sr$_x$CuO$_4$ \\cite{AMH97}. \nHaving all the necessary theoretical tools in place, we now address \nthe questions raised in the introduction. \n\n\n\\section{Analysis of NMR experiments}\n\n\\subsection{YBa$_2$Cu$_4$O$_8$}\n\\label{1248}\n\nYBa$_2$Cu$_4$O$_8$ is a benchmark system for our analysis, \nsince Curro {\\it et al.} \\cite{Cur97} have measured the \nrelaxation times $T_1$ and $T_{2G}$ over a wide temperature range from 80K to \n750K, and in particular between $T_*approx 200$ K and $T_{cr}approx 500$ K.\nWe are therefore able to extract the relevant parameters which we discussed \nabove from a self-consistent fit of Eqs.(\\ref{T1T})- (\\ref{constr3}) to the \n$T_1$ and $T_{2G}$ data. In doing so, we assume that the constraints given by \nEqs.(\\ref{constr1})-(\\ref{constr3}) for YBa$_2$Cu$_3$O$_7$ are also valid for \nYBa$_2$Cu$_4$O$_8$. \nSince we can determine $\\xi$ and $\\omega_{sf}$ independently, we are able to \nstudy the effect of both an incommensurate magnetic response, and of \nnon-universal \nlattice corrections, on the scaling law $\\omega_{sf} = \\hat{c}/\\xi^z $.\n\nWe first reconsider the case of a commensurate structure of $\\chi$, \nand set $\\Lambda=2\\sqrt{\\pi}$, i.e. we choose a momentum cut-off which \ncorresponds to the linear size of the BZ.\n We present the temperature dependence of both $\\omega_{sf}$ and $\\xi^{-1}$, \n which results from a self-consistent fit to the experimental data, in \nFig.~\\ref{temp}.\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{NMRFig1.ai}\n\\end{center}\n\\caption{ The temperature dependence of $\\omega_{sf}$ and \n $\\xi^{-1}$ in YBa$_2$Cu$_4$O$_8$ for the commensurate case with $\\Lambda=2 \n\\pi^{1/2}$.}\n\\label{temp}\n\\end{figure}\nBetween $T_*=200$ K and $T_{cr}=500$ K, $\\omega_{sf}$ and $\\xi^{-1}$ clearly \nscale linearly with temperature. In order to study the extent to which \n$\\omega_{sf}$ and $\\xi$ obey the above scaling law, we plot in \nFig.~\\ref{omsf_xi}, $\\ln(\\omega_{sf})$ as a function of $\\ln(\\xi)$ (lower \ncurve). \n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{NMRFig2.ai}\n\\end{center}\n\\caption{$ln(\\omega_{sf})$ as a function of $ln(\\xi)$ for the commensurate \ncase. \nUpper \ndata set: $\\Lambda=2 \\pi^{1/2}/4$; \nLower data set: $\\Lambda=2\\pi^{1/2}$. Solid line: z=1; dashed line: z=2.}\n\\label{omsf_xi}\n\\end{figure}\n The dynamical scaling range is of course too limited to\n prove the existence of a scaling relation. However, assuming $\\omega_{\\rm \nsf} \n\\propto \\xi^{-z}$,\nwe find $z\\approx 1$ below $500\\, {\\rm K}$, which we identify with $T_{\\rm \ncr}$\nand $z\\approx 2$ above $T_{\\rm cr}$. \nThese results are consistent with an earlier analysis of the scaling \nbehavior~\\cite{Sok93,BP95} and the microscopic \nscenario of Ref.~\\cite{SPS97}.\nTo study the effect of stronger lattice corrections, \nwe decreased $\\Lambda$ to $\\Lambda=2\\sqrt{\\pi}/4$ (upper curve in \nFig.~\\ref{omsf_xi} which for clarity is offset). Our conclusions concerning \nthe scaling \nrelations remain unchanged.\nWe can thus conclude that the dynamical scaling behavior of spin excitations \nis robust against even sizeable lattice corrections, as long as the structure \nof \n$\\chi''$ is commensurate.\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{NMRFig3.ai}\n\\end{center}\n\\caption{$ln(\\omega_{sf})$ as a function of $ln(\\xi)$ for the incommensurate \ncase.\nLower data set: $\\Lambda=2 \\pi^{1/2}$; \nUpper data set: $\\Lambda=2\\pi^{1/2}/15$. Solid line: z=1; dashed line: z=0.75.}\n\\label{omsf_xi_inc}\n\\end{figure}\n\nWe next examine the effect of incommensuration on the dynamical scaling. It was \nearlier argued that\nthe magnetic response in YBa$_2$Cu$_4$O$_8$ should be very similar\n to YBa$_2$Cu$_3$O$_{6.8}$~\\cite{BP95}. Using the observed doping dependence \nof the incommensuration in YBa$_2$Cu$_3$O$_{6+x}$, we estimate the \nincommensuration in YBa$_2$Cu$_4$O$_8$\nto be $\\delta=0.23$. Setting $\\Lambda=2\\sqrt{\\pi}$ we plot in \nFig.~\\ref{omsf_xi_inc} $\\ln(\\omega_{sf})$ as a function of $\\ln(\\xi)$ (lower \ncurve) and find, following the same argumentation as above,\n that the dynamical scaling exponent below $T_{\\rm cr} \\approx 500$ K is now \n$z\\approx 0.75$. \n Increasing the strength of the lattice corrections by decreasing $\\Lambda$ we \nfind that $z$ \nis increased. However, in order to obtain again $z \\approx 1$ \n(the upper curve in Fig.~\\ref{omsf_xi_inc}), which one would expect from the \nINS data on La$_{2-x}$Sr$_x$CuO$_4$, we need a very small\n momentum cut-off, $\\Lambda \\approx 2 \\sqrt{\\pi}/ 15$. Such a small cut-off \nof order $O(1/\\xi)$ is inconsistent with the continuum theory which\nis the basis for a dynamical scaling approach.\nWe therefore conclude that a spatially homogeneous incommensurate \nmagnetic response is in\n contradiction with the available NMR data.\nThe physical origin of this sensitivity of the magnetic response of a \nhomogeneously incommensurate\nsystem, compared to a locally commensurate one, results from the fact that the \nformer effectively\ndecreases the spatial extent of the spin-spin\ncorrelations, thus increasing the role of large values of $\\delta{\\bf q} \\sim \n\\Lambda$.\nIt should also be noted that our arguments using lattice \ncorrections to discriminate between these two scenarios works only for \nintermediate values of the correlation length.\n For very large $\\xi$ the system should be insensitive to the cut-off procedure \nregardless of\nwhether it is commensurate or incommensurate.\n\n In what follows we assume that the low frequency magnetic response measured in \nan NMR experiment is indeed locally commensurate which implies that $z=1$ \nscaling prevails between $T_*$ and $T_{cr}$. In Sec.~\\ref{mag_stripes} we \npresent a possible theoretical scenario to resolve this apparent contradiction \nbetween \nthe incommensuration seen in INS experiments and our conclusions\nbased on the available NMR data.\nFinally, using the commensurate form of Eq.(\\ref{chi}), \nand a momentum cut-off $\\Lambda= 2 \\sqrt{\\pi}$ we find the following \nparameters from the solution of the above self-consistent equations:\n$\\alpha= 10 eV^{-1}$, $A_{ab}=20.2\\, kOe/\\mu_B$, $A_{c}=-182.8\\, \nkOe/\\mu_B$, and $B=45.7 \\, kOe/\\mu_B$ for YBa$_2$Cu$_4$O$_8$. \n\n\\subsection{YBa$_2$Cu$_3$O$_7$}\n\\label{1237}\n\nSince it was earlier argued that for the optimally doped YBa$_2$Cu$_3$O$_7$, \n$T_{cr} \n\\approx 125$ K, only slightly above $T_c$, there is no significant \ntemperature \ndependence of the relaxation rates between $T_c$ and $T_{cr}$. We will \ntherefore perform our\n analysis of the NMR data only at $T_{cr}$. Assuming that $\\xi(T_{cr} ) = 2$, \nand setting \n$\\Lambda=2\\sqrt{\\pi}$, \nwe utilize $T_{2G}$ data by Itoh {\\it et al.}~\\cite{Itoh94} and Stern {\\it et \nal.}~\\cite{Stern95}, \nas well as $T_{1c}$ data by Hammel {\\it et al.}~\\cite{Ham89} to find the \nfollowing hyperfine coupling constants $A_{ab}=27.8 \\, kOe/\\mu_B$, \n$A_{c}=-175.2\\, kOe/\\mu_B$, and $B=43.9 \\, kOe/\\mu_B$. Note that these values \nagree well with the hyperfine coupling constants we extracted for \nYBa$_2$Cu$_4$O$_8$.\nWith $1/T_{2G} = 10^{4}$ s$^{-1}$ and $T_{1c}T \\approx 0.15$ Ks at $T_{cr}$, \nwe \nobtain $\\alpha=18.5$ eV$^{-1}$, \n$\\omega_{sf} \\approx 19.8 $ meV and $\\hat{c} \\approx 39.6$ meV. \nIt turns out that the above parameter set is rather robust against changes in \n$\\Lambda$. \nWhen $\\Lambda$ is decreased from $\\Lambda=2\\sqrt{\\pi}$ to \n$\\Lambda=2\\sqrt{\\pi}/4$, $A_c$ and $B$ \ndecrease by about 1.5\\%, $A_{ab}$ increases by about 8.5\\%, $\\alpha$ increases \nby 11\\%, \nand $\\omega_{sf}$ decreases by about 15\\%. A change of the momentum cut-off by \na \nfactor of 4, \nwhich decreases the area of integration in the BZ by a factor of 16, thus \nleads \nonly to moderate changes in the parameter set. However, as we show in \nSec.~\\ref{INS}, the corresponding\n changes in the local susceptibility at high frequencies are much more \ndramatic.\n\n\n\\subsection{YBa$_2$Cu$_3$O$_{6.63}$}\n\\label{1236}\nIn what follows we show that, though only a limited set of NMR data on \nYBa$_2$Cu$_3$O$_{6.63}$\nis available, we nevertheless reach the same conclusions regarding the scaling \nbehavior as in \nYBa$_2$Cu$_4$O$_8$.\nIn particular, since no NMR data on YBa$_2$Cu$_3$O$_{6.63}$ are available above \n$T=300$ K, and since we do not know the exact value of $T_{cr}$, we cannot \nextract the parameter set from a self-consistent fit to the NMR data as\n we did in the case of YBa$_2$Cu$_4$O$_8$. \nHowever, it is in this doping range that INS \ndata are available and therefore a consistency check between the \nparameter sets extracted from NMR and INS measurements might be the most \npromising.\nIt turns out that though we are not able to perform a fully self-consistent \nfit, \nwe can still extract the parameter set if we make several additional \nassumptions \nwhich\n are supported by our results on YBa$_2$Cu$_4$O$_8$.\nFirst, we assume that $z=1$ scaling is present between $T_*$ and $T_{cr}$ and \nthat we can describe the temperature dependence of $\\xi$ and the relation \nbetween $\\omega_{sf}$ and $\\xi$ by Eqs.(\\ref{scale1}) and (\\ref{xi}), \nrespectively. \nSecond, we assume that the hyperfine coupling constants are only weakly doping \ndependent, \nand that to good approximation we can use the same constants we extracted from \nthe analysis of \nYBa$_2$Cu$_3$O$_7$ for YBa$_2$Cu$_3$O$_{6.63}$. \nThird, we need an estimate for $T_{cr}$, which is unknown for \nYBa$_2$Cu$_3$O$_{6.63}$. \nHowever, since $T_{cr} \\approx 500$ K for \nYBa$_2$Cu$_4$O$_8$, and since $T_{cr}$ increases with decreasing doping, \nwe assume $T_{cr} \\approx 550-650$ K for YBa$_2$Cu$_3$O$_{6.63}$. \nTo demonstrate the effect of the uncertainty in the latter assumption, we \ncalculate \nhe parameters $\\alpha, \\omega_{sf}(T)$ and $\\xi(T)$ for $T_{cr}=650$ K and \n$T_{cr}=550$ K as well as for two different values of the momentum cut-off \n$\\Lambda$.\n The resulting parameter sets based on experimental data by Takigawa {\\it et \nal.} \\cite{Tak91} \nare shown in Table~\\ref{O663}. \n\\begin{table} [t]\n\\begin{tabular}{ccccc}\n & $\\alpha$ (eV$^{-1}$) & $\\hat{c}$ (meV) & $b$ ($10^{-7}$ K$^2$) & \n$\\xi_0^{-2}$ ($10^{-2}$) \\\\[0.2cm] \\hline \\\\\n$T_{cr}=650$ K & 11.0 & 68.1 & 5.1 & 3.5 \\\\\n$\\Lambda=2\\sqrt{\\pi} $ & & & & \\\\ \\hline \\\\\n$T_{cr}=650$ K & 12.2 & 61.2 & 5.0 & 3.9 \\\\\n$\\Lambda=2\\sqrt{\\pi}/4$ & & & & \\\\ \\hline \\\\\n$T_{cr}=550$ K & 13.0 & 67.3 & 6.7 & 4.7 \\\\\n$\\Lambda=2\\sqrt{\\pi}$ & & & & \\\\ \\hline \\\\\n$T_{cr}=550$ K & 14.5 & 59.9 & 6.5 & 5.3 \\\\\n$\\Lambda=2\\sqrt{\\pi}/4$ & & & & \\\\\n\\hline \\\\\n\\end{tabular}\n\\caption{Parameter sets for two different values of $T_{cr}$ and $\\Lambda$.}\n\\label{O663}\n\\end{table}\n Note that the parameters in Table~\\ref{O663} are rather robust against \nlarge variations in $T_{cr}$ and/or $\\Lambda$ and only differ at the most by \nabout 30\\%. \nIn Fig.~\\ref{NMR_O663} we compare our theoretical results for \n the relaxation rates, $1/T_1$ and $1/T_{2G}$, \n with the experimental data by Takigawa {\\it et al.}~\\cite{Tak91}.\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{NMRFig4.ai}\n\\end{center}\n\\caption{Comparison of our theoretical results for the temperature dependence \nof \n$1/T_1T$ (upper figure), and $1/T_{2G}$ (lower figure) in \nYBa$_2$Cu$_3$O$_{6.63}$ \nwith the \nexperimental data by Takigawa {\\it et al.}~\\protect\\cite{Tak91}.}\n\\label{NMR_O663}\n\\end{figure}\nWe find that the temperature dependence of the relaxation rates calculated for \neach of the four different parameter sets in Table~\\ref{O663}, is practically \nindistinguishable; \nwe therefore only plot the results for the parameter set with $T_{cr}=650$ K \nand $\\Lambda=2\\sqrt{\\pi}$. \nThis result is consistent with the robustness of the scaling behavior in the\n commensurate case against non-universal lattice corrections as found for \nYBa$_2$Cu$_4$O$_8$.\nWe find a consistent description \nof the experimental data for YBa$_2$Cu$_3$O$_{6.63}$ using the assumption of \na \n$z=1$ scaling\n relationship and local\ncommensuration in the temperature regime $T_*\\approx 230\\, {\\rm K} < T < 300\\, \n{\\rm K} $, and, as was the case for YBa$_2$Cu$_4$O$_8$, assuming commensurate \nbehavior leads to results which are practically independent of non-universal \nlattice corrections.\n\n \n\n\\section{Inelastic Neutron Scattering}\n\\label{INS}\n\nIn this section we address the following two questions:(i) What is the \neffect \nof lattice corrections on the local susceptibility, and (ii) can one describe \nthe available INS data with the parameter set extracted in the previous section \n? \n\nAs we discussed in the introduction we expect lattice corrections to lead to \nsubstantial corrections of $\\chi''_{loc}$ at frequencies larger than \n$\\omega_{sf}$. Above this energy scale we also expect that the typical energy, \n$\\Delta_{\\rm sw}$, of propagating spin waves,\nwhich, at low frequencies are completely overdamped by particle hole \nexcitations, \ncomes into play, leading to a modified form of the spin susceptibility \n\\begin{equation}\n\\chi({\\bf q},\\omega)= { \\alpha \\xi^2 \\over 1 + \\xi^2({\\bf q}-{\\bf Q})^2\n- i \\omega/\\omega_{sf} - (\\omega/\\Delta_{\\rm sw})^2 } \n\\label{chiQ} \\ .\n\\end{equation}\nFor $\\omega < \\Delta_{\\rm sw}^2/\\omega_{\\rm sf}$ no sign of a propagating peak \nexists due to the diffusive character of the spin excitations.\nFor $\\omega > \\Delta_{\\rm sw}^2/\\omega_{\\rm sf}$ and ${\\bf q}={\\bf Q}$ the \nconsequence of\na propagating mode is a pronounced pole in the excitation spectrum if \n$\\Delta_{\\rm sw} < \\omega_{\\rm sf}$ and a soft upper cut off in energy\nif $\\Delta_{\\rm sw} > \\omega_{\\rm sf}$. The form of $\\chi({\\bf q},\\omega)$ in \nEq.(\\ref{chiQ}) \ncan be shown to describe the propagating spin mode in YBa$_2$Cu$_3$O$_{6.5}$ \nabove $T_c$ \\cite{Morr98}.\n\nUsing the parameter sets (1) and (2) in Table~\\ref{O663}, extracted in the \nprevious section from NMR experiments,\nwe plot in Fig.~\\ref{chi_loc} the local susceptibility of \nYBa$_2$Cu$_3$O$_{6.63}$ for two different values of $\\Lambda$.\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{NMRFig5.ai}\n\\end{center}\n\\caption{The local susceptibility $\\chi_{loc}''(\\omega)$ as a function \nof frequency for two different values of $\\Lambda$.}\n\\label{chi_loc}\n\\end{figure}\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{NMRFig6.ai}\n\\end{center}\n\\caption{$\\chi''({\\bf Q},\\omega)$ for YBa$_2$Cu$_3$O$_{6.63}$. {\\it (a)} Solid \nline: $\\chi''({\\bf Q},\\omega)$ calculated with parameter set (2) in \nTable~\\protect\\ref{O663} from NMR experiments. {\\it (b)} Dashed line: \n$\\chi''({\\bf Q},\\omega)$ from {\\it (a)} multiplied by an overall factor of 2.3. \nThe experimental data are from Fong {\\it et al.}~\\protect\\cite{Fong96}.}\n\\label{INS_NMR_com}\n\\end{figure}\nWe clearly see that upon decreasing the momentum cut-off, the maximum in \n$\\chi''_{loc}$ \nmoves towards lower frequencies. Experimentally, however, the intensity at \nhigher frequencies \ndrops much faster than in Fig.~\\ref{chi_loc}, even for \n$\\Lambda=2\\sqrt{\\pi}/4$ \\cite{Dai99}. \nIn order to explain this discrepancy, we note that at higher frequencies, the \ndominant contribution to $\\chi''_{loc}$ comes from regions in momentum space \nwhich \nare far away from the peak position at $(\\pi,\\pi)$. In these regions, \n$\\chi''({\\bf q}, \\omega)$ is only weakly momentum dependent in which case its \ncontribution might be easily attributed\n to the experimental background. In other words, we believe that the origin of \nthe \ndiscrepancy lies in an underestimate of the experimental intensity at\n higher frequencies due to the problems one has in resolving it from the \nbackground \\cite{Dai_pc}.\n\nIn Fig.~\\ref{INS_NMR_com} we plot the calculated INS intensity, i.e., \n$\\chi''({\\bf Q}, \\omega)$ as a function of frequency at ${\\bf Q}=(\\pi,\\pi)$. \nSince the calculated intensity is practically the same for all four parameter \nsets in Table~\\ref{O663}, we only present $\\chi''({\\bf Q}, \\omega)$ calculated \nwith the second parameter set in Table~\\ref{O663} (solid line).\nIn the same figure, we also include the experimental data by Fong {\\it et al.} \nfor YBa$_2$Cu$_3$O$_{6.7}$ \\cite{Fong96}.\nFor the temperature range of interest, this material is the closest match for \nYBa$_2$Cu$_3$O$_{6.63}$.\n Calculating the overall INS intensity with the parameters extracted from NMR \nexperiments in the previous section, \nwe find that we underestimate the experimental INS intensity by about 56 \\%, or \na factor of 2.3 (the dashed line shows the calculated INS intensity, multiplied \nby an overall factor of 2.3). \nGiven the uncertainties in the experimental determination of the \nabsolute scale of $\\chi''$, we believe that the above result represents \nreasonable agreement between the INS and NMR data and thus demonstrates \nconsistency in the description of spin excitations based on these two \nexperimental techniques. \n\n\n\\section{Spin and Charge Inhomogeneities}\n\\label{mag_stripes}\n\nWe saw in Sec.~\\ref{1248} that under the assumption of $z=1$ scaling, supported \nby \nrecent INS experiments \\cite{AMH97}, a locally incommensurate magnetic response \nis \ninconsistent with the available experimental \nNMR data. We therefore concluded that NMR data \nsupport a locally commensurate magnetic structure. \nThe question thus arises of how one can understand \na locally commensurate, but globally incommensurate\nmagnetic response as seen by INS? It has been suggested \nthat charge stripes separated by an average \ndistance $l_0=2\\pi/\\delta$\nare the origin of the incommensurate magnetic \nresponse seen in INS experiments~\\cite{stripes}. For a large part\nof the sample this would leave \nthe locally commensurate structure intact. \nExcept for the Nd-doped 214 compounds \\cite{tran} and probably for the\nparticular doping value $x=\\frac{1}{8}$ in La$_{2-x}$Sr$_x$CuO$_2$, \nthese stripes are believed to be dynamic rather than static in nature. \n \nConsidering a spatial charge variation:\n\\begin{equation}\n\\rho(r)=\\rho_0+\\delta\\rho(r) \\ ,\n\\end{equation}\nwhere $\\rho_0$ is independent of $r$.\nThe question arises whether the system is in a regime with $\\rho_0 \\ll \n\\delta\\rho(r) $,\nas suggested in Ref.~\\cite{stripes}, or rather characterized by rather small \ncharge\nvariations, i.e. $\\delta\\rho(r)< \\rho_0 $.\nThe particular appeal of the former assumption of strong charge modulations is \nan immediate explanation of the doping dependence of the position of the \nincommensurate\npeak, $\\delta=2x$. Stripes with one hole every second lattice site in a hole \nfree\nantiferromagnetic environment move closer together upon doping, leading to the \nabove \n$x$-dependence of $\\delta$. Also, the exceptional behavior of various \nmagnetic \nand\ntransport properties for the doping value $x=\\frac{1}{8}$ can be easily \nunderstood in terms \nof a stable commensurate arrangement of such stripes and the underlying \nlattice.\nAnother argument in favor of this scenario are the recent results by Vojta and \nSachdev \\cite{Voj99}\nwho investigated the mean field behavior of a system with competing magnetic \nand long range Coulomb interactions and found areas with a strong charge \nmodulation \nseparated by regions with barely disturbed antiferromagnetic correlations.\n\nNevertheless, there are several conceptional problems \nwith such a \nstrong charge modulation, which lead us to present some arguments favoring a \nmore \nmoderate\ncharge variation, $\\delta\\rho(r)< \\rho_0 $,\nunder circumstances that the transverse mobility of the charge carriers with \nrespect to the averaged\nposition of the stripes is large.\n First, uncorrelated spatial stripes of width $r_0$, which fluctuate \ntransversely\nover a distance $d$, will give rise to a \nbroadening \n$ d/l_0^2$ of the magnetic peaks observed by INS. \nThe experimental INS results, however, provide strong support for a\nscenario in which the width of the magnetic peaks is determined \nby $\\xi^{-1}$ \\cite{AMH97}.\n Thus, in order to observe \nseparated incommensurate peaks, we need at least $\\xi \\gg d$ to establish\na well defined antiphase domain wall, implying $d \\ll l_0$. \n In other words, assuming uncorrelated stripes, the width $d$ over which \nstripes \n fluctuate\n must be very small to account for the existing INS data. \nSuch \"stiff\" but uncorrelated stripes seem to be in contradiction to the notion \nof\nstripes as a dynamical entity.\nTherefore, we arrive at the conclusion that spatial stripe \nfluctuations, if they exist, must be strongly correlated.\n\nSecond, at the characteristic temperatures where stripes with $\\rho_0 \\ll \n\\delta\\rho(r) $ \nappear, strong modifications of the resistivity and other transport \nproperties\nare expected to occur. None or only \n moderate changes of this kind have been observed and \nno indication for a dimensional crossover from quasi one-dimensional\nto quasi two-dimensional dynamics seems to be present.\n\nThird, for low frequencies,\nthe spin excitations in doped cuprates are overdamped rather than propagating\nspin waves. The suppression of the spin damping upon opening\nof a quasiparticle gap in the superconducting as well as in the\npseudogap state implies that the dominant source of the spin damping, $\\gamma$,\nare particle-hole excitations. Assuming weak charge modulations, we then find \n${\\rm Im} \\chi^{-1}(\\omega) \\propto \\gamma \\omega$, in agreement with the \nresults\nof INS experiments \\cite{AMH97}. \n\nOn the other hand, rigid stripes separated by one dimensional charge carriers, \nwhich are effectively bosonic in character, \nlead to strong deviations from the above frequency dependence of \n${\\rm Im} \\chi^{-1}(\\omega) $\\cite{ACN,JS_1}, in disagreement with INS \nexperiments. Thus, ${\\rm Im} \\chi^{-1}(\\omega) $ does not arise\nfrom spatially varying\nstripes even though they certainly affect the spin degrees of freedom and \ncan cause the decay of these excitations. \n \n\nFinally, NMR and other magnetic measurements \n on various cuprates which provide strong support\nfor spatial inhomogeneities\\cite{CHO,Bor,PCH,Hunt,MHJ,Haase} suggest\nmostly \nspatially varying {\\em spin} degrees of freedom but not necessarily a \nstrong\n modulation of the charge background. \n \nThe above points suggest that the inhomogeneities observed in \nthe high-$T_c$ cuprates might be predominantly magnetic in \norigin with only moderate modulations of the charge density.\nThis is not unrealistic since in strongly\ncorrelated systems small variations in the charge density can bring about \nsubstantial changes in the magnetic properties of the system. \nThis scenario of a predominantly magnetic character of the inhomogeneities, \nleading to magnetic stripes and clusters, is\nschematically presented in Fig.~\\ref{mag_dom}. \n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{NMRFig7.ai}\n\\end{center}\n\\caption{ Schematic picture of charge and spin inhomogeneities.}\n\\label{mag_dom}\n\\end{figure}\nNote, that the magnetic stripes, in \nwhich the electronic charge\ndensity is lower than that in the magnetic clusters, represent\nmagnetic domain walls with a phase slip of $\\pi$ in the ordering of the\nspins.\nTheoretically, we believe that this more homogeneous charge arrangement is a \nresult\nof quantum fluctuations beyond the mean field level investigated in \nRef.~\\cite{Voj99}.\n\nThe formation of magnetic clusters and stripes requires competing\ninteractions on different length scales. Besides the local Coulomb\ninteraction, we need to identify a longer length scale interaction. It\nwas argued earlier that in the cuprate superconductors an increase of the \nmagnetic correlation\nlength leads to strong vertex corrections, which in turn give rise to a\ngradient coupling between the fermionic and spin degrees of freedom\n\\cite{vertex}. \nFor a system with\nan ordered ground state, this takes the form of a dipole-dipole coupling\nwhich has been\nshown to cause inhomogeneities and domain formation in\ntwo dimensional magnetic systems.\nThe length scale of this gradient coupling should roughly\nspeaking be the magnetic correlation length, $\\xi$. We thus need a\nminimum value of $\\xi$ to (a) generate the gradient coupling, and to (b)\nobserve the incommensuration. The temperature at which incommensuration\nshould appear is roughly set by $\\xi(T) \\approx l_0 = 4$, where the\nlast equality arises from the momentum position of the incommensurate\npeaks. Since, as we argued earlier, $\\xi(T_{cr}) \\approx 2$, $T_{cr}$ thus\npresents an upper bound for the formation of magnetic clusters. \nThis anomalous coupling is of course enhanced once magnetic clusters are \nformed; the creation of magnetic inhomogeneities is a self-consistent process. \nThe magnetic inhomogeneity scenario presented here is admittedly very \nqualitative \nand further microscopic investigations will be required to verify or disprove \nit.\n \n\n\n\\section{Conclusions}\n\\label{concl}\nIn this communication,\nwe investigated whether it is possible\nto understand theoretically INS and NMR data\nwithin a single theoretical framework.\nBased on the observation of \nan incommensurate structure of the magnetic response by \nINS experiments~\\cite{Dai98,AMH97},\n we considered whether this reflects a locally or globally\n incommensurate ordering. \nOn analyzing NMR data on YBa$_2$Cu$_4$O$_{8}$ we found with the condition of \n$z=1$ \nscaling that a \nhomogeneous incommensuration\nis, within the framework given by Eq.(\\ref{chi}), inconsistent with \nthe available\nexperimental data.\n We thus concluded that the local magnetic structure as probed by NMR \nis likely commensurate.\n\nWe then investigated the effect of lattice corrections on both \nthe parameter sets extracted from NMR data and \non the local susceptibility measured\n in INS experiments. We found that while the \nresulting corrections to the NMR parameters are only weak, the most \npronounced effect of such corrections appears in the local \nsusceptibility determined in INS measurements at frequencies above \n$\\omega_{sf}$. \nSpecifically, we found that decreasing the momentum cut-off leads \nto a suppression of $\\chi''_{loc}$ \nat higher frequencies, thus \nimproving the agreement with the \nexperimental data. We argued that the \nremaining discrepancies \ncan be explained by an experimental underestimate \nof the INS intensity at higher frequencies. \nFurthermore, we quantitatively compared INS and NMR data and found agreement\nwithin a factor of 2.\n Given the large uncertainties in resolving $\\chi''$ from the large\nbackground and in determining the absolute intensity of $\\chi''$,\nwe believe that this result demonstrates \nthat a consistent description of INS and NMR data can be obtained using \nthe expression for $\\chi$ given in Eq.(\\ref{chi}).\n\nFinally, we discussed a spin and charge inhomogeneity scenario\nto reconcile the local\ncommensurations, as seen in NMR experiments, and\nthe incommensurate peaks, seen by INS.\nThis scenario, even though similar in spirit to earlier\ncharge stripe pictures, is based on a moderate to weak modulation\n of the charge density,\ncausing a pronounced inhomogeneity\nin the magnetic properties and leading to magnetically coupled clusters\n separated by weakly correlated stripes.\n\nWe would like to thank A.V. Chubukov, P. Dai, B. Keimer, T. Mason, \nH. Mook, R. Stern, C.P. Slichter and B. Stojkovic for valuable discussions.\n This work has been supported in part by the Science and Technology\n Center for Superconductivity through NSF-grant DMR91-20000 (D.K.M.), \nand by DOE at Los Alamos (D.P.).\n\n\\begin{thebibliography}{99}\n\\bibitem{general} D. Pines, Z. Phys. B{\\bf 103}, 129 (1997); Proc. of\nthe NATO ASI on {\\em The Gap Symmetry and Fluctuations in High T$_c$\nSuperconductors}, J. Bok and G. Deutscher, eds., Plenum Pub. (1998).\n%\n \\bibitem{DJS} D. J. Scalapino, Phys. Rep. {\\bf 250}, 329 (1995).\n %\n %\n\\bibitem{NMRreview} C. P. Slichter, in { Strongly Correlated\n Electron Systems}, ed. K. S. Bedell { et al.} (Addison-Wesley,\n Reading, MA,1994).\n%\n\\bibitem{Mason1} T. E. Mason, cond-mat/9812287\n \\bibitem{Bourges1} Ph. Bourges, cond-mat/9901333; \n Ph. Bourges, Y. Sidis, H. F. Fong, B. Keimer, L. P. Regnault,\n J. Bossy, A. S. Ivanov, D. L. Milius, I. A. Aksay, cond-mat/9902067.\n%\n\\bibitem{Tra92} J.M. Tranquada {\\it et al.}, Phys. Rev.\n B {\\bf 46}, 5561 (1992).\n%\n\\bibitem{Dai97} P. Dai {\\it et al.}, preprint, \ncond-mat 9712311; H. Mook {\\it et al.}, preprint, cond-mat 9712326.\n%\n\\bibitem{Dai98} P. Dai {\\it et al.}, Phys. Rev. Lett.\n {\\bf 80}, 1738 (1998).\n\\bibitem{FN1} Due to the absence of a corresponding elastic Bragg peak\n in neutron and X-ray scattering, stripes,\n if present at all in YBa$_2$Cu$_3$O$_{6+x}$, have to be dynamical in \ncharacter.\n\\bibitem{MMP90} A. Millis, H. Monien, and D. Pines, Phys.\n Rev. B {\\bf 42}, 1671 (1990).\n%\n%\n\\bibitem{Zha96} Y. Zha, V. Barzykin and D. Pines, Phys. Rev. B \n{\\bf 54}, 7561\n(1996).\n\\bibitem{Sok93} A. Sokol and D. Pines, Phys. Rev. Lett.\n {\\bf 71}, 2813 (1993).\n\\bibitem{BP95} V. Barzykin and D. Pines, Phys. Rev. B {\\bf 52}, \n 13585 (1995).\n%\n\\bibitem{Cur97} N. J. Curro, T. Imai, C.P. Slichter, and \nB. Dabrowski, Phys. Rev. B {\\bf 56}, 877 (1997).\n% \n\\bibitem{AMH97} G. Aeppli, T. E. Mason, \nS. M. Hayden, H. A. Mook, J. Kulda, Science {\\bf 278}, 1432 (1997).\n%\n\\bibitem{HR} R. Hlubina and T.M. Rice, Phys. Rev. B {\\bf 51}, 9253\n (1995); {\\em ibid} {\\bf 52}, 13043 (1995). \n%\n\\bibitem{SP96} B. P. Stojkovi\\'c and D. Pines, Phys. Rev. Lett.\n {\\bf 76}, 811 (1996); Phys. Rev. B {\\bf 55},\n 8576 (1997).\n\\bibitem{IM98} L. B. Ioffe and A. J. Millis, Phys. Rev. B {\\bf 58},\n11631 (1998).\n\\bibitem{Sto_pc} B. Stojkovic, private communication\n%\n\\bibitem{Barrett} S. E. Barrett, {\\em et al.} Phys. Rev. Lett.\n {\\bf 66}, 108 (1991).\n\\bibitem{SPS97} J. Schmalian, D. Pines, and B. Stojkovi\\'c,\n Phys. Rev. Lett. {\\bf 80}, 3839\n (1998); {\\em ibid} to appear in Phys. Rev. B.\n\\bibitem{Sch98} J. Schmalian, preprint, cond-mat/9810041.\n\\bibitem{Itoh94} Y. Itoh {\\it et al.}, J. Phys. Soc. \nJpn {\\bf 63}m 1455 (1994).\n\\bibitem{Stern95} R. Stern, M. Mali, J. Roos, and \nD. Brinkmann, Phys. Rev. \nB {\\bf 51}, 15478 (1995).\n\\bibitem{Ham89} P.C. Hammel {\\it et al.}, Phys. Rev. Lett {\\bf 63}, 1992 \n(1989).\n\\bibitem{Tak91} M. Takigawa {\\it et al.}, Phys. Rev. B {\\bf 43}, 247 (1991).\n\\bibitem{Morr98} D.K. Morr and D. Pines, Phys. Rev. Lett. {\\bf 81}, 1086 \n(1998).\n%\n\\bibitem{Dai99} P. Dai, {\\it et al.}, Science {\\bf 284}, 1344 (1999).\n%.\n\\bibitem{Dai_pc} P. Dai, private communication.\n%\n\\bibitem{Fong96} H.F. Fong {\\it et al.} Phys. Rev. B {\\bf 54}, 6708 (1996).\n\\bibitem{stripes}See, for example, J. Zaanen and \nO. Gunnarson, Phys. Rev B {\\bf 40}, 7391 (1991);\nV. J. Emery and S. A. Kivelson, Physica C {\\bf 263},\n 44 (1996); O. Zachar, S. A. Kivelson and\nV. J. Emery, Phys. Rev. B {\\bf 57}, 1422 (1995).\n\\bibitem{tran} J. M. Tranquada, cond-mat-/9802043.\n\\bibitem{Voj99} M. Vojta and S. Sachdev, preprint, cond-mat/9906104.\n\\bibitem{ACN} A. H. Castro Neto, Z. Phys. B{\\bf 103}, 185 (1997).\n\\bibitem{JS_1}J. Schmalian ({\\em unpublished}).\n\\bibitem{CHO} J. H. Cho, F. C. Chou, and D. C. Johnston, Phys. Rev. Lett. {\\bf \n70}, 222 (1993).\n\\bibitem{Bor} F. Borsa {\\em et al.}, Phys. Rev. B {\\bf 52}, 7334 (1995).\n\\bibitem{PCH} B. J. Suh, P. C. Hammel, Y. Yoshinary, J. D. Thomson, J. L. \nSarrao,\nand Z. Fisk, Phys. Rev. Lett. {\\bf 81}, 2791 (1998).;P. C. Hammel, B. J. Suh, \nJ. L. Sarrao,\nand Z. Fisk, cond-mat/9809096.\n\\bibitem{Hunt}A. W. Hunt, P. M. Singer, K. R. Thurber, and\nT. Imai, Phys. Rev. Lett. {\\bf 81}, 5209 (1998).\n\\bibitem{MHJ}M.-H. Julien, F. Borsa, P. Caretta, M. Horvatic, C. Berthier, and\nC. T. Lin, cond-mat/9903005.\n\\bibitem{Haase} J. Haase {\\em et al.} preprint.\n\\bibitem{vertex} J.R. Schrieffer, J. Low Temp. Phys. {\\bf 99}, 397 (1995); A.V. \nChubukov and D.K. Morr, Phys. Rep. {\\bf 288}, 355 (1997); J. Schmalian, D. \nPines, and B. Stojkovic, Phys. Rev. Lett. {\\bf 80}, 3839 (1998). \n\\end{thebibliography}\n\\end{document}\n"
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[
{
"name": "cond-mat0002164.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{general} D. Pines, Z. Phys. B{\\bf 103}, 129 (1997); Proc. of\nthe NATO ASI on {\\em The Gap Symmetry and Fluctuations in High T$_c$\nSuperconductors}, J. Bok and G. Deutscher, eds., Plenum Pub. (1998).\n%\n \\bibitem{DJS} D. J. Scalapino, Phys. Rep. {\\bf 250}, 329 (1995).\n %\n %\n\\bibitem{NMRreview} C. P. Slichter, in { Strongly Correlated\n Electron Systems}, ed. K. S. Bedell { et al.} (Addison-Wesley,\n Reading, MA,1994).\n%\n\\bibitem{Mason1} T. E. Mason, cond-mat/9812287\n \\bibitem{Bourges1} Ph. Bourges, cond-mat/9901333; \n Ph. Bourges, Y. Sidis, H. F. Fong, B. Keimer, L. P. Regnault,\n J. Bossy, A. S. Ivanov, D. L. Milius, I. A. Aksay, cond-mat/9902067.\n%\n\\bibitem{Tra92} J.M. Tranquada {\\it et al.}, Phys. Rev.\n B {\\bf 46}, 5561 (1992).\n%\n\\bibitem{Dai97} P. Dai {\\it et al.}, preprint, \ncond-mat 9712311; H. Mook {\\it et al.}, preprint, cond-mat 9712326.\n%\n\\bibitem{Dai98} P. Dai {\\it et al.}, Phys. Rev. Lett.\n {\\bf 80}, 1738 (1998).\n\\bibitem{FN1} Due to the absence of a corresponding elastic Bragg peak\n in neutron and X-ray scattering, stripes,\n if present at all in YBa$_2$Cu$_3$O$_{6+x}$, have to be dynamical in \ncharacter.\n\\bibitem{MMP90} A. Millis, H. Monien, and D. Pines, Phys.\n Rev. B {\\bf 42}, 1671 (1990).\n%\n%\n\\bibitem{Zha96} Y. Zha, V. Barzykin and D. Pines, Phys. Rev. B \n{\\bf 54}, 7561\n(1996).\n\\bibitem{Sok93} A. Sokol and D. Pines, Phys. Rev. Lett.\n {\\bf 71}, 2813 (1993).\n\\bibitem{BP95} V. Barzykin and D. Pines, Phys. Rev. B {\\bf 52}, \n 13585 (1995).\n%\n\\bibitem{Cur97} N. J. Curro, T. Imai, C.P. Slichter, and \nB. Dabrowski, Phys. Rev. B {\\bf 56}, 877 (1997).\n% \n\\bibitem{AMH97} G. Aeppli, T. E. Mason, \nS. M. Hayden, H. A. Mook, J. Kulda, Science {\\bf 278}, 1432 (1997).\n%\n\\bibitem{HR} R. Hlubina and T.M. Rice, Phys. Rev. B {\\bf 51}, 9253\n (1995); {\\em ibid} {\\bf 52}, 13043 (1995). \n%\n\\bibitem{SP96} B. P. Stojkovi\\'c and D. Pines, Phys. Rev. Lett.\n {\\bf 76}, 811 (1996); Phys. Rev. B {\\bf 55},\n 8576 (1997).\n\\bibitem{IM98} L. B. Ioffe and A. J. Millis, Phys. Rev. B {\\bf 58},\n11631 (1998).\n\\bibitem{Sto_pc} B. Stojkovic, private communication\n%\n\\bibitem{Barrett} S. E. Barrett, {\\em et al.} Phys. Rev. Lett.\n {\\bf 66}, 108 (1991).\n\\bibitem{SPS97} J. Schmalian, D. Pines, and B. Stojkovi\\'c,\n Phys. Rev. Lett. {\\bf 80}, 3839\n (1998); {\\em ibid} to appear in Phys. Rev. B.\n\\bibitem{Sch98} J. Schmalian, preprint, cond-mat/9810041.\n\\bibitem{Itoh94} Y. Itoh {\\it et al.}, J. Phys. Soc. \nJpn {\\bf 63}m 1455 (1994).\n\\bibitem{Stern95} R. Stern, M. Mali, J. Roos, and \nD. Brinkmann, Phys. Rev. \nB {\\bf 51}, 15478 (1995).\n\\bibitem{Ham89} P.C. Hammel {\\it et al.}, Phys. Rev. Lett {\\bf 63}, 1992 \n(1989).\n\\bibitem{Tak91} M. Takigawa {\\it et al.}, Phys. Rev. B {\\bf 43}, 247 (1991).\n\\bibitem{Morr98} D.K. Morr and D. Pines, Phys. Rev. Lett. {\\bf 81}, 1086 \n(1998).\n%\n\\bibitem{Dai99} P. Dai, {\\it et al.}, Science {\\bf 284}, 1344 (1999).\n%.\n\\bibitem{Dai_pc} P. Dai, private communication.\n%\n\\bibitem{Fong96} H.F. Fong {\\it et al.} Phys. Rev. B {\\bf 54}, 6708 (1996).\n\\bibitem{stripes}See, for example, J. Zaanen and \nO. Gunnarson, Phys. Rev B {\\bf 40}, 7391 (1991);\nV. J. Emery and S. A. Kivelson, Physica C {\\bf 263},\n 44 (1996); O. Zachar, S. A. Kivelson and\nV. J. Emery, Phys. Rev. B {\\bf 57}, 1422 (1995).\n\\bibitem{tran} J. M. Tranquada, cond-mat-/9802043.\n\\bibitem{Voj99} M. Vojta and S. Sachdev, preprint, cond-mat/9906104.\n\\bibitem{ACN} A. H. Castro Neto, Z. Phys. B{\\bf 103}, 185 (1997).\n\\bibitem{JS_1}J. Schmalian ({\\em unpublished}).\n\\bibitem{CHO} J. H. Cho, F. C. Chou, and D. C. Johnston, Phys. Rev. Lett. {\\bf \n70}, 222 (1993).\n\\bibitem{Bor} F. Borsa {\\em et al.}, Phys. Rev. B {\\bf 52}, 7334 (1995).\n\\bibitem{PCH} B. J. Suh, P. C. Hammel, Y. Yoshinary, J. D. Thomson, J. L. \nSarrao,\nand Z. Fisk, Phys. Rev. Lett. {\\bf 81}, 2791 (1998).;P. C. Hammel, B. J. Suh, \nJ. L. Sarrao,\nand Z. Fisk, cond-mat/9809096.\n\\bibitem{Hunt}A. W. Hunt, P. M. Singer, K. R. Thurber, and\nT. Imai, Phys. Rev. Lett. {\\bf 81}, 5209 (1998).\n\\bibitem{MHJ}M.-H. Julien, F. Borsa, P. Caretta, M. Horvatic, C. Berthier, and\nC. T. Lin, cond-mat/9903005.\n\\bibitem{Haase} J. Haase {\\em et al.} preprint.\n\\bibitem{vertex} J.R. Schrieffer, J. Low Temp. Phys. {\\bf 99}, 397 (1995); A.V. \nChubukov and D.K. Morr, Phys. Rep. {\\bf 288}, 355 (1997); J. Schmalian, D. \nPines, and B. Stojkovic, Phys. Rev. Lett. {\\bf 80}, 3839 (1998). \n\\end{thebibliography}"
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cond-mat0002165
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Coexistent State of Charge Density Wave and Spin Density Wave in One-Dimensional Quarter Filled Band Systems under Magnetic Fields
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[
{
"name": "lett.tex",
"string": "%\n%\n% The effect of the spin on the magnetic breakdown\n% 1998. 12. 2\n% \n% revised 1999 4 1\n% \n% \n% \n%\n% \n%\n\n\n%\\documentstyle[seceq,twocolumn,short]{jpsj}\n%%\\documentstyle[seceq]{jpsj}\n\n\\documentstyle[twocolumn,epsf]{jpsj}\n%\\documentstyle[preprint,epsf]{jpsj}\n\n%%%\\def\\sf{\\rm}\n%%%\\renewcommand\\figureheight[1]{\\vspace{24pt}\\mbox{\\rule{0cm}{#1}}}\n\n\\def\\runtitle{\nCoexistent State of Charge Density Wave and Spin Density Wave in\nOne-Dimensional Quarter \nFilled Band Systems\n% under Magnetic Fields\n}\n\\def\\runauthor{Keita {\\sc Kishigi} \nand Yasumasa {\\sc Hasegawa} \n}\n\n\\title{\nCoexistent State of Charge Density Wave and Spin Density Wave in\nOne-Dimensional Quarter \nFilled Band Systems under Magnetic Fields\n}\n\n\n\\author\n{\nKeita {\\sc Kishigi}\n\\footnote{E-mail: kishigi@sci.himeji-tech.ac.jp}\n and Yasumasa {\\sc Hasegawa} \n}\n\n\\inst\n{\nFaculty of Science, Himeji Institute of Technology, Akou-gun, Hyogo\n678-1297, Japan \n}\n\n\\recdate\n{\n\\today\n%\\tomorrow\n}\n\n\\abst\n{\nWe theoretically study how the \ncoexistent state of the charge density wave and \nthe spin density wave in the one-dimensional \nquarter filled band is \nenhanced by magnetic fields. \nWe found that \nwhen the correlation between electrons is strong \nthe spin density wave state is suppressed under high magnetic fields, \nwhereas the charge density wave state still remains. \nThis will be observed in experiments such as the X-ray \nmeasurement. \n}\n\n\\kword\n{\none-dimensional quarter filled band, charge density wave, spin density wave,\nquasi-one-dimensional organic conductors, \nPauil paramagnetic limit, strongly correlated system \n}\n\n\\begin{document}\n\\sloppy\n\\maketitle\n%\\pagestyle{empty}\n\\section{Introduction}\nIt is found that \nthe system with the one-dimensional quarter filled band \nbecomes the coexistent state of the charge \ndensity wave (CDW) and \nspin density wave (SDW) due to the \ninterplay between the on-site Coulomb interaction ($U$) and the \ninter-site Coulomb interaction ($V$) by recent \ntheoretical works.\\cite{seofukuyama,nobuko,nobuko2,Mazumdar} \nThe inter-site Coulomb interaction plays \nimportant role of the charge ordering, and \nthe ground state is the coexistent state of \n$2k_{\\rm F}$-SDW and $4k_{\\rm F}$-CDW,\\cite{seofukuyama} \nwhere $k_{\\rm F}$ is Fermi wave vector, $k_{\\rm F}=\\pi /4a$ and \n$a$ is the lattice constant. \nFurthermore, when the next nearest neighbor and the dimerization of the \nenergy band are considered, it has been indicated that \n$2k_{\\rm F}$-SDW and $2k_{\\rm F}$-CDW \ncoexist.\\cite{nobuko,nobuko2} \n\nQuasi-one dimensional organic conductors \nsuch as (TMTSF)$_2$$X$ and (TMTTF)$_2$$X$ \n($X$=ClO$_4$, PF$_6$, AsF$_6$, ReO$_4$, Br, SCN, etc.) \nare known as the one-dimensional quarter filled band and \nexhibit many kinds of ground state, \nfor example, spin-Peierls, \nSDW, \nsuperconductivity.\\cite{review,jerome} \nIn (TMTSF)$_2$PF$_6$, the incommensurate SDW is occurred at $T=12$ K, \nwhere the wave vector is (0.5, 0.24,-0.06) by \nNMR measurement.\\cite{takahashi,delrieu} \nRecently, from the X-ray measurement, Pouget and Ravy argue \nthe coexistence of $2k_{\\rm F}$-SDW and \n$2k_{\\rm F}$-CDW,\\cite{pouget} which has been \ntheoretically explained by Kobayashi et al.\\cite{nobuko,nobuko2} \nmentioned above \nand Mazumdar et al.\\cite{Mazumdar}. \n\nOn the other hand, in (TMTTF)$_2$$X$, (X=Br and SCN), \nit is known that the ground state is the antiferromagnetic \nphase understood as Mott-Hubbard inslator phase due to \nthe dimerization and the quarter filling. \nIt is clear that the wave vector of the SDW \nis commesurate, (0.5, 0.25,0) from the measurements of \n$^{13}$C-NMR\\cite{barthel} and $^1$H-NMR\\cite{nakamura}. \nFrom the angle depenence of satelite peak positions of \n$^1$H-NMR\\cite{nakamura,nakamura2}, \nthe alignment of the spin moment along the conductive axis (a-axis) \nbecomes ($\\uparrow,0,\\downarrow,0$), which \ncorresponds to the recent calculational results\\cite{seofukuyama,tanemura}. \nIn (TMTTF)$_2$Br, $4k_{\\rm F}$-CDW accompanied by $2k_{\\rm F}$-SDW \nis found in X-ray measurments.\\cite{pouget} \nThis can be explained by Seo and Fukuyama\\cite{seofukuyama} \nby using the extended Hubbard model. \n\n\nWhen the pressure is applied, the commensurate antiferromagnetic phase \nin (TMTTF)$_2$Br at \nambient pressure changes to \nthe incommensurate SDW phase such as (TMTSF)$_2$$X$.\\cite{klemme}\nIt is originated to the increasing of the hopping transfer integral, $t$. \nIn the case of small $t$, as the exchange interaction is strongly\ninfluenced, the state becomes Mott antiferromagnetic state. \nWhen $t$ becomes large, the system becomes the SDW phase due to \nthe Peierls instability of the Fermi surface. \nThe difference between \n(TMTTF)$_2$$X$ and (TMTSF)$_2$$X$ is whether the \ncharge or spin ordering is localized or not. \nThis difference is attributed that \n$U/t$ in (TMTTF)$_2$$X$ is \nlarger than that in (TMTSF)$_2$$X$. \nIt is indicated that $U/t\\simeq 5.0$ (3.0) in (TMTTF)$_2$$X$ \n((TMTSF)$_2$$X$) since $t$ in (TMTTF)$_2$$X$ ((TMTSF)$_2$$X$) are \nabout 0.2 (0.3) eV by the extended Huckel band calculations.\n\\cite{band,band2,band3,band4}\nWe consider the system in (TMTTF)$_2$$X$ ((TMTSF)$_2$$X$) \nas strongly (non-strongly) correlated. \n\nIn the one-dimensional system, \nwhen the magnetic field ($H$) is applied to \n$c$-axis, the amplitudes of the charge density or \nthe spin moment along the $c$-axis in the CDW or SDW state by coupling of\nelectrons with same spins are suppressed due to \nPauli paramagnetic limit field ($H_p$),\\cite{pauli,pauli2,Mckenzie} \nwhere $H_p\\simeq \\Delta (0)/\\sqrt{2}\\mu_{\\rm B}$, \n$\\Delta(0)$ is the amplitude of the \nenergy gap at $H=0$ and \n$\\mu_{\\rm B}=e\\hbar/2m_{0}c$ is the Bohr magneton. \nThe energy band is splitted by \nZeeman effect, so that \nthe original wave vector at $H=0$ \nbecomes the not good nesting vector. \nIn the case of the magnetic field applied along \nthe $a$- or $b$-axis, the CDW and SDW are not broken, \nbecause the nesting vector\nis unchanged by Zeeman effect.\nIn other words, \nthe CDW and SDW are not influenced when \nthe magnetic field is applied perpendicular to \nthe easy axis. \n%easy axis in the CDW or SDW state \n%made by the nesting of coupling by up-spin and down-spin, \n%the CDW and SDW are not broken, \n%because the nesting vector \n%is unchanged by Zeeman effect. \nThis picture is for the weak coupling system. \n\n\n\nIn the strong coupling system, \nthe $H$-dependence of the \nantiferromagnetic state by one-dimensional Ising model has been \nstudied in the mean field approximation.\\cite{ising} \nThe amplitude of the spin moment \nalong the $c$-axis of the antiferromagnetic state \nunder magnetic fields applied along the $a$-axis \nis easily obtained, which \nobey \n\\begin{eqnarray}\nS_z(H,j)/S_z(0,j)=\\sqrt{1-(H/H_x^0)^2},\n\\end{eqnarray}\nwhere \n$S_z(H,j)$ is the amplitude of the spin moment \nalong the $c$-axis at \n$j$ site at $H=0$ and $H_x^0$ is the critical field at which the \nordering of the antiferromagnetic state disappear.\\cite{ising} \nSince the spin of electrons are tilted to \n$a$-axis by the magnetic field, \n$S_z$ is smaller upon increasing $H_x$, \nfinally, $S_z$ becomes zero. \n\n\nIn the case of magnetic fields applied to the $c$-axis, \nthe antiferromagnetic state along the $c$-axis is kept, \nbecause if the spin is tilted to $a$-$b$ plane, this tilted state \nis not unstable in weak fields. \nHowever, the paramagnetic state becomes more\nstable \nat higher critical field, $H_z^0$, \nwhere $H_z^0=H_x^0$, \\cite{ising} \n\n\nWe try to analyze \nthe coexistent state \nin the two cases when the electron correlation is strong or not, \nsince the {\\it coexistent} phase of CDW and SDW \nunder magnetic fields dose not have been studied \nalthough the state of CDW or SDW under magnetic fields has been studied. \nWhen $U/t$ and $V/t$ are large ($\\sim 5.0$), \nwe consider that the system is strongly correlated, \nbecause the charge and spin are \nlocalized as shown in Figs. 1 and 2. \nOn the other hand, \nin small $U/t$ and $V/t$ ($\\sim 1.5$), the \ncorrelation between electrons is not strong, \nwhere there are small amplitudes of charge \ndensity and spin moment as shown in \nFig. 7, 8 and 9. \nWe calculate to compare the strong coupling system with \nthe non-strong coupling system by using of \ntwo sets of the values of $U/t$ and $V/t$. \n\n\nIn this paper, \nwe calculate the self-consistent solutions at $T=0$ for \nthe one-dimensional band model under the magnetic field \nperpendicular to (or parallel to) the a-axis \nbased on the mean field approximation. \nWe use the one-dimensional quarter filled extended \nHubbard model, where the effect of \nthe dimerization do not be considered to \nbe simplified problems. \n%However, \n%the essences are not lost.\n \n\\section{Formulation}\n\n\nWe treat the one-dimensional extended \nHubbard model, \n\n\\begin{eqnarray}\n\\hat{\\cal H}&=&\\hat{\\cal K}+\\hat{\\cal U}+\\hat{\\cal V}, \\\\\n\\hat{\\cal K}&=&t\\sum_{i,\\sigma}(C^{\\dagger}_{i,\\sigma} C_{i+1,\\sigma}+\nh.c.)\n-\\frac{\\mu_{\\rm B}gH_{j}}{2}\\sum_{i, \\sigma}n_{i,\\sigma},\\\\\n%=\\sum_{{\\bf k},\\sigma}, \\\\\n\\hat{\\cal U}&=&U\\sum_{i}n_{i, \\uparrow}n_{i, \\downarrow}, \\\\ \n\\hat{\\cal V}&=&V\\sum_{i,\\sigma,\\sigma^{\\prime}}n_{i,\\sigma}n_{i+1,\\sigma^{\\prime}}, \n\\end{eqnarray}\nwhere $C^{\\dagger}_{i,\\sigma}$ is the creation \noperator of $\\sigma$ spin electron at $i$ site, \n$n_{i,\\sigma}$ is the number operator, $g=2$, \n$i=1,\\cdots,N_{\\rm S}$, $N_{\\rm S}$ is the \nnumber of the total sites and $\\sigma =\\uparrow$ and $\\downarrow$. \nWhen the magnetic field is \napplied to (x or z)-axis ($H_x$ or $H_z$), \n$H_{j}=H\\hat{\\sigma}_{j}$ ($j=x$ and $z$), where \n$H$ is the strength of the magnetic field and \n$\\hat{\\sigma}_{j}$ is Pauli spin matrix. \nIn this model, the filling of electrons is 1/4. \n%In eqs. (4) and (5), $U$ and $V$ are \n%the on-site and inter-site Coulomb interaction, respectively. \n\n%$\\mu_{\\rm B}$ is the Bohr magneton, \n%and ${\\bf Q}=(0,\\pi /b^{\\prime})$ \n%is the lattice potential vector. \n%In the above Hamiltonian, the Brillouin zone is \n%$-\\pi/a\\leq k_x<\\pi/a$, $-\\pi/b^{\\prime}\\leq k_y<\\pi/b^{\\prime}$. \n\n\nThe interaction term, $\\hat{\\cal U}$ and \n$\\hat{\\cal V}$ are treated in mean field\napproximation as\n\n\\begin{eqnarray}\n\\hat{\\cal U}_{\\rm M}&=&\\sum_{k_{x}}\n\\sum_{Q}\\{\n\\rho_{\\uparrow}(Q)\nC^{\\dagger}(k_{x},\n\\downarrow) \nC(k_{x}-Q,\\downarrow) \\nonumber \\\\\n&+&\\rho^{*}_{\\downarrow}(Q)\nC^{\\dagger}(k_{x}-Q,\n\\uparrow) \nC(k_{x},\\uparrow)\\} \\nonumber \\\\\n&-&\\frac{1}{I}\\sum_{Q}\\rho_{\\uparrow}(Q)\\rho^{*}_{\\downarrow}(Q), \\\\\n\\hat{\\cal V}_{\\rm M}&=&(\\frac{V}{U})\\sum_{k_{x},\n\\sigma,\\sigma^{\\prime}}\\sum_{Q}e^{-iQa}\\{\n\\rho_{\\sigma}(Q)\nC^{\\dagger}(k_{x}, \n\\sigma^{\\prime}) \nC(k_{x}-Q,\n\\sigma^{\\prime}) \\nonumber \\\\\n&+&\\rho_{\\sigma^{\\prime}}^{*}(Q)\nC^{\\dagger}(k_{x}, \n\\sigma) \nC(k_{x}-Q,\n\\sigma)\\} \\nonumber \\\\\n&-&\\frac{V}{IU}\\sum_{Q,\\sigma,\\sigma^{\\prime}}e^{-iQa}\\rho_{\\sigma}(Q)\n\\rho_{\\sigma^{\\prime}}^{*}(Q),\n\\end{eqnarray}\nwhere $I=U/N_{\\rm S}$. \nThe self-consistent equation for the order parameter $\\rho_{\\sigma}(Q)$ \nis given by \n\\begin{eqnarray}\n\\rho_{\\sigma}(Q)&=&I\\sum_{k_{x}}\n<C^{\\dagger}(k_{x},\n\\sigma)\nC(k_{x}-Q,\n\\sigma)>.\n\\end{eqnarray}\nWe use the mean field, $\\rho_{\\sigma}(Q)$, by \nthe coupling between electrons with same spins. \nIn order to simplify, we do not consider the case of \nthe mean field, \n$\\bar{\\rho_{\\sigma}}(Q)=I\\sum_{k_{x}}<C^{\\dagger}(k_{x},\\sigma)C(k_{x}-Q,\\bar{\\sigma})>$, by the coupling of electrons with opposite spin. \n\n\n\nWe limit the sum of the wave vector as $Q=q,2q,3q$ and $4q$ \n($q=2k_{\\rm F}$), because \nthe wave vectors of $2k_{\\rm F}=\\pi /2a$ and its higher harmonics \nshould be considered due to the nesting of the Fermi surface \nin the one-dimensional quarter filled band. \nWe can obtain the self-consistent solutions from eq. (8) by \nusing eigenvectors obtained by diagonalizing \n$\\hat{\\cal K}+\\hat{\\cal U}_{\\rm M}+\\hat{\\cal V}_{\\rm M}$, \nwhich becomes $8\\times 8$ matrix. \nThe electron density at $j$ site, $n(j)$, and the \nspin moment at $j$ site, $S_z(j)$, are given by \n\\begin{eqnarray}\nn(j)&=&\\frac{1}{U}\\sum_{Q,\\sigma}\\rho_{\\sigma}(Q)e^{iQja}, \\\\\nS_z(j)&=&\\frac{1}{2U}\\sum_{Q}(\\rho_{\\uparrow}(Q)-\\rho_{\\downarrow}(Q))e^{iQja}.\n\\end{eqnarray}\nThe notation in this paper follows Seo and Fukuyama.\\cite{seofukuyama} \nWe can calculate \nthe total energy, $E$,\n\\begin{eqnarray}\nE=\\sum_{i=1,\\sigma}\\epsilon_{i,\\sigma},\n\\end{eqnarray}\nwhere the sum is limited to electron filling and \n$\\epsilon_{i,\\sigma}$ is an eigenvalue. \nFrom the ordered state energy ($E_{\\rm OS}$) and \nthe normal state energy ($E_{\\rm N}$), \nthe energy gain ($E_{\\rm g}$) can be obtained \nby $E_{\\rm g}=E_{\\rm OS}-E_{\\rm N}$. \n\n\nThe Pauli spin susceptibility, $\\chi$, is \n$\\mu_{\\rm B}^2N(0)$ ($N(0)$ is the density of state \non the Fermi energy and \n$N(0)=N_{\\rm S}/(4\\pi t\\sin ak_{\\rm F})$),\\cite{pauli,pauli2,Mckenzie} \nand the energy gain from \nthe normal state is given by \n$-\\chi H^2$. \nWhen $E_{\\rm g}=-\\chi H_p^2$, \nthe Pauli limit field, $h_p$, is given by \n\\begin{eqnarray}\nh_p=2\\sqrt{\\frac{-\\pi E_{\\rm g}}{\\sqrt{2}N_{\\rm S}}},\n\\end{eqnarray}\nwhere $h_p=\\mu_{\\rm B}gH_p/2t$. \n\n\n\n%We calculate the self-consistent eq. () by using the eigenvectors \n%obtained by diagonalizing $\\hat{\\cal K}+\\hat{\\cal U_{\\rm M}}$+\n%$\\hat{\\cal V_{\\rm M}}$. \n%we obtain eigenvalue, $E_{j}$, \n%where index $j$ includes the spin index. \n%Under the condition of the fixed total electron number, $N$, \n%the ground state energy per site is calculated by \n%\\begin{eqnarray}\n%F(N,H)=\\frac{1}{N_s}\\sum^{N}_{j=1}E_{j}, \n%\\end{eqnarray}\n%where $N_s$ is the total site number. \n%In the case of the chemical potential fixed, \n%\\begin{eqnarray}\n%\\Omega(h,\\mu)=\\frac{1}{2N_s}\\sum_{\\epsilon_{j}-\\mu<0}\n%(\\epsilon_{j}-\\mu), \n%\\end{eqnarray}\n%where the chemical potential, $\\mu$, is decided by \n%the Fermi energy \n%at $2/3$-filling and $h=0$. \n%\\end{eqnarray}\n%For example, as $a\\simeq 10\\AA$ and $b\\simeq 10\\AA$, \n%$h=p/q$ is about 1/400 when $H\\simeq 10$ T. \n%We calculate at \n%higher field ($H\\sim 100$ T, $h\\sim 1/40$). \n\n\n%In order to see the effect of the Zeeman term clearly, \n%we define $\\widetilde{g}$ as $\\widetilde{g}h\\equiv\n%\\frac{1}{2}g\\mu_{\\rm B}H/t_a$. \n%$-\\frac{1}{2}g\\mu_{\\rm B}H\\sigma=-t_a\\widetilde{g}h\\sigma$. \n\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=8cm\n\\epsfbox{fig1a.eps}\n\\caption{2S$_z$ as a function of $v$ at $H=0$ \n}\n\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig1b.eps}\n\\caption{$\\delta$ as a function of $v$ at $H=0$ \n}\n%\\label{fig:1}\n\\end{figure}\n\n\\section{Results and Discussions}\n\\subsection{Strong coupling}\nFirst, we show the result at $H=0$ \nat $U/t=5.0$, whose large value \nmeans the strongly correlated system. \nFigs. 1 and 2 are $S_z$ and $\\delta$ \nas a function of $V/t$ \nat $U/t=5.0$. \nAt $0\\leq V\\leq 0.392$, \nthe antiferromagnetic ordering \n(($\\uparrow$,$\\uparrow$,$\\downarrow$,$\\downarrow$), i.e., \n$S_z(1)=S_z(2)=-S_z(3)=-S_z(4)$) \nis stabilized and there is no charge ordering. \nThe spin ordering of \n($\\uparrow$,$\\uparrow$,$\\downarrow$,$\\downarrow$) has \nthe wave vector of $2k_{\\rm F}$. \nAbove $V/t=0.392$, \nthe spin ordering becomes \n($\\uparrow$,0,$\\downarrow$,0) \n($S_z(1)=-S_z(3)$, $S_z(2)=S_z(4)=0$) \nand the charge ordering ($\\delta$,-$\\delta$,$\\delta$,-$\\delta$) exist, \nwhere \n$n(1)=n(3)$=0.5+$\\delta$, $n(2)=n(4)$=0.5-$\\delta$, \nwhich can be seen in Figs. 1 and 2. \nThese ($\\uparrow$,0,$\\downarrow$,0) and ($\\delta$,$-\\delta$,$\\delta$,$-\\delta$) \nmean $2k_{\\rm F}$-SDW and $4k_{\\rm F}$-CDW, respectively. \nBy including the inter-site Coulomb interaction, \n$4k_{\\rm F}$-CDW is induced. \n\n\nWhen $U/t$ and $V/t$ are smaller than ($\\sim 1.5$), \nwe understand that the system with small $S_z$ is \nthe SDW transition due to Peierls instability \nof the Fermi surface. \nThe spin and charge orderings are localized \nif $U/t$ and $V/t$ become larger than ($\\sim 4.0$) since \nthe amplitudes of $S_z$ and $n$ are saturated. \nThis state with the localized spin \nand charge orderings is \nMott antiferromagnetic state due to the \nlarger values of $U/t$ and $V/t$. \nThese are the same results as Seo and Fukuyama.\\cite{seofukuyama} \n\n\n\nNext, we show \n$S_z$ and $n$ at $H\\neq0$ and $U/t=V/t=5.0$ \nby using the ground state, ($\\uparrow$,0,$\\downarrow$,0) \nand ($\\delta$,-$\\delta$,$\\delta$,-$\\delta$), \nin the strong coupling system. \nWhen the magnetic field is applied along \n$x$-axis ($h_x=\\mu_{\\rm B}gH_{x}/2t$), \nthe antiferromagnetic state is gradually suppressed up to \nthe critical field ($h_x^{c}=2.4$) and above $h_x^{c}$ \nthe spin ordering becomes $(0, 0, 0, 0)$, \nwhereas the charge ordering is unchanged, \nas shown in Figs. 3 and 4, where \n$S_z(1)=-S_z(3)$, $S_z(2)=S_z(4)$ and \n$n(1)=n(3)$=0.5+$\\delta$, $n(2)=n(4)$=0.5-$\\delta$. \nThe $h_x$-dependence of the amplitude of \n$S_z$ is in good agreement with eq. (1) when we set \n$H_x^0$ as $H_x^c=2th_x^c/\\mu_{\\rm B}g$, \nwhich is shown by solid lines \nin Fig. 3. \n\n\nIn the case of $h_z=\\mu_{\\rm B}gH_{z}/2t\\neq 0$, \nthe alignment of the spin moment, ($\\uparrow$,0,$\\downarrow$,0) when \n$h_z^{c}=0$ is kept as $h_z^{c}$ increases, but, \nthe system becomes \n($\\downarrow$,0,$\\downarrow$,0) \nat $h_z^{c}=2.4$, \nas shown in Fig.5. \nThis $h_z^{c}$ has the same value of $h_x^{c}$. \nThe charge ordering is not changed upon \nincreasing $h_z$, as shown in Fig. 6, where \n$n(1)=n(3)$=0.5+$\\delta$, $n(2)=n(4)$=0.5-$\\delta$. \n%This $h_z^{c}$ is due to the Pauli limit, which is \n%in good agreement with $h_p$, because \n%we estimate $h_p\\simeq 2.6$ by using eq. (9) \n%and \n%$E_{\\rm g}$ at $U/t=V/t=5.0$ and $H=0$. \nThese ($\\downarrow$,0,$\\downarrow$,0) and \n($\\delta$,$-\\delta$,$\\delta$,$-\\delta$) are $4k_{\\rm F}$-SDW \nand $4k_{\\rm F}$-CDW, that is, \nabove $h_z^{c}$ the system becomes \nparamagnetic state to stay to be \nlocalized due to larger $U/t$ and $V/t$. \n\n\nThe $h_x$- and $h_z$-\ndependences of the spin ordering in \nour results can be understood \nby the mean field solutions for \nstrong coupled Ising \nmodel mentioned \nin the introduction.\\cite{ising}\n\nThe charge ordering is unchanged by the magnetic \nfield. The magnetic field is not contributed \nto localized electrons made by \nlarger $V/t$. \n\n\n\n%It has been shown that the angles of the \n\n\n%suppression of $\\beta\\pm\\alpha$ oscillations \n%are not the same in the dHvA experiment of \n%$\\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ \n%by Meyer et al.\\cite{Meyer}, where \n%the angle of the applied magnetic field is fixed perpendicular \n%to the conductive plane. \n\n\n%\\begin{figure}\n%%\\figureheight{0cm}\n%\\leavevmode\n%%\\epsfxsize=8.7cm\n%\\epsfxsize=8.5cm\n%\\epsfbox{ftaNm1.eps}\n%\\caption{The FTAs of \n%$f_\\alpha$ and $f_\\beta$ \n%in $M(N,H)$ as a function of $\\widetilde{g}$ for \n%$m_{\\alpha}/m_{0}=0.65, m_{\\beta}/m_{0}=1.0, \\epsilon^{b}_{\\alpha}=754$ and \n%$\\epsilon^{b}_{\\beta}=0$. \n%}\n%\\end{figure}\n\n%\\begin{figure}\n%\\leavevmode\n%\\epsfxsize=8.5cm\n%\\epsfbox{ftaNm1b.eps}\n%\\caption{The FTAs of \n%$f_{2\\alpha}$ and $f_{2\\beta}$ \n%in $M(N,H)$ as a function of $\\widetilde{g}$ for \n%$m_{\\alpha}/m_{0}=0.65, m_{\\beta}/m_{0}=1.0, \\epsilon^{b}_{\\alpha}=754$ and \n%$\\epsilon^{b}_{\\beta}=0$. \n%}\n%\\end{figure}\n\n%\\begin{figure}\n%\\leavevmode\n%\\epsfxsize=8.5cm\n%\\epsfbox{ftaNm1c.eps}\n%\\caption{The FTAs of \n%$f_{\\beta-\\alpha}$ and $f_{\\beta+\\alpha}$ \n%in $M(N,H)$ as a function of $\\widetilde{g}$ for \n%$m_{\\alpha}/m_{0}=0.65, m_{\\beta}/m_{0}=1.0, \\epsilon^{b}_{\\alpha}=754$ and \n%$\\epsilon^{b}_{\\beta}=0$. \n%}\n%\\end{figure}\n\n\n\n\n\n\n\n%\\begin{figure}\n%\\leavevmode\n%\\epsfxsize=8.5cm\n%\\epsfbox{ftam2.eps}\n%\\caption{The FTAs of \n%$f_{2\\beta-\\alpha}$ and \n%$f_{2\\beta+\\alpha}$ \n%in $M(N,H)$ as a function of $\\widetilde{g}$ for \n%$m_{\\alpha}/m_{0}=0.65, m_{\\beta}/m_{0}=1.0, \\epsilon^{b}_{\\alpha}=754$ and \n%$\\epsilon^{b}_{\\beta}=0$. \n%}\n%\\end{figure}\n\n\n\n\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig7.eps}\n\\caption{\nWhen $U/t=V/t=5.0$, \n2S$_z$ as a function of $h_x$. \nThe solid lines are written by \neq. (1). \n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig8.eps}\n\\caption{\nWhen $U/t=V/t=5.0$, \n$\\delta$ as a function of $h_x$. \n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig9.eps}\n\\caption{\nWhen $U/t=V/t=5.0$, \n2S$_z$ as a function of $h_z$. \n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig10.eps}\n\\caption{\nWhen $U/t=V/t=5.0$, \n$\\delta$ as a function of $h_z$. \n}\n%\\label{fig:1}\n\\end{figure}\n\n\\subsection{Non-Strong coupling}\nWe calculated the solutions as a function of $V/t$ at \n$U/t=1.5$. There were two solutions \n(($\\uparrow$,$\\uparrow$,$\\downarrow$,$\\downarrow$) and \n($\\uparrow$,0,$\\downarrow$,0) and \n($\\delta$,$-\\delta$,$\\delta$,$-\\delta$)), \nwhich are shown \nin Figs 7, 8 and 9. \nThese are for the state with \n$2k_{\\rm F}$-SDW and the coexistent state with \n$2k_{\\rm F}$-SDW and $4k_{\\rm F}$-CDW, respectively. \nIt is found that these amplitudes of $S_z$ and \n$\\delta$ are small. \nIn the region of \n$0\\leq V/t\\leq 1.5$, \nthe energies with these solutions are nearly \nthe same, namely, these states are degenerate. \nTherefore, we analyze \n$h_x$- and $h_z$-dependences of $S_z$ and $n$ \nfor two solutions. \n\n\nWe calculate the case of $U/t=V/t=1.5$ \nat $H \\neq0$. \nOn using \n($\\uparrow$,$\\uparrow$,$\\downarrow$,$\\downarrow$) when $H=0$, \nfor $h_{z}$ \nthis state is changed to \n($\\downarrow$,$\\downarrow$,$\\downarrow$,$\\downarrow$) at \n$h_{z}^{c}=0.0155$, but, \nfor $h_{x}$, $S_z$ is unchanged, which are shown \nin Figs. 10 and 11.\nThis suppression for $h_{z}$ comes from \nPauli paramagnetic limit since \n$h_{z}^{c}$ corresponds to $h_p\\simeq 0.0125$ \nobtained by using eq. (12) and $E_{\\rm g}/N_{\\rm S}\\simeq0.0000175$ \ncalculated at \n$U/t=V/t=1.5$ and $H=0$. \nFor $h_{x}$, there is no Pauli limit, \nbecause the nesting vector of the SDW is not \naffected by Zeeman splitting. \nIn Fig. 11, $S_z$ linearly increases as $h_z$ \nincreases, which means that \nthe paramagnetic state is stabilized by the magnetic \nfield. \n\n\nFor ($\\uparrow$,0,$\\downarrow$,0) and \n($\\delta$,-$\\delta$,$\\delta$,-$\\delta$), \nwhen the magnetic field is applied along $x$-axis, \nthe spin and charge ordering do not change, as shown in \nFigs. 12 and 13. \nHowever, for $h_{z}$, both orderings of \nthe spin and charge disappear at $h_{z}^{c}=0.0155$, \nwhich is corresponding to Pauli paramagnetic limit field, \n$h_p\\simeq 0.0125$, as shown in \nFigs. 14 and 15. \nThis is due to the effect of \nPauli limit, too. \nThe coexistent state of \n$2k_{\\rm F}$-SDW and $4k_{\\rm F}$-CDW \nwith ($\\uparrow$,0,$\\downarrow$,0) \nand \n($\\delta$,-$\\delta$,$\\delta$,-$\\delta$) is \nchanged to paramagnetic state \nwith ($\\downarrow$,$\\downarrow$,$\\downarrow$,$\\downarrow$) \nand (0,0,0,0). \nIt is seen that the linear increasing of $S_z$ \nabove $h_{z}^{c}$. \n\n\n%When $U/t$ and $V/t$ are small, \n%$4k_{\\rm F}$-CDW can not be obtained, because \n%electrons are not localized.\n\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig11.eps}\n\\caption{By using of \n($\\uparrow$,$\\uparrow$,$\\downarrow$,$\\downarrow$) \nat $U/t=1.5$ and $V/t=0$, \n$2S_z$ as a function of $v$ at $H=0$. \n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig12.eps}\n\\caption{By using of \n($\\uparrow$,0,$\\downarrow$,0) \nat $U/t=1.5$ and $V/t=0$, \n$2S_z$ as a function of $v$ at $H=0$. \n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig13.eps}\n\\caption{By using of \n($\\uparrow$,0,$\\downarrow$,0) \nat $U/t=1.5$ and $V/t=0$, \n$\\delta$ as a function of $v$ at $H=0$. \n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig21.eps}\n\\caption{\nWhen $U/t=V/t=1.5$, \n$2S_z$ as a function of $h_x$ by using of \n($\\uparrow$,$\\uparrow$,$\\downarrow$,$\\downarrow$) \nat $h_x=0$.\n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig22.eps}\n\\caption{\nWhen $U/t=V/t=1.5$, \n$2S_z$ as a function of $h_z$ by using of \n($\\uparrow$,$\\uparrow$,$\\downarrow$,$\\downarrow$) \nat $h_z=0$.\n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig23.eps}\n\\caption{\nWhen $U/t=V/t=1.5$, \n$2S_z$ as a function of $h_x$, \nby using of \n($\\uparrow$,0,$\\downarrow$,0) \nat $h_x=0$.\n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig24.eps}\n\\caption{\nWhen $U/t=V/t=1.5$, \n$\\delta$ as a function of $h_x$, \nby using of \n($\\uparrow$,0,$\\downarrow$,0) \nat $h_x=0$.\n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig25.eps}\n\\caption{\nWhen $U/t=V/t=1.5$, \n$2S_z$ as a function of $h_z$, \nby using of \n($\\uparrow$,0,$\\downarrow$,0) \nat $h_z=0$.\n}\n%\\label{fig:1}\n\\end{figure}\n\n\\begin{figure}\n%\\figureheight{0cm}\n\\leavevmode\n\\epsfxsize=7cm\n\\epsfbox{fig26.eps}\n\\caption{\nWhen $U/t=V/t=1.5$, \n$\\delta$ as a function of $h_z$, \nby using of \n($\\uparrow$,0,$\\downarrow$,0) \nat $h_z=0$.\n}\n%\\label{fig:1}\n\\end{figure}\n\n%\\subsection{Comparison strong coupling with non-strong coupling}\n%In the strong coupling case, although \n%the spin ordering is affected by the magnetic field, \n%the charge ordering is not unchanged. \n%When the correlation between electrons is \n%not large, both orderings of the spin and charge \n%disappear at the same time under the magnetic \n%field along $z$-axis. \n\n\\subsection{Comparison with Experiments}\nThe coexistent state with \nCDW and SDW is realized due to the \ninter-site Coulomb interaction even if \nthe correlation between electrons \nis strong or not.\n\nIn the non-localized SDW system, \nwhen the magnetic field is applied parallel to \nthe easy axis, both orderings of SDW and CDW disappear at \nthe critical field of the Pauli paramagnetic limit. \nIn the case of the magnetic field perpendicular to \neasy axis, both orderings of \nCDW and SDW are unchanged. \nIn (TMTSF)$_2$$X$, for example, since \nthe easy axis is $b$-axis, it is expected \nthat when the magnetic \nfield is applied to $b$-axis \nboth orderings become to disorder at \nPauli limit field. \n\n\nOn the other hand, \nin the localized antiferromagnetic state such \nas (TMTTF)$_2$$X$, the charge ordering is not \nsuppressed in both cases of \nthe magnetic field applied to parallel to and \nperpendicular to the easy axis. \nThus, \n$4k_{\\rm F}$-CDW may be \nobserved from the X-ray measurement even if \nthe magnetic field is applied to $a$, $b$ and $c$-axis. \nIt is a means of finding whether the system \nis strongly correlated or not. \n\nIn (DCNQI)$_2$Ag, which are strongly correlated system such \nas (TMTTF)$_2$$X$, $4k_{\\rm F}$-CDW has been observed \nat zero magnetic field.\\cite{hiraki,moret} \nEven under high fields, the charge ordering \nshould be appeared.\n\n\n\\section{Conclusions}\nWe theoretically study the coexistent state \nof CDW and SDW under the magnetic field. \nAs a result, in the case of the strongly coupling system, \nalthough the spin ordering is suppressed at \nhigh fields, the charge ordering still remains. \nWhen the coupling is not so large, \nthe CDW and the SDW disappear at Pauli paramagnetic limit field. \n\nThese features of the coexisitent state of \nCDW and SDW under \nmagnetic fields should be observed in \nthe strongly correlated system (non-strongly correlated system) \nsuch as (TMTTF)$_2$$X$ and (DCNQI)$_2$Ag ((TMTSF)$_2$$X$). \n\n\n%both systems with and without the magnetic breakdown \n%cannot be described by \n%the spin reduction factor for the single-band (eq. (4)). \n%Even in the two-band model, however, \n%the spin-splitting-zero condition for fundamental \n%frequencies ($\\alpha$ and $\\beta$ oscillations) \n%in $M(N,H)$ in no magnetic breakdown system is given by eq. (4).\n%We expect that \n%the $\\widetilde{g}$-dependences of the amplitudes of these oscillations \n%may be observed in the dHvA-experiment of \n%tilting magnetic field \n%in two-dimensional multi-band system such as Sr$_2$RuO$_4$ \n%and quasi-two-dimensional organic conductors. \n\n%\\newpage\n\n\\section{Acknowledgment}\n\n%One of the authors (K. K.) thanks S. Uji for useful discussions.\n%and T. Osada\n%for useful information on experiments.\n%We are also indebted to T. Mori for information on his band\n%calculation.\nOne of the authors (K. K.) would like to thank \nT. Sakai for valuable discussions.\n%We would like to \n%thank Y. Yoshida and R. Settai for sending us the \n%information of their recent experimental results. \nK. K. was partially supported by Grant-in-Aid for\nJSPS Fellows from the Ministry of\nEducation, Science, Sports and Culture.\nK. K.\nwas financially supported by the Research Fellowships\nof the Japan Society\nfor the Promotion of Science for Young Scientists.\n\n\n\n\n\\begin{thebibliography}{99}\n\n\n\n\\bibitem{seofukuyama}\nH.Seo and H. Fukuyama: J. Phys. Soc. Jpn. {\\bf 66}\n(1997) 1249.\n\n\\bibitem{nobuko}\nN. Kobayashi and M. Ogata: J. Phys. Soc. Jpn. {\\bf 66}\n(1997) 3356.\n\n\\bibitem{nobuko2}\nN. Kobayashi, M. Ogata and K. Yonemitsu: \nJ. Phys. Soc. Jpn. \n{\\bf 67} (1998) 1098.\n\n\n\\bibitem{Mazumdar}\nS. Mazumdar, S. Rammasesha, R. Torsten Clay and \nDavid K. Campbell: \nPhys. Rev. Lett. {\\bf 82} (1999) 1522.\n\n\\bibitem{review}\nFor a review, see: T. Ishiguro, K. Yamaji, and G. Saito: {\\it\nOrganic \nSuperconductors}\n(Springer-Verlag, Berlin 1998).\n\n\\bibitem{jerome}\nFor a review, see D. Jerome: {\\it\nOrganic \nConductors} ed J. P. Farges\n(Marcel Deckker, New York, 1994).\n\n\n\n%\\bibitem{nad}\n%F. Nad', P. Monceau and J. M. Fabre: Eur. Phys. J. B {\\bf 3} \n%(1998) 301. \n\n\n\\bibitem{takahashi}\nT. Takahashi, Y. Maniwa, H. Kawamura and \nG. Saito: J. Phys. Soc. Jpn. {\\bf 55} \n(1986) 1364.\n\n\\bibitem{delrieu}\nJ. M. Delrieu, M. Roger, Z. Toffano and A. Moradpour: \nJ. Phys. (Paris) {\\bf 47} \n(1986) 839. \n\n\n\\bibitem{pouget}\nJ. P. Pouget and S. Ravy: \nSynth. Met. {\\bf 85} (1997) 1523.\n\n\n\\bibitem{barthel}\nE. Barthel, G. Quirion, P. Wzietek, D. Jerome, J. B. Christensen, \nM. Joregensen and K. Bechgaard: \nEurophys. Lett. {\\bf 21} \n(1993) 87. \n\n\\bibitem{nakamura}\nT. Nakamura, T. Nobutoki, Y. Kobayashi, T. Takahashi and \nG. Saito: Synth. Met. {\\bf 70} (1995) 1293.\n\n\n\n\\bibitem{nakamura2}\nT. Nakamura, R. Kinami, Takahashi and \nG. Saito: Synth. Met. {\\bf 86} (1997) 2053.\n\n\\bibitem{tanemura}\nN. Tanemura and Y. Suzumura: J. Phys. Soc. Jpn. {\\bf 65}\n(1996) 1792.\n\n\\bibitem{klemme}\nB. J. Klemme, S. E. Brown, P. Wzietek, G. Kriza, P. Batail, \nD. Jerome and J. M. Fabre: Phys. Rev. Lett. {\\bf 75} (1995) 2408.\n\n\\bibitem{band}\nT. Mori, A. Kobayashi, Y. Sasaki and H. Kobayashi: \nChem. Lett. (1982) 1923.\n\n\\bibitem{band2}\nP. M. Grant: J. Phys. Colloq. C{\\bf 3} (1983) 847.\n\n\\bibitem{band3}\nT. Mori, A. Kobayashi, Y. Sasaki and H. Kobayashi, G. Saito and \nH. Inokuchi: Bull. \nChem. Soc. Jpn. {\\bf 57} (1984) 627.\n\n\\bibitem{band4}\nL. Ducasse, M. Abderrabba, J. Hoarau, M. Pesquer, \nB. Gallois and J. Gaultier: J. Phys. C{\\bf 19} (1986) 3805.\n\n\\bibitem{pauli}\nB. S. Shandrasekhar: Appl. Phys. Lett. {\\bf 1} (1962) 7.\n\n\n\\bibitem{pauli2}\nA. M. Clogston: Phys. Rev. Lett. {\\bf 9} (1962) 266.\n\n\\bibitem{Mckenzie}\nFor study of the Pauli limit of CDW in recent, \nRoss H. McKenzie cond-mat/9706235.\n\n\n\\bibitem{ising}\nV. Yu. Irkhin and A. A. Katanin: Phys. Rev. B{\\bf 58} (1998) 5509.\n\n\\bibitem{hiraki}\nK. Hiraki and K. Kanoda: Phys. Rev. B{\\bf 54} (1996) 17276.\n\n\\bibitem{moret}\nR. Moret, P. Erk, S. Hunig and J. U. Von Shultz:\nJ. Phys. {\\bf 49} (1988) 1925.\n\n\n%\\bibitem{sasaki}\n%T. Sasaki and T. Fukase: Phys. Rev. B{\\bf 59} (1999) 13872.\n\n\n\n%\\bibitem{kishigi}\n%K. Machida et al.: Synth. Met. {\\bf 70} (1995) 853; Phys.\n%K. Machida, K. Kishigi and Y. Hori: \n%Phys. Rev. {\\bf B 51}(1995) 8946.\n%K. Machida, K. Kishigi, and Y. Hori: Synth. Met. {\\bf 70} (1995) 853; Phys.\n%Rev. {\\bf B 51}(1995) 8946.\n\n\n%\\bibitem{kishigi2}\n%K. Kishigi, M. Nakano, K. Machida, and Y. Hori: J. Phys. Soc. Jpn. {\\bf 64} \n%(1995) 3043. \n%K. Kishigi et al.: J. Phys. Soc. Jpn. {\\bf 64} \n%(1995) 3043. \n\n\n%\\bibitem{kishigi3}\n%K. Kishigi: J. Phys. Soc. Jpn. {\\bf 66} \n%(1997) 910. \n\n\n\n\n\n\n\n\n%\\bibitem{sasaki}\n%T. Sasaki and T. Fukase: Phys. Rev. B{\\bf 59} \n%(1999) 13872.\n\n\n\n\\end{thebibliography}\n\n\n\n\n\n\n\n\n\\end{document}\n\n\n"
}
] |
[
{
"name": "cond-mat0002165.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\n\n\\bibitem{seofukuyama}\nH.Seo and H. Fukuyama: J. Phys. Soc. Jpn. {\\bf 66}\n(1997) 1249.\n\n\\bibitem{nobuko}\nN. Kobayashi and M. Ogata: J. Phys. Soc. Jpn. {\\bf 66}\n(1997) 3356.\n\n\\bibitem{nobuko2}\nN. Kobayashi, M. Ogata and K. Yonemitsu: \nJ. Phys. Soc. Jpn. \n{\\bf 67} (1998) 1098.\n\n\n\\bibitem{Mazumdar}\nS. Mazumdar, S. Rammasesha, R. Torsten Clay and \nDavid K. Campbell: \nPhys. Rev. Lett. {\\bf 82} (1999) 1522.\n\n\\bibitem{review}\nFor a review, see: T. Ishiguro, K. Yamaji, and G. Saito: {\\it\nOrganic \nSuperconductors}\n(Springer-Verlag, Berlin 1998).\n\n\\bibitem{jerome}\nFor a review, see D. Jerome: {\\it\nOrganic \nConductors} ed J. P. Farges\n(Marcel Deckker, New York, 1994).\n\n\n\n%\\bibitem{nad}\n%F. Nad', P. Monceau and J. M. Fabre: Eur. Phys. J. B {\\bf 3} \n%(1998) 301. \n\n\n\\bibitem{takahashi}\nT. Takahashi, Y. Maniwa, H. Kawamura and \nG. Saito: J. Phys. Soc. Jpn. {\\bf 55} \n(1986) 1364.\n\n\\bibitem{delrieu}\nJ. M. Delrieu, M. Roger, Z. Toffano and A. Moradpour: \nJ. Phys. (Paris) {\\bf 47} \n(1986) 839. \n\n\n\\bibitem{pouget}\nJ. P. Pouget and S. Ravy: \nSynth. Met. {\\bf 85} (1997) 1523.\n\n\n\\bibitem{barthel}\nE. Barthel, G. Quirion, P. Wzietek, D. Jerome, J. B. Christensen, \nM. Joregensen and K. Bechgaard: \nEurophys. Lett. {\\bf 21} \n(1993) 87. \n\n\\bibitem{nakamura}\nT. Nakamura, T. Nobutoki, Y. Kobayashi, T. Takahashi and \nG. Saito: Synth. Met. {\\bf 70} (1995) 1293.\n\n\n\n\\bibitem{nakamura2}\nT. Nakamura, R. Kinami, Takahashi and \nG. Saito: Synth. Met. {\\bf 86} (1997) 2053.\n\n\\bibitem{tanemura}\nN. Tanemura and Y. Suzumura: J. Phys. Soc. Jpn. {\\bf 65}\n(1996) 1792.\n\n\\bibitem{klemme}\nB. J. Klemme, S. E. Brown, P. Wzietek, G. Kriza, P. Batail, \nD. Jerome and J. M. Fabre: Phys. Rev. Lett. {\\bf 75} (1995) 2408.\n\n\\bibitem{band}\nT. Mori, A. Kobayashi, Y. Sasaki and H. Kobayashi: \nChem. Lett. (1982) 1923.\n\n\\bibitem{band2}\nP. M. Grant: J. Phys. Colloq. C{\\bf 3} (1983) 847.\n\n\\bibitem{band3}\nT. Mori, A. Kobayashi, Y. Sasaki and H. Kobayashi, G. Saito and \nH. Inokuchi: Bull. \nChem. Soc. Jpn. {\\bf 57} (1984) 627.\n\n\\bibitem{band4}\nL. Ducasse, M. Abderrabba, J. Hoarau, M. Pesquer, \nB. Gallois and J. Gaultier: J. Phys. C{\\bf 19} (1986) 3805.\n\n\\bibitem{pauli}\nB. S. Shandrasekhar: Appl. Phys. Lett. {\\bf 1} (1962) 7.\n\n\n\\bibitem{pauli2}\nA. M. Clogston: Phys. Rev. Lett. {\\bf 9} (1962) 266.\n\n\\bibitem{Mckenzie}\nFor study of the Pauli limit of CDW in recent, \nRoss H. McKenzie cond-mat/9706235.\n\n\n\\bibitem{ising}\nV. Yu. Irkhin and A. A. Katanin: Phys. Rev. B{\\bf 58} (1998) 5509.\n\n\\bibitem{hiraki}\nK. Hiraki and K. Kanoda: Phys. Rev. B{\\bf 54} (1996) 17276.\n\n\\bibitem{moret}\nR. Moret, P. Erk, S. Hunig and J. U. Von Shultz:\nJ. Phys. {\\bf 49} (1988) 1925.\n\n\n%\\bibitem{sasaki}\n%T. Sasaki and T. Fukase: Phys. Rev. B{\\bf 59} (1999) 13872.\n\n\n\n%\\bibitem{kishigi}\n%K. Machida et al.: Synth. Met. {\\bf 70} (1995) 853; Phys.\n%K. Machida, K. Kishigi and Y. Hori: \n%Phys. Rev. {\\bf B 51}(1995) 8946.\n%K. Machida, K. Kishigi, and Y. Hori: Synth. Met. {\\bf 70} (1995) 853; Phys.\n%Rev. {\\bf B 51}(1995) 8946.\n\n\n%\\bibitem{kishigi2}\n%K. Kishigi, M. Nakano, K. Machida, and Y. Hori: J. Phys. Soc. Jpn. {\\bf 64} \n%(1995) 3043. \n%K. Kishigi et al.: J. Phys. Soc. Jpn. {\\bf 64} \n%(1995) 3043. \n\n\n%\\bibitem{kishigi3}\n%K. Kishigi: J. Phys. Soc. Jpn. {\\bf 66} \n%(1997) 910. \n\n\n\n\n\n\n\n\n%\\bibitem{sasaki}\n%T. Sasaki and T. Fukase: Phys. Rev. B{\\bf 59} \n%(1999) 13872.\n\n\n\n\\end{thebibliography}"
}
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cond-mat0002166
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Vesicle propulsion in haptotaxis : a local model
|
[
{
"author": "Isabelle Cantat"
},
{
"author": "Chaouqi Misbah and Yukio Saito"
}
] |
[
{
"name": "haptlocweb.tex",
"string": "%\\documentstyle[floats,twocolumn,prl,aps]{revtex}\n\\documentstyle[floats,aps]{revtex}\n \\newcommand{\\drp}[2]{\\frac{\\partial #1}{\\partial #2}}\n \\newcommand{\\vv}[1]{{\\bf \\hat {#1}}}\n\\input epsf\n\\input rotate\n\\input psfig.sty\n\n\\newcommand{\\tens}[1]{\\overline{\\overline{#1}}}\n \n\\begin{document}\n\\draft\n\\title{Vesicle propulsion in haptotaxis : a local model}\n\n\\author{Isabelle Cantat, Chaouqi Misbah and Yukio Saito}\n\\address{Laboratoire de Spectrom\\'etrie Physique, Universit\\'e Joseph Fourier\n (CNRS),\nGrenoble I, B.P. 87, Saint-Martin d'H\\`eres, 38402 Cedex, France\n}\n\\address{Department of Physics, Keio University, \n 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan\n }\n\\author{\\parbox{397pt}{\\vglue 0.3cm \n\\small\nWe study theoretically vesicle locomotion due to haptotaxis. \n Haptotaxis is referred to motion induced by an adhesion gradient\n on a substrate. The problem is solved\n within a local approximation where a Rayleigh-type dissipation\n is adopted. The dynamical model is akin to the Rousse model\n for polymers. A powerful gauge-field invariant formulation\nis used to solve a dynamical model which includes a kind\nof dissipation due to bond breaking/restoring with the substrate.\nFor a stationary situation where the vesicle acquires\na constant drift velocity, we formulate the propulsion problem\nin terms of a nonlinear eigenvalue (the a priori unknown drift velocity)\none of Barenblat-Zeldovitch type. A counting argument shows\nthat the velocity belongs to a discrete set. For a relatively tense\nvesicle, we provide an analytical expression for the drift velocity\nas a function of relevant parameters. We find good agreement\nwith the full numerical solution. Despite the oversimplification of the model\nit allows the identification of a relevant quantity, namely the \nadhesion length, which turns out to be crucial also in the nonlocal\nmodel in the presence of hydrodynamics, a situation on which\nwe have recently \nreported [I. Cantat, and C. Misbah, Phys. Rev. Lett. {\\bf 83}, 235 (1999)]\nand which constitutes the subject of a forthcoming extensive study.\n}}\n\\author{\\vskip 0.2 cm PACS numbers 87.22.Bt, 87.45 -k, 47.55 -Dz}\n\n\\maketitle\n\n\\section{Introduction}\n Phospholipidic vesicles constitute a simple model of cytoplasmic\n membranes of real cells. A simple model due to Helfrich\\cite{Helfrich73}\n based on curvature energy has accounted for a variety of equilibrium\n shapes. The model is based \n on a minimal energy principle\\cite{Lipowsky}. Some of the shapes (the so-called discocytes) bear \n strong resemblance with that of an erythrocyte (the red blood cell).\n Additionally,\n analysis of flickering (temporal small fluctuations\n around a given shape) of an erythrocyte by Brochard and\n Lennon\\cite{Brochard75} has been quite successfully\ndescribed by the Helfrich model including hydrodynamics dissipation.\n The vesicle model has seemed then as a natural\n candidate, at\n least in a first stage,\n for dealing with more complex entities such\n as those encountered in the realm of biology.\n In that context, however, most of the features are of\n nonequilibrium dissipative nature. Very recently \n several \n theoretical\\cite{Prost96,Kraus96,Durand97,Cantat99,Cantat99a,Seifert99,Cantat99b} \n and experimental\\cite{Nardi99} investigations have\n been directed along that line. \n\nWe are interested here in vesicle migration, a question\non which we have given recently a brief account\\cite{Durand97,Cantat99}.\n Despite the very complex biochemical behaviour of a cell, cells may\n also exhibit behaviours where simple physical concepts may be evoked. \n It is\n well documented that, for example, the migration of the pronephric duct cells in salamanders\n is regulated by haptotaxis. Haptotaxis is a terminology that is used\n to express the following fact: when adhesive molecules\n are present in increasing amounts along an extracellular\n matrix (or simply on a substrate in {\\it vitro}), a cell\n that was constantly making and breaking adhesion with such\n a molecule would migrate from a region of low\n concentration to an area where that adhesive molecule\n was more highly concentrated\\cite{Carter67,Curtis69}.\n There are also evidences that cell migration during embryo development may be guided by\n an adhesion gradient. In other words cell migration\n is here guided by a purely external physical\n factor, while the internal structure (the cytoskeletton)\n is quite unaffected on the time scale of interest.\n This feature drastically differs from that of a cell\n belonging to the immune system where the cytoskeletton plays\n a decisive role\\cite{cytoskeletton}. Despite\n the fact that the cytoskeletton in pure haptotaxis does not\n undergo a structural change as is the case during cell\n crawling, \n the problem remains very much involved\n since the cell cytoskeletton dissipation \n may come to the fore as well as an intricate\n bond breaking and restoring with the substrate. We shall\n consider here a pure vesicle moving in haptotaxis. Our belief\n is that advancement in this field can be achieved only by\n the progressive refinement of concepts.\n\n We consider a vesicle moving along the substrate thanks\n to an adhesion gradient. As the vesicle moves, it generates\n hydrodynamics flow both inside and outside. \n Hydrodynamics induces nonlocal interactions leading\n\t to an effective coupling of two\n\t\t distinct regions on the vesicle.\n\t\t In addition,\n the two\n monolayers that form the phospholipidic\n membrane might slide one relative to the other.\n Finally during motion the vesicle forms new bonds ahead and destroys\n others behind, and this process of bond breaking and restoring\n may be so slow that it may dominate dynamics (see later).\n\n This paper should be regarded as using a very simplistic\n view in the hope of introducing the concepts of migration\n and to exhibit in a transparent\n fashion the way the problem is addressed. We shall keep\n the description as simple as possible. That is to say: (i) we\n ignore nonlocality due to hydrodynamics --incorporation\n of hydrodynamics was briefly discussed in \\cite{Cantat99} and\n will be the subject of a forthcoming paper--, (ii) we confine\n ourselves to a 2D geometry. \n%We shall\n% come back to these points in the section devoted to the discussion.\n Some kind of dissipation due to bond breaking and restoring\n is introduced in our model. The adoption of a local\n model (no hydrodynamics) allows one to quite easily obtain analytical\n results and thus to extract some\n key ingredients about migration --especially the role\n of the adhesion area (length in 2D)-- which \n turns out to be crucial also when hydrodynamics is\n included. \n\n The scheme of this paper is organized as follows. In Section II\n we write down the equations of motion and comment them. In section\n III we present a forward time integration and present the main\n result. Section IV presents the solution\n of the stationary system in a form of a nonlinear eigenvalue\n problem, where the drift velocity is the eigenvalue. In section V\n we give an analytical solution. A conclusion\n and a discussion is presented in section VI.\n\n\n\n\n\\section{Equation of motion}\n\n\\subsection{Parameterization}\nWe consider an adhering vesicle, deposited on a flat substratum which is\n oriented \nby its normal vector ${\\bf \\hat y}$ (Fig.1).\nThe x-axis is along \nthe wall and represents the direction of vesicle motion occurring by convention \nfrom left to right. \nAs stated before we confine ourselves to two dimensions.\nThat is to say, the vesicle morphology is invariant in the z-direction, similar \nto a tubular vesicle. \n\n%\\begin{figure}[htb]\n% \\centerline{\n% \\psfig{figure=fig3/nota1.eps,width=8cm,angle=-90}}\n% \\caption{{\\it Notations used in the text.}}\n% \\label{nota1}\n%\\end{figure}\n\nThe interaction between\nthe vesicle and the substrate is taken into account by introducing\nan adhesion potential. The range of the potential in realistic\nsituations (typically several $nm$) is small in comparison \nto the vesicle size (several $\\mu m$), so that it is\njustified in practice to consider a contact potential, unless\notherwise indicated (see later).\nThe energy interaction is then zero if $y > 0$ and is non vanishing only close\nto contact (if $y=0$). \nAt the junction point between the free part of the vesicle (whose\nlength is denoted as $L^*$) and the adhered part (with length\n$L_{adh}$), the potential undergoes an abrupt change.\nThe contact between the membrane and substrate occurs\nat two well-defined points $x_1$ at the left and $x_2$ at the right.\nThese parameters are related to the total length of the curve $L$ by \n$L= L_{adh}+L^*= (x_2-x_1)+L^*$.\nWe use an intrinsic representation of the vesicle contour by\nintroducing $\\psi(s)$,\nthe angle between the outward normal and the y-axis, and $s$\nthe arc length,\nas shown on Fig. 1.\nWe only need to consider the function $\\psi(s,t)$ from $s=0$ to $s=L^*$ \ncorresponding \nto the contact points $x_1$ and $x_2$, respectively. Because\nof the contact potential character the adhesion length is completely fixed\nif the two contact points $x_1$ and $x_2$ are known.\nThus the vesicle shape and its dynamical properties (like the \npropulsion velocity) are known if $x_1$, $x_2$ and the \nfunction $\\psi(s,t)$ are determined. The demand that\nthe parametrization of the vesicle be compatible with the \nadhesion on the substrate it fulfilled by the \ntwo geometrical constraints :\n(i) the distance between both contact points, $x_2-x_1$, must\ncoincide with the adhesion length $L_{adh}=x _2-x_1$, (ii) \ntheir vertical coordinates $y_1$ and $y_2$ must have\nthe same value, \n $y_2 -y_1=0$. These two constraints can be expressed in terms\nof $\\psi(s,t)$. For that purpose we use the relations\n\\begin{equation}\n\\drp{x}{s}=\\cos \\psi \\; , \\;\\;\\;\\;\\; \\drp{y}{s}=-\\sin \\psi \\; ,\n\\label{dxds}\n\\end{equation}\nwhich allow us to write the two constraints in the \nfollowing form :\n \\begin{eqnarray}\n\\int_0^{L^*} \\drp{x}{s} \\, ds &=& \\int_0^{L^*} \\cos \\psi(s)\\, ds = x_2-x_1 = \nL_{adh} \\; , \\label{cos} \\\\\n\\int_0^{L^*} \\drp{y}{s}\\, ds &=& \\int_0^{L*} - \\sin \\psi(s) \\, ds = y_2-y_1 = 0 \n\\; .\n\\label{sin}\n \\end{eqnarray}\nThese are the geometrical constraints. In order to describe vesicle\ndynamics, we need a dynamical equation for the evolution\nof $\\psi(s,t)$. A movement of the vesicle (due for example\nto an adhesion gradient) is limited\nby dissipation (such as hydrodynamics etc...). The vesicle\nreacts to any deviation from equilibrium by its internal forces\n(bending, possible stretching --or resistance to stretching--). \nLet us first discuss these forces.\n\\subsection{Energy and forces}\n\nAll the relevant membrane properties are summarized in the following energy, \nexpressed in 2D, with the \ndimension of an energy per unit length : \n\\begin{equation}\nE=\\int_{\\cal C} \\kappa \\frac{(c-c_s)^2}{2} ds-\\int_{x_1}^{x_2}w(x)\\,dx\n+\\int_{\\cal C} \\zeta(s) ds\n+p\\, S \\; .\n\\label{Fad}\n\\end{equation}\nThe first term is the well known Helfrich curvature energy,\n with the rigidity $\\kappa$, the curvature $c=\\partial \\psi/ \\partial s$\nand the spontaneous curvature $c_s$ \\cite{Helfrich73}. \nBecause of the 2D-specific \nconservation law $\\int c \\,ds=2\\pi$, any curve displacement leaves unchanged\n the energy terms associated to the \n spontaneous curvature. We can thus omit\nthe term associated with $c_s$.\nThe second term expresses the adhesion energy and is only integrated on the \nadhered part of the curve. \nAs we are concerned with an inhomogeneous substratum, the contact potential \ndepends on the variable\n $x$ and is denoted by \n$-w(x)$ (with $w>0$, meaning that \nadhesion is favorable).\nFinally the last two terms ensure the length and surface conservation, \nrespectively. \nThe membrane is a two dimensional {\\it incompressible} fluid. \nThe phospholipid exchanges \nwith the solvent is virtually absent, and %so that\nthe area per molecule on the vesicle remains constant.\nThis \nleads to the local length conservation (in the 2D language). \nThe variable $\\zeta$ is a local Lagrange multiplier which enforces\nthe arc length\n$ds$ to a constant value. \nThe enclosed surface $S$ conservation is a consequence of the membrane \nimpermeability and fluid incompressibility. \nIt is ensured by the global Lagrange multiplier \n$p$.\nThe interpretation of $\\zeta$ and $p$ as a tension and a pressure will be \ndiscussed latter.\n \nThe functional\nderivative of the energy (\\ref{Fad}) provides\nus with the various forces acting on the membrane. \n \n\\vspace{0.2cm} \n\\noindent {\\bf (i) Curvature forces} \\\\\n Under the assumption that the membrane is completely flat on the adhered part, \nwe reduce\n the integration domain of the first energetic term in eq.(\\ref{Fad}) to \n$[0,L^*]$.\n Making use of the relations ${\\bf t}= {\\bf r}'$ and ${\\bf n}=- (1/c) {\\bf r}''$ \n (the prime designates derivative with respect to $s$) \n %that \n which are the membrane tangential and normal unit vectors, we obtain for \n the curvature energy $E_c$\n \\begin{equation}\n E_c={\\kappa \\over 2}\\int_0^{L^*} \\left ( \\drp{^2{\\bf r}}{s^2} \\right )^2 ds \\; \n.\n \\end{equation}\n \n When taking the functional derivative of $E_c$ with respect\n to the position, care must be taken. \n Indeed the arc length element must also undergo a variation.\nA convenient formulation avoiding confusion rests on the introduction\nof a general\n parametrization $a \\in [0,1]$, related to $s$\nby the metric $g=(ds/da)^2$, and is time-independent. \nThe energy expression becomes\\cite{Cantat99a,Cantatthes}\n\\begin{equation}\n E_c={\\kappa \\over 2} \\int_0^1 \\left[ \\left( \\drp{^2 {\\bf r}}{a^2} \\right )^2 \n -{1 \\over g} \\left( \\drp{^2 {\\bf r}}{a^2} \\drp{{\\bf r}}{a}\\right )^2 \\right ] \ng^{-3/2} da \\; . \n\\label{Fdea} \n\\end{equation} \nThe functional derivative, though straightforward, may\nbe too lengthy if one does not take\ncare in regrouping adequately various\nterms as explained in \\cite{Cantat99a}.\nThe result can be written in a simple form:\n\\begin{equation}\n{\\bf f}_c= -{\\delta E_c \\over \\delta {\\bf r}(s)}= \\kappa \\left (\\drp{^2 c}{s^2} \n+ {c^3 \\over 2 }\\right )\n \\, {\\bf n} \\; .\n\\label{fc} \n\\end{equation}\n\nThe curvature force is, as expected, free of any tangential contribution. \n The first term in eq.(\\ref{fc}), involving the second derivative of the\n curvature,\n tends to keep curvature repartition as homogeneous \nas possible. It is also present in 3D under the more complicated form of the \nLaplace-Beltrami operator. \nThe second term proportional to $c^3$ is in contrast \n 2D-specific. It tends to increase\nthe size of any convex shape. Note that in 3D the \ncurvature energy is scale invariant, which implies a \nvanishing curvature force of this type on a sphere. \nThe difference between 2D and 3D can be explained in the following way. \nLet us consider a finite cylinder of length $H$ and radius $R \\ll H$,\nwhich constitutes a good approximation \nfor an infinitely long cylinder. \nIn order to make the cylinder \"closer\" \nto a sphere, which is the corresponding 3D equilibrium shape,\nthe curvature force would tend to increase the radius and \ndecrease the length so as to bring the cylinder shape as close\nas possible to a sphere.\n This gives an intuitive picture of the\n$c^3$ term in 2D. \\\\\n\nIn the discussion above we did omit the boundary contribution\nwhen taking the functional derivative. Since this point is a bit subtle\nwe have postponed it to the end of this section.\n\n\\vspace{0.2cm}\n\\noindent {\\bf (ii) Length and surface constraints} \\\\\nOn the free part of the curve, the force which is associated\nto the first Lagrange multiplier $\\zeta$ is obtained\nupon functional derivation of \n$E_l=\\int \\zeta ds$. The result is\n\\begin{equation}\n{\\bf f}_l=-\\delta E_l/\\delta {\\bf r}(s)= -c \\, \\zeta(s)\\,{\\bf n} +{d \\zeta \\over \nds} \\, {\\bf t} \\; .\n\\label{fl}\n\\end{equation}\nThe normal component is easily identified as a Laplace pressure, whereas \nthe tangential one looks like a Marangoni force (which is encountered\nwhen surface tension is inhomogeneous). \nHowever, $\\zeta$ is not exactly similar to a surface tension as\nfor an interface\nbetween two fluids. \nThe \"tension\" $\\zeta$ is not an intrinsic property of the membrane. It \nadapts itself to the other forces in order to maintain the local length\nfixed. \nIn other words,\nthe problem is implicitly written in a thermodynamical ensemble with fixed \nlength. This differs from \nthe usual problem for fluid or solid surfaces\nwhere the surface tension is fixed instead. \n Thus $\\zeta$ is a variable that must be determined self consistently \nas a Lagrange multiplier, by use of the constraint equation (see \nappendix in Ref.\\cite{Csahok99a}) :\n\\begin{equation}\n0=\\drp{(ds)}{t}=\\left ( \\drp{v_t}{s}+c v_n \\right ) ds \\; .\n\\label{divv}\n\\end{equation}\nThis relation (\\ref{divv}) simply expresses the condition of vanishing velocity \ndivergence on the curved contour\nof the vesicle, \nwhich is precisely the incompressibility condition in the 2D fluid\nconstituting the membrane (written here in one dimension). A more\nintuitive way of viewing expression (\\ref{divv}) is presented on\nFig.2. \n%\\begin{figure}[h]\n% \\centerline{\n% \\psfig{figure=fig3/divv3.eps,width=9cm,angle=-90}}\n% \\caption{A geometrical explanation of the\n% arclength variation with time.}\n% \\label{divergence}\n% \\end{figure}\n%%%>>>>>\n% In equations in Fig.2, changes are necessary:\n%\tA\"B\"= ds(1 + c v_n dt)\n%\tA'B'=ds(1 + c Vn dt) + ds d(Vt)/ds dt\n%\td(ds)/dt = ds (d(Vt)/ds + cVn\n%%%<<<<<\nThe Marangoni term is the only tangential term among all membrane forces \n(see eqs. \\ref{fc},\\ref{fl},\\ref{fs}). It is seen from (\\ref{fl})\nthat the Lagrange multiplier must be uniform at equilibrium.\nFor sake of simplicity and in order to get more insight \n%%%>>>>>towards %%%<<<\ninto analytical understanding, a uniform \nvalue will be assigned to $\\zeta$, even out-of-equilibrium. A discussion\nof this point will be presented in section VI. \nThis assumption implies some consequences on dynamics (and especially\non the tangential velocity) which will be presented in \nsection \\ref{dynamics}.\n\n\\vspace{0.2cm}\nFinally we have to consider the force associated to $E_s=p S$ :\n\\begin{equation}\n{\\bf f}_s=-\\delta E_s/\\delta {\\bf r}(s)= -p\\,{\\bf n} \\; .\n\\label{fs}\n\\end{equation}\nThe Lagrange multiplier $p$ depends only on time; it enforces\na constant \narea. \nTwo physical interpretations can be invoked depending on the \n situation under consideration. \n Either we consider an impermeable membrane, \nand $p$ would be \nthe hydrostatic pressure difference \nbetween outside and inside; or we choose \na model of permeable membrane and $p$ would play\nthe role of an osmotic pressure. Both models \nare equivalent as long as we do not consider \n%%%>>>>> hydrodynamics %%%<<<<<\nhydrodynamic flows.\n\n\\vspace{0.2cm}\n\\noindent {\\bf (iii) Adhesion forces and boundary terms} \\\\\nThe functional derivative induces boundary terms at each contact point. \nThe additional variation $\\delta E_c^b$ and $\\delta E_{w,l}^b$ for the \ncurvature, adhesion and tension \nenergies, \n associated with a small displacement $\\delta{\\bf r}$ of the \ncontact points is given by (see \\cite{Cantat99a,Cantatthes})\n\\begin{eqnarray}\n&&\\delta E_c^b = \\left [\\delta \\dot{{\\bf r}}\n\\cdot\\left ( -{\\kappa c \\over \\sqrt{g}} {\\bf n}\\right ) \\right ]^{L^*}_0 +\\left \n[\\delta {\\bf r}\\cdot\\left (\n\\kappa \\drp{c}{s}{\\bf n} -{\\kappa c^2 \\over 2}{\\bf t}\\right ) \\right ]^{L^*}_0 \\; , \n\\label{courbord} \\\\\n&&\\delta E_{w,l}^b=\\left [\\rule{0cm}{0.4cm}\\delta {\\bf r}\\cdot\\left \n(\\rule{0cm}{0.4cm}\n\\zeta {\\bf t} + (\\zeta - w(x)) \\vv{x}\\right ) \\right ]^{L^*}_0 \\; .\n\\label{adhbord}\n\\end{eqnarray}\nFollowing the definition of these boundary points, they remain\non the substrate.\nThus,\nthe accessible values for $\\delta{\\bf r}$ is then reduced to $\\delta{\\bf r} \n\\propto {\\bf \\hat{x}}$. \nAdditionally, in order to keep the curvature energy finite, we impose a \nvanishing \nvalue for the contact angle $\\phi$ between \nthe membrane and the substrate (see Fig.3). Within our formulation, \nthis constraint \ndoes not follow from the energy minimization and has thus to be added into \nthe physical model. More precisely, at the discontinuity point (say $x_2$)\none has to add to the Helfrich energy a term of the form \n$\\kappa (\\Delta\\psi/\\Delta s)^2$ which informs\nus on how would the vesicle on the adhered part feels, so to\nspeak, the behavior of the vesicle at the junction\npoint on the right side. Across the contact point of a vanishing\nextent, $\\Delta s\\rightarrow 0$, while the angle, if it had\nto have another value than zero, would make a jump leading\nto an abnormally increasing curvature energy. We must then\nimpose a vanishing contact angle.\nThese various conditions (motion along the wall and a\n vanishing contact angle) lead to\n${\\bf n}=-{\\bf \\hat{y}}$, ${\\bf t}=-{\\bf \\hat{x}}$ and \n$\\delta \\dot{{\\bf r}} \\propto {\\bf \\hat{x}}$. It follows then that\nthe term proportional to $\\delta \\dot{{\\bf r}}$ in eq.(\\ref{courbord}) vanishes\nautomatically. \nThe second term becomes $\\kappa/2 \\left (c_2^2 \\delta x_{2}- c_1^2 \\delta \nx_{1}\\right )$\nwith $c_1$ and $c_2$ the curvatures at the left and right contact points. \nThese terms are counterbalanced by adhesion and tension terms \n(eq.(\\ref{adhbord})) leading to the relation \n\\begin{equation}\n\\frac{\\delta E}{\\delta x_i}=\\mp \\left ( \\kappa \\frac{c_i^2}{2}-w(x_i) \\right ) \n{\\bf \\hat{x}} \\; ,\n\\label{Fbord}\n\\end{equation}\nwhere the $-$ and $+$ signs refer to the rear and fore \ncontact points represented\nby the subscript $i=1,2$.\nAt equilibrium, we recover here the relation $c=\\sqrt{2 w/\\kappa}$ \n\\cite{Seifert91}.\n\nThe energy variation given by eq.(\\ref{Fbord}) can not really be identified as a \nphysical force. \nIt corresponds indeed to a {\\it geometrical} point displacement. The \"force\" \norientation is here\nparallel to the substrate, whereas the real force acting on the contact point, \nconsidered as a material \npoint, is expected to be normal to the substrate.\nAs we have seen above the curvature forces are indeed normal\nwhen applied to a an adjacent piece of the membrane (see eq. \\ref{fc}).\nThe present \"force\" has the meaning of how much energy would be involved\nin displacing the contact point from one position to another.\nThat geometrical point is by its very nature sitting on the substrate,\nso that the \"force\" associated with its displacement is naturally tangential.\n\n% \\begin{figure}\n% \\centerline{\n% \\psfig{figure=fig3/adhloc.eps,width=6cm,angle=-90}}\n% \\caption{\\protect Force equilibrium at the fore contact point in the small \n%rigidity limit. }\n% \\label{adhloc}\n% \\end{figure}\n\nWe find it worthwhile to make a short digression. Suppose that \nthe angle is not fixed to zero as we did above. More precisely\nsuppose that the rigidity is so small or the adhesion\nis so large (see below what does\nthis mean) then the vesicle will be so tense that it would look\nlike a droplet outside some length scale $\\ell$ to be determined\nbelow (of course within that scale, which is sufficiently\nclose to the substrate, the matching must be tangential). If we\ndo not assume a value\nfor the angle (that is no relation between ${\\bf n}$ or ${\\bf t}$ with\n${\\bf \\hat x}$ and ${\\bf \\hat y}$), and set $\\delta {\\bf r} \\propto \n{\\bf \\hat x}$ we find from (\\ref{Fbord}) that $c=0$ at the contact\n(which means a straight line at the contact) and that the angle\nbetween that line and the substrate obeys\n% and \n\\begin{eqnarray}\nw_i&=&\\left (1-cos(\\phi_i)\\right )\\zeta \\nonumber\\\\\n\\phi_i &\\sim &\\sqrt{{2 w_i \\over \\zeta}} \\mbox{ ( for small angles )}\n\\end{eqnarray}\nwhich is nothing but the Young condition. We have\nneglected $\\kappa \\partial c/\\partial s$ in comparison\nto $w$. The justification is as follows. $\\kappa \\partial c/\\partial s\\sim\n\\kappa c_0/\\ell$, where $c_0$ is the true contact curvature\ngiven by $\\sqrt {2 w/ \\kappa}$. The approximation\nis legitimate provided that the length scale $\\ell \\gg \\sqrt{\\kappa/w}$.\nThe length $\\sqrt{\\kappa/w}$ is the radius of curvature at contact.\nIf the scale of interest is outside that internal region, then\nthe droplet limit is justified.\nIt must be emphasized however that \nthe effective contact angle is not an intrinsic property of the adhered\nmembrane, as for a droplet, but it is linked to other parameters\n(rigidity, the vesicle scale--on which depends $\\zeta$--, etc... ).\nIn particular, \nthe tension $\\zeta$ is fixed by the reduced volume, which is a global property \nof the \nvesicle : different vesicles of the same phospholipid composition,\nbut with different sizes, may have \ndifferent \ncontact angle on the same substrate. \n\n\n\\subsection{Dynamical equation}\\label{dynamics}\n\nAn important point which must be emphazised when dealing with dynamics\nis the identification of the dissipation sources.\n%The dynamical behavior of the vesicle is entirely governed by dissipative \n%processes. \n%This is easily checked by estimating \n%the Reynolds number\n %$Re=R \\rho V / \\eta \\sim 10^{-3}$, with $\\rho \\sim 1 g/cm^3$ the fluid density,\n %$\\eta\\sim 10^{-2}Pois$ its dynamical viscosity, $R \\sim 10 \\mu m$ the vesicle \n%size and $V \\sim 10 \\mu m/s$ a typical velocity. \n%The membrane and the surrounding fluid inertia are completely negligible and \n%the injected energy\n%is instantaneously dissipated in the various degrees of freedom. \nThese are the following: (i) the dissipation in the \nmembrane\nvia molecule rotations (very much like liquid crystals where\ndissipation is characterized by the Leslie coefficient), (ii) hydrodynamics\nflows inside and outside the vesicle, (iii) friction between the \nmonolayers, and (iv) bond breaking and restoring with the substrate.\nIt is well known that dissipation associated with rotation (internal\ndissipation) is negligible in practice\\cite{Brochard75}, and\nfor free vesicles (no substrate) hydrodynamics seems to be the\nmost important dissipation. Hydrodynamics induces nonlocal \ninteractions\\cite{Cantat99}\nand this will be dealt with extensively in a forthcoming paper.\nOur wish in this paper is to present a pedestrian model, namely a local one,\nwhich allows for a complete analytical solution that will\nhelp to identify some key ingredients in the migration process.\nA specific dissipation with the substrate will be introduced later.\nFor the moment we confine our description to the free \nvesicle case.\nThe local model to be presented here is similar to the so-called\nRousse model \\cite{deGennes} in the community of polymers. Indeed, for\na one dimensional contour in a three dimensional space dynamics\nbecomes local even in the presence of hydrodynamics\\cite{remarkrousse}.\n\nThe best way to introduce the dynamical law is to consider a dissipation \nfunction, proportional \nat each point to the square of the velocity : \n\\begin{equation}\nF_d=\\frac{\\eta}{2}\\int |{\\bf v}|^2 ds \\; .\n\\label{FD}\n\\end{equation}\nThe coefficient $\\eta$ is here an effective viscosity and has the dimension of \na viscosity per unit length.\nIts numerical value is estimated by $\\eta = \\eta_{wat}/ R \\sim 10^2 kg m^{-2} \ns^{-1}$, \nwith $\\eta_{wat}$ the water viscosity and $R$ a typical vesicle size.\n\nNeglecting inertial terms, the Euler-Lagrange equations become then \n\\begin{equation}\n-{\\delta E \\over \\delta {\\bf r}} ={\\delta F_d \\over \\delta {\\bf v}} \n\\hspace{0.5cm} \n\\Rightarrow \\hspace{0.5cm} \\eta {\\bf v}= {\\bf f} \\; . \n\\label{dyn}\n \\end{equation}\nAs expected, we find a {\\it local} proportionality between the membrane\nvelocity ${\\bf v}$ and the membrane force ${\\bf f}$, which is a nonlinear\nfunction of position.\nIn the present picture where the effective tension $\\zeta$ is space-independent\nno tangential force appears so that physics will only fix the normal\nvelocity, while the tangential velocity has no physical\nmeaning as described below.\n\n\\vspace{0.2cm}\n\\noindent {\\bf (i) Normal velocity}\\\\\nThe normal membrane force is \n (see eq. (\\ref{fc}, \\ref{fl}, \\ref{fs})) :\n \\begin{equation}\nf_n=\\kappa \\left(\\drp{^2c}{s^2}+\\frac{c^3}{2} \\right)-c \\zeta-p \\; .\n\\label{force}\n \\end{equation}\nFrom the dynamical law (\\ref{dyn}) and the membrane forces expression \n(\\ref{force}) we obtain the \nnormal velocity as a function of $\\psi(s)$ :\n\\begin{equation}\nv_n(s)=f_n=\\frac{\\kappa}{\\eta} \\left[\\drp{^3 \\psi}{s^3}+\\frac{1}{2} \\left \n(\\drp{\\psi}{s} \\right)^3\n-\\frac{\\zeta}{\\kappa}\\, \\drp{\\psi}{s}\n -\\frac{p}{\\kappa} \\right] \\; .\n \\label{vnloc}\n\\end{equation} \nIt is convenient to write the dynamical equation in terms of the angle\n$\\psi$ and not the curvature $c$. The reason is that the \nboundary conditions are written naturally as a function of $\\psi$ (tangential\nmatching, $\\psi=\\pm \\pi$, and contact curvature $\\partial \\psi/\\partial s=\n\\sqrt{2w/\\kappa}$). \n\n\n\\vspace{0.2 cm}\n\\noindent {\\bf (ii) Tangential velocity}\\\\\nThere is only one tangential contribution to the membrane forces, $\\partial \n\\zeta /\\partial s$, which \nis zero with the assumption of a uniform tension (see eq. \\ref{fl}). \n%This implies vanishing tangential forces, and \n%consequently \n% vanishing tangential velocities for the membrane. \n% \\begin{figure}[h]\n% \\centerline{\n% \\psfig{figure=fig3/jauge.eps,width=3.5cm,angle=-90}}\n% \\caption{Translation of a circle obtained with a purely normal motion. The rigth part is dilated, \n% whereas the left part is contracted. } %%\n% \\label{jauge}\n% \\end{figure}\nThis implies that only the total length is conserved, and not the local one. \nA dilatation of a part of the membrane is then permitted, as long as the \nremaining part of the vesicle \nis contracted in order to keep the total length unchanged (see Fig.4). \nWithin this approximation, there is no energy variation associated to \ntangential motion, and\ntherefore no forces. \nIn other words we consider the vesicle contour as a mathematical curve, loosing \nthe concept\nof density : only the shape matters, \nindependently of the points distribution on the curve.\n\n%%This situation is equivalent to the case where we would assume \n%%that only the normal velocity contributes to dissipation.\n%%In that case the tangential force would have a contribution\n%%force $\\partial\\zeta /\\partial s$, but since in the functional\n%%derivative of dissipation provides only a normal part, this implies\n%%automatically that $\\zeta=$constant.\n%%\nWe could equivalently assume that only the normal velocity contributes to dissipation.\nIn that case the dissipation function (\\ref{FD}) would take\nthe form $F_{dn}=\\eta/2 \\, \\int |v_n|^2 ds$\nand the equation of motion (\\ref{dyn}) becomes $ {\\bf f} = \\eta v_n {\\bf n}$. The tangential force\n$\\partial\\zeta /\\partial s$ must then vanish and we get automatically that $\\zeta=$constant.\n%%\n Thus our assumption\nof a global Lagrange multiplier can also be viewed as the result of a %%\ndissipation due uniquely to normal displacements. \n%This also implies that \n%there is no way that allows one to fulfill a local constraint (eq.(\\ref{divv})),\n%a consequence of the absence of influence from the tangential velocity.\n%Thus the only constraint is global (the total length) and there is\n%no information whatsoever on the tangential velocity of physical\n%nature.\n\n%% Il me semble que ce passage va mieux un peu plus loin. Sinon on \n%% disait plusieurs fois la meme chose \n%%It must be emphasized that even if the tangential velocity had\n%%a physical meaning,\n%%the knowledge of the normal\n%%velocity is sufficient to describe the vesicle dynamics.\n%%In that case the tangential velocity would\n%%be coupled to the normal one, and one can in principle\n%%determine the normal one by eliminating (at least formally)\n%%the tangential part. It is by this way that the physical tangential\n%%velocity would affect the physics. \n\n\nAs we have already mentioned, tangential displacements do not \ninduce a geometrical change. If the tangential\nvelocity has no physical meaning (as is the case with\na constant $\\zeta$) its choice\nshould not affect the physics. We are thus at liberty to choose one\nwhich is convenient (very much like a gauge-field invariant formulation in\nelectrodynamics).\nThe choice of a gauge is\ninterpreted as a reparametrization of the curve. \nAs seen below the tangential\nvelocity is fixed by the normal velocity once the gauge\nis specified, but there is naturally no feed back\nof that tangential velocity on the normal one (the physical one).\n%%%\nThis is the crucial difference between this non physical tangential velocity and \na physical one that may arise in the general case (as discussed\nin a forthcoming paper).\nIt must be noted however, that whether a tangential velocity \nis of physical nature or not,\nthe knowledge of the normal\nvelocity is sufficient to describe vesicle dynamics.\nIt is thus only via its influence on the normal\nvelocity that a physical tangential velocity would affect the physics (see\nbelow).\nStill in that case we can introduce a second tangential\nvelocity of geometrical nature that corresponds\nto the displacement of the representative points\nof the curve and not to the {\\it material ones} which are\naffected by the physical tangential velocity.\n%%%\n\nIn the present model \nthe most convenient parametrization requires a homogeneous points distribution\nalong the free \npart of the curve, which is expressed as\n$d/dt \\, (s(a)/L^*)=0$. This provides \nthe expression for the \"non physical \ntangential velocity\" (see appendix in Ref.\\cite{Csahok99a}) : \n\\begin{equation}\nv_t(s)=v_t(0)-\\int_0^{s} c \\, v_n \\, ds + {s \\over L^*}\n\\left( \\int_0^{L^*} c \\, v_n \\, ds + v_t(L^*)-v_t(0) \\right) \\; .\n\\label{vt}\n\\end{equation}\nIf the free length $L^*$ remains constant during the motion, as happens for \na stationary regime, eq.(\\ref{vt}) fixing the gauge imposes nothing \nbut \na constant distance between two consecutive points on the vesicle. \nThe local length conservation (\\ref{divv}), which\nis physical, seems then to be implied by a gauge!. In reality, once\nwe have adopted a contant tension --implying a vanishing tangential physical\nvelocity--, any point distribution is of purely geometrical nature,\nand we could impose another gauge than the above one, without affecting\nthe physics; the above tangential velocity does not act\non the normal velocity. Had we considereed $\\zeta$ to be non contant,\nwe would then have obtained a physical tangential velocity, which\nwould act on the normal one; use of (\\ref{divv}) fixes $\\zeta (s)$ \nwhich in turn acts on the value of $v_n$, and then on physics.\n%But that condition is of {\\it physical nature}!.\n%This example illustrates the difference between a physical \n%tangential velocity and the gauge velocity (\\ref{vt}).\n%In reality, the local length conservation results from a \n%reparametrization and \n% {\\it has nothing to do}\n%with the physical condition of incompressibility.\n%The physical constraint has to be included in the calculation upon introduction\n%of a local Lagrange parameter. In order that the incompressibility condition \n%(\\ref{divv})\n %be satisfied all over the curve, $\\zeta$ must in general depend on $s$. \n%The obtained local tension then affects dynamics through\n%its contribution in the {\\it normal velocity} (see\n%eq. (\\ref{vnloc})).\nIn the simplistic model we adopt,\nthe tangential velocity is determined\n{\\it a posteriori}, independently of the normal velocity. That is why\nit is only a non physical reparametrization, a \"gauge\". \nA remark is in order :\nin this situation the question of a rolling or sliding motion does\nnot make sense, \nsince both motions differ only by a tangential velocity. \n\nThe membrane velocity is given as a function of $\\psi(a)$, $a$ being the \nauxiliary parametrization of the free part of the vesicle,\nrunning from one contact point to the other. \nIn order to obtain a closed system we need a relation between the evolution of \n$\\psi(a)$ and the velocities. \nThe temporal derivative of $\\psi$, for a given $a$, is presented in the appendix \nof Ref.\\cite{Csahok99a}\n\\begin{equation}\n\\left . {\\partial \\psi \\over\\partial t } \\right ) _a= c\\, v_t -\\drp{v_n}{s}\\; .\n\\label{dpsi}\n\\end{equation}\n\nThe last step is the determination of the boundary conditions at the\ncontact with the substrate.\n\n\\vspace{0.2 cm}\n {\\bf (iii) Contact points velocity} \\\\\n The motion of the contact point is %%%>>>>> govern %%%<<<<<\ngoverned by a binding/unbinding \nmechanism, implying\na dissipation law that differs from the bulk dissipation. \nThe most natural way for introducing a dissipation law is\nthe following (with $\\Gamma$ a phenomenological dissipation\ncoefficient)\n\\begin{equation}\n\\Gamma\\frac{dx_i}{dt}=-\\frac{\\delta E}{\\delta x_i}\\; .\n\\end{equation}\nUsing the energy variation (\\ref{Fbord}) we get the following dynamical law, \nwith \n$w_i=w(x_i)$ and $v_i=dx_i/dt$\n\\begin{equation}\nc_1=\\sqrt{\\frac{2 w_1 + \\Gamma v_1}{\\kappa}} \\; , \\; c_2=\\sqrt{\\frac{2 w_2 - \n\\Gamma v_2}{\\kappa}}\\; ,\n\\label{cheq}\n \\end{equation}\nwhich constitute the dynamical boundary conditions. \nThese out of equilibrium values for the curvature are quite intuitive : \nthe unbinding delay at the rear point induces a larger curvature than at \nequilibrium, \nwhereas the binding delay at the fore point induces a smaller curvature. \n \n \n \n\\section{Transient behavior}\n\n \n\nThe formalism presented in the first part lends itself very well \nto analytical computation and stationary shape determination, as will appear \nin the following paragraphs. \nNevertheless, having access to the transient process is highly desirable. \nIn particularly, it checks the dynamical stability of an eventual stationary \nbehavior, \nobtained after a relaxation. \nThe successive vesicle profiles are determined by a direct numerical \n implementation of the dynamical equations (\\ref{vnloc}), (\\ref{vt}) and \n(\\ref{dpsi}). Unfortunately, numerical instabilities are difficult to avoid\n around each contact point (due to a contact\nadhesion potential). A smoother model, without discontinuities, \nis more convenient for such\n an approach. For this reason, in this paragraph devoted to transient processes, \nthe adhesion potential \n will be supposed to be of small, but non vanishing,\nrange.\nWe rapidly summarize below the small technical changes arising from this model \nmodification.\nThe chosen potential profile is \n\\begin{equation}\nw({\\bf r}) = w_0 (1 + u_0 x)(\\frac{y_0^4}{y^4}-\\frac{2y_0^2}{y^2})\\; ,\n\\label{ad-pot}\n\\end{equation}\nwith the new length $y_0$ fixing the characteristic distance between \nthe substrate and the membrane, and $\\hat w(x)=-w_0 (1 + u_0 x)$ the minimum of \nthe potential interaction, occurring for\n$y=y_0$. It plays the role of the previous $w(x)$. It depends linearly on $x$ \nwith an adhesion gradient $u_0$.\nThe distance $y_0$ is chosen of the order of $10 n m$, which is small enough in \nregard of the vesicle size\nto introduce only small variations between both models. \nIn this case the parametrization is performed on the whole closed curve, and the \nboundary terms (eqs. \\ref{cheq}) are no more relevant. \nThe adhesion forces ${\\bf f}_w$ are obtained by functional derivation of $\\int \nw({\\bf r}) ds$ \nleading to \n\\begin{equation}\n{\\bf f}_w = -(c w + \\nabla w \\cdot {\\bf n} ) {\\bf n} \\; .\n\\label{dynlaw2}\n\\end{equation}\n\n\nAdditionally the gauge condition fixing the tangential velocity eq. (\\ref{vt}) \nis simplified :\nthe velocity $v_t(0)$ is supposed to be zero, without loss of generality, so \nthe first term \n disappears ; \nthe last term is proportional to the length variation of the total parametrized \ncurve, which is \nzero because we consider the complete profile and no more the free part of the \ncurve. \nThus we obtain for the gauge, replacing eq. (\\ref{vt}) :\n\\begin{equation}\nv_t = - \\int_0^s ds v_n c \\; .\n\\label{vt2}\n\\end{equation}\nUsing equations (\\ref{dynlaw2}) and (\\ref{vt2}) we finally get the dynamical \nequation for ${\\bf r}$\n\\begin{equation}\n\\frac{\\partial {\\bf r}(a,t)}{\\partial t} =\n\\left[ \\kappa \\left(\n\\frac{d^2 c}{ds^2} + \\frac{1}{2} c^3 \\right) -c w - ({\\bf \\nabla} w \\cdot {\\bf \nn})\n- p - \\zeta c \\right] {\\bf n}+ v_t {\\bf t} \\; .\n\\label{drdt2}\n\\end{equation}\nThe Lagrangian multipliers are determined from the following constraint\nequations :\n\\begin{eqnarray}\n&&{dL \\over dt} = \\int c v_n ds =0 \\; , \\\\\n&&{dS \\over dt} = \\int v_n ds =0 \\; .\n\\label{constr}\n\\end{eqnarray}\nThe normal component of the velocity in eq. (\\ref{drdt2}) will be denoted by \nconvention as $v_n=v_n^0 - p - c \\zeta$.\nWith this notation the equation (\\ref{constr}) appears as \na very simple linear equation system in $\\zeta$ and $p$. Its solution provides \nthe pressure and tension values :\n\\begin{eqnarray}\n\\zeta = \\frac{\\langle c v_n^0 \\rangle -\\langle c \\rangle \\langle v_n^0 \\rangle}\n{\\langle c^2 \\rangle - \\langle c \\rangle^2 } \\; , \\\\\np= -\\zeta \\langle c \\rangle + \\langle v_n^0 \\rangle \\; .\n\\label{zeta-p}\n\\end{eqnarray}\nwith the average defined by\n\\begin{equation}\n\\langle \\cdots \\rangle \\equiv \\frac{1}{L} \\int_0^L ds \\cdots.\n\\label{av}\n\\end{equation}\n\nWe are now in a position to deal with \nthe numerical anlysis. The dynamics is overdamped and \nfor this reason {\\it local in time}. Starting from % times???\nan arbitrary profile, forward time integration\nprovides us with the vesicle\nevolution. We have checked that (i) for a free vesicle (no substrate)\nthe shape (with no external force) tends\ntowards that obtained by direct energy minimization, (ii) we have\nalso checked that for a homogeneous substrate an arbitrary initial\nshape evolves after some time to the shapes obtained in \\cite{Seifert91}\nby direct %%%>>>>>\nenergy %%%%<<<<<\nminimization. \n\nLet us now turn to the non-equilibrium situation ensured by an adhesion\ngradient. Starting from an initial shape, the vesicle \nacquires a non-symmetric shape and moves in the gradient\ndirection. After transients have decayed the vesicle acquires\na permanent regime with a constant velocity. Figure 5\nshows the shape evolution.\n\n% \\begin{figure}\n% \\centerline{\n% \\psfig{figure=fig3/fig4a.epsi,width=8cm,angle=-90}}\n% \\caption{\\protect Successive vesicle profiles. The first one %%%>>>>> \n%with open circles %%%<<<<<\n% is an arbitrarily \n% chosen initial shape. It relaxes to a permanent shape %%%>>>>>\n%marked by filled circles %%%<<<<<\n%on a inhomogeneous \n%substrate. }\n% \\label{relaxa}\n % \\end{figure}\n\n\n\\section{Stationary motion : direct numerical solution} \n\nThe formulation of our problem in terms of a direct stationary\nproblem is very convenient both for a systematic study of the \nvelocity evolution as a function of various parameters.\nIt will also allow us to present a simple analytical solution.\nIt is convenient here to come back to the \ncontact potential model. \nFor a vesicle which has attained a stationary shape and velocity $V$\nthe equations become steady with $V$ as an unknown\nparameter.\n\n\n For a stationary motion along the x-axis, normal and tangential velocities can \nbe written as\nfunctions of the angle $\\psi$ and of the translational velocity $V$ :\n\\begin{eqnarray}\n&&v_n=V \\vv{x}\\cdot\\vv{n}= V \\sin \\psi \\; , \\label{vndeV} \\\\\n&&v_t=V \\vv{x}\\cdot\\vv{t}= V \\cos \\psi \\; .\n\\end{eqnarray}\nThe shape and velocity are entirely determined from the relation between normal \nvelocities \nand forces. \nThe equation of motion is obtained from eqs. (\\ref{vnloc},\\ref{vndeV}):\n\\begin{equation}\nV \\sin \\psi =\\frac{\\kappa}{\\eta} \\left[\\drp{^3 \\psi}{s^3}+\\frac{1}{2} \\left \n(\\drp{\\psi}{s} \\right)^3\n-\\frac{\\zeta}{\\kappa}\\,\\drp{\\psi}{s}\n -\\frac{p}{\\kappa} \\right] \\; . \n\\label{eqmv}\n \\end{equation}\n \nLet us present briefly a counting argument showing that the problem %% argument that ???\nis well defined.\n Equation (\\ref{eqmv}) is a %%%>>>>> non linear %%%<<<<<\nnonlinear third order differential equation for \n$\\psi$, \n with 3 parameters to be determined : $\\zeta$, $p$ and $V$. So we need 6 \n\"informations\".\n \nWe have the following equations at our disposal :\\\\\n$\\bullet$ 2 geometrical constraints (eqs. (\\ref{cos}) and (\\ref{sin})) \n:\n\\begin{equation}\n\\int_0^{L^*} \\cos \\psi(s)\\, ds = L_{adh}\\, , \\, \n \\int_0^{L*} - \\sin \\psi(s) \\, ds = 0 \n \\label{sincosbis}\n\\end{equation} \n$\\bullet$ 4 boundary equations corresponding\nto the contact angles and their first derivatives\n(the dynamical contact curvatures $c_1$ and $c_2$ given by equation \n(\\ref{cheq})) \n\\begin{equation}\n\\psi_1\\equiv \\psi(s=0)=-\\pi \\, , \\hspace{0.3cm} \\psi_2\\equiv \\psi(s=L^*)=\\pi \\, \n, \\hspace{0.3cm}\n\\left . \\drp{\\psi}{s} \\right)_{s=0}= c_1 \\, , \\hspace{0.3cm}\n \\left .\\drp{\\psi}{s}\\right)_{s=L^*}=c_2\n\\label{limites}\n\\end{equation} \n$\\bullet$ 1 equation ensuring that \nthe enclosed surface\nis equal to the prescribed area. \\\\\n$\\bullet$ 1 equation ensuring that the total length of the curve \nis precisely the prescribed one, $L$, which is related to the two other\nlengths by\n\\begin{equation}\nL=L^*+L_{adh} \\; . \n\\label{consL}\n\\end{equation}\n\n\\vspace{0.3cm}\n\nThere are thus 8 conditions, for only 6 informations needed.\nThe system seems then to be overdetermined. This is not the case. Indeed\nit must be noted that the problem involves additional \nunknowns which are $L^*$ and $L_{adh}$. So in reality \nwe have 8 unknown parameters as well. The problem is thus well defined.\n \nOnce the shape is determined we must in principle evaluate the area\nand change the parameter $p$ until the area coincides with \nthe prescribed one. But since the area is a conjugate variable\nto $p$ we can fix $p$ --which is more convenient--\nand this will fix some area that is treated as free (not imposed\nin advance). \nAdditionally we are at liberty to prescribe $L$ \n(that fixes some length scale). \n$L_{adh}$ can then be determined if $L^*$ is known; \n$L_{adh}$ can thus be removed from the problem \nupon using eq.(\\ref{consL}). The first constraint (\\ref{sincosbis}) \nbecomes then \n\\begin{equation}\n\\int_0^{L^*} \\cos \\psi(s)\\, ds = L-L^* \\; .\n\\label{coster}\n\\end{equation}\nIn other words prescribing the total length to $L$ and the pressure\nto $p$ lowers the number of unknowns by two. This is so because\nwe do not want to have a specific area, and that $L^*$ and $L_{adh}$\nare not independent if we treat the total length as known.\nThat is to say we have \nfinally 6 fixed boundary conditions or constraints (\\ref{sincosbis}-\\ref{limites})\nand six parameters which are \n$L^*$, $V$ and $\\zeta$, plus three constants of integration\ndue to the third order differential equation (\\ref{eqmv}). \n\n\n\nA convenient way to solve a differential equation of order $n$ is to transform it \ninto a set of $n$ first order coupled differential equations. For that purpose we set\n $f_1=\\psi$, $\\dot{f_1}=f_2$ and $\\dot{f_2}=f_3$ %%%>>>>>\nwhere the dot stands for $\\partial / \\partial s$.\n%%%<<<<\nEquation (\\ref{eqmv}) then provides us with the expression for $\\dot{f_3}$ \n$\\equiv \\partial ^3 \\psi /\\partial s^3$\nas a function of $f_1$, $f_2$ :\n\\begin{equation}\n\\dot{f_3} =\\eta/\\kappa \\, P_2 \\sin f_1 -f_2^3/2 -p/ \\kappa + f_2 P_3 / \\kappa \n\\equiv F \\; .\n \\end{equation} \n In order to make visible the quantities which are treated as unknown parameters\n we shall use the symbols $P_i$ (with $i=1,2...$). As stated above\n there are \nthree parameters $L^*=P_1$, $V=P_2$ \nand $\\zeta=P_3$.\nSolution of a set of three equations involves three integration factors.\nThis means that we have 6 unknowns, as argued in the last paragraph.\nFour physical conditions\nat the two end points (see eq. (\\ref{limites})) are known. Two constraints\nare imposed \n(eq. (\\ref{sincosbis})), and this makes the problem well posed.\nNote that conditions (\\ref{sincosbis}) have\nan integral form. We find it convenient\nto rewrite them in a differential form.\nIt is easy to realize that by setting\n\\begin{equation}\nf_4(s)\\equiv \\int_0^s \\sin \\psi(s')ds'\\; ,\\; f_5(s)\\equiv \\int_0^s \\cos \n\\psi(s')ds'\\; ,\n\\label{f45}\n\\end{equation} \nwe can write\n\\begin{equation}\n\\dot{f_4}= \\sin \\psi \\; =\\sin f_1 \\; ,\\; \\dot{f_5}= \\cos \\psi \\; =\\cos f_1 \n\\; .\n\\label{df45}\n\\end{equation}\nThese two functions obviously obey\n$f_4(0)=0$, $f_5(0)=0$ , whereas at the second boundary we must impose\n$f_4(L^*)=0$ and $f_5(L^*)=L-L^*$ in order to fulfill the two constraints \n(\\ref{sincosbis}, \\ref{coster}). This trick is performed\nat the expense of two additional functions $f_4$ and $f_5$ (whose\ndeterminations involve two integration constants). We have thus augmented our system by 2\ndifferential equations of first order. The two additional integration constants are\nprecisely fixed by the demand $f_4(0)=0$, $f_5(0)=0$, whereas the conditions\n$f_4(L^*)=0$ and $f_5(L^*)=L-L^*$ are\nsubstituted to (\\ref{sincosbis}). Finally\nthe shooting NAG code used here requires to invoke the boundary conditions\nfor each function $f_i$, with $i\\le 5$. The boundary conditions for each function is invoked\nabove, except for $f_3$ which represents the second derivative of $\\psi$. This\nquantity is not known at the boundaries and there is no constraint to be\nimposed on it. Let $P_4$ and $P_5$ denote the values of $f_3$ at the two end\npoints.\nWe can thus invoke the boundary conditions of $f_3$ and the boundary values\nare quantities which are to be determined. That is to say we introduce two\nconditions with two additional unknown parameters. We have then in total\nten unknowns and ten conditions. Cast into this form our formulation can\nstraightforwardly be implemented into a NAG code (code D02HBF).\n\nIn summary the problem to be solved can be written in a standard boundary value\nproblem with unknown parameters :\n\\begin{equation}\n\\begin{array}{lll} \n\\dot{f_1} = f_2 \\; ,\\; &f_1(0)= -\\pi \\; ,\\;& f_1(P_1)= \\pi \\\\\n\\dot{f_2} = f_3 \\; ,\\; &f_2(0)=c_1 \\; ,\\; &f_2(P_1)=c_2 \\\\\n\\dot{f_3} = F \\; ,\\; &f_3(0)= P_4 \\; ,\\; &f_3(P_1)= P_5\\\\\n\\dot{f_4} = \\sin f_1 \\; ,\\; &f_4(0)= 0 \\; ,\\;& f_4(P_1)= 0\\\\\n\\dot{f_5} = \\cos f_1\\; ,\\;& f_5(0)= 0 \\; ,\\; &f_5(P_1)= L-P_1\n\\end{array}\n\\label{systeme}\n\\end{equation}\n\nOnce the problem is solved the vesicle shape is obtained \nby making use of\nequations (\\ref{dxds}).\n\n% \\begin{figure}\n% \\centerline{\n% \\psfig{figure=fig3/adhvit.epsi,width=8cm,angle=-90}}\n%% \\caption{\\protect Out of equilibrium adhering vesicle profiles.\n% $V$ is measured in units of $100\\mu m$\n%and $W$ in units of $10^{-4} mJ/m^2$.}\n% \\label{adhvit}\n% \\end{figure}\n \n \nIf $w_1=w_2$, the vesicle is at equilibrium on a homogeneous substrate and \none obviously expects a vanishing velocity. This comes out automatically\nfrom the above formulation. If we were interested from the beginning \nin an equilibrium problem, we would then not have introduced $V$ as\nan unknown parameter. In that case because the profile is symmetric the second\ncondition (\\ref{sincosbis}) is automatically satisfied, and we would then be left with with\nnine conditions for nine unknowns.\n\nWhen $w_1 \\neq w_2$, there is no equilibrium solution for the \nvesicle, which has to move towards the stronger adhesion region.\nIf we impose a vanishing velocity there is no way to fulfill\nthe second condition (\\ref{sincosbis}) (a typical profile would\nbe the one shown on Fig.7) where starting from one end we arrive\nat the other end at a different height. Arriving at the same height\ncan be achieved only for a specific velocity (or at most\na discrete set of solutions), the one we are seeking. Thus the second condition\nof eq.(\\ref{sincosbis}) (which is parametrized by the set of $P_i$) can\nbe viewed as 'quantization' condition. This is a nonlinear eigenvalue\nproblem of Barenblat-Zeldovitch type.\n%\\begin{figure}[htb]\n% \\centerline{\n% \\psfig{figure=fig3/pieds.eps,width=4cm,angle=-90}}\n% \\caption{{\\it }}\n% \\label{pieds}\n%\\end{figure}\n\nThe numerical solution reveals an out-of-equilibrium shape which \nis significantly different from \nthe equilibrium one, as shown \non Fig.6. We note that the curvature in front of the vesicle\nis higher\nthan the one behind. The reason is that the adhesion energy\nis higher in the front part, so that the curvature/adhesion balance allows\na higher curvature (the vesicle looses curvature energy at the expense\nof a stronger adhesion).\n\n\n\\section{Stationary motion : an analytical solution}\n\n\n%\\begin{figure}[h]\n% \\centerline{\n% \\psfig{figure=fig3/vthvnu.epsi,width=7cm,angle=-90}}\n% \\caption{Evolution of the vesicle velocity as a function of the adhesion \n%difference.}\n% \\label{vthvnu}\n% \\end{figure}\n \nThe advantage offered by the simplistic\npicture of our model is the possibility to provide analytical results\nand thus to shed light on the physical processes that are involved\nin the problem of vesicle propulsion.\nIt turns out that the equation of motion (\\ref{eqmv}), if multiplied by \n$\\partial^2\\psi/\\partial s^2$, \npossesses practically a first integral \n\\begin{equation}\nV\\int_0^{L^*} \\drp{^2\\psi}{s^2} \\sin(\\psi) ds = {\\kappa \\over \n\\eta} \n\\left( [(\\partial c/ \\partial s )^2/2]^1_2 +{1 \\over 2} \n[\\,c^4/4\\,]^1_2-\\frac{\\zeta}{\\kappa}\n[\\,c^2/2\\,]^1_2-\\frac{p}{\\kappa}[\\,c\\,]^1_2 \\right) \\; .\n\\end{equation}\nEach r.h.s. term has an explicit form as a function of the contact \ncurvatures (which are \nknown), and of their\nfirst derivatives, which have only negligible contribution for swelled vesicle.\nThe l.h.s. term can be evaluated for a vesicle shape close to a circle. \nThe calculation is sent into appendix and leads to \n\\begin{equation} \n\\int \\drp{^2\\psi}{s^2} sin(\\psi) ds =4 \\pi^2 \\frac{L_{adh}}{L^2} \\; .\n\\label{denom}\n\\end{equation}\nUsing the dynamical values for the contact curvature given by eq.(\\ref{cheq}),\nwe obtain\nan explicit expression for the velocity \n\\begin{equation} \n%%%>>>>>V = \\frac{L^2 }{(2\\pi)^2\\eta L_{adh}}\nV = \\frac{L^2 \\kappa }{(2\\pi)^2\\eta L_{adh}}\n%%%<<<<<\n\\left({1 \\over 2} [\\,c^4/4\\,]^1_2-\\frac{\\zeta}{\\kappa}\n[\\,c^2/2\\,]^1_2-\\frac{p}{\\kappa}[\\,c\\,]^1_2 \\right)\\; .\n\\label{velo}\n\\end{equation}\nIn the simple case where $\\Gamma =0 $ (no dissipation\nassociated with the substrate), expression (\\ref{velo}) becomes \nexplicit \nand provides a good agreement with numerical solution (see Fig.8).\nThe analytical expression for the velocity involves only known parameters, \nexcept %%%>>>>>\n$L_{adh}$ and\n%%%<<<<< ??\n $\\zeta$. For the comparison between numerical and analytical results, we \ntook their\n numerical values. \n\n\n\\vspace{0.2cm}\n{\\bf Limit ${\\bf \\delta w \\ll 1}$}\n\\vspace{0.2cm}\n\nAnother interesting limiting case concerns the small adhesion difference. \nExpansion of the numerator in eq.(\\ref{velo}) to leading order in $\\delta w$\nyields\n\\begin{equation}\nV \\simeq {\\delta w\\over \\eta/A+\\Gamma}\\;\\;\\;\\;\nA\\equiv {w\\over \\kappa }\n{R^2 \\over L_{adh}} \\left [1- { p\\over w} \\sqrt{\\kappa \\over 2 w}-{\\zeta \\over w} \\right]\n\\label{anal}\n\\end{equation}\n%%%>>>>>\nwhere $R=L/2 \\pi$.\n%%%<<<<< ?\nThe influence of the two dissipation coefficients appears then clearly. \nIt depends on the quantity $A$, proportional to the ratio $R/L_{adh}$.\nThe bulk dissipation increases with the adhesion length, which seems to be a \nvery \nrobust result, as encountered in the model including hydrodynamics \ndissipation \\cite{Cantat99b}.\nThe local dissipation represented\nby $\\Gamma$ does not depend on the adhesion length. \nOnly the two contact points\nmatter. Note also that the effective dissipation is $\\eta/A+\\Gamma$.\nThe bulk dissipation $\\eta/A$ and the contact one $\\Gamma$ play\na role of resistances (in an electric analogy) which\nwould be mounted in series. \n\n\n\\section{Discussion and conclusion}\nThis paper has given a first extensive presentation of the problem\nof vesicle migration in haptotaxis. We have %, for %%%>>>>\n%the\n%%%<<<<<\n%sake of simplicity,\nreduced as much as possible the complexity of the problem in order\nto gain some analytical approximate results.\nFor that purpose we have neglected hydrodynamics which \ninduce nonlocal interactions, and adopted a local model\nof the Rousse type. The full dynamical problem\nhas been solved by adopting a powerful gauge-field\ninvariant formulation. The dynamical code could\naccount for the transient and the evolution towards\na steady-state solution. In that context an introduction\nof an adhesion potential with a finite, albeit small, range\nhas proven to be necessary in order to circumvent \nnumerical instabilities related to the motion \nof the contact point. This code has the advantage\nof dealing with various problems not leading necessarily to\npermanent motions. For a stationary situation we could cast the problem\ninto a standard boundary value one where the migration velocity\nappeared as an eigenvalue. This problem is akin\nto the nonlinear eigenvalue problem of Barenblat-Zeldovitch type.\nA counting argument showed us that the velocity should belong\nto a discrete set, only one of them has been identified; we \nspeculate that the solution is unique. The problem could\nbe systematically solved in a fully intrinsic representation\nof the contour. For a rather tense vesicle we have provided\nan analytical solution which is in a good agreement with the numerical\none. We have identified the role played by the adhesion length\nin selecting the magnitude of the migration velocity even\nif no dissipation with the substrate is included. We have\nalso shown that the bond breaking/restoring dissipation\nand the (effective) bulk one are additive in a way which is\n%%%>>>>>very much like if they were like \nanalogous to the problem of %%%<<<<<\nelectrical resistances in series. Bulk\ndissipation dominates when %%%>>>>>>\nthe ratio of %%%<<<<<\nthe bulk dissipation coefficient\n%%%>>>>>over \nto %%%<<<<<\nthe contact one exceeds a certain %%%>>>>> ratio, \nlimit, %%%<<<<<\nwhich\ndepends in an intricate manner on various\nparameters. \nFor real situations, vesicles, and cells in general, are\nsuspended in aqueous solutions. It is therefore highly important\nto include hydrodynamics. Moreover the Lagrange multiplier\n$\\zeta$ is a local quantity. We have recently given a brief\naccount on these questions\\cite{Cantat99b}. An extensive discussion\nwill be presented in the near future.\n\n\n\n\\appendix\n\n\n\\section{Derivation of equation 44} % \\ref{denom}\n%%%>>>>> In the following equations, many \"ds\"'s are missing. I add them. %%%<<<<<\n\n$$\nD=\\int_0^{L^*} \\psi'' \\sin \\psi ds = -\\int_0^{L^*} (\\psi')^2 \\cos\\psi ds$$\nWe write $\\psi = 2\\pi s/L^* - \\pi + \\epsilon$, which implies to first order \nin \n$\\epsilon '$\n\\begin{eqnarray}\nD&=&-\\left ({2 \\pi \\over L^*}\\right)^2 \\int_0^{L^*}\\cos\\psi ds - {4 \\pi \\over \nL^*}\n \\int_0^{L^*}\\epsilon ' \\cos (2\\pi s/L^* - \\pi + \\epsilon) ds \\nonumber \\\\\n &=&-\\left ({2 \\pi \\over L^*}\\right)^2 L_{adh}+ {4 \\pi \\over L^*}\n\\int_0^{L^*}\\epsilon ' \\cos (2\\pi s/L^* + \\epsilon) ds \\nonumber \n\\end{eqnarray}\nWe then make use of the following relation:\n$$ {d \\over ds}(\\sin \\psi)= -\\left({2 \\pi \\over L^*}+\\epsilon '\\right) \\cos(2\\pi \ns/L^*+ \\epsilon)$$\nThe integral between $0$ and $L^*$ of the l.h.s. trem vanishes. We then\nobtain\n$$\\int_0^{L^*}\\epsilon ' \\cos (2\\pi s/L^* + \\epsilon) ds = {2 \\pi \\over \nL^*}\\int_0^{L^*}\\cos \\psi ds\n= {2 \\pi \\over L^*} L_{adh}$$\nThe sought after relation has then the form\n$$D= \\left ({2 \\pi \\over L^*}\\right)^2 L_{adh}$$\n\n%%%>>>>> In the bibliography in Ref.19) the sentence \"... at least ion ...\"\n% should be altered \" ... at least in ...\".\n%%%<<<<<\n\\begin{thebibliography}{10}\n\n\\bibitem{Helfrich73}\nW. Helfrich, Z. Naturforsch. C {\\bf 28}, 693 (1973).\n\n\\bibitem{Lipowsky}\n{\\em Structure and Dynamics of Membranes, Handbook of Biological Physics},\n edited by R. Lipowsky and E. Sackmann (Elsevier, North-Holland, 1995).\n\n\\bibitem{Brochard75}\nF. Brochard and J.-F. Lennon, J. Phys. France {\\bf 36}, 1035 (1975).\n\n\\bibitem{Prost96}\nJ. Prost and R. Bruinsma, Europhys. Lett. {\\bf 33}, 321 (1996).\n\n\\bibitem{Kraus96}\nM. Kraus, W. Wintz, U. Seifert, and R. Lipowsky, Phys. Rev. Lett. {\\bf 77},\n 3685 (96).\n\n\\bibitem{Durand97}\nI. Durand {\\it et~al.}, Phys. Rev. E {\\bf 56}, 3776 (1997).\n\n\\bibitem{Cantat99}\nI. Cantat and C. Misbah, Phys. Rev. Lett. {\\bf 83}, 235 (1999).\n\n\\bibitem{Cantat99a}\nI. Cantat and C. Misbah, in {\\em Transport and structures, their competitive\n roles in biophysics and chemistry}, edited by S. Mueller, J. Parisi, W. Zimmermann.\n (Springer, Berlin, 1999), Vol.~532, p.\\ 93.\n\n\\bibitem{Seifert99}\nU. Seifert, Phys. Rev. Lett. {\\bf 83}, 876 (1999).\n\n\\bibitem{Cantat99b}\nI. Cantat and C. Misbah, Phys. Rev. Lett. {\\bf 83}, 880 (1999).\n\n\\bibitem{Nardi99}\nJ. Nardi, R. Bruinsma, and E. Sackmann, Phys. Rev. Lett. {\\bf 82}, 5168\n (1999).\n\n\\bibitem{Carter67}\nS.~B. Carter, Nature {\\bf 213}, 256 (1967).\n\n\\bibitem{Curtis69}\nA.~S.~G. Curtis, J. Embryol. Exp. Morphol. {\\bf 22}, 305 (1969).\n\n\\bibitem{cytoskeletton}\nAs a leucocyte starts to crawl, part of its fluid cytoplasm turns rigid in a\n kind of sol-gel transition. The neutrophil extends a flat protrusion (leading\n lamella) that attaches to the underlying substrate, primarily through the\n action of membrane-adhesion proteins. The adhesion with the substrate\n provides a traction force that enables the cell to pull itself forward. The\n molecular motors provide a driving force to the actin filament that pushes\n the cell forward.\n\n\\bibitem{Cantatthes}\nI. Cantat, Ph.D. thesis, Th\\`ese de doctorat de l'universit\\'e Grenoble I,\n Grenoble, 1999.\n\n\\bibitem{Csahok99a}\nZ. Csah\\'ok, C. Misbah, and A. Valance, Physica D {\\bf 128}, 87 (1999).\n\n\\bibitem{Seifert91}\nU. Seifert, Phys. Rev. A {\\bf 43}, 6803 (1991).\n\n\\bibitem{deGennes}\nP. de~Gennes, {\\em Scaling concepts in polymer physics} (Cornell University\n Press, London, 1979).\n\n\\bibitem{remarkrousse}\nThe reason for locality is the following. For a true surface in 3D any motion\n on some scale $R$ induces a perturbation at least on that scale that couples\n different points on the surface. For 1D object in 3D circulation of the\n liquid can wander around the line (or polymer like geometry) without needing\n to affect distant points, leading thereby to local dynamics.\n\n\\end{thebibliography}\n\n\n\\newpage \n\\noindent Fig. 1 Notations used in the text. \\\\ %nota1\nFig. 2 A geometrical explanation of the arc length variation with time. \\\\ %divergence\nFig. 3 Force equilibrium at the fore contact point in the small rigidity limit. \\\\%adhloc\nFig. 4 Translation of a circle obtained with a purely normal motion. The right part is dilated, \n whereas the left part is contracted.\\\\ %jauge\nFig. 5 Successive vesicle profiles. The first one with open circles is an arbitrarily \n chosen initial shape. It relaxes to a permanent shape marked by filled circles on a inhomogeneous \n substrate. \\\\%relaxa\nFig. 6 Out of equilibrium adhering vesicle profiles. $V$ is measured in units of $100\\mu m$\n and $W$ in units of $10^{-4} mJ/m^2$.\\\\ %adhvit\nFig. 7 Geometrical constraint on the curve. \\\\%pieds \nFig. 8 Evolution of the vesicle velocity as a function of the adhesion difference.%vthvnu\n\n\n\n\\end{document}\n\n\n\n"
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[
{
"name": "cond-mat0002166.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{Helfrich73}\nW. Helfrich, Z. Naturforsch. C {\\bf 28}, 693 (1973).\n\n\\bibitem{Lipowsky}\n{\\em Structure and Dynamics of Membranes, Handbook of Biological Physics},\n edited by R. Lipowsky and E. Sackmann (Elsevier, North-Holland, 1995).\n\n\\bibitem{Brochard75}\nF. Brochard and J.-F. Lennon, J. Phys. France {\\bf 36}, 1035 (1975).\n\n\\bibitem{Prost96}\nJ. Prost and R. Bruinsma, Europhys. Lett. {\\bf 33}, 321 (1996).\n\n\\bibitem{Kraus96}\nM. Kraus, W. Wintz, U. Seifert, and R. Lipowsky, Phys. Rev. Lett. {\\bf 77},\n 3685 (96).\n\n\\bibitem{Durand97}\nI. Durand {\\it et~al.}, Phys. Rev. E {\\bf 56}, 3776 (1997).\n\n\\bibitem{Cantat99}\nI. Cantat and C. Misbah, Phys. Rev. Lett. {\\bf 83}, 235 (1999).\n\n\\bibitem{Cantat99a}\nI. Cantat and C. Misbah, in {\\em Transport and structures, their competitive\n roles in biophysics and chemistry}, edited by S. Mueller, J. Parisi, W. Zimmermann.\n (Springer, Berlin, 1999), Vol.~532, p.\\ 93.\n\n\\bibitem{Seifert99}\nU. Seifert, Phys. Rev. Lett. {\\bf 83}, 876 (1999).\n\n\\bibitem{Cantat99b}\nI. Cantat and C. Misbah, Phys. Rev. Lett. {\\bf 83}, 880 (1999).\n\n\\bibitem{Nardi99}\nJ. Nardi, R. Bruinsma, and E. Sackmann, Phys. Rev. Lett. {\\bf 82}, 5168\n (1999).\n\n\\bibitem{Carter67}\nS.~B. Carter, Nature {\\bf 213}, 256 (1967).\n\n\\bibitem{Curtis69}\nA.~S.~G. Curtis, J. Embryol. Exp. Morphol. {\\bf 22}, 305 (1969).\n\n\\bibitem{cytoskeletton}\nAs a leucocyte starts to crawl, part of its fluid cytoplasm turns rigid in a\n kind of sol-gel transition. The neutrophil extends a flat protrusion (leading\n lamella) that attaches to the underlying substrate, primarily through the\n action of membrane-adhesion proteins. The adhesion with the substrate\n provides a traction force that enables the cell to pull itself forward. The\n molecular motors provide a driving force to the actin filament that pushes\n the cell forward.\n\n\\bibitem{Cantatthes}\nI. Cantat, Ph.D. thesis, Th\\`ese de doctorat de l'universit\\'e Grenoble I,\n Grenoble, 1999.\n\n\\bibitem{Csahok99a}\nZ. Csah\\'ok, C. Misbah, and A. Valance, Physica D {\\bf 128}, 87 (1999).\n\n\\bibitem{Seifert91}\nU. Seifert, Phys. Rev. A {\\bf 43}, 6803 (1991).\n\n\\bibitem{deGennes}\nP. de~Gennes, {\\em Scaling concepts in polymer physics} (Cornell University\n Press, London, 1979).\n\n\\bibitem{remarkrousse}\nThe reason for locality is the following. For a true surface in 3D any motion\n on some scale $R$ induces a perturbation at least on that scale that couples\n different points on the surface. For 1D object in 3D circulation of the\n liquid can wander around the line (or polymer like geometry) without needing\n to affect distant points, leading thereby to local dynamics.\n\n\\end{thebibliography}"
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cond-mat0002167
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Analysis of soft optical modes in hexagonal BaTiO$_3$:\\ transference of perovskite local distortions
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"author": "Jorge \\'I\\~niguez~\\cite{email} and Alberto Garc\\'{\\i}a"
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We have performed detailed first-principles calculations to determine the eigenvectors of the zone-center modes of hexagonal BaTiO$_3$ and shown that the experimentally relevant low-energy modes (including the non-polar instability) can be represented as suitable combinations of basic local polar distortions associated with the instability of the cubic perovskite phase. The hexagonal structure provides a testing ground for the analysis of the influence of the stacking of TiO$_6$ octahedra: the occurrence of relatively high-energy chains of dipoles highlights the importance of local effects related to the coherent hybridization enhancement between Ti and O ions. Our results provide simple heuristic rules which could be useful for the analysis of related compounds.
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"name": "paper.tex",
"string": "\\documentstyle[aps,prl,twocolumn]{revtex}\n\n\\def\\chain{\\rm Ti$\\Rightarrow$O--Ti$\\Rightarrow$O}\n\n\\begin{document}\n\n%%%%%% PP Comment out next two lines for preprint\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname\n@twocolumnfalse\\endcsname\n\n\\title{Analysis of soft optical modes in hexagonal BaTiO$_3$:\\\\\ntransference of perovskite local distortions}\n\n\n\\author{Jorge \\'I\\~niguez~\\cite{email} and Alberto Garc\\'{\\i}a}\n\\address{\nDepartamento de F\\'{\\i}sica Aplicada II, Universidad del Pa\\'{\\i}s\nVasco, Apdo. 644, 48080 Bilbao, Spain}\n\n\\author{J.M. P\\'erez-Mato}\n\\address{ \nDepartamento de F\\'{\\i}sica de la Materia Condensada, \nUniversidad del Pa\\'{\\i}s\nVasco, Apdo. 644, 48080 Bilbao, Spain}\n\n\\maketitle\n\n\\begin{abstract}\nWe have performed detailed first-principles calculations to determine\nthe eigenvectors of the zone-center modes of hexagonal BaTiO$_3$ and\nshown that the experimentally relevant low-energy modes (including the\nnon-polar instability) can be represented as suitable combinations of\nbasic local polar distortions associated with the instability of the\ncubic perovskite phase. The hexagonal structure provides a testing\nground for the analysis of the influence of the stacking of TiO$_6$\noctahedra: the occurrence of relatively high-energy chains of dipoles\nhighlights the importance of local effects related to the coherent\nhybridization enhancement between Ti and O ions. Our results provide\nsimple heuristic rules which could be useful for the analysis of\nrelated compounds.\n\\end{abstract}\n\n\\pacs{PACS 63.20.Dj, 63.90.+t, \n 77.80.Bh, 77.90.+k}\n\n% 63.20.Dj Phonon states and bands, normal modes, and\n% phonon dispersion\n% 63.90.+t Other topics in lattice dynamics (restricted to\n% new topics in section 63)\n% 77.80.Bh Phase transitions and Curie point\n% 77.90.+k Other topics in dielectrics, piezoelectrics,\n% and ferroelectrics and their properties\n% (restricted to new topics in section 77)\n\n\n\n\n%%PP%% Comment out these two lines for preprint...\n\\vskip1pc\n]\n\n\\narrowtext\n\\marginparwidth 2.7in\n\\marginparsep 0.5in\n\nBarium titanate has two structural polymorphs: the cubic perovskite\ntype (c-BT) and its hexagonal modification (h-BT) with six formula\nunits per unit cell. While c-BT has been one of the best studied\nferroelectric materials for decades~\\cite{lines-glass}, most of the\nwork on the structural and dielectric properties of h-BT is quite\nrecent~\\cite{h-BT}. As shown in Fig.~\\ref{fig1}, h-BT is also composed\nof TiO$_6$ groups, albeit with a different stacking than the\nperovskite form. It seems well established that the hexagonal\npolymorph undergoes two zone-center structural phase transitions: at\n$222$~K from the high temperature $P6_3/mmc$ hexagonal phase to a second\nnon-polar $C222_1$ phase, and at $74$~K to a ferroelectric $P2_1$\nphase. The first transition is associated with the softening of an optical\nmode and the second attributed to a shear strain instability; but a\ndetailed analysis is lacking due to the absence of structural\ninformation on the two low-symmetry phases, and little is known about\nthe microscopic origin of the instabilities.\n\nOn the other hand, the discovery of a giant LO-TO splitting in h-BT by\nInoue {\\sl et al.}~\\cite{h-BT-pol} suggested that its ferroelectric\nmodes have the same origin as those of c-BT. In the cubic phase the\nferroelectric instabilities can be essentially described as chains of\ndipoles that originate in the movement of Ti ions relative to their\nsurrounding O$_6$ octahedra with a minor distortion of the\nlater. Additional evidence in support of this view is provided by the\nsuccessful use of polar local modes by Zhong {\\it et al.}~\\cite{zvr}\nin the construction of an effective Hamiltonian for c-BT.\n%\nIn view of the basic structural similarities between the cubic and\nhexagonal forms of BaTiO$_3$, it is meaningful to ask whether the\nlocal modes that describe the unstable branches in c-BT can somehow be\ntransferred to h-BT and serve as a basis to discuss the low-energy\ndistortions of the structure.\n\nHere we show the results of first-principles calculations that provide\nfor the first time structural information on the low-symmetry phases\nof the hexagonal polymorph of BaTiO$_3$ and reveal the microscopic\nnature of the modes. Our analysis proves that the structure of the\nexperimentally found zone-center optical soft~\\cite{soft} modes in\nh-BT is indeed characterized by the same distortions of the TiO$_6$\noctahedra that are relevant in c-BT, leading to similar chains of\ndipoles in the hexagonal structure.\n%\n\nIn h-BT, the experimentally found optical soft modes are: the\nzone-center instability that drives the phase transition at $222$~K\nand transforms according to the $E_{2u}$ irreducible representation\n(irrep) of $6/mmm$, and the $A_{2u}$ ferroelectric mode that softens\n(though remaining stable) in the temperature range of the $C222_1$\nphase and is responsible for the giant LO-TO splitting. Our\ncalculations agree with this experimental evidence. We performed a\nfull ab-initio~\\cite{tech-det} relaxation of the thirty-atom h-BT\nstructure, resulting in lattice parameters $a$=$10.68$ and\n$c$=$26.053$ a.u. (to be compared with experimental values of 10.77\nand 26.451~\\cite{exp-cell}, respectively). The five free internal\ncoordinates are also in excellent agreement with the experimentally\ndetermined ones (within $1\\%$). After computing the force-constant\nmatrix at $\\Gamma$ and diagonalizing it within the subspaces of the\nappropriate symmetries, we found an unstable E$_{2u}$\nmode~\\cite{praga} and a soft but not unstable A$_{2u}$ mode. [It\nshould be noted that if the calculations are performed using the\nexperimental lattice parameters (i.e., at a larger cell volume), the\nA$_{2u}$ mode is found to be unstable (and the E$_{2u}$ instability is\nmore pronounced). The fact that the A$_{2u}$ mode is very close to\nbeing unstable could be particularly relevant for the phase transition\nat $74$~K.]\n\nA first analysis of the eigenvectors for both soft modes reveals that\nthe Ba contribution is small (around $10\\%$ of the total mode norm,\ncompared to $4\\%$ in the perovskite soft mode). Moreover, we checked\nthat if the Ba ions are frozen at their high-symmetry positions the\nmodes are still soft, so we do not consider them the following\ndiscussion, and focus on the distortions of the TiO$_6$ groups.\n\nTo make the comparison to the perovskite quantitative, let us consider\nthe polar deformations of the TiO$_6$ groups of c-BT. These are shown\nin Fig.~\\ref{fig2}, where we assume that the Ti ion is located at the\norigin of coordinates, so only the displacement patterns of the O ions\nneed to be considered. For each spatial direction $\\alpha$=$x,y,z$ we\nhave two symmetry-adapted distortions denoted by $\\hat{s}_{1,\\alpha}$\nand $\\hat{s}_{2,\\alpha}$ and transforming according to the $T_{1u}$\n(vector like) irrep of $m3m$, the point group of the regular octahedra\nof c-BT. In terms of this basis, the tetragonal ferroelectric\ndistortion in c-BT (along $x$ for concreteness) can be written as\n$0.69\\hat{s}_{1x}+0.73\\hat{s}_{2x}$, with the distorted octahedra\nexhibiting point group symmetry $4mm$~\\cite{more-m3m}. In h-BT there\nare two kinds of octahedra: those centered around Ti ions at $2a$\nWyckoff positions with $\\bar{3}m$ point symmetry (denoted by\nTiO$_6$(1) in Fig.~\\ref{fig1}) and those arranged around Ti ions at\n$4f$ Wyckoff positions with $3m$ point symmetry (TiO$_6$(2) in the\nfigure)~\\cite{hex-polar}. The octahedra in the first set are\ncoordinated in the same way as in c-BT, i.e., by sharing O ions with\nsix other octahedra. Those in the second set are linked to other $1+3$\noctahedra by sharing one O$_3$ face and single O ions respectively.\n\nDue to the low (as compared to the case of c-BT) symmetry of the\noctahedra in h-BT, their distortions associated to general $E_{2u}$\nand $A_{2u}$ modes can be decomposed in a relatively large number of\nsymmetry-adapted displacement patterns. Among all the possible ones,\nwe restrict ourselves to those of c-BT type and check if they can\nactually account for the structure of the soft\nmodes~\\cite{we-can}. For instance, a general $A_{2u}$ distortion\nleads the crystal to a phase with space group $P6_3mc$, in which the\nTiO$_6$(1) groups reduce their point symmetry to $3m$ (see\nTable~\\ref{tab1}). As shown in Fig.~\\ref{fig3}.b, c-BT distortions in\nthe $s_{ix}=s_{iy}=s_{iz}$ component combination (Ti ions move towards\nO$_3$ faces as in the rhombohedric phase of c-BT) produce this\nsymmetry breaking. In the case of an $E_{2u}$ distortion, TiO$_6$(1)\nreduces its point symmetry to $2$, and the appropriate c-BT mode has\nthe form $s_{ix}=-s_{iy}$ (orthorhombic), as shown in\nFig.~\\ref{fig3}.c.\n\nFor the two soft modes of h-BT, we considered separately the various\nclasses of octahedra, computed from our ab-initio eigenvectors the\ndisplacement of the O ions relative to the Ti ion, and performed a\nprojection of the resulting distortion field into the c-BT type\nsymmetry-adapted modes. The results (last column of Table~\\ref{tab1})\npresent two main features: First, almost $100\\%$ of the total\nstructural change associated with both soft modes can be described in\nterms of the c-BT type polar distortions (normalization is chosen in\nsuch a way that, for instance, for the first row in Table~\\ref{tab1}\nwe have $(0.62^2+0.78^2)\\times100=99.3\\%$). Second, the components\n$s_1;s_2$ are always similar in magnitude to those of c-BT\n($0.69;0.73$) and present a positive $s_1/s_2$ ratio, which implies\nthat the O$_6$ octahedral cage moves almost rigidly relative to the Ti\nion also in h-BT. We have also computed the Born effective charge\nassociated with the ferroelectric $A_{2u}$ soft mode and found it\nunusually large ($Z^*=11.29$), further confirming the relation with\nthe (rhombohedric) ferroelectric instability of c-BT (for which\n$Z^*=9.956$)~\\cite{z}.\n\nWe have proved then that at a local level the soft modes in h-BT can\nbe described by the same distortion vectors that determine the c-BT\npolar instability. In the crystal as a whole, these local polar\ndistortions lead to chains of dipoles, which points at the $E_{2u}$\ninstability and the softness of the $A_{2u}$ mode of h-BT as being\ncaused by Coulomb destabilizing forces, as it happens in the cubic\nperovskite. The ferroelectric $A_{2u}$ soft mode, polarized along\n$z'$, is roughly depicted in Fig.~\\ref{fig1}. In the $E_{2u}$\ndistortion the chains of dipoles lay on the $x'y'$ plane and alternate\nin orientation with a zero net polarization (the resulting $C222_1$\nphase could be informally considered {\\sl anti-ferroelectric} rather\nthan paraelectric).\n%\n\nFrom first-principles studies of the c-BT phase it is known that\nparallel dipole chains are very weakly coupled, so that a transverse\nmodulation of a chain-like instability is not energetically relevant,\nand, therefore, unstable TO normal modes exist almost in the whole\nBrillouin Zone (BZ)~\\cite{kx}. [The only exception are $k$ points near\n${\\bf k}_R = \\frac{2\\pi}{a}(1,1,1)$, for which we have an anti-phase\nmodulation of the Ti displacements (Ti$\\Rightarrow$O$\\Leftarrow$Ti--O)\nin the three spatial directions, so the long-range destabilizing\nforces are always canceled.] If this view is taken to its logical\nconclusion, we could expect to find more zone-center soft modes in\nh-BT, corresponding to the other possible distributions of chains of\ndipoles. Table~\\ref{tab2} enumerates all the possibilities. Apart from\nthe already discussed $A_{2u}$ and $E_{2u}$ modes, our ab-initio\ncalculations show that there is one $E_{1g}$ mode that is indeed\nrather low in energy, while the ferroelectric $E_{1u}$ and the\n$E_{2g}$ modes that are dominated by the movement of Ti ions are quite\nhard. In order to explain this result, let us remark that for the\n$E_{2u}$ and $A_{2u}$ soft modes the distortion is such that if an O\nion is approached by one of its two Ti neighbors the second Ti ion\nmoves away from it. This reflects the hybridization of the Ti $3d$ and\nO $2p$ electronic states, which has been shown to be essential for the\noccurrence of the c-BT ferroelectric\ninstability~\\cite{cohen-ghosez}. It can be checked that any other\nzone-center arrangement of the chains of dipoles results in either two\nTi ions approaching one O ion (for example, if the two Ti ions in one\nof the O$_3-$Ti$-$O$_3-$Ti$-$O$_3$ groups depicted in Fig.~\\ref{fig1}\nmove in the same way in the $x'y'$ plane, there is at least one oxygen\nof the shared face that is approached by both) or in the second\ntitanium not moving away from an oxygen. In the former case ($E_{1u}$\nand $E_{2g}$) the effect of the hybridization is lost and the\ncorresponding modes are hard. In the latter ($E_{1g}$ and $B_{1g}$),\nthe hardening is not as strong. Thus, we conclude that the particular\nstacking of the TiO$_6$ groups in h-BT causes (through this local\neffect) the relatively high energy of some chain-like distortions. We\ncan then formulate two ``rules of thumb'' for the characterization of\na given locally polar distortion as low-energy: First ($\\cal R$1):\n``There need to be chains of dipoles (without regard for their\ntransverse modulation)''. Second ($\\cal R$2): ``The distribution of\nsuch chains must lead to a {\\sl coherent} hybridization enhancement\nbetween Ti and O ions, where the word {\\sl coherent} means that the\ndestabilizing effect is lost when two Ti ions approach the same O\nion''. These heuristic rules could be used to predict the occurrence\nof locally polar soft modes at other $k$ points of the BZ of h-BT, as\nwell as in other structures with TiO$_6$ octahedra as basic building\nblocks.\n\nIn summary, first-principles calculations of the character of the\nzone-center modes of hexagonal BaTiO$_3$ support the physically\nappealing idea that the experimentally relevant soft modes (including\nthe non-polar instability) can be represented as combinations of local\npolar distortions transferred directly from the cubic perovskite form\nof the compound. Our results lead also to heuristic rules that provide\ninsight into the influence of the arrangement of TiO$_6$ octahedra on\nthe low-energy dynamics of a structure~\\cite{more-details}.\n\nThis work was supported in part by the UPV research grant\n060.310-EA149/95 and by the Spanish Ministry of Education grant\nPB97-0598. J.I. acknowledges financial support from the Basque\nregional government.\n\n\\begin{references}\n\n\\bibitem[*]{email} Electronic address: wdbingoj@lg.ehu.es\n\n\\bibitem{lines-glass} M. E. Lines and A. M. Glass, {\\sl Principles and\nApplications of Ferroelectrics and Related Materials} (Clarendon\nPress, Oxford) 1977.\n\n\\bibitem{h-BT} M. Yamaguchi, K. Inoue, T. Yagi, and Y. Akishige,\nPhys. Rev. Lett. {\\bf 74}, 2126 (1995); M. Yamaguchi, M. Watanabe,\nK. Inoue, Y. Akishige, and T. Yagi, Phys. Rev. Lett. {\\bf 75}, 1399\n(1995); Y. Akishige, J. Kor. Phys. Soc. {\\bf 27}, S81 (1994).\n\n\\bibitem{h-BT-pol} K. Inoue, M. Wada, and A. Yamanaka,\nJ. Kor. Phys. Soc. {\\bf 29}, S721 (1996).\n\n\\bibitem{zvr} W. Zhong, D. Vanderbilt, and K.M. Rabe,\nPhys. Rev. Lett. {\\bf 72}, 3618 (1994).\n\n\\bibitem{soft} We use the term {\\sl soft} to refer to instable modes\nas well as low-energy (though maybe stable) modes.\n\n\\bibitem{tech-det} We have used the local-density approximation and\nthe pseudopotential approach with Vanderbilt's ultrasoft\npseudopotentials. The electron wave functions were expanded in a plane\nwave basis with an energy cut-off of 25 Rydberg, and the BZ sums were\ncalculated by a $3\\times3\\times(2(+0.5))$ Monkhorst-Pack special\nk-point mesh.\n\n\\bibitem{exp-cell} J. Akimoto, Y. Gotoh, and Y. Oosawa, Acta Cryst. C\n{\\bf 50}, 160 (1994).\n\n\\bibitem{praga} A complete ab-initio study of the energy surface\ncorresponding to the $E_{2u}$ instability can be found in J.\n\\'I\\~niguez, A. Garc\\'{\\i}a, and J.M. P\\'erez-Mato, to appear in\nFerroelectrics.\n\n\\bibitem{more-m3m} A rhombohedral distortion ($3m$) would be\nrepresented by equal amplitudes of the three components of $\\hat{s}_1$\nand $\\hat{s}_2$ (maintaining the same $(0.69,0.73)$ mix), and an\northorhombic one ($222$) by equal amplitudes for two of the components\nand zero for the third.\n\n\\bibitem{hex-polar} In this case, the octahedra exhibit a polar\ndistortion even in the $P6_3/mmc$ phase of h-BT: both Ti ions in a\nO$_3-$Ti$-$O$_3-$Ti$-$O$_3$ set are displaced along $z'$ towards the\nO$_3$ faces on the side (which agrees with the $\\cal R$2 heuristic\nrule in the text),\nproducing two opposite local dipoles and no net polarization.\n\n\\bibitem{we-can} It can be proven (by group-theoretical\nconsiderations) that polar distortions located at both $2a$ and $4f$\nWyckoff positions produce zone-center modes of the desired symmetries.\n\n\\bibitem{z} Born effective charges are calculated following the Berry's\nphase approach as in W. Zhong, R.D. King-Smith, and D. Vanderbilt,\nPhys. Rev. Lett. {\\bf 72}, 3618 (1994).\n\n\\bibitem{kx} Ph. Ghosez, E. Cockayne, U. V. Waghmare, and K.M. Rabe,\nPhys. Rev. B {\\bf 60}, 836 (1999). For example, the modulation given\nby ${\\bf k}_{\\rm X}=\\frac{2\\pi}{a}(1,0,0)$ is compatible with chains\nof dipoles polarized along $y$ and $z$ directions, so two (degenerate)\ninstabilities exist at ${\\bf k}_{\\rm X}$.\n\n\\bibitem{cohen-ghosez} R.E. Cohen, Nature {\\bf 358}, 136 (1992).\nCohen showed that this hybridization reduces the ``repulsive'' effect\nof short-range forces and favors the ferroelectric phases of c-BT. The\nforce-constants calculated in Ref.~\\protect\\cite{kx} suggest that this\nhybridization effect becomes a short-range repulsion when an O ion is\napproached by two Ti ions. [The force-constant coupling the\nlongitudinal displacement of two neighboring Ti ions is $-0.0672$\n(total) $=-0.0368$ (long-range) $-0.0304$ (short-range) a.u.. This is\nthe main contribution to the instability for in-phase displacements of\nthe Ti ions, but for anti-phase displacements both short- and\nlong-range contributions become repulsive.]\n\n\\bibitem{more-details} A more detailed account of the work presented\nhere, as well as other results (such as the existence of a previously\nunreported zone-center soft Rigid Unit Mode) will be presented\nelsewhere.\n\n\\end{references}\n\n\\clearpage\n\n%% Tables\n\n\\begin{table}\n\\begin{tabular}{lllll}\nMode & Octahedra & Final & $s_{i,\\alpha}$ & Projection \\\\\n & type & symmetry & distortion & components \\\\\n\\tableline\n$E_{2u}$ &TiO$_6$(1)$\\bar{3}m$ & 2 & $s_x=-s_y$ & 0.62; 0.78 \\\\\n &TiO$_6$(2)$3m$ & 1 & $s_x,s_y,s_z$ & 0.05;0.01 ($x$) \\\\\n & & & & 0.44;0.50 ($y$) \\\\\n & & & & 0.51;0.53 ($z$) \\\\\n & & & & 0.68; 0.73 \\\\\n\\hline\n$A_{2u}$ &TiO$_6$(1)$\\bar{3}m$ & $3m$ & $s_x=s_y=s_z$ & 0.67; 0.70 \\\\\n &TiO$_6$(2)$3m$ & $3m$ & $s_x=s_y=s_z$ & 0.63; 0.74 \\\\\n\\end{tabular}\n\\vskip .5cm\n\\caption{Symmetry breakings of the h-BT phase TiO$_6$ groups\nassociated to the $A_{2u}$ and $E_{2u}$ soft modes. The fourth column\nshows the combinations of symmetry-adapted c-BT type distortions that\nare compatible with the symmetry reduction (it applies to both $s_1$\nand $s_2$).\n%\nThe last column shows the projections of the normalized total\ndistortion of the TiO$_6$ groups onto the $\\hat{s}$ modes of the\nsecond column, in the form $s_1;s_2$.\n%\nThe $E_{2u}$ mode removes all symmetry elements from the TiO$_6$(2)\noctahedra, and any combination of $s_{\\alpha}$ is possible. For this\ncase we have\nlisted in the last column the $s_1$ and $s_2$ projections along each\nof the three spatial directions as well as the modulus.}\n\\label{tab1}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{lllllll}\nLayer & $A_{2u}^f$ & $B_{1g}$ & $E_{1g}$ & $E_{1u}^f$ & $E_{2g}$ &\n$E_{2u}$ \\\\\n\\tableline\nTi(2) & { }$\\Uparrow$ & { }$\\Uparrow$ & $\\Rightarrow$ &\n$\\Rightarrow^*$ & $\\Rightarrow^*$ & $\\Rightarrow$ \\\\\nTi(2) & { }$\\Uparrow$ & { }$\\Uparrow$ & $\\Leftarrow$ &\n$\\Rightarrow^*$ & $\\Rightarrow^*$ & $\\Leftarrow$ \\\\\nTi(1) & $\\uparrow$ & & & $\\rightarrow$ & & $\\rightarrow$ \\\\\nTi(2) & { }{ }$\\Uparrow$ & { }{ }$\\Downarrow$ & $\\Rightarrow$ &\n$\\Rightarrow^*$ & $\\Leftarrow^*$ & $\\Leftarrow$ \\\\\nTi(2) & { }{ }$\\Uparrow$ & { }{ }$\\Downarrow$ & $\\Leftarrow$ &\n$\\Rightarrow^*$ & $\\Leftarrow^*$ & $\\Rightarrow$ \\\\\nTi(1) & $\\uparrow$ & & & $\\rightarrow$ & & $\\leftarrow$ \\\\\n\\tableline\n & 0.0074 & ($\\geq$ 0.046) &0.0105 &0.3305 &0.2543 &-0.0123\\\\\n\\end{tabular}\n\\vskip .5cm\n\\caption{Symbolic description of the possible zone-center chains of\ndipoles in h-BT, classified in terms of irreps of $6/mmm$. The layers\nof Ti ions are represented along the $z'$ direction as in\nFig.~\\protect\\ref{fig1} (Ti(1) and Ti(2) refer to Ti ions in\nTiO$_6$(1) and TiO$_6$(2) groups respectively). Arrows indicate the\norientation of the dipoles (horizontal ones symbolize any direction in\nthe $x'y'$ plane), and those set in the same type are symmetry\nrelated. The superscript $f$ marks the ferroelectric\nmodes. Displacement patterns that violate $\\cal R$2 (see text) are\nmarked with an asterisk. The bottom line shows the mode force\nconstants in atomic units (for $B_{1g}$ an unambiguous assignment\ncannot be made).}\n\\label{tab2}\n\\end{table}\n\n\n\\begin{figure}\n\\caption{Unit cell of the $P6_3/mmc$ phase of h-BT. For the sake of\nsimplicity, only Ti and O ions are shown. The distributions of local\ndipoles corresponding to the $E_{2u}$ and $A_{2u}$ soft modes are\nindicated.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Polar (vector like) deformations of the $m3m$ (regular)\noctahedra of c-BT. Only the modes polarized along $x$ are\nindicated. The $y$ and $z$ sets are analogous.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Combinations of $\\hat{s}_{i,\\alpha}$ modes that represent the\nsymmetry breaking of the octahedra in h-BT by the soft modes. Panel\na)~shows a conveniently oriented octahedron which will be assumed to\nhave $\\bar{3}m$ (resp. $3m$) symmetry. Oxygen ions are labeled as in\nFig.~\\protect\\ref{fig2} and the cartesian axes with origin in the Ti\nion are indicated. Panel b)~shows the $s_x=s_y=s_z$ distortion that\nbreaks the 2-fold axes (resp. no symmetry element) and is related to\nthe $A_{2u}$ mode. In panel c), a $s_x=-s_y$ distortion results in a 2\n(resp. 1) point-symmetry, as appropriate for the $E_{2u}$ mode. Only\n$\\hat{s}_1$ modes are shown, but the same combinations apply to\n$\\hat{s}_2$.}\n\\label{fig3}\n\\end{figure}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002167.extracted_bib",
"string": "\\bibitem[*]{email} Electronic address: wdbingoj@lg.ehu.es\n\n\n\\bibitem{lines-glass} M. E. Lines and A. M. Glass, {\\sl Principles and\nApplications of Ferroelectrics and Related Materials} (Clarendon\nPress, Oxford) 1977.\n\n\n\\bibitem{h-BT} M. Yamaguchi, K. Inoue, T. Yagi, and Y. Akishige,\nPhys. Rev. Lett. {\\bf 74}, 2126 (1995); M. Yamaguchi, M. Watanabe,\nK. Inoue, Y. Akishige, and T. Yagi, Phys. Rev. Lett. {\\bf 75}, 1399\n(1995); Y. Akishige, J. Kor. Phys. Soc. {\\bf 27}, S81 (1994).\n\n\n\\bibitem{h-BT-pol} K. Inoue, M. Wada, and A. Yamanaka,\nJ. Kor. Phys. Soc. {\\bf 29}, S721 (1996).\n\n\n\\bibitem{zvr} W. Zhong, D. Vanderbilt, and K.M. Rabe,\nPhys. Rev. Lett. {\\bf 72}, 3618 (1994).\n\n\n\\bibitem{soft} We use the term {\\sl soft} to refer to instable modes\nas well as low-energy (though maybe stable) modes.\n\n\n\\bibitem{tech-det} We have used the local-density approximation and\nthe pseudopotential approach with Vanderbilt's ultrasoft\npseudopotentials. The electron wave functions were expanded in a plane\nwave basis with an energy cut-off of 25 Rydberg, and the BZ sums were\ncalculated by a $3\\times3\\times(2(+0.5))$ Monkhorst-Pack special\nk-point mesh.\n\n\n\\bibitem{exp-cell} J. Akimoto, Y. Gotoh, and Y. Oosawa, Acta Cryst. C\n{\\bf 50}, 160 (1994).\n\n\n\\bibitem{praga} A complete ab-initio study of the energy surface\ncorresponding to the $E_{2u}$ instability can be found in J.\n\\'I\\~niguez, A. Garc\\'{\\i}a, and J.M. P\\'erez-Mato, to appear in\nFerroelectrics.\n\n\n\\bibitem{more-m3m} A rhombohedral distortion ($3m$) would be\nrepresented by equal amplitudes of the three components of $\\hat{s}_1$\nand $\\hat{s}_2$ (maintaining the same $(0.69,0.73)$ mix), and an\northorhombic one ($222$) by equal amplitudes for two of the components\nand zero for the third.\n\n\n\\bibitem{hex-polar} In this case, the octahedra exhibit a polar\ndistortion even in the $P6_3/mmc$ phase of h-BT: both Ti ions in a\nO$_3-$Ti$-$O$_3-$Ti$-$O$_3$ set are displaced along $z'$ towards the\nO$_3$ faces on the side (which agrees with the $\\cal R$2 heuristic\nrule in the text),\nproducing two opposite local dipoles and no net polarization.\n\n\n\\bibitem{we-can} It can be proven (by group-theoretical\nconsiderations) that polar distortions located at both $2a$ and $4f$\nWyckoff positions produce zone-center modes of the desired symmetries.\n\n\n\\bibitem{z} Born effective charges are calculated following the Berry's\nphase approach as in W. Zhong, R.D. King-Smith, and D. Vanderbilt,\nPhys. Rev. Lett. {\\bf 72}, 3618 (1994).\n\n\n\\bibitem{kx} Ph. Ghosez, E. Cockayne, U. V. Waghmare, and K.M. Rabe,\nPhys. Rev. B {\\bf 60}, 836 (1999). For example, the modulation given\nby ${\\bf k}_{\\rm X}=\\frac{2\\pi}{a}(1,0,0)$ is compatible with chains\nof dipoles polarized along $y$ and $z$ directions, so two (degenerate)\ninstabilities exist at ${\\bf k}_{\\rm X}$.\n\n\n\\bibitem{cohen-ghosez} R.E. Cohen, Nature {\\bf 358}, 136 (1992).\nCohen showed that this hybridization reduces the ``repulsive'' effect\nof short-range forces and favors the ferroelectric phases of c-BT. The\nforce-constants calculated in Ref.~\\protect\\cite{kx} suggest that this\nhybridization effect becomes a short-range repulsion when an O ion is\napproached by two Ti ions. [The force-constant coupling the\nlongitudinal displacement of two neighboring Ti ions is $-0.0672$\n(total) $=-0.0368$ (long-range) $-0.0304$ (short-range) a.u.. This is\nthe main contribution to the instability for in-phase displacements of\nthe Ti ions, but for anti-phase displacements both short- and\nlong-range contributions become repulsive.]\n\n\n\\bibitem{more-details} A more detailed account of the work presented\nhere, as well as other results (such as the existence of a previously\nunreported zone-center soft Rigid Unit Mode) will be presented\nelsewhere.\n\n"
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cond-mat0002168
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[
{
"name": "PRE.tex",
"string": "\\documentclass[a4paper,10pt]{article}\n%\n\\usepackage[latin1]{inputenc}\n%\\usepackage[italian]{babel}\n\\usepackage{graphicx}\n%\n%\n\\begin{document}\n\\begin{titlepage}\n\\begin{center}\n\\Large{\\bf{Canonical solution of a system of long-range\n interacting rotators on a lattice}}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n\\large{Alessandro Campa$^1$, Andrea Giansanti$^{2, \\S}$\n and Daniele Moroni$^2$}\n\\end{center}\n\\vspace{1cm}\n\\begin{center}\n\\normalsize{$^1$Physics Laboratory, Istituto Superiore di Sanit\\`a and\nINFN Sezione di Roma1, Gruppo Collegato Sanit\\`a\\\\Viale Regina\nElena 299, 00161 Roma, Italy}\n\\end{center}\n\\begin{center}\n\\normalsize{$^2$Physics Department, Universit\\`a di Roma ``La Sapienza'' and\nINFM Unit\\`a di Roma1, \\\\ Piazzale Aldo Moro 2, 00185 Roma, Italy}\n\\end{center}\n\\vspace{0.5cm}\n\\begin{center}\n\\emph{(13 April 2000, accepted for publication in Physical Review E)}\n\\end{center}\n\\vspace{1cm}\n\\begin{center}\n\\large{\\bf {Abstract}}\n\\end{center}\n\\vspace{1cm}\n\\small{\nThe canonical partition function of a system of rotators (classical\nX-Y spins) on a lattice, coupled by terms decaying as the inverse of\ntheir distance to the power $\\alpha$, is analytically computed. It is\nalso shown how to\ncompute a rescaling function that allows to reduce the model, for any\n$d$-dimensional lattice and for any $\\alpha<d$, to the mean field\n($\\alpha=0$) model.\n}\n{\\bf PACS}: 05.20.-y, 05.70.Ce, 05.10.-a\n\n\\vspace{2cm}\n$^{\\S}$ \\small{Author to whom correspondence should be addressed; electronic address:\nAndrea.Giansanti@roma1.infn.it}\n\\end{titlepage}\n\\section{Introduction}\nLet us consider the following classical hamiltonian model of a system\nof rotators:\n\\begin{equation} \\label{num1}\n H=\\frac{1}{2} \\sum_{i=1}^N L_i^2 \\, + \\, \\frac{1}{2}\n\\sum_{i,j=1}^N\\left[1-\\cos(\\theta_i-\\theta_j)\\right]=K+V\\, .\n\\end{equation}\nThe potential energy $V$ is not thermodynamically stable and the ensemble\naveraged energy\ndensity $U=\\langle \\frac{H}{N}\\rangle$ diverges in the thermodynamic\nlimit (TL) \\cite{ruelle}.\nIf the potential energy term is divided by $N$, then the \nenergy density becomes intensive and it \nis bounded as $N$ goes to infinity.\n\nIndeed, dynamics and thermodynamics of the $1/N$ rescaled model\nhas been extensively investigated \\cite{LRR}; in particular, Ruffo and\nAntoni, who called it the hamiltonian mean field X-Y model (HMF),\nsolved it in the canonical ensemble, and compared the\ntheoretical caloric ($T\\, vs\\, U$) and magnetization ($M\\, vs\\, U$)\ncurves with those obtained from a microcanonical simulation \\cite{AR}.\n\nHere we consider a generalization of model (\\ref{num1}):\n\\begin{equation} \\label{num2}\n H=\\frac{1}{2} \\sum_{i=1}^{N} L_i^2 \\, + \\, \\frac{1}{2}\n\\sum_{i\\neq j}^{N} \\frac {1-\\cos(\\theta_i-\\theta_j)}\n{r_{ij}^\\alpha} \\, .\n\\end{equation}\nThe rotators are placed at the sites of a lattice \nand the interaction between rotators $i$ and $j$ decays as the inverse of\ntheir distance to the power $\\alpha$.\n\nA onedimensional version of model (\\ref{num2}) has been studied by \nAnteneodo and Tsallis \\cite{AT}, who have numerically measured the largest\nLyapounov exponent, as a function of $N$ and $\\alpha$.\nThrough a rescaling factor $N^{*}=\\frac {N^{1-\\alpha}-1}{1-\\alpha}$ \nAnteneodo and Tsallis showed that their results\ncoincide with those previously obtained for the HMF ($\\alpha=0$) model;\nthis rescaling could then give a well defined TL to model (\\ref{num2}). \n\nIn a recent paper Tamarit and Anteneodo, using a rescaling\nfactor $\\tilde{N}= 2^{\\alpha}\\frac {N^{1-\\alpha}-1}{1-\\alpha}$, have shown\nthat the caloric and magnetization curves of model (\\ref{num2}) in one\ndimension collapse onto the curves \nof the HMF model \\cite{TA}.\nThis universality emerges plotting $T/\\tilde{N}$ as a function of\n$H/N\\tilde{N}$\nand $M$ as a function of $H/N\\tilde{N}$, from molecular dynamics simulation\nof model (\\ref{num2}) for different $N$ and $\\alpha$ values. These authors\nconjecture that \nthe results they obtained in the onedimensional case might be general,\nvalid in any dimension $d$ and for $\\alpha <d$, as suggested also\nin \\cite{TA}.\n\\section{Partition function}\nIn this work, inspired by \\cite{AR} and \\cite{TA}, we analytically compute\nthe partition function of an $\\tilde{N}$-rescaled model (\\ref{num2})\nfor any $d$ and $\\alpha <d$.\nIn formula (\\ref{exactn}) we give the right expression of the rescaling\nfunction $\\tilde{N}$, to obtain universal state curves for all lattice\nmodels with long range ($\\alpha < d$) interactions.\n\nLet us now rewrite the rescaled version of Hamiltonian (\\ref{num2}):\n\\begin{eqnarray}\\label{num3}\nH&=&\\frac{1}{2} \\sum_{i=1}^N L_i^2 \\, + \\, \\frac{1}{2 \\tilde{N}}\n\\sum_{i,j=1}^N \\frac {1-\\cos(\\theta_i-\\theta_j)}\n{r_{ij}^\\alpha} \\nonumber \\\\\n&&- h_x \\sum_{i=1}^N m_{ix} - h_y\\sum_{i=1}^N m_{iy} \\, ,\n\\end{eqnarray}\nwhere we have introduced an external magnetic field\n$\\mathbf{h}=(h_x,h_y)$ of modulus $h$, that makes possible to compute\nthe magnetization.\nThe indexes $i,j$ label the sites of a $d$-dimensional generic\nlattice; $r_{ij}$ is the distance between them, with periodic\nboundary conditions and nearest image convention (the definition of\n$r_{ii}$ will be given shortly); $\\alpha \\geq 0$.\nAt each site a classical rotator (X-Y spin) of unit\nmomentum of inertia is represented by conjugate canonical\ncoordinates $(L_i,\\theta_i)$, where the $L_i$'s are angular\nmomenta, and the $\\theta_i$'s $\\in [0,2\\pi )$ are the angles of\nrotation on a family of parallel planes,\neach one defined at each lattice point; $x$ and $y$ refer to the\ncomponents of boldface twodimensional vectors defined over these planes.\nTo each lattice site a spin vector\n\\begin{equation}\n\\mathbf{m}_i=(m_{ix},m_{iy})=(\\cos \\theta_i,\\sin \\theta_i)\n\\end{equation}\nis associated, and the total magnetization is given by:\n\\begin{equation}\n\\mathbf{M}=(M_x,M_y)=\\frac{1}{N}\\sum_{i=1}^N\\mathbf{m}_i \\, .\n\\end{equation}\nNote in (\\ref{num3}) the rescaling factor $\\tilde{N}$ in front of the\npotential energy term, now written as a free double sum over both indexes.\n$\\tilde{N}$ should be regarded as an unknown function of\n$N,\\alpha,d$ and the geometry of the lattice, with the\nfundamental property of making\n\\begin{equation}\\label{Ntildedefi}\n\\frac{1}{\\tilde{N}}\\sum_{j,j \\neq i} \\frac{1}{r_{ij}^\\alpha}\n\\end{equation}\nan intensive quantity; this guarantees the thermodynamic stability of\nthe potential. We also note that the sum in (\\ref{Ntildedefi}) is\nindependent of the origin $i$ because of periodic conditions.\nTo reproduce the usual HMF it is also necessary that\n$\\tilde{N}(N,\\alpha=0,d)=N$.\nThe constraint $i\\neq j$ over the double sum is removed defining\n$r_{ii}^\\alpha = 1/b$, a finite number. Since the numerator\n$1-\\cos(\\theta_i-\\theta_j)$ is zero for $i=j$ the choice of $b$ is\nfree. The removal of the constraint allows to introduce the distance\nmatrix $R'_{ij}=\\frac{1}{r_{ij}^\\alpha}$; the diagonalization of\nsuch matrix is the key point to obtain, in the computation of the\npartition function, known integrals in the variables $\\theta_i$.\n\nAs usual the partition function factorizes in a kinetic\npart $Z_K=\\left(\\frac{2 \\pi}{\\beta} \\right)^{\\frac{N}{2}}$, where\n$\\beta=1/k_B T$, and a potential part $Z_V$. After defining\n$R_{ij}=\\frac{\\beta}{2 \\tilde{N}} R'_{ij}$,\n$\\mathbf{B}=\\beta \\mathbf{h}$, $C=\\exp\\left(-\\frac{\\beta}{2 \\tilde{N}}\n\\sum_{ij} \\frac {1}{r_{ij}^\\alpha}\\right)$, the potential part\ncan be written as:\n\\begin{equation}\\label{Zmatr}\nZ_V=C \\int_{-\\pi}^{\\pi}\n d^N\\theta\\, \\exp\\left[\\sum_{i,j,\\mu}m_{i\\mu}R_{ij}\nm_{j\\mu} + \\sum_iB_{\\mu}m_{i\\mu} \\right],\n\\end{equation}\nwhere $\\mu=x,y$. Diagonalizing the symmetric matrix $R=(R_{ij})$ with the\nunitary matrix\n$U$ such that $R=U^TDU$, $D=(R_i\\delta_{ij})$, where $R_i$ are the\neigenvalues of $R$, we can write the first part of the exponent in\n(\\ref{Zmatr}) as:\n\\begin{equation}\\label{diago}\n\\sum_{ij}\\left(m_{ix}R_{ij}m_{jx} + m_{iy}R_{ij}m_{jy}\\right) =\n\\sum_i\\left(n_{ix}^2R_i + n_{iy}^2R_i\\right),\n\\end{equation}\nwhere $n_{i\\mu}=\\sum_jU_{ij}m_{j\\mu}$. In order\nto apply the gaussian transformation:\n\\begin{equation}\\label{gauss}\ne^{aS^2}=\\frac{1}{\\sqrt{4\\pi a}}\\int_{-\\infty}^{+\\infty}dz e^{-\\frac{z^2}{4a}+Sz}\n\\,\\,\\,\\, a>0\n\\end{equation}\nto each term of the sum in the right hand side of (\\ref{diago}), each\n$R_i$ must be positive. The spectrum can be explicitly\ncomputed using a $d$-dimensional Fourier transform of matrix $R$, the\neigenvalues being labelled by vectors of the reciprocal lattice. These\neigenvalues are trivially related to those of matrix $R'$. A study of the\nspectrum of $R'$ in the limit $N \\rightarrow \\infty$ and for $b=0$ shows\nthat: when $\\alpha > d$ each element of the spectrum converges to a finite\nquantity, the least eigenvalues being negative and of order one in modulus;\nwhen $\\alpha < d$ a part of the spectrum converges to a finite quantity,\nanother part diverges to $+ \\infty$, at most as $\\tilde{N}$. However this\nlast part consists of a\nfraction of the total number of eigenvalues which goes to zero in the limit\n$N \\rightarrow \\infty$. The least eigenvalue is still negative and of order\none in modulus.\nThen part of the spectrum is negative, but it is easily seen that it is\nshifted by $b$. Thus calling $p$ the least eigenvalues of $R'$ for $b=0$\nand choosing\n\\begin{equation}\\label{defbi}\nb=-p + \\epsilon \\quad \\epsilon > 0 \\quad,\n\\end{equation}\nwe have that with this $b$ the whole spectrum of $R'$ (and therefore that\nof $R$) becomes positive. Then for each $i=1,\\cdots,N$,\n$\\mu=x,y$ we can apply (\\ref{gauss}) with the correspondence\n$a\\rightarrow R_i$, $S\\rightarrow n_{i\\mu}$, $z\\rightarrow z_{i\\mu}$.\nPerforming the integrals over variables $\\theta_i$ and using the\ntransformation $z_{i\\mu}=2\\sum_j(UR)_{ij}\\Psi_{j\\mu}$ with Jacobian\n$2^N\\det R$, we can rewrite the partition function as:\n\\begin{eqnarray}\nZ&=&CZ_K\\frac{\\det R}{\\pi^N}\\int_{-\\infty}^{+\\infty} d^N\\Psi_x\n d^N\\Psi_y \\\\\n&& e^{ N\\left[-\\sum_{ij\\mu} \\Psi_{i\\mu} \\frac{R_{ij}}\n {N}\\Psi_{j\\mu} +\\frac{1}{N} \\sum_l \\ln \\left( 2 \\pi I_0\\left( |2 \\sum_j R_{lj}\n \\mathbf{\\Psi}_j+\\mathbf{B}|\\right) \\right) \\right] } \\nonumber\n\\end{eqnarray}\nwhere $I_0$ is the zeroth order modified Bessel function.\nThe isolation of the $N$ factor in the exponential prepares the object\nfor the use of the saddle point method. The quantity in square brackets\nis intensive. Double sums in the first two terms are compensated by\n$R/N=(\\beta/2 N \\tilde{N}) R'$ and the last sum has $1/N$ in front of\nit. The argument of $I_0$ is also intensive because involves a\nterm of the form $\\sum_j R_{lj}=(\\beta / 2 \\tilde{N}) \\sum_j R'_{lj}$.\nIf we call $f(w)$ the function in square brackets, where\n$w=(\\Psi_{1x},\\cdots,\\Psi_{Nx},\\Psi_{1y},\\cdots,\\Psi_{Ny})$, then\nthe application of the method requires the following three conditions:\n$f(w)$ admits a stationary point $w_0$; $w_0$ is a simple stationary\npoint, i.e., $\\det He f|_0\\neq 0$, where $He f|_0$ is the hessian matrix\nof $f$ in $w_0$; the path of integration can be\ndeformed (generally going into ${\\cal C}^{2N}$) into a path that\npasses through $w_0$ following the steepest descent of $f(w)$ and such\nthat $f(w)<f(w_0)$ throughout the all path. If the point $w_0$ is a\nmaximum no deformation is necessary and the method is also called the\nLaplace method. Since, as we show below, $w_0$ is indeed a real-valued\nmaximum, we readily obtain for the free energy per particle $F$:\n\\begin{eqnarray}\\label{fren}\n-\\beta F&=&\\lim_{N\\rightarrow \\infty}\\frac{\\ln Z}{N}=\n\\lim_{N\\rightarrow \\infty} \\{ \\frac{1}{2}\\ln\n\\left( \\frac{2\\pi}{\\beta}\\right) -\\frac{\\beta}{2\\tilde{N}}\\sum_{j}\n\\frac{1}{r_{ij}^\\alpha} \\nonumber \\\\\n&&+ \\max_{w}[f(w)] +\\frac{1}{N}\\ln \\frac{\\det R}{\\sqrt\n{\\det \\left( -\\frac{N}{2} He f|_0\\right)}} \\} \\, .\n\\end{eqnarray}\nThe stationary point $w_0$ is given by the vector\n$(\\Psi_x,\\cdots,\\Psi_x,\\Psi_y,\\cdots,\\Psi_y)$, homogeneous on the lattice\nsites. Defining $\\mathbf{\\Psi}=\n(\\Psi_x,\\Psi_y)$, its direction is that of $\\mathbf{B}$, and its modulus\n$\\Psi$ is given by the solution of:\n\\begin{equation}\\label{psieq}\n\\Psi =\\frac{I_1}{I_0}\\left( \\beta\\left[A\\Psi +h\\right] \\right) \\, ,\n\\end{equation}\nwith\n\\begin{equation}\\label{ascal}\nA=\\frac{1}{\\tilde{N}} \\sum_j R'_{ij}=\\frac{1}{\\tilde{N}} \\left[ b +\n \\sum_{j\\neq i}\\frac{1}{r_{ij}^\\alpha} \\right] \\, ,\n\\end{equation}\nand where $I_1$ is the first order modified Bessel function. In (\\ref{ascal})\n$A$ does not depend on $i$ because of the periodic boundary conditions.\nWe note that when $h=0$ we have\ninfinitely many degenerate solutions, since only the modulus $\\Psi$ is\ndetermined. Evaluation of the elements of the hessian matrix at the\nstationary point gives:\n\\begin{equation}\\label{hesmat}\n\\left. -\\frac{N}{2} \\frac{\\partial^2 f}{\\partial \\Psi_{i\\mu}\n\\partial \\Psi_{j\\nu}}\n\\right|_0 = \nR_{ij} \\delta_{\\mu \\nu} - (R^2)_{ij} g_{\\mu \\nu}(w_0)\n\\end{equation}\nwhere we do not give the explicit expression of $g_{\\mu \\nu}(w_0)$.\nAs we will see shortly, the eigenvalues analysis of the hessian matrix\n(\\ref{hesmat}) shows that the stationary point $w_0$ is a maximum. Then,\nLaplace method applies and Eq. (\\ref{fren}) is valid. However, only in the\nlong range case ($\\alpha<d$) the last term in the rightmost side of\n(\\ref{fren}) is zero; when $\\alpha>d$ its expression does not appear to\nbe manageable. We will comment on this point later. Restricting then to\n$\\alpha<d$, and computing the derivative of (\\ref{fren}) with\nrespect to the magnetic field we find that the magnetization\n$M=\\langle|\\mathbf{M}|\\rangle$ is given by the solution $\\Psi$ of\n(\\ref{psieq}). Then the internal energy $U$ is given by:\n\\begin{equation}\\label{energy}\nU=\\frac{\\partial (\\beta F)}{\\partial \\beta} = \\frac{1}{2\\beta} +\n\\frac{A}{2}(1-M^2) -hM \\, .\n\\end{equation}\nEquations (\\ref{psieq}) and (\\ref{energy}) are the same as those\nof HMF, as soon as a proper $\\tilde{N}$ rescaling gives\n\\begin{equation}\\label{rescal}\nA=\\frac{1}{\\tilde{N}} \\sum_j R'_{ij}=\\frac{1}{\\tilde{N}} \\left[\n b + \\sum_{j\\neq i}\\frac{1}{r_{ij}^\\alpha} \\right] = 1 \\, .\n\\end{equation}\nNow, from equations (\\ref{hesmat}) and (\\ref{rescal}), and calling\n$\\lambda_n$ the eigenvalues of $R'$, we find that, choosing $\\mathbf{B}$\nalong one of the coordinate axes, the eigenvalues of the hessian matrix\nat the stationary point are given by:\n\\begin{eqnarray}\\label{eighes}\n\\chi^{(1)}_n&=&\\frac{\\beta}{2}\\frac{\\lambda_n}{\\tilde{N}}\\left[1-\n\\left(\\beta-\\Psi^2\\beta-\\frac{\\Psi}{\\Psi+h}\\right)\n\\frac{\\lambda_n}{\\tilde{N}}\\right] \\\\\n\\chi^{(2)}_n&=&\\frac{\\beta}{2}\\frac{\\lambda_n}{\\tilde{N}}\\left[1-\n\\frac{\\Psi}{\\Psi+h}\\frac{\\lambda_n}{\\tilde{N}}\\right] \\quad \\quad\nn=1,\\cdots,N\n\\nonumber\n\\end{eqnarray}\nFollowing our previous analysis we have that:\n\\begin{equation}\n\\frac{\\epsilon}{\\tilde{N}}\\leq \\frac{\\lambda_n}{\\tilde{N}}\\leq 1 \\, .\n\\end{equation}\nThen we immediately see that $\\chi^{(2)}_n$ are all positive for any\n$\\beta$ and $h$; for $\\chi^{(1)}_n$ we need to include $\\Psi (\\beta,h)$ from\n(\\ref{psieq}). We have checked numerically that the quantity in round brackets\nin (\\ref{eighes}) is always smaller than $1$, and therefore $\\chi^{(1)}_n$\nare also all positive. From (\\ref{eighes}) we can derive an expression for\nthe determinant of matrix (\\ref{hesmat}). It is given by:\n\\begin{eqnarray}\\label{dethes}\n\\frac{1}{N}&&\\ln \\det \\left( -\\frac{N}{2} He f|_0\\right)=\\frac{2}{N}\\ln \\det R\n\\nonumber \\\\ +&&\\frac{1}{N}\\sum_{n=1}^N \\{ \\ln \\left[1-\n\\left(\\beta-\\Psi^2\\beta-\\frac{\\Psi}{\\Psi+h}\\right)\n\\frac{\\lambda_n}{\\tilde{N}}\\right]\\nonumber \\\\ +&&\\ln \\left[1-\n\\frac{\\Psi}{\\Psi+h} \\frac{\\lambda_n}{\\tilde{N}}\\right]\n\\} \\, .\n\\end{eqnarray}\nWhen $\\alpha<d$ most of $\\frac{\\lambda_n}{\\tilde{N}}$ go to zero for\n$N\\rightarrow \\infty$, then the sum in (\\ref{dethes}) is\neffectively constituted by the terms with the remaining\n$\\frac{\\lambda_n}{\\tilde{N}}$. These terms are a fraction of $N$ that, as\nwe already pointed out, goes to zero when $N\\rightarrow \\infty$. If we call\n$N'(N)$ this fraction, then the sum in (\\ref{dethes}) can be bounded\nfrom above by $\\frac{N'}{N}c \\rightarrow 0$ for $N\\rightarrow \\infty$, where\n$c$ is a finite number. Therefore the last term in (\\ref{fren}) is zero.\nWhen $\\alpha>d$ all terms contribute to the sum in (\\ref{dethes}), and we\ncan not give a meaningful expression for (\\ref{fren}). At the end of the\ncalculations we can let $\\epsilon \\rightarrow 0$ in (\\ref{defbi}).\n\nThen we have shown that any model with $\\alpha<d$ on any lattice is\nequivalent to HMF. From (\\ref{rescal}) we get an exact expression\nfor $\\tilde{N}$:\n\\begin{equation}\\label{exactn}\n\\tilde{N} = -p + \\sum_{j\\neq i}\\frac{1}{r_{ij}^\\alpha} \\, .\n\\end{equation}\nWe have made a microcanonical simulation of Hamiltonian\n(\\ref{num3}) on a threedimensional simple cubic lattice\nin zero magnetic field, using a fourth\norder simplectic algorithm \\cite{yosh} with time step $0.02$, selected to have\nrelative energy fluctuations not exceeding $1/10^6$. We have chosen a fixed\n$N=343=7^3$, and have simulated various energy densities\n$H/N$ and various $\\alpha<3$. In Fig. 1 we show that the numerical\ncaloric curves collapse onto the universal HMF curve. The kind of results\nshown in \\cite{TA} for a onedimensional lattice, where a slightly\ndifferent $\\tilde{N}$ has been used.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm]{fig1.eps}\n\\end{center}\n\\caption{The full line gives the canonical theoretical caloric curve\n(temperature $T$ $vs$ energy density $U$) for\nlong range rotators compared with the microcanonical simulation of a\nthreedimensional simple cubic lattice for three different $\\alpha$ values:\n$0.75$ (open circles), $1.5$ (diamonds) and $2.25$ (crosses). Note that\nin spite of the size of the system, still not very large (side with $7$\nlattice sites), the results already follow very well the theoretical\ncurve.}\n\\label{Fig. 1}\n\\end{figure}\n\n\\section{Conclusions}\nGoing back to the beginning of our discussion:\nit is now clear that model (\\ref{num2}) completely reduces to\nmodel (\\ref{num1}) for $\\alpha=0$. In model (\\ref{num1}) the range of the\ninteractions is infinite; each rotator interacts with\nall the others and with the same intensity. To get\na well defined TL it is sufficient to divide\n$V$ in (\\ref{num1}) by $N$, the total numbers of rotators.\nIt is then possible to compute caloric and magnetization\ncurves \\cite{AR}; the spatial arrangement of the rotators has no effect on\nthem since the intensity of the interaction is the same for each couple\nof rotators. In this work\nwe have shown that, when considering model (\\ref{num2}), it is possible\nto take into account the spatial $d$-dimensional arrangement\nof the rotators and the decaying of their mutual interaction\nthrough a factor $\\tilde{N}$, which is computable for any\nperiodic lattice and any $\\alpha < d$. Dividing by $\\tilde{N}$\nthe potential energy in (\\ref{num2}), the model gets a well defined\nTL and it is possible to compute state curves which become those\nof the HMF model with a proper normalization of the constant $A$\nin (\\ref{rescal}). The HMF ($\\alpha =0$) model has revealed peculiar \nequilibrium and nonequilibrium properties \\cite{LRR}, namely:\nensemble inequivalence, metastability, collective oscillations,\nanomalous diffusion and interesting chaotic properties, both in\nthe ferromagnetic and antiferromagnetic case. On the basis of\nthe thermodynamical equivalence here established it would be\ninteresting to investigate the $\\alpha$ dependence of all these\nproperties. The study of the Lyapounov exponents in \\cite{AT} \nis the first in this direction.\n\\section{Aknowledgments}\nA. G. warmly thanks C. Tsallis for having suggested the study\nof the long range interacting rotators.\n\n\n\\begin{thebibliography}{9}\n\\bibitem{ruelle}\nD. Ruelle, {\\it Statistical mechanics: rigorous results},\n(Addison-Wesley, New York 1989).\n\\bibitem{LRR}\nV. Latora, A. Rapisarda, and S. Ruffo, cond-mat/0001010,\nto appear in Prog. Theor. Phys. Supplement.\n\\bibitem{AR}\nM. Antoni and S. Ruffo, Phys. Rev. E {\\bf 52}, 2361 (1995).\n\\bibitem{AT}\nC. Anteneodo and C. Tsallis, Phys. Rev. Lett. {\\bf 80}, 5313 (1998).\n\\bibitem{TA}\nF. Tamarit and C. Anteneodo, Phys. Rev. Lett. {\\bf 84}, 208 (2000).\n\\bibitem{yosh}\nH. Yoshida, Phys. Lett. A {\\bf 150}, 262 (1990).\n\\end{thebibliography}\n\\end{document}\n\n\n"
}
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[
{
"name": "cond-mat0002168.extracted_bib",
"string": "\\begin{thebibliography}{9}\n\\bibitem{ruelle}\nD. Ruelle, {\\it Statistical mechanics: rigorous results},\n(Addison-Wesley, New York 1989).\n\\bibitem{LRR}\nV. Latora, A. Rapisarda, and S. Ruffo, cond-mat/0001010,\nto appear in Prog. Theor. Phys. Supplement.\n\\bibitem{AR}\nM. Antoni and S. Ruffo, Phys. Rev. E {\\bf 52}, 2361 (1995).\n\\bibitem{AT}\nC. Anteneodo and C. Tsallis, Phys. Rev. Lett. {\\bf 80}, 5313 (1998).\n\\bibitem{TA}\nF. Tamarit and C. Anteneodo, Phys. Rev. Lett. {\\bf 84}, 208 (2000).\n\\bibitem{yosh}\nH. Yoshida, Phys. Lett. A {\\bf 150}, 262 (1990).\n\\end{thebibliography}"
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cond-mat0002169
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Boundary-induced phase transitions in traffic flow
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[
{
"author": "V. Popkov$^{1,2}$"
},
{
"author": "L. Santen$^{3}$"
},
{
"author": "A. Schadschneider$^{3}$"
},
{
"author": "and G. M. Sch\\\"utz $^{1}$"
}
] |
Boundary-induced phase transitions are one of the surprising phenomena appearing in nonequilibrium systems. These transitions have been found in driven systems, especially the asymmetric simple exclusion process. However, so far no direct observations of this phenomenon in real systems exists. Here we present evidence for the appearance of such a nonequilibrium phase transition in traffic flow occurring on highways in the vicinity of on- and off-ramps. Measurements on a German motorway close to Cologne show a first-order nonequilibrium phase transition between a free-flow phase and a congested phase. It is induced by the interplay of density waves (caused by an on-ramp) and a shock wave moving on the motorway. The full phase diagram, including the effect of off-ramps, is explored using computer simulations and suggests means to optimize the capacity of a traffic network.
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[
{
"name": "prl_submit.tex",
"string": "\\documentstyle[prl,aps,floats,epsf,psfig]{revtex}\n\n\\def\\ADD#1{{\\bf{#1}}\\marginpar{$\\longleftarrow$ {\\bf ADD!}}}\n\n\\newcommand{\\bs}{\\bigskip}\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\bel}[1]{\\begin{equation}\\label{#1}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\bea}{\\begin{eqnarray}}\n\\newcommand{\\ba}{\\begin{array}}\n\\newcommand{\\eea}{\\end{eqnarray}}\n\\newcommand{\\ea}{\\end{array}}\n\\newcommand{\\hfour}{\\hspace*{4mm}}\n\\newcommand{\\bra}[1]{\\mbox{$\\langle \\, {#1}\\, |$}}\n\\newcommand{\\ket}[1]{\\mbox{$| \\, {#1}\\, \\rangle$}}\n\\newcommand{\\exval}[1]{\\mbox{$\\langle \\, {#1}\\, \\rangle$}}\n\n\n\\begin{document}\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse%\n\\endcsname\n\n\\title{Boundary-induced phase transitions in traffic flow}\n% %\n\\author{V. Popkov$^{1,2}$, L. Santen$^{3}$, A. Schadschneider$^{3}$,\nand G. M. Sch\\\"utz $^{1}$}\n\n\\address{$1$ Institut f\\\"ur Festk\\\"orperforschung, Forschungszentrum\nJ\\\"ulich, 52425 J\\\"ulich, Germany}\n\\address{$2$ Institute for Low Temperature Physics, 310164 Kharkov, Ukraine}\n\\address{$3$ Laboratoire de Physique Statistique, Ecole Normale\nSup{\\'{e}}rieure, 24, rue Lhomond, F-75231 Paris Cedex 05, France}\n\\address{$4$ Institut f\\\"ur Theoretische Physik, Universit\\\"at zu K\\\"oln,\nD-50937 K\\\"oln}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}Boundary-induced phase transitions are one of the \nsurprising phenomena appearing in nonequilibrium systems. \nThese transitions have been found in driven systems, especially\nthe asymmetric simple exclusion process.\nHowever, so far no direct observations of this phenomenon in real\nsystems exists. \nHere we present evidence for the appearance of such a nonequilibrium\nphase transition in traffic flow occurring on highways in the\nvicinity of on- and off-ramps. Measurements on a German motorway close to\nCologne show a first-order nonequilibrium phase transition between \na free-flow phase and a congested phase. It is induced by \nthe interplay of density waves (caused by an on-ramp) and a shock wave \nmoving on the motorway. The full phase diagram, including the effect of \noff-ramps, is explored using computer simulations and suggests means\nto optimize the capacity of a traffic network.\n\\end{abstract}\n\\pacs{PACS numbers: 05.40.+j, 82.20.Mj, 02.50.Ga}\n]\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nOne-dimensional physical systems with short-ranged interactions in thermal\nequilibrium do not exhibit phase transitions. This is no longer true if the\naction of external forces sets up a steady mass transport and drives the \nsystem out of equilibrium. Then boundary conditions, usually insignificant \nfor an equilibrium system, can induce nonequilibrium phase transitions. \nMoreover, such phase transitions may occur in a rather wide class of \ndriven complex systems, including biological and sociological mechanisms \ninvolving many interacting agents. \nIn spite of the importance of this phenomenon and a number of theoretical \nstudies \\cite{Krug91,Gunter93,Oerd98,Kolo98}, it has never been observed \ndirectly. So far only indirect experimental evidence for a \nboundary-induced phase transition exists in older studies of the \nkinetics of biopolymerization on nucleic acid templates\n\\cite{MacD69,Schu97}.\n\nIn the present work we report the first direct observation of a\nboundary-induced phase transition in traffic flow.\nAnalysis of traffic data sets\ntaken on a motorway near Cologne exhibits transitions from free-flow \ntraffic to congested traffic, caused by a boundary effect, viz. the\npresence of an on-ramp. These transitions are characterized by a\ndiscontinuous reduction of the average speed \\cite{Neub99}.\n\nVehicular traffic on a motorway is controlled by a mixture of bulk and\nboundary effects caused by on- and off-ramps, varying number of lanes, speed\nlimits, leaving aside temporary effects of changing weather\nconditions and other non-permanent factors.\nThe fundamental characteristic of the bulk motion is the stationary\nflow-density diagram, i.e.\\ the fundamental diagram,\nwhich incorporates the collective\neffects of individual drivers behavior such as trying to\nmaintain an optimal high speed while keeping safety precautions.\nThe qualitative shape\nof the flow-density diagram $j(\\rho)$ is largely independent of the precise\ndetails of the road and hence amenable to numerical analysis using either\nstochastic lattice gas models or partial differential equations\n\\cite{CSS99,Hell}.\n\nA by now well-established lattice gas model for traffic\nflow, the cellular automaton model by Nagel and Schreckenberg \\cite{NS},\nreproduces empirical traffic data \\cite{Hall86} rather well. \nFig.~\\ref{fig_fund} shows simulation data for the fundamental \ndiagram obtained from the Nagel-Schreckenberg (NaSch) model.\nThis has to be compared to\nmeasurements of the flow $j$ taken with the help of detectors on the\nmotorway A1 near Cologne which show a maximum of about 2000 vehicles/hour\nat a density of about $\\rho^\\ast= 20$ vehicles per lane and km \\cite{Neub99}.\nAt densities below $\\rho^\\ast$ one observes free flow, while\nfor larger densities one observes congested traffic.\n\\begin{figure}[h]\n\\centerline{\\psfig{figure=fig1.ps,height=5cm}}\n\\caption{\\protect{Fundamental diagram (flow-density-relation) as modeled by\nthe NaSch model with $v_{max}=4, \\ p=0.25$,\ntime step =$1.0$ sec, lattice spacing =$7.5$ m. The system has 3200 sites,\nand the flux is averaged over $10^6$ lattice updates. The\nbroken and full lines indicate the slope which defines the \ncollective velocity of spontaneous local traffic jams and the shock\nvelocity, respectively.}}\n\\label{fig_fund}\n\\end{figure}\n\nIn addition to the density dependence of the flow\ntwo important characteristics are derived directly from the fundamental\ndiagram:\nthe shock velocity of a `domain wall' \\cite{dowalldef}\nbetween two stationary regions of densities $\\rho^-,\\rho^+$\n\\be\nv_{shock} = \\frac{j(\\rho^+) - j(\\rho^-)}{\\rho^+ - \\rho^-},\n\\label{v_shock}\n\\ee\nobtained from mass conservation, and the collective velocity\n\\bel{v_c}\nv_c = \\frac{\\partial j(\\rho) }{\\partial \\rho}\n\\ee\nwhich is the velocity of the center of mass of a local perturbation\nin a homogeneous, stationary background of density $\\rho$.\nBoth velocities are readily observed in real traffic.\nThe collective velocity $v_c$ describes the upstream movement of a local,\ncompact jam. In the density range $25 \\dots 130$ cars/km\n$v_c$ ranges from approximately $-10$ km/hour to $-20$ km/hour \n(Fig.~\\ref{fig_fund}) which has to be\ncompared with the empirically observed value $v\\approx -15$ km/hour\n\\cite{NS,KR_Rapid}.\nThe shock velocity is the velocity of the\nupstream front of an extended, stable traffic jam.\nThe formation of a stable shock is usually a boundary-driven\nprocess, caused by a `bottleneck' on a road. Bottlenecks on a highway\narise from a reduction in the number of lanes and from on-ramps where\nadditional cars enter a road \\cite{Daganzo,schorsch}.\n\nThe experimental data considered here (see\nFig.~\\ref{fig_ramp} for the relevant part of the highway) show\nboundary effects caused by the presence of an on-ramp.\nFar upstream from the on-ramp, free flow of \nvehicles with density $\\rho^{-}$ and flow $j^-\\equiv j(\\rho^{-})$\nis maintained. Just before the on-ramp the vehicle density is $\\rho^{+}$ \nwith corresponding flow $j^+\\equiv j(\\rho^{+})$. Note that no experimental\ndata are available for $\\rho^-$, $j^-$ and $\\rho^+$, $j^+$ as well as \nthe activity of the ramp. The only data come from a detector located\nupstream from the on-ramp \\cite{detector_distance} which measures a \ntraffic density $\\hat{\\rho}$ and the corresponding flow $\\hat{\\jmath}$.\n\\begin{figure}[h]\n\\centerline{\\psfig{figure=fig2.ps,height=3.4cm}}\n\\caption{\\protect{Schematic road design of a highway with an on-ramp\nwhere cars enter the road. The arrows indicate the direction of the flow.\nThe detector measures the local bulk density $\\hat{\\rho}$ and bulk flow\n$\\hat{\\jmath}$.}}\n\\label{fig_ramp}\n\\end{figure}\n\nNext the effects of the on-ramp are considered.\nCars entering the motorway cause the mainstream of vehicles\nto slow down locally. Therefore, the vehicle density\njust before the on-ramp increases to $\\rho^{+}>\\rho^{-} $.\nThen a shock, formed at the on-ramp, will propagate\nwith {\\em mean} velocity $v_{shock}$ (see (\\ref{v_shock})).\nDepending on the sign of $v_{shock}$, two scenarios are possible:\\\\\n1) $v_{shock}>0$ (i.e. $j^{+}> j^{-}$): In this case\nthe shock propagates (on average) downstream towards the on-ramp. \nOnly by fluctuations a brief upstream motion is possible. \nTherefore the detector will measure a traffic density \n$\\hat{\\rho}=\\rho^{-}$ and flow $\\hat{\\jmath}= j^{-}$.\\\\\n2) $v_{shock}<0$ (i.e. $j^{+}< j^{-}$): \nThe shock wave starts propagating with the mean velocity\n$v_{shock}$ upstream, thus expanding the congested traffic region with\ndensity $\\rho^{+}$. The detector will now measure $\\hat{\\rho}=\\rho^{+}$ \nand flow $\\hat{\\jmath}= j^{+}$.\n\nLet us now discuss the transition between these two scenarios.\nSuppose one starts with a situation where $j^{+}> j^{-}$\nis realized. If now the far-upstream-density $\\rho^{-}$ increases \nit will reach a critical point $\\rho_{crit} < \\rho^\\ast$ above which\n$j^{-}> j^{+}$,\ni.e., the free flow upstream $j^{-}$ prevails over the flow $j^{+}$ which \nthe `bottleneck', i.e. the on-ramp, is able to support.\nAt this point shock wave velocity $v_{shock}$ will change sign\n(see (\\ref{v_shock})) and the shock starts traveling upstream.\nAs a result, the stationary bulk density $\\hat{\\rho}$ measured by\nthe detector upstream from the on-ramp will change discontinuously \nfrom the critical value $\\rho_{crit}$ to $\\rho^{+}$. This marks a\nnonequilibrium phase transition of first order with respect\nto order parameter $\\hat{\\rho}$. The discontinuous change \nof $\\hat{\\rho}$ leads also an abrupt reduction of the local velocity.\nNotice that the flow $\\hat{\\jmath}=j^{+}$\nthrough the on-ramp (then also measured by the detector) will stay\n{\\em independent} of the free flow upstream from the congested \nregion $j^{-}$ as long as the condition $j^- > j^+$ holds. \n\nEmpirically this phenomenon can be seen in the traffic data taken from \nmeasurements at the detector D1 on the motorway A1 close to Cologne \n\\cite{Neub99}. Fig.~\\ref{fig_time} shows a typical time series of the \none-minute velocity averages. One can clearly see the sharp drop of the \nvelocity at about 8 a.m.\n\n\\begin{figure}[h]\n\\centerline{\\psfig{figure=fig5.ps,height=5.3cm}}\n\\caption{\\protect{Time series of the velocity. Each data point represents \nan one-minute average of the speed. Shown are empirical data \n(from [7]) in comparison with the results of computer simulations\nof a simplified model (see text).}}\n\\label{fig_time}\n\\end{figure}\n\nAlso the measurements of the flow versus local density,\ni.e.\\ the fundamental diagram (Fig.~\\ref{fig_measur}), support our\ninterpretation. Two branches can be distinguished. \nThe increasing part corresponds to an almost linear rise of the flow \nwith density in the free-flow regime \\cite{KR_Rapid}. \nIn accordance with our considerations this part of the flow diagram is \nnot affected by the presence of the on-ramp at all and one \nmeasures $\\hat{\\jmath} = j^{-}$ which is the actual upstream flow. \nThe second branch are measurements taken during congested traffic hours, \nthe transition period being omitted for better statistics. The transition \nfrom free flow to congested traffic is characterized by a discontinuous \nreduction of the local velocity. However, as predicted above\nthe flow does not change significantly in the congested regime.\nIn contrast, in local measurements large density fluctuations can\nbe observed. Therefore in this regime the density does not take the \nconstant value $\\rho^{+}$ as suggested by the argument given above,\nbut varies from 20 veh/km/lane to 80 veh/km/lane (see Fig.~\\ref{fig_measur}).\n\nOne should stress here that congested traffic data are usually not easy\nto interpret, because the traffic conditions (mean inflow and outflow of cars\non the on- and off-ramps, and so the bulk mean flow) are changing in time.\nAccording to our arguments, in a congested regime the detector\nmeasures $j^{+}$, {\\it solely} due to the on-ramp activity.\nTherefore, $j^{+}(t) = j(\\rho^{+}(t)) < j^{-} (t)$ must be satisfied.\nDuring times of very dense traffic one expects always cars ready to\nenter the motorway at the on-ramp, thus guaranteeing\na sufficient and approximately constant on-ramp activity.\nThe measured flow is constant over long periods of time which is\nin agreement with the notion that the transition is due a stable\ntraffic jam. Spontaneously emerging and decaying jams would lead to the\nobservation of a non-constant flow.\n\n\\begin{figure}[h]\n\\centerline{\\psfig{figure=fig3.eps,height=5.5cm}}\n\\caption{\\protect{Measurements of the flow versus local density before an\non-ramp on the motorway A1 close to Cologne. The detector is located at a\ndistance 1 km upstream from on-ramp. Data are collected over a period of 12\ndays.}}\n\\label{fig_measur}\n\\end{figure}\n\nThe use of our approach is not limited to a qualitative explanation of \nthe traffic data. Beyond that it can also be used to calculate the \nphase diagrams of systems with open boundary conditions for a large \nclass of traffic models.\nWe modeled a section \nof a road with on-ramp on the left and off-ramp (on-ramp) on the right \nusing the NaSch cellular automaton \\cite{NS}.\nWe modify the basic model by using open boundary conditions with injection \nof cars at the left boundary (corresponding to in-flow into the road \nsegment) and removal of cars at the right boundary (corresponding to \noutflow). Therefore it can also be regarded as a generalization\nof the asymmetric simple exclusion process \\cite{ASEP} to particles\nwith higher velocity.\n\nDuring the simulations local measurements of the velocity have\nbeen performed analogous to the experimental setup.\nFor comparison the results of the computer simulations have been\nincluded in Fig.~\\ref{fig_time}. Note that even the quantitive\nagreement with the empirical data is very good. This has been achieved \nby using a finer discretization of the model, i.e.\\ the \nlength of the cell is considered as $l=2.5m$. The results were\nobtained for $L=960$, $p=0.25$ and $v_{max}=13$. We kept the input \nprobability $\\alpha = 0.65$ constant. Then the free-flow part is \nobtained using $\\beta = 1.$ and the congested part using \n$\\beta = 0.55$. The transition was observed at ten minutes after \nwe reduced the output probability. The ``detector'' was located at \nthe link from site $480$ to $481$.\n\nFig.~\\ref{fig_phase} shows the full phase diagram of the NaSch model\nwith open boundary conditions. It describes the stationary bulk \ndensity $\\hat{\\rho}$ as a function of the far-upstream in-flow boundary \ndensity $\\rho^-$ and the effective right boundary density $\\rho^+$. \nFor the case of an on-ramp (or shrinking road width etc.) at the right \nboundary corresponds to the situation discussed above. Here, the density \nis increased locally to $\\rho^+ >\\rho^-$. In agreement with the \nempirical observation we find a line of first order transitions \nfrom a free flow (FF) phase with bulk density $\\hat{\\rho}=\\rho^-$ to a \ncongested (CT) phase with $\\hat{\\rho}=\\rho^+$. On this line \n$v_{shock}$ changes sign.\n\n\n\\begin{figure}[h]\n\\centerline{\\psfig{figure=fig4.ps,height=5.3cm}}\n\\caption{\\protect{Phase diagram of the NaSch model with open\nboundaries for $p=0.25, v_{max}=4$. Cars enter the road from a reservoir of\ndensity $\\overline{\\rho^{-}}$, inducing the upstream-density $\\rho^-$ discussed\nin the text. At the right boundary cars leave the road into a reservoir of\ndensity $\\overline{\\rho^{+}}$, leading to the on-ramp density $\\rho^+$.\nThe solid (dashed) curve denotes the theoretical prediction for\nthe first (second) order transitions lines obtained from the\nnumerically determined flow-density relation. The points\nrepresent phase transition points for the simulated system of size 3200.\nThe phases are: free flow (FF), congested traffic (CT), maximal flow (MF)\nphase.}}\n\\label{fig_phase}\n\\end{figure}\n\nThe case of an off-ramp (or expansion of road space etc.) leads to a \nlocal decrease $\\rho^+ <\\rho^-$ of the density. Here the collective \nvelocity $v_c $ (\\ref{v_c}) plays a prominent role. \nAs long as $v_c$ is positive (i.e. in the free-flow regime\n$\\rho^-< \\rho^\\ast$, see Fig.~\\ref{fig_fund}), perturbations caused by a \nsmall increase of the upstream boundary density $\\rho^-$ gradually spread\ninto the bulk, rendering $\\hat{\\rho}=\\rho^-$ (FF regime).\nAt $\\rho^-= \\rho^\\ast$, $v_c$ changes sign \\cite{negative_v_c}\nand an overfeeding effect occurs: a perturbation from the upstream\nboundary does not spread into the bulk \\cite{Gunter93,Kolo98} and therefore \nfurther increase of the upstream boundary density does not increase the \nbulk density. The system enters the maximal flow (MF) phase with\nconstant bulk density $\\hat{\\rho}=\\rho^\\ast$ and flow\n$j(\\rho^\\ast)=j_{max}$. The transition to the MF phase is of second \norder, because $\\hat{\\rho}$ changes continuously across the phase\ntransition point.\n\nThe existence of a maximal flow phase was not emphasized in the\ncontext of traffic flow up to now. At the same time, it is the most\ndesirable phase, carrying the maximal possible throughput of vehicles\n$j_{max}$. \nFor practical purposes our observations may directly be used in\norder to operate a highway in the optimal regime. E.g.\\ the flow near a\nlane reduction could be increased significantly if the traffic state\nat the entry would allow to attain the maximal possible flow \nof the bottleneck. This could be achieved by controlling the density \nfar upstream, e.g.\\ by closing temporally an on-ramp, such \nthat the cars still enter the bottleneck with high velocity.\n\nWe stress that the stationary phase diagram Fig.~\\ref{fig_phase} is \ngeneric in the sense that it is determined solely by the macroscopic\nflow-density relation.\nThe number of lanes of the road, the distribution of individual\noptimal velocities, speed limits, and other details enter only\nin so far as they determine the exact values characterizing the flow-density\nrelation for that particular road.\nWe also note that throughout the paper we assumed the external \nconditions to vary slowly, so that the system has enough time to readjust \nto its new stationary state. Experimenting with different cellular\ntraffic models in a real time scale shows that the typical time to reach a\nstationary state in a road segment of about 1.2 km is\nof the order of 3-5 min,\nwhich is reasonably small. Close to phase transitions lines,\nhowever, where the shock velocity vanishes, this time diverges\nand intrinsically non-stationary dynamic phenomena \\cite{SH,RH} take\nthe lead.\n\nIn conclusion, we have shown that traffic data collected on German\nmotorways provide evidence for a boundary-induced nonequilibrium phase\ntransition of first order from the free flowing to the congested\nphase. The features of this phenomenon are readily understood in terms\nof the flow-density diagram. The dynamical mechanism leading to this\ntransition is an interplay of shocks and local fluctuations caused by\nan on-ramp. Full investigation of a cellular automaton model for\ntraffic flow reproduces this phase transition, but also exhibits a\nricher phase diagram with an interesting maximal flow phase.\nThese results are not only important from the point of view of\nnonequilibrium physics, but also suggest new mechanisms\nof traffic control.\n\n\n{\\bf Acknowledgments}: We thank Lutz Neubert for useful discussions and\nhelp in producing Figs.~\\ref{fig_time} and\n\\ref{fig_measur}. L.~S. acknowledges support from the Deutsche\nForschungsgemeinschaft under Grant No. SA864/1-1.\n\n\n\\bibliographystyle{unsrt}\n\\begin{thebibliography}{99}\n\n\\bibitem{Krug91}\nKrug, J., Phys. Rev. Lett. {\\bf 67}, 1882 (1991).\n\n\\bibitem{Gunter93}\n Sch\\\"utz, G. and Domany, E., J. Stat. Phys. {\\bf 72}, 277 (1993).\n\n\\bibitem{Oerd98}\nOerding, K., and Janssen, H.K., Phys. Rev. E {\\bf 58}, 1446 (1998).\n\n\\bibitem{Kolo98}\nKolomeisky, A.B., Sch\\\"utz, G.M., Kolomeisky, E.B. and Straley, J.P.,\nJ.Phys. A {\\bf 31}, 6911 (1998).\n\n\\bibitem{MacD69}\nMacDonald, J.T. and Gibbs J.H., Biopolymers {\\bf 7}, 707 (1969).\n\n\\bibitem{Schu97}\nSch\\\"utz, G.M., Int. J. Mod. Phys. B {\\bf 11}, 197 (1997).\n\n\\bibitem{Neub99}\n Neubert, L., Santen, L., Schadschneider, A. and Schreckenberg, M.,\n Phys. Rev. E {\\bf 60}, 6480 (1999).\n\n\\bibitem{CSS99}\nChowdhury, D., Santen, L., and Schadschneider, A.,\nCurr. Sci. {\\bf 77}, 411 (1999) and Physics Reports (in press).\n\n\n\\bibitem{Hell}\nHelbing, D., {\\em Verkehrsdynamik: Neue Physikalische Modellierungskonzepte}\n(in German) (Springer, Berlin, 1997).\n\n\n\\bibitem{NS}\nNagel, K. and Schreckenberg, M., J. Phys. I France {\\bf 2}, 2221 (1992).\n\n\\bibitem{Hall86}\nHall, F.L., Allen, B.L. and Gunter, M.A., Transp. Res. A {\\bf 20}, 197 (1986).\n\n\\bibitem{dowalldef}\nIn nonequilibrium systems a domain wall is an object connecting\ntwo possible stationary states.\n\n\\bibitem{KR_Rapid} Kerner, B.S. and Rehborn, H., Phys. Rev. E {\\bf 53},\n R4275 (1997). For the situation studied in this paper the front\nvelocity of the jam can be identified with its center-of-mass velocity.\n\n\n\\bibitem{Daganzo} Daganzo, C.F., Cassidy, M.J. and Bertini R.L.,\nTransp. Res. A {\\bf 33}, 365 (1999).\n\n\\bibitem{schorsch} G.\\ Diedrich, L.\\ Santen, A.\\ Schadschneider and\nJ.\\ Zittartz, Int.\\ J.\\ Mod.\\ Phys.\\ C (in press)\n\n\\bibitem{detector_distance} The distance between the detector and the on-ramp\nshould be large enough, so that the on-ramp fluctuations are not\n measured directly. In our case,\nthe detector is located approximately 1 km upstream from on-ramp.\n\n\\bibitem{ASEP} see e.g.\\ G.M. Sch\\\"utz, {\\it Exactly solvable models for\nmany-body systems far from equilibrium}, to appear in {\\it Phase Transitions\nand Critical Phenomena}, C. Domb und J. Lebowitz (eds.), \n(Academic Press, London, 2000); T.M. Liggett, {\\it Stochastic Interacting\nSystems: Contact, Voter and Exclusion Processes} (Springer, Berlin, 1999);\nand the contributions by B.\\ Derrida, M.R\\ Evans and S.A.~Janowsky,\nJ.L.~Lebowitz in V. Privman (ed.), {\\it Nonequilibrium Statistical \nMechanics in One Dimension}, (Cambridge University Press, Cambridge, 1997).\n\n\n\\bibitem{negative_v_c} In this case the upstream\nentrance to the road itself acts as a `dynamical' bottleneck with maximal\ncapacity $j_{max}$.\n\n\\bibitem{SH} Kerner, B.S. and Rehborn, H., Phys. Rev. Lett. {\\bf 79},\n 4030 (1997).\n\n\\bibitem{RH} Lee, H.Y., Lee, H.-W. and Kim, D.,\nPhys. Rev. Lett. {\\bf 81}, 1130 (1998).\n\n\\end{thebibliography}\n\\end{document}\n\n"
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[
{
"name": "cond-mat0002169.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Krug91}\nKrug, J., Phys. Rev. Lett. {\\bf 67}, 1882 (1991).\n\n\\bibitem{Gunter93}\n Sch\\\"utz, G. and Domany, E., J. Stat. Phys. {\\bf 72}, 277 (1993).\n\n\\bibitem{Oerd98}\nOerding, K., and Janssen, H.K., Phys. Rev. E {\\bf 58}, 1446 (1998).\n\n\\bibitem{Kolo98}\nKolomeisky, A.B., Sch\\\"utz, G.M., Kolomeisky, E.B. and Straley, J.P.,\nJ.Phys. A {\\bf 31}, 6911 (1998).\n\n\\bibitem{MacD69}\nMacDonald, J.T. and Gibbs J.H., Biopolymers {\\bf 7}, 707 (1969).\n\n\\bibitem{Schu97}\nSch\\\"utz, G.M., Int. J. Mod. Phys. B {\\bf 11}, 197 (1997).\n\n\\bibitem{Neub99}\n Neubert, L., Santen, L., Schadschneider, A. and Schreckenberg, M.,\n Phys. Rev. E {\\bf 60}, 6480 (1999).\n\n\\bibitem{CSS99}\nChowdhury, D., Santen, L., and Schadschneider, A.,\nCurr. Sci. {\\bf 77}, 411 (1999) and Physics Reports (in press).\n\n\n\\bibitem{Hell}\nHelbing, D., {\\em Verkehrsdynamik: Neue Physikalische Modellierungskonzepte}\n(in German) (Springer, Berlin, 1997).\n\n\n\\bibitem{NS}\nNagel, K. and Schreckenberg, M., J. Phys. I France {\\bf 2}, 2221 (1992).\n\n\\bibitem{Hall86}\nHall, F.L., Allen, B.L. and Gunter, M.A., Transp. Res. A {\\bf 20}, 197 (1986).\n\n\\bibitem{dowalldef}\nIn nonequilibrium systems a domain wall is an object connecting\ntwo possible stationary states.\n\n\\bibitem{KR_Rapid} Kerner, B.S. and Rehborn, H., Phys. Rev. E {\\bf 53},\n R4275 (1997). For the situation studied in this paper the front\nvelocity of the jam can be identified with its center-of-mass velocity.\n\n\n\\bibitem{Daganzo} Daganzo, C.F., Cassidy, M.J. and Bertini R.L.,\nTransp. Res. A {\\bf 33}, 365 (1999).\n\n\\bibitem{schorsch} G.\\ Diedrich, L.\\ Santen, A.\\ Schadschneider and\nJ.\\ Zittartz, Int.\\ J.\\ Mod.\\ Phys.\\ C (in press)\n\n\\bibitem{detector_distance} The distance between the detector and the on-ramp\nshould be large enough, so that the on-ramp fluctuations are not\n measured directly. In our case,\nthe detector is located approximately 1 km upstream from on-ramp.\n\n\\bibitem{ASEP} see e.g.\\ G.M. Sch\\\"utz, {\\it Exactly solvable models for\nmany-body systems far from equilibrium}, to appear in {\\it Phase Transitions\nand Critical Phenomena}, C. Domb und J. Lebowitz (eds.), \n(Academic Press, London, 2000); T.M. Liggett, {\\it Stochastic Interacting\nSystems: Contact, Voter and Exclusion Processes} (Springer, Berlin, 1999);\nand the contributions by B.\\ Derrida, M.R\\ Evans and S.A.~Janowsky,\nJ.L.~Lebowitz in V. Privman (ed.), {\\it Nonequilibrium Statistical \nMechanics in One Dimension}, (Cambridge University Press, Cambridge, 1997).\n\n\n\\bibitem{negative_v_c} In this case the upstream\nentrance to the road itself acts as a `dynamical' bottleneck with maximal\ncapacity $j_{max}$.\n\n\\bibitem{SH} Kerner, B.S. and Rehborn, H., Phys. Rev. Lett. {\\bf 79},\n 4030 (1997).\n\n\\bibitem{RH} Lee, H.Y., Lee, H.-W. and Kim, D.,\nPhys. Rev. Lett. {\\bf 81}, 1130 (1998).\n\n\\end{thebibliography}"
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cond-mat0002170
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Some remarks on the Lieb-Schultz-Mattis theorem and its extension to higher dimensions
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[
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"author": "G. Misguich$^*$"
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"author": "C. Lhuillier\\thanks{Laboratoire de Physique Th{\\'e}orique des Liquides UMR 7600 of CNRS. Universit\\'e P. et M. Curie, case 121, 4 Place Jussieu, 75252 Paris Cedex.}"
}
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The extension of the Lieb-Schultz-Mattis theorem to dimensions larger than one is discussed. It is explained why the variational wave-function built by the previous authors is of no help to prove the theorem in dimension larger than one. The short range R.V.B. picture of Sutherland, Rokhsar and Kivelson, Read and Chakraborty gives a strong support to the assertion that the theorem is indeed valid in any dimension. Some illustrations of the general ideas are displayed on exact spectra. PACS numbers: 71.10.Fd; 75.10.Jm; 75.40.-s; 75.50.Ee; 75.60.Ej; 75.70.Ak
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"name": "LSMcondmat.tex",
"string": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[dvips]{graphics}\n\\usepackage{a4wide}\n\\begin{document}\n\n%______________________________________________________________________________\n\n\\title{Some remarks on the Lieb-Schultz-Mattis theorem and its\n\textension to higher dimensions}\n\n\\author{G. Misguich$^*$, C. Lhuillier\\thanks{Laboratoire de Physique\n\tTh{\\'e}orique des Liquides UMR 7600 of CNRS. Universit\\'e P.\n\tet M. Curie, case 121, 4 Place Jussieu, 75252 Paris Cedex.}\n\t}\n\\maketitle \n\\bibliographystyle{prsty}\n\n%______________________________________________________________________________\n\n\\abstract{\nThe extension of the Lieb-Schultz-Mattis theorem to dimensions larger\nthan one is discussed. It is explained why the variational\nwave-function built by the previous authors is of no help to prove the\ntheorem in dimension larger than one. The short range R.V.B. picture\nof Sutherland, Rokhsar and Kivelson, Read and Chakraborty gives a\nstrong support to the assertion that the theorem is indeed valid in\nany dimension. Some illustrations of the general ideas are displayed\non exact spectra.\nPACS numbers: 71.10.Fd; 75.10.Jm; 75.40.-s; 75.50.Ee; 75.60.Ej; 75.70.Ak\n}\n\n\\maketitle\n%______________________________________________________________________________\n\\begin{section}{Introduction}\n\nIn 1961 Lieb, Schultz and Mattis proved that a spin 1/2\nantiferromagnetic periodic chain of length $L$ has a low energy\nexcitation of order ${\\cal O} (1/L)$~\\cite{lsm61}. This theorem\n(called in the following LSMA) was then extended by Lieb and Affleck\nto {\\em odd-integer} spin but fails for integer ones~\\cite{al86}. It\nstates that SU(2) invariant Hamiltonians with odd-integer spins in the\nunit cell, either have gapless excitations or degenerate ground-states\nin the thermodynamic limit. The authors suggested that it might be\nextended to higher space dimensions, but up to now, no complete\nargument has been worked out~\\cite{a88}.\n\nIn this paper we revisit the method used by LSMA (construction of a\nvariational excited state) and the physical meaning of the unitary\noperator involved in this construction. This sheds some light on the\nreason why the LSMA excited state is generally not a low energy\nexcitation in space dimension larger than one (as for example in the\ncase of N\\'eel order on the triangular lattice), and how qualitatively\none might try to build a truly low energy excited state. We then study\nan alternative wave-function very much in the spirit of the resonating\nvalence-bond (RVB) states of Sutherland~\\cite{s88}, Rokhsar and\nKivelson~\\cite{rk88}, and Read and Chakraborty~\\cite{rc89}. In this\nlast framework, we show that the above-mentioned statement is indeed\ntrue in any dimension, and discuss the quantum numbers of these quasi\ndegenerate ground-states.\n\n\\end{section}\n\\begin{section}{The LSMA Theorem}\n\n To analyze LSMA theorem in dimension larger than one a dichotomy\n could be done between situations {\\em with} $T=0$ {\\em long-range\n order and symmetry breaking order parameters} on one hand, and\n systems {\\em without long-range order} on the other.\n\n\\begin{itemize}\n\n\\item In the first case the theorem is trivially true: a\nsymmetry-breaking situation necessarily implies in the thermodynamic\nlimit the mixing of states belonging to different irreducible\nrepresentations of the Hamiltonian and/or lattice symmetry group and\nthus a degeneracy of the ground-state.\n\n {\\em i)} In the case of N{\\'e}el long-range order, there are both\n degenerate states (which form the true thermodynamic ground-state)\n and gapless excitations. The gapless excitations, the\n antiferromagnetic magnons, are the Goldstone mode associated to the\n broken continuous SU(2) symmetry. Whereas it is well known that the\n softest magnons scale as ${\\cal O} (1/L)$, it is often asserted that\n these quasi-particle first excitations are the first excited levels\n of the multi-particle spectra. This is indeed false: the \"T=0 N\\'eel\n ground-state\" (or \"vacuum of excitations\") is itself a linear\n superposition of $\\sim N^{\\alpha}$ eigen-states of the $SU(2)$\n invariant hamiltonian which collapse to the ground-state as ${\\cal O}\n (1/L^2)$ ($\\alpha$ is the number of sub-lattices of the N{\\'e}el\n state, $L$ is the linear size of the sample and $N$ the number of\n sites of the sample)~\\cite{f89,nz89,bllp94}.\n\n {\\em ii)} In the case of a discrete broken symmetry, as for example the\n space symmetry breaking associated to long-range dimer or plaquette\n order, or in the case of $T$-symmetry breaking associated to\n long-range order in chirality, the vacuum of excitations is also\n degenerate in the thermodynamic limit. The number of quasi\n degenerate states is related to the dimension of the broken symmetry\n group, the nature of the order parameter and is independent of the\n sample size~\\cite{mlbw99}. The collapse of these levels on\n the ground-state is supposed to be exponential with the size of the\n lattice.\n\n\\item It is only when there is no long-range order, a\nspin gap and when all spin correlations are short-ranged that the\nexistence of a ground-state degeneracy is non-trivial. In that case,\nthe theorem states that the state(s) degenerate with the absolute\nground-state in the thermodynamic limit have wave vector $\\pi$ with\nrespect to the ground-state. More generally, in dimension larger than\n1, we will argue that the states which collapse to the ground-state\nhave wave vector ${\\bf k}_{A_{i}} $ (wave vectors joining the\ncenter to the middle of the sides of the Brillouin\nzone)~\\footnote{This result supposes that the ground-state wave vector\nis zero. It is zero in all situations we have been looking at. If a\nsystem had a non-zero momentum ground-state, it would be degenerate\nand the theorem true anyway !}. These low lying levels collapse to\nthe absolute ground-state exponentially with $L$.\n\n\\end{itemize}\n\\end{section}\n\n\\begin{section}{The LSMA variational state}\n \n\\begin{subsection}\n{Finite size spectra and Twisted Boundary Conditions.}\n\nThe spectrum of a given Hamiltonian on a finite sample depends on the\nboundary conditions: these can be free, periodic or twisted. The LSMA\ntheorem is valid for periodic boundary conditions. For a complete\nunderstanding of the nature of the LSMA variational state it is useful\nto study how a given exact spectrum evolves under a twist of the\nboundary conditions.\n\nLet us consider a finite sample in $d$ dimensions, described by $d$\nvectors ${\\bf T}_j$. Generalized twisted boundary conditions are\ndefined by the choice of the twist axis (here the $z$ axis in the\noriginal ${\\cal B}_0$ spin frame) and a set of $d$ angles $\\phi_j$ as:\n\n\\begin{equation}\n{\\bf S}({\\bf R}_i+{\\bf T}_j)=e^{i\\phi_jS^z({\\bf R}_i)}\n\t{\\bf S}({\\bf R}_i)\n\te^{-i\\phi_jS_{z}({\\bf R}_i)}\n\\label{twbc}\n\\end{equation}\n\nFrom now on, and for the sake of simplicity, we will develop the\nalgebra on simple 1-dimensional models (extrapolation to larger\ndimensions or more complicated $SU(2)$ invariant Hamiltonians is just\na problem of notations). As an example, let us consider the nearest\nneighbor Heisenberg Hamiltonian with periodic boundary conditions:\n\n\\begin{equation}\n{\\cal H}_0=\\sum_{n=0}^{L-1} {\\bf S}_n\\cdot{\\bf S}_{n+1 [L]}\n\\end{equation}\nA twist $\\phi$ in the boundary conditions\nimplies the calculation of the eigenstates of\n\\begin{eqnarray}\n{\\cal H}_{\\phi}&=&{\\bf S}_{L-1}\\cdot\n\\left(e^{i\\phi S^z_0}{\\bf S}_0e^{-i\\phi S^z_0}\\right)+\n\\sum_{n=0}^{L-2} {\\bf S}_n\\cdot{\\bf S}_{n+1} \\nonumber \\\\\n&=&{\\cal H}_0\n+\\frac{1}{2} \\left( ( e^{i\\phi} -1) S_{L-1}^-S_0^++{\\rm h.c.}\\right)\n\\label{Hphi}\n\\end{eqnarray} \n\nUnder an adiabatic twist of the boundary conditions the spectrum of\n${\\cal H}_\\phi$ evolves periodically with a period $2\\pi$, as the\nboundary conditions (Eq.~\\ref{twbc}). But the eigenstates evolution\nmight be more complicated: a unique spin-$\\frac{1}{2}$ wave function\nacquires a phase factor $-1$ under a $2\\pi$ twist. And there is no\nguaranty that the the ground-state of ${\\cal H}_{\\phi =0}$\nadiabatically transforms into the ground-state of ${\\cal H}_{\\phi =\n2\\pi}$. As we will show below, this is generally not the case and the\ntrue period of the eigenstates is $4\\pi$.\n\n\\end{subsection}\n\n\\begin{subsection}{Twisted Boundary Conditions and Translational\nInvariance }\n\nTo follow adiabatically an eigenstate while twisting the boundary\nconditions may be difficult if there are level crossings during the\ntwist. The only way to do it in an unambiguous way is to follow a\ngiven one-dimensional representation of the symmetry group during the\ntwist: in such a representation the levels are non degenerate and\nnever cross. The ground-state of the Hamiltonian with periodic\nboundary conditions indeed belongs to such a\nrepresentation~\\footnote{Up to now, in any antiferromagnetic models we\nhave studied, the ground-sate of an even number of spins belongs to\nthe trivial representation of the total group (the only exceptions are\nassociated to special pathological behavior of very small samples): it\nis a state with total spin zero, zero momentum, invariant in any\noperation of the point symmetry group. This makes sense and appears\nas a powerful extension of Marshall theorem for bipartite lattices.\nIf the ground-state were to belong systematically to a\nmultidimensional representation, as it might be the case for chiral\nspin liquids, then the LSM theorem would again be trivially true.}.\n\nSuch a program is not easy in the framework of Eq.~\\ref{Hphi}, which\nbreaks the translational symmetry of the problem. But the\ntranslational invariance can be restored by a unitary transformation\nrotating the spin frame at each lattice site. Let us call ${\\cal\nB}_{\\phi}$, the new frame deduced from the original ${\\cal B}_0$ by a\nspatially dependent twist described by the unitary operator:\n\n\\begin{equation}\n U(\\phi)= \\exp( i\\frac{\\phi}{L} {\\sum_{n=0}^{L-1} n S^z_n})\n\\label{unit}\n\\end{equation}\n\nIn this new frame, the twisted Heisenberg Hamiltonian reads:\n\\begin{equation}\n\\tilde{{\\cal H}}_\\phi=U(\\phi) {\\cal H}_{\\phi} U(\\phi)^{-1};\n\\end{equation}\n(in this equation and in the following, we put a tilde on each quantity\nmeasured in the ${\\cal B}_\\phi$ frame).\nThis unitary transformation is chosen so that the boundary term(s) in\nEq.~\\ref{Hphi} disappear(s):\n\\begin{equation}\n\\tilde{{\\cal H}}_\\phi=\n{\\sum_{n=0}^{L-1} \\left[{S}_n^z{S}_{n+1 [L]}^z\n+\\frac{1}{2}\\left(e^{i\\phi/L}{S}_n^-{S}_{n+1 [L]}^++{\\rm h.c.}\\right)\\right]}\n\\end{equation}\n\\begin{equation}\n\t\\tilde{{\\cal H}}_\\phi=\n\t{\\cal H}_{0}\n\t+\\frac{1}{2}{\\sum_{n=0}^{L-1}\\left[(e^{i\\phi/L}-1) {S}_n^-{S}_{n+1\n[L]}^++{\\rm h.c.}\\right]}.\n\\label{h1h0}\n\\end{equation}\n$\\tilde{{\\cal H}}_\\phi$ is translation invariant ($ \\left[\\tilde{{\\cal\nH}}_\\phi, {\\cal T}\\right]=0$, where ${\\cal T}$ is the operator for\none-step translations) and its spectrum is indeed identical to the\nspectrum of ${\\cal H}_\\phi$. We can now define irreducible\nrepresentations of the translation group labelled by their wave\nvectors in the ${\\cal B}_\\phi$ frame and adiabatically follow a given\neigenstate of the momentum in the successive ${\\cal B}_\\phi$ frames\nwhile increasing $\\phi$ from $0$ to $ 4 \\pi$ (see example in\nFig.~\\ref{twistedspectrum}).\n\nFor a given twist $\\phi$, the zero-momentum eigenstate\n$\\left|\\tilde{\\psi}_\\phi ^{k=0}\\right>$ of $\\tilde{\\cal H}_\\phi$ in\nthe ${\\cal B}_\\phi$ frame has for expression in the ${\\cal B}_0$\nframe:\n\\begin{equation}\n\\left|\\psi_{\\phi}^0\\right>=U^{-1}(\\phi)\\left|\\tilde{\\psi}_\\phi^{k=0}\\right>.\n\\label{wavef}\n\\end{equation}\n For an arbitrary twist, $\\left|\\psi_{\\phi}^0\\right>$\n does not describe a spatially\nhomogeneous state in the ${\\cal B}_0$ frame.\n\n We will now\nshow that for a $2 \\pi$ twist, the trivial representation of\nthe translations in the ${\\cal B}_{2 \\pi}$ frame ($\n\\left|\\tilde{\\psi}_{2\\pi}^{k=0}\\right>$), has \nmomentum $ { \\bf k}_{A_i}$ in the ${\\cal B}_0$ frame. Following\nAffleck \\cite{a88}, this is easily done by noting that for odd-integer\nspins $U(2\\pi)$ anti-commutes with ${\\cal T}$, as soon as the number\nof rows in the transverse direction is an odd\ninteger~\\cite{lsm61,a88}. This proves that\n$\\left|\\psi_{2\\pi}^0\\right> $ defined by Eq. \\ref{wavef} takes a phase\nfactor $-1$ in one-step translation along the twist direction and thus\nhas a momentum $ { \\bf k}_{A_i}$ in the ${\\cal B}_0$ frame.\n\nAn example of such an adiabatic continuation is shown in\nFig.~\\ref{twistedspectrum}. The spectrum of the multi-spin exchange\nHamiltonian on a small losange ($ 4 \\times 5$) is displayed as a\nfunction of the twist angle. The boundary conditions are twisted in\nthe direction of length $L=4$ (the number of rows is odd). We can\nnote the above-mentioned properties:\n\\begin{itemize}\n\\item The spectrum is periodic in $\\phi$ of period $2\\pi$.\n\\item The eigenstates of $\\tilde{\\cal H}_\\phi$ and ${\\cal T}$\nevolve with a period\n$4\\pi$.\n\\item For a $2\\pi$ twist, the zero momentum eigenstate of $\\tilde{\\cal\nH}_{2\\pi}$\nin the ${\\cal B}_{2\\pi}$ frame has a momentum $ { \\bf\nk}_{A_i}$ in the ${\\cal B}_0$ frame (compare the spectra and\nlabels for twists $0$ and $2\\pi$)\n\\end{itemize}\n\n\n\n\n\\end{subsection}\n\n\\begin{subsection}{The LSMA variational state revisited}\nThe proof of LSMA theorem relies on the construction of a low lying\nexcited state for the problem with periodic boundary conditions. Let\nus call $\\left|\\psi_0\\right>$ the exact ground-state of this problem.\nThe LSMA excited state is obtained by the action of the unitary\noperator $U(2\\pi)$ (Eq.~\\ref{unit}) on $\\left|\\psi_0\\right>$:\n\\begin{equation}\n\\left|\\theta^{LSMA}_{2\\pi}\\right>=U(2\\pi)\\left|\\psi_0\\right>\n\\label{LSMA}\n\\end{equation}\n\nAs already mentioned, in specific geometries,\n $U(2 \\pi)$ anti-commutes with ${\\cal T}$\nand $\\left| \\theta ^{LSMA} _{2\\pi}\\right >$ has momentum $\\pi$ with\nrespect to $\\left| \\phi _0\\right>$. Contrary to the states described\nabove, $ \\left | \\theta ^{LSMA} _{2\\pi}\\right >$ \nis not an eigenstate of\n${\\cal H}_0$. Its variational energy can be computed with\nelementary algebra as:\n\\begin{equation}\n\\left< \\theta ^{LSMA} _{2\\pi}\\right|\n{\\cal H}_0\\left| \\theta ^{LSMA} _{2\\pi}\\right>\n=\n\\left<\\psi_0\\right|\\tilde{{\\cal H}}_{\\phi=2\\pi}\\left|\\psi_0\\right>\n.\n\\label{scalarproduct}\n\\end{equation} \n\nUsing Eq.~\\ref{h1h0} (generalized to d dimensions) we see that $\\left|\n\\theta ^{LSMA} _{2\\pi} \\right>$ variational energy is equal to the\nexact ground-state energy $\\left<\\psi_0\\right| {\\cal\nH}_0\\left|\\psi_0\\right>$, plus a correction of the order of ${\\cal\nO}(N\\times(e^{2i\\pi/L}-1))$, that is ${\\cal O}(N/L^2)$. This gives,\nin one dimension, an energy ${\\cal O}(1/L)$ and achieves the proof of\nthe LSM theorem. The number of correction terms does not allow to\nextend the proof to higher space dimensions~\\footnote{ Only systems\nwith aspect ratios going to zero as the size goes to infinity have\ndegenerate states reminding of the one-dimensional\nproblem~\\cite{a88}.}.\n\nA simple counter-example showing that $\\left| \\theta ^{LSMA}\n_{2\\pi}\\right >$ is generally not a low lying excitation is given by\nthe spectra of the Heisenberg hamiltonian on the triangular lattice:\nin this situation the exact lowest excitation energy (and thus the\nLSMA variational energy) in the $ { \\bf k}_{A_i}$ sector is indeed\n${\\cal O}(1)$: it is associated to the corresponding magnon which has\na large energy in this three-sublattice ordered\nstructure~\\cite{bllp94}.\n\nThe point of view developed in the previous subsection provides\nanother way to look at the right-hand side of Eq.~\\ref{scalarproduct}.\n$\\left|\\psi_0\\right>$ might be seen as a variational guess to solve\nthe twisted boundary conditions problem using the $ {\\cal B}_{2\\pi}$\nframe, and then equivalently $\\left| \\theta ^{LSMA} _{2\\pi} \\right>$\nas the solution of the same physical problem in the ${\\cal B}_{0}$\nframe ($ {\\cal H}_{\\phi=2\\pi} = {\\cal H}_0$). This new point of\nview enlights the weakness of the LSMA wave function (in $d>1$) and\nhow we might try to overcome it.\n\nLet us for the moment assume that $\\phi$ is equal to $2\\pi-\\epsilon$.\nIn the $ {\\cal B}_{0}$ frame, the perturbation to the periodic problem\nonly appears at a boundary of dimension (d-1): but the variational\nsolution $\\left| \\theta ^{LSMA} _{2\\pi - \\epsilon} \\right>$ smears out\nthe spin response on the entire system. This would be a sensible\nsolution in the case of long range order, where the system shows\nstiffness and sensitivity to the boundary conditions. On the\ncontrary, in the present case, where spin-spin correlations have a\nfinite range $\\xi$, it seems reasonable to expect that the boundary\nperturbation does not propagate at a distance much larger than $\\xi$\nfrom the boundary. The LSMA solution is thus probably very far from\noptimal.\n\nWe might thus expect to find in the $ {\\cal B}_{0}$ frame, a solution\nwith energy lower than $\\left| \\theta ^{LSMA} _{2\\pi} \\right>$ \n by perturbing $\\left|\n\\psi _{0}\\right>$ only $locally$ at the boundary. For such a wave\nfunction, Equation~(\\ref{Hphi}) (and its generalization to $d$\ndimensions) implies a distance in energy from the ground-state of the\norder of $ \\epsilon^2 \\, \\xi \\, L^{d-1}$. We could then speculate\nthat this difference in energy might be tuned to zero by an\nappropriate choice of the small free parameter $\\epsilon$. But indeed\nsuch a reasoning involves a difficult and may be pathological limit.\nIn the following paragraph we thus propose a new variational wave\nfunction for the low lying ${\\bf k}_A$ excited state: in the\ntranslationally broken picture ($ {\\cal B}_{0}$), this excited state\ndiffers very little from $\\left|\n\\psi _{0}\\right>$ and only in the vicinity of the boundary defect.\n\nAs an illustration of the present analysis, we can look at the\nevolution of the low lying levels of the spectrum of the MSE\nhamiltonian under a twist $\\phi$ of the boundary conditions\n(Fig.~\\ref{twistedspectrum}). In spite of the very small size of the\nsample ($4\\times 5$), one clearly sees that the exact ground-state\nenergy does not increase as $\\phi ^2$ but more or less as $\\phi^4$\n(see Fig.~\\ref{twistedspectrumphi4}). For small enough $\\phi$ , this\nstate does not present stiffness to twisted boundary conditions: this\nis exactly what is expected from a Spin-Liquid.\n\nFor comparison one can compute the variational energy of the state which\ninterpolates between $\\left|\\psi _{0}\\right>$ and\n$\\left|\\theta^{LSMA}_{2\\pi}\\right>$. It is defined as:\n\\begin{equation}\n\\left|\\theta_{\\phi}\\right>=U^{-1}(\\phi)\\left|\\psi_0\\right>.\n\\label{LSM2}\n\\end{equation}\nIts variational energy can be rewritten as a linear combination of\n2-body and 4-body observables of $\\left|\\psi _{0}\\right>$ multiplied\nby cosines of $\\phi/L$. It thus increases as $\\phi ^2$ and has a non\nzero stiffness, which explains why it is a bad approximation of the\nexact state.\n\nRemark: Contrary to the the $N=36$ spectrum of Fig.~\\ref{sp36} (where\n$ \\xi > L$), the sample in Fig.~\\ref{twistedspectrum} does not display\nall the features expected from a Spin-Liquid spectrum. The\ncorrelations at a distance 4 are still not completely negligeable:\nthis explains the rapid increase in the ground-state energy of ${\\cal\nH}_\\phi$ for $\\phi\n\\ge 0.2$. As a consequence, the ordering of the eigen-levels of\n${\\cal H}_0$ is different from the thermodynamic limit: in particular\nthe ground-state in the $ {\\bf k}_A$ sector is not the first excited\nstate of the $4\\times 5$ spectrum, as it is in the $6 \\times 6$\nexample of Fig.~\\ref{sp36}. The evolution under a twist of the $N=36$\nspectrum, would have been much more pedagogic: it is for the moment\ntoo expensive in computer time.\n\nAt that point it is interesting to discuss Oshikawa's approach of this\nquestion~\\cite{o99}. In place of twisting the boundary conditions,\nOshikawa suppose that the quasi particles of the problem have a\nfictitious charge. He inserts a magnetic flux in the hole of the torus\ndefined by the boundary conditions (see Fig.~\\ref{torus}) and then\nincreases the flux from $0$ to $2 \\pi$. This is a procedure\nabsolutely similar to our twist of the boundary conditions in the spin\nproblem, and the adiabatic insertion of a $2 \\pi$ flux brings the\nsystem from the $\\left|\\psi_0\\right>$ state to the\n$\\left|\\tilde{\\psi}_{k=0}\\right>$ of $\\tilde{\\cal H}_{2 \\pi}$ defined\nin subsection 3.2. At that point he does the implicit hypothesis that\n{\\it the gap of the system does not close in the operation}. He thus\nautomatically arrives to the conclusion that the ground-state is\ndegenerate.\n\nOur approach gives a complementary insight in the physics of such a\nsystem: a Spin-Liquid, with its absence of stiffness is not sensitive\nto a twist of the boundary conditions (in the thermodynamic limit).\nThus the ground-state energy of ${\\cal H}_{\\phi}$ in the translation\ninvariant sector does not depend on the twist $\\phi$, which implies\nthat the ground-state of $ {\\cal H}_0$ is degenerate (with wave\nvectors $0$ and $ {\\bf k}_{A_i}$ in the ${\\cal B}_0$ frame). In the\nfollowing paragraph, using a specific mathematical definition of the\nSpin-Liquid state, we will exhibit a variational wave-function giving\na strong support to the LSMA conjecture.\n\\end{subsection}\n\\end{section}\n\n\\begin{section}{The short range RVB picture of the first excited states}\n\nIn this part we use the main ideas of Sutherland~\\cite{s88}, Read and\nChakraborty~\\cite{rc89} to build an explicit variational wave-function\northogonal to the ground-state and collapsing to it in the\nthermodynamic limit. We then show that these first excited states\nhave momentum ${\\bf k}_{A_{i}}$ with respect to the ground-state.\n\nSutherland first showed that the zero-temperature observables of a\nnearest neighbor resonating valence bond wave-function can be computed\nthanks to the classical properties of a gas of loops. In the quantum\nproblem, the loops appear formally when scalar product of wave-\nfunctions (or matrix elements of spin permutations) are written in\nterms of dimer coverings. The Sutherland nearest neighbor resonating\nvalence-bond (NNRVB) wave-function description can be mapped to the\nhigh temperature disordered phase of the classical loop model: its\ncorrelation length is finite and the weight of long loops is\nexponentially decreasing with their length.\n\nOur own reasoning rests on the following {\\bf basic assumption (A)}:\nThe ground-state $\\left| \\psi _{0}\\right>$ is a R.V.B. state, and the\nlong loops weight in the norm $\\left<\\psi_0\\right|\\left.\\psi_0\\right>$\ndecreases exponentially with the loop length. This last requirement\nimplies {\\it the exponential decrease of all multi-point correlations\nwith distance}, the reverse proposal might equally be true but its\nproof is less obvious~\\footnote{Miscellaneous remarks: i) a dimerized\nstate does not fulfill property (A) because it has dimer-dimer\nlong-range order (4-point correlation function). ii) the Spin-Liquid\nobserved on the kagom\\'e lattice might not obey assumption A but it\nhas most probably gapless singlet excitations. }.\n\nThe steps of the demonstration are as follows:\n\\begin{itemize}\n\n\\item Choice of a dimer basis.\nDimer decompositions are a bit uneasy because dimer coverings are not\northogonal and the entire family of dimer coverings is\nover-complete. So we must suppose that in a first time we have\nextracted a non orthogonal basis of independent dimer coverings called\ngenerically $\\left|C\\right>$. This basis should respect the\ntranslational invariance of the problem: which means that if\n$\\left|C\\right>$ is a basis vector, so is ${\\cal T}\\left|C\\right>$\n(where ${\\cal T}$ is any unit step translation)\n\n\\item Decomposition of the translation invariant ground-state.\n We have understood previously that our problem is a boundary problem.\n Let us draw a cut $\\Delta$ encircling the torus created by periodic\n boundary conditions (see Fig.~\\ref{torus}). This hyper-surface of\n dimension $d-1$ cuts bonds of the lattice but there is no sites\n sitting on it. The position of the cut is arbitrary; but we may\n decide in order to follow closely our previous discussion to put it\n between spin $L-1$ and spin 0 of each row of the lattice. In the\n decomposition of $\\left| \\psi _{0}\\right>$ on the dimer basis, let us\n sort the coverings in two sets $\\Delta_{+}$ and $\\Delta_{-}$\n according to the parity $\\Pi_{\\Delta}$ of the number of bounds going\n across the cut. This leads to a formal decomposition of $\\left| \\psi\n _{0}\\right>$ in two parts:\n\\begin{equation}\n\\left| \\psi_0\\right> =\\left|\\psi_0^+\\right> + \\left|\\psi_0^-\\right>\n\\end{equation}\nwhere the vectors $\\left|\\psi_0^{\\pm}\\right>$ belong respectively to\nthe sets $\\Delta_{\\pm}$.\n\nLet us now consider a finite system with an odd number of rows along\nthe direction of the cut, and a one-step translation ${\\cal\nT}_{\\Delta}$ that crosses the cut. If $\\left|C\\right>$ belongs to\n$\\Delta_{+}$ , ${\\cal T}_{\\Delta}\\left|C\\right>$ belongs to\n$\\Delta_{-}$ and reversely (Property (B)). The translational\ninvariance of $\\left| \\psi _{0}\\right>$ thus implies that both $\\left|\n\\psi _{0}^+\\right>$ and $\\left| \\psi _{0}^-\\right>$ are simultaneously\nnon zero. With the assumption $(A)$, it is easy to show that the two\ncomponents $\\left|\\psi_0^{\\pm}\\right> $ are orthogonal.\n\\item Excited state.\nOne can thus build the state\n\\begin{equation}\n\\left|\\psi_{1,\\Delta}\\right> =\\left|\\psi_{0}^+\\right> - \\; \\left|\\psi_{0}^-\\right>.\n\\end{equation}\nUsing the fact that $\\left| \\psi _{0}\\right>$ has momentum zero,\nProperty (B) implies that $\\left| \\psi _{1, \\Delta}\\right>$ has a\nmomentum $k_{A}$ in the direction of ${\\cal T}_{\\Delta}$. It seems\nreasonable to think that this property demonstrated to be true for\nsamples with an odd number of rows is valid in the thermodynamic limit\nfor any samples (this is clearly indicated by small size exact\ndiagonalizations, see Fig.~\\ref{sp36} for the $6 \\times 6$ sample).\nRemark: $\\left|\\psi_{1,\\Delta}\\right>$ is obtained from $\\left| \\psi\n_{0}\\right>$ by changing the sign of the dimers crossing the boundary:\nin the case of a short range RVB state it is exactly the kind of local\nperturbation we were searching for in the previous paragraph.\n\\item\nThe demonstration is achieved by proving that $\\left| \\psi\n_{1,\\Delta}\\right>$ and $\\left| \\psi _{0}\\right>$ have the same short\nrange correlations and thus the same energy in the thermodynamic\nlimit. More precisely only the phases of the {\\em long}\nloops~\\footnote{Loops with non-zero winding number around the torus.}\nin the expression of the energy $\\left<\\psi_0\\right|{\\cal\nH}\\left|\\psi_0\\right>$ change sign and thus the variational energy of\n$\\left|\\psi_{1,\\Delta}\\right>$ only differs from the energy of\n$\\left|\\psi_0\\right>$ by terms of the order of $ {\\cal\nO}(exp(-L/\\xi))$ where $\\xi$ is the characteristic length of the\nloops. Such a construction can be done for any main direction of the\nlattice, which proves that the degeneracy of the ground-state in the\nthermodynamic limit is $2^d$ (a result already obtained by Read and\nChakraborty, without reference to the wave vectors of the quasi\ndegenerate ground-states)\n\n\\end{itemize}\n\nThis completes our assertion that the LSMA conjecture is indeed valid\nin a very large number of situations whatever the dimension of the\nlattice.\n\n\\section{Miscellaneous remarks}\n\\begin{itemize}\n\\item\nWe thus arrive at the conclusion that in the absence of long range\norder there is, strictly speaking, symmetry breaking of one step\ntranslations. Long range order in any of these antiferromagnetic\nsystems implies symmetry breaking, BUT the reverse is false, the\nsymmetry breaking described here does not imply long range order in a\n{\\it local} order parameter~\\cite{mlbw99} . This property is of\ntopological origin: the observation of this symmetry breaking would\nneed sensitivity to a global observable or to boundary conditions\n(edge states).\n\\item\nIt should be noticed that the results on the $6 \\times 6$ sample seem\nto indicate that the conserved symmetry of the ground-state is in fact\n${\\cal T}\\Sigma$, where $\\Sigma$ is the reflexion through a plane\ncontaining ${\\bf k}_{A}$.\n\\end{itemize}\n\n\n\n{\\bf Acknowledgments:} We acknowledge very fruitful discussions with\nS. Sachdev and C. Henley. One of us (C.L.) thanks the hospitality of\nI.T.P. and the organisors of the Quantum Magnetism program. Special\nthanks are due to I. Affleck and D. Mattis whose vivid interest have\nprompted these remarks. This research was supported in part by the\nNational Science Foundation under Grant No. PHY94-07194, and by the\nCNRS and the Institut de D\\'eveloppement des Recherches en\nInformatique Scientifique under contracts 994091-990076.\n\\end{section}\n\\begin{figure}\n\t\\begin{center}\n\t\\resizebox{11cm}{!}{\n\t\\includegraphics{sp36m210.eps}}\n\t\\end{center}\n\n\t\\caption[99] { First eigenstates of the multiple-spin exchange\n\tmodel on a $6 \\times 6$ sites sample (ref.~\\cite{mlbw99},\n\tparameters $J_2=-2$ and $J_4=1$; the system is in a\n\tSpin-Liquid phase). Eigenstates with total spin $S=0$ and\n\t$S=1$ are displayed. The symbols represent the spatial\n\tquantum numbers ( $\\mathcal{R}_\\theta$, and $\\sigma=1$ are the\n\tphase factors taken by the many-body wave function in a\n\trotation of the lattice of an angle $\\theta$ or in a reflexion\n\tsymmetry.). The ground-state ($E=-142.867$), belongs to the\n\ttrivial representation of the space group: ${\\bf k}={\\bf 0}$,\n\t$\\mathcal{R}_{\\frac {2 \\pi}{3}}=1$, $\\mathcal{R}_{\\pi}=1$,\n\t$\\sigma=1$. The first excited states have wave-vectors ${\\bf\n\tk}_{A_i}$ (three-fold degeneracy). The finite size scaling\n\tstrongly indicates that these states collapse to the absolute\n\tground-state in the thermodynamic limit. The third and fourth\n\teigen-levels in the $S=0$ spin sector do probably not collapse\n\tto the absolute ground-state~\\cite{mlbw99}. They may be\n\tdegenerate in the thermodynamic limit (4-fold degeneracy) and\n\tdescribe an $S=0$ bound-state just below the continuum of\n\ttriplet excitations.}\\label{sp36}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\\resizebox{10.5cm}{!}{\n\t\t\\includegraphics{twist.eps}}\n\t\\end{center}\n\n\t\\caption{ Low-lying spectrum of the $4 \\times 5$ sample of the\n\tmultiple-spin exchange model (same parameters as in\n\tFig.~\\ref{sp36}), as a function of the twist $\\phi$. Squares\n\tare ${\\bf k}={\\bf 0}$ states in the ${\\cal B}_\\phi$ frame and\n\tcircles stand for states with momentum ${\\bf k}_{ A_1}$\n\t(momentum $\\pi$ in the even direction).}\n\t\\label{twistedspectrum}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\\resizebox{8.5cm}{!}{\n\t\t\\includegraphics{twistphi4.eps}\n\t\t}\n\t\\end{center}\n\n\t\\caption{ Ground-state energy as a function of $\\phi^4$ in the\n\tvicinity of $\\phi=0$ (same parameters as in\n\tFig.~\\ref{twistedspectrum}). The energy is not quadratic\n\t but rather proportional to\n\t$\\phi^4$ (vanishingly small stiffness). \n The line is a guide to the eye.}\\label{twistedspectrumphi4}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\t\\begin{center}\n\t\\resizebox{6cm}{!}{\\includegraphics{torus2.eps}}\n\t\\end{center} \\caption{2-torus with one cut $\\Delta$.} \\label{torus} \n\\end{figure}\n\n%______________________________________________________________________________\n\\begin{thebibliography}{11}\n\n\\bibitem{lsm61}\nE.~H.~Lieb, T.~D.~Schultz, and D.~C.~Mattis., Ann. Phys. (N.Y) {\\bf 16}, 407\n (1961).\n\n\\bibitem{al86}\nI.~Affleck and E.~Lieb, Lett. Math. Phys. {\\bf 12}, 57 (1986).\n\n\\bibitem{a88}\nI.~Affleck, Phys. Rev. B {\\bf 37}, 5186 (1988).\n\n\\bibitem{s88}\nB.~Sutherland, Phys. Rev. B {\\bf 37}, 3786 (1988).\n\n\\bibitem{rk88}\nD.~Rokhsar and S.~Kivelson, Phys. Rev. Lett. {\\bf 61}, 2376 (1988).\n\n\\bibitem{rc89}\nN.~Read and B.~Chakraborty, Phys. Rev. B {\\bf 40}, 7133 (1989).\n\n\\bibitem{f89}\nD.~Fisher, Phys. Rev. B {\\bf 39}, 11783 (1989).\n\n\\bibitem{nz89}\nH.~Neuberger and T.~Ziman, Phys. Rev. B {\\bf 39}, 2608 (1989).\n\n\\bibitem{bllp94}\nB.~Bernu, P.~Lecheminant, C.~Lhuillier, and L.~Pierre, Phys. Rev. B {\\bf 50},\n 10048 (1994).\n\n\\bibitem{mlbw99}\nG.~Misguich, C.~Lhuillier, B.~Bernu, and C.~Waldtmann, Phys. Rev. B {\\bf 60},\n 1064 (1999).\n\n\\bibitem{o99}\nM.~Oshikawa, cond-mat/9911137 (1999).\n\n\\end{thebibliography}\n%______________________________________________________________________________\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002170.extracted_bib",
"string": "\\begin{thebibliography}{11}\n\n\\bibitem{lsm61}\nE.~H.~Lieb, T.~D.~Schultz, and D.~C.~Mattis., Ann. Phys. (N.Y) {\\bf 16}, 407\n (1961).\n\n\\bibitem{al86}\nI.~Affleck and E.~Lieb, Lett. Math. Phys. {\\bf 12}, 57 (1986).\n\n\\bibitem{a88}\nI.~Affleck, Phys. Rev. B {\\bf 37}, 5186 (1988).\n\n\\bibitem{s88}\nB.~Sutherland, Phys. Rev. B {\\bf 37}, 3786 (1988).\n\n\\bibitem{rk88}\nD.~Rokhsar and S.~Kivelson, Phys. Rev. Lett. {\\bf 61}, 2376 (1988).\n\n\\bibitem{rc89}\nN.~Read and B.~Chakraborty, Phys. Rev. B {\\bf 40}, 7133 (1989).\n\n\\bibitem{f89}\nD.~Fisher, Phys. Rev. B {\\bf 39}, 11783 (1989).\n\n\\bibitem{nz89}\nH.~Neuberger and T.~Ziman, Phys. Rev. B {\\bf 39}, 2608 (1989).\n\n\\bibitem{bllp94}\nB.~Bernu, P.~Lecheminant, C.~Lhuillier, and L.~Pierre, Phys. Rev. B {\\bf 50},\n 10048 (1994).\n\n\\bibitem{mlbw99}\nG.~Misguich, C.~Lhuillier, B.~Bernu, and C.~Waldtmann, Phys. Rev. B {\\bf 60},\n 1064 (1999).\n\n\\bibitem{o99}\nM.~Oshikawa, cond-mat/9911137 (1999).\n\n\\end{thebibliography}"
}
] |
cond-mat0002171
|
[
{
"author": "F. Minardi"
},
{
"author": "C. Fort"
},
{
"author": "P. Maddaloni"
},
{
"author": "and M. Inguscio"
}
] |
[
{
"name": "erice.tex",
"string": "\n\\documentclass{edbk} \n% \\usepackage{edbkps}\n \\usepackage{epsfig}\n\n\n%\\documentstyle{edbk} % Computer Modern fonts\n%\\documentstyle[edbkps]{edbk} %For PostScript fonts\n%\\documentstyle[m-times,edbkps]{edbk} % For PostScript and MathTimes\n\n\\setcounter{secnumdepth}{0}\n%\\setcounter{tocdepth}{0}\n%\\kluwerbib\n\\normallatexbib\n%\\bibliographystyle{apalike} \n\n\\begin{document}\n\n\\articletitle[]{Production Of Multiple $^{87}$Rb Condensates And Atom Lasers By Rf\nCoupling }\n\n\\chaptitlerunninghead{Production Of Multiple $^{87}$Rb Condensates And Atom\n Lasers...}\n\n\\author{F. Minardi, C. Fort, P. Maddaloni, and M.\n Inguscio}\n\n\\affil{ INFM -- European Laboratory for Non Linear Spectroscopy\n (L.E.N.S.) -- Dipartimento di Fisica dell'Universit\\`a di Firenze\n L.go E. Fermi 2, I-50125 Firenze, Italy}\n\n\n\\section{Introduction}\n\n\nOne of the major goals in the study of Bose--Einstein condensation\n(BEC) in dilute atomic gases has been the realization and development\nof atom lasers. An atom laser may be understood as a source of\ncoherent matter waves. One can extract coherent matter waves from a\nmagnetically trapped Bose condensate. Schemes to couple the atomic\nbeam out of the magnetic trap have been demonstrated\n\\cite{1ketterle,2bloch,3anderson,4phillips}. In the experiments of\n\\cite{1ketterle,2bloch} the output coupling is performed by the\napplication of a radio--frequency (rf) field that induces atomic\ntransitions to untrapped Zeeman states. The atom laser described in\n\\cite{3anderson} is based on the Josephson tunnelling of an\noptically trapped condensate and in \\cite{4phillips} a two--photon\nRaman process is described that allows directional output coupling\nfrom a trapped condensate. \n\n\nCharacterizing the output coupler is\nnecessary to understand the atom laser itself. The literature dealing\nwith theoretical descriptions of output couplers for Bose--Einstein\ncondensates has focused both on the use of rf transitions \\cite{5,6,7,8} and\nRaman processes \\cite{9}. \n\n\nRf output coupling is based on single-- or multi--\nstep transitions between trapped and untrapped atomic states. As a\nconsequence, a rich phenomenology arises that include the observation\nof \"multiple\" condensates corresponding to atoms in different atomic\nstates. These may display varied dynamical behaviour while in the\ntrap. Also, both pulsed and continuous output--coupled coherent matter\nbeams have been observed. The phenomenology is made even more varied\nby the possibility of out--coupling solely under gravity and also of\nmagnetically pushed out beams. The apparatus operated by the Florence\ngroup \\cite{10} offers the possibility to investigate various aspects of\noutput--coupling achieved by rf transitions of atoms in a magnetically\ntrapped $^{87}$Rb.\n\n\n\\section{experimental production of the condensate}\n\n\nWe bring a $^{87}$Rb sample to condensation using the now standard\ntechnique of combining laser cooling and trapping in a double\nmagneto--optical trap (MOT) and evaporative cooling in a\nmagneto--static trap. Our apparatus had been originally designed for\npotassium, as presented by C. Fort in this book \\cite{11}. \n\nOur double MOT set--up consists of two cells connected in the\nhorizontal plane by a 40~cm long transfer tube with an inner diameter\nof 1.1~cm. We maintain a differential pressure between the two cells\nin order to optimize conditions in the first cell for rapid loading of\nthe MOT ($10^{-9}$~Torr) while the pressure in the second cell is\nsufficiently low ($10^{-11}$~Torr) to allow for the long trapping\ntimes in the magnetic trap necessary for efficient evaporative\ncooling. \n\n\nLaser light for the MOTs is provided by a cw Ti:sapphire\nlaser (Coherent model 899-21) pumped with 8~W of light coming from an\nAr$^+$ laser. The total optical power of the Ti:sapphire laser on the\nRb D$_2$ transition at 780~nm is 500~mW. The laser frequency is locked\nto the saturated absorption signal obtained in a rubidium vapour cell.\nThe laser beam is then split into four parts each of which is\nfrequency and intensity controlled by means of double pass through an\nAOM: two beams are red detuned respect to the $F=2 \\rightarrow F'=3$\natomic resonance and provide the cooling light for the two MOTs.\nAnother beam, resonant on the $F=2 \\rightarrow F'=3$ cycling\ntransition, is used both for the transfer of cold atoms from the first\nto the second MOT and for resonant absorption imaging in the second\ncell. Finally a beam, resonant with the $F=2 \\rightarrow F'=2$\ntransition, optically pumps the atoms in the low field seeking\n$F=2$,~$m_F=2$ state immediately before switching on the\nmagneto--static trap. 5~mW of repumping light for the two MOTs\nresonant on the $F=1 \\rightarrow F'=2$ transition are provided by a\ndiode laser (SDL-5401-G1) mounted in external cavity configuration. \n\n\nIn the first MOT, with 150~mW of cooling light split into three\nretroreflected beams (2~cm diameter), we can load $10^9$ atoms within\na few seconds. However, every 300~ms we switch off the trapping\nfields of the first MOT and we flash on the \"push\" beam (1~ms\nduration, few mW) in order to accelerate a fraction of atoms through\nthe transfer tube into the second cell. Permanent magnets placed\naround the tube generate an hexapole magnetic field that guides the\natoms during the transfer. In the second cell the atoms are recaptured\nby the second MOT which is operated with six independent beams\n(diameter=1~cm) each with 10~mW of power. The overall transfer\nefficiency between the two MOTs is $\\sim30$\\%, and after 50 shots we\nhave typically loaded $1.2 \\cdot 10^9$ atoms in the second MOT. The final\npart of laser cooling in the second MOT is devoted to maximizing the\ndensity and minimizing the temperature just before loading the\nmagnetic trap. Firstly the atomic density is increased with 30~ms of\nCompressed--MOT \\cite{13} and this is followed by 8~ms of optical\nmolasses to reduce the temperature. Soon after, we optically pump the\natoms into the low-field seeking $F=2$,~$m_F=2$ state by shining the\n$\\sigma^+$--polarized $F=2 \\rightarrow F'=2$ beam for 200~$\\mu s$,\ntogether with the repumping light. At this point we switch on the\nmagneto--static trap in the second cell where we perform evaporative\ncooling of the atoms. \n\n\n\\begin{figure}[h,t,b]\n\\begin{displaymath}\n\\epsfxsize=7cm\n\\epsfbox{image1.ps}\n\\end{displaymath}\n\\caption{4-coils magnetic trap: Q1 and Q2 coils produce the quadrupole\nfield, the curvature coil (C) provides the axial confinement and the\nantibias coil (A) reduces the bias field thus increasing the radial\nconfinement.}\n\\label{mag}\n\\end{figure}\n\n\nThe magneto--static trap is created by passing DC current through\n4--coils (see Fig.\\ref{mag}), which gives rise to a cigar--shaped\nharmonic magnetic potential elongated along the $z$ symmetry axis\n(Ioffe--Pritchard type). Our magnetic trap is inspired by the scheme\nfirst introduced in \\cite{12}, but is operated with a higher current.\nThe coils are made from 1/8--inch, water cooled, copper tube. The\nthree identical coils consist of 15 windings with diameters ranging\nfrom 3~cm to 6~cm. The fourth coil consists of 6 windings with a\ndiameter of 12~cm. The coils Q1 and Q2 (Fig.\\ref{mag}) generate a\nquadrupole field symmetric around the vertical $y$--axis, and in this\ndirection the measured field gradient is 10~mT/(A $\\cdot$ m). These\ntwo coils operated together at low current ($\\sim$10~A) also provide\nthe quadrupole field for operation of the MOT. The {\\it curvature} (C)\nand {\\it antibias} (A) coils produce opposing fields in the $z$\ndirection. The modulus of the magnetic field during magnetic trapping\nhas a minimum displaced by 5~mm from the center of the quadrupole\nfield (toward the curvature coil) and the axial field curvature is\n0.46~T/(A$\\cdot$m$^2$). The coils are connected in series and fed by a\nHewlett Packard 6681A power supply. By means of MosFET switches we\ncan, however, disconnect coils A and C. The maximum current is 240~A,\ncorresponding to an axial frequency of $\\nu_z$=13 Hz for atoms trapped\nin the $F=2$,~$m_F=2$ state. The radial frequency $\\nu_r$ can be\nadjusted by tuning B$_b$=min(|B|), the bias field at the center of the\ntrap: $\\nu_r$=(2.18/(B$_b$[T])$^{1/2}$)~Hz. With typical operating\nvalues of B$_b$ from 0.14~mT to 0.18~mT, $\\nu_r$ ranges from 160~Hz to\n180~Hz. In addition, a set of three orthogonal pairs of Helmholtz\ncoils provide compensation for stray magnetic fields.\n\n\nThe transfer of atoms from the MOT to the magneto-static trap is\ncomplicated by the fact that the MOT (centered at the minimum of the\nquadrupole field) and the minimum of the harmonic magnetic trap are\n5~mm apart. The transfer of atoms from the MOT to the magnetic trap\nconsists of a few steps. We first load the atoms in a purely\nquadrupole field with a gradient of 0.7~T/m (I=70~A), roughly\ncorresponding to the ``mode-matching'' condition (magnetic potential\nenergy equals the kinetic energy) which ensures minimum losses in the\nphase--space density. Then we adiabatically increase the gradient to\n2.4~T/m by ramping the current to its maximum value I=240~A in 400~ms.\nFinally, the quadrupole potential is adiabatically (750~ms)\ntransformed into the harmonic one by passing the current also through\nthe {\\it curvature} (C) and {\\it antibias} (A) coils, hence moving the\natoms 5~mm in the $z$ direction. At the end of this procedure, 30\\% of\natoms have been transferred from the MOT into the harmonic magnetic\ntrap and we start rf forced evaporative cooling with $4 \\cdot 10^8$ atoms\nat 500~$\\mu$K. We estimate the elastic collision rate to be $\\gamma\n\\sim$30~s$^{-1}$ and this, combined with the measured lifetime in the\nmagnetic trap of 60~s, gives a ratio of \"good\" to \"bad\" collisions of\n$\\sim$1800. This is sufficiently high to perform the evaporative\ncooling and reach BEC. \n\n\nThe rf field driving the evaporative cooling is generated by means of\na 10~turn coil of diameter 1--inch placed 3~cm from the center of the\ntrap in the $x$ direction and fed by a synthesiser (Stanford Research\nDS345). The rf field is first ramped for 20~s with a exponential--like\nlaw from 20~MHz to a value which is only 100~kHz above the frequency\n that empties the trap, $\\nu_{rf}^{0}=\\mu_0$B$_b/2\\hbar$. Then a 5~s\nlinear ramp takes the rf closer to $\\nu_{rf}^{0}$: the BEC transition\ntakes place roughly 5~kHz above $\\nu_{rf}^{0}$. \n\n\nWe analyse the atomic cloud using resonant absorption imaging. The\natomic sample is released by switching off the current through the\ntrapping coils in 1~ms. The cloud then falls freely under gravity and\nafter a delay of up to 25~ms, we flash a probe beam, resonant with the\n$F=2 \\rightarrow F'=3$ transition, for 150~$\\mu$s and at one tenth of\nthe saturation intensity. The shadow cast by the cloud is imaged onto\na CCD array (pixel size=24~$\\mu$m) with two lenses, giving a\nmagnification of 6. However our resolution is 7~$\\mu$m, due to the\ndiffraction limit of the first lens (f=60~mm, N.A.=0.28). We process\nthree images to obtain the two dimensional column density\n\\~{n}($y,z$)=$\\int$ n($x,y,z$)~d$x$. The column density is then fitted\nassuming that n($x,y,z$) is the sum of a gaussian distribution\ncorresponding to the uncondensed fraction and an inverted parabola,\nwhich is solution of the Gross--Pitaevskii equation in the\nThomas--Fermi approximation (condensed fraction). The effect of free\nexpansion, which is trivial for the gaussian part, is taken into\naccount also for the condensate as a rescaling of the cloud radii,\naccording to \\cite{14}. The temperature is obtained from the gaussian\nwidths of the thermal cloud.\n\nWe observe the BEC transition at a temperature T$_c$=200~nK with 2$\n\\cdot 10^5$ atoms, the peak density n$_c$ being 7$ \\cdot\n10^{19}$~m$^{-3}$. The number of condensed atoms shows fluctuations of\n20\\% from shot to shot. We may attribute this to thermal fluctuations\nof the magnetic trap coils giving rise to fluctuation of the magnetic\nfield.\n\n\n\\begin{figure}[h,t,b]\n\\begin{displaymath}\n\\epsfxsize=10cm\n\\epsfbox{image2.ps}\n\\end{displaymath}\n\\caption{Absorption imaging picture of the atomic cloud after a time of\nflight of 20~ms. From the left to the right the rf ramp was stopped\nrespectively at 0.94~MHz, 0.92~MHz, 0.90~MHz and 0.88~MHz. The\nsymmetric expansion in the first image is typical of a thermal\ncloud. By further evaporation the high density peak characteristic of\nthe condensate emerges, finally giving rise to a pure condensate.}\n\\label{trans}\n\\end{figure}\n\n\n\\section{Radio-frequency output coupling}\n\n\nAfter producing the condensate in the $F=2$,~$m_F=2$ state, the same\nrf field used for evaporation is also employed to coherently transfer\nthe condensate into different $m_F$ states of the $F=2$ level.\nMultistep transitions take place at low magnetic field where the\nZeeman effect is approximately linear. This means that the rf field\ncouples all the Zeeman sublevels of $F=2$. Two of these, $m_F=2$ and\n$m_F=1$, are low--field seeking states and stay trapped. $m_F=0$ is\nuntrapped and falls freely under gravity, while $m_F=-1$ and $m_F=-2$\nare high--field seeking states and are repelled from the trap.\nDifferent regimes may be investigated by changing the duration and\namplitude of the rf field. The absorption imaging with a resonant beam\ntuned on the $F=2 \\rightarrow F'=3$ transition allows us to detect at\nthe same time all Zeeman sublevels of the $F=2$ state.\n\n\nIt is worth noting that the spatial extent of the condensate results\nin a broadening of the rf resonance. Due to their delocalization\ndensity atoms experience a magnetic field that is non--uniform over\ntheir spatial extent. Our condensate is typically 40~$\\mu$m in the axial\ndirection and 4~$\\mu$m in the radial one. The corresponding resonance\nbroadening is of the order of 1~kHz. This means that rf pulses shorter\nthan 0.2~ms interact with all the atomic cloud, while for longer\npulses, and sufficiently small amplitudes, only a slice of the\ncondensate will be in resonance with the rf\nfield.\n\n\n\\section{pulsed regime}\n\n\nWe investigate the regime of ``pulsed'' coupling characterized by rf\npulses shorter than 0.3~ms. In particular, Fig.\\ref{pulse} shows the\neffect of a pulse of 10 cycles at $\\sim$1.2~MHz (B$_b$=0.17~mT) with\nan amplitude B$_{rf}$=7~$\\mu$T. After the rf pulse, we leave the\nmagnetic trap on for a time $\\Delta$t and then switch off the trap,\nthus allowing the atoms to expand and fall under gravity for 15~ms.\nPictures from the left to the right correspond to trap times after the\nrf pulse of $\\Delta$t=2,3,4,5 and 6~ms.\n\n\n\\begin{figure}[h,t,b]\n\\begin{displaymath}\n\\epsfxsize=6cm\n\\epsfbox{image3.ps}\n\\end{displaymath}\n\\caption{Time of flight images of multiple condensates produced by a\n rf pulse. Taken after 15~ms of free fall, the pictures refer to\n different evolution times $\\Delta$t in the magnetic trap,\n respectively 2,3,4,5 and 6~ms. The $m_F=2$ condensate is below the\n center of oscillation of the $m_F=1$, opposite to what one expects\n considering the gravitational sagging. This can be explained by the\n presence of a gradient during the switching off of the magnetic\n field}\n\\label{pulse}\n\\end{figure}\n\n\nThree distinct condensates are visible (Fig.\\ref{pulse}): we observe\nthat one is simply falling freely in the gravitational field and hence\nwe attribute to the condensate atoms being in the $m_F=0$ state. The\nother two condensates initially overlap and then separate. However, we\npoint out that the pictures are always taken after an expansion in the\ngravitational field. The initial position of the condensate in the\ntrap may be found by applying the equation of motion for free--fall\nunder gravity.\n\n\nLeaving the magnetic field on for longer times after the rf pulse\nallows us to identify the condensates in different $m_F$ state by\ntheir different center of mass oscillation frequency in the trap.\nConsidering that the images are taken after a free fall expansion of\nt$_{exp}$=15~ms, one can deduce the oscillation amplitude in the trap,\n{\\it a}, from the observed oscillation amplitude, A, by using the\nrelation\n\\begin{equation}\nA=a\\sqrt{1+\\omega_i^2 t_{exp}^2}\n\\end{equation}\nwhere $\\omega_i$ is the oscillation frequency for atoms in the $m_F=i$\nlevel.\n\n\nFrom Fig.\\ref{pulse} we note that the position of the center of mass\nof the condensate in $m_F=2$ is fixed (at the level of resolution)\nwhile the condensate in $m_F=1$ oscillates at the radial frequency of\nthe corresponding trap potential, with a measured amplitude of {\\it\n a}=$8.7 \\pm 0.4$~$\\mu$m. This can be explained by considering the\ndifferent trapping potentials experienced by the two condensates. The\ntotal potential results from the sum of the magnetic and the\ngravitational potentials, so that the minima for the two states in the\nvertical direction are displaced by $\\Delta y$=g/$\\omega_r^2$\n(``sagging''). With the experimental parameter of Fig.\\ref{pulse}\n($\\omega_r=2 \\pi(171 \\pm 4$~Hz) for atoms in $F=2$,~$m_F=2$ state)\n$\\Delta$y equals $8.5 \\pm 0.4$~$\\mu$m. This is in very good agreement with\nthe measured center of mass oscillation amplitude. The $m_F=1$ condensate\nis produced at rest in the equilibrium position of the $m_F=2$ condensate\nand begins to oscillate around its own potential minimum with an\namplitude equal to\n$\\Delta$y.\n\n\nFig.\\ref{pulse} shows the situation where the rf pulse is adjusted to\nequally populate the two trapped states, $m_F=1$ and $m_F=2$. In\ngeneral, the relative population in different Zeeman sublevels can be\ndetermined by varying the duration of the rf pulse. This is clearly\nillustrated in Fig.\\ref{popo}, where the relative population of the\n$m_F=2,1$ and 0 condensates are shown as a function of the pulse\nduration.\n\n\n\\begin{figure}[h,t,b]\n\\begin{displaymath}\n\\epsfxsize=8cm\n\\epsfbox{image4.eps}\n\\end{displaymath}\n\\caption{Measured population fraction of $m_F=2$ (solid circle), $m_F=1$\n(open circle) and $m_F=0$ (triangle) states as a function of rf pulse\nduration. The curves correspond to the calculated Rabi oscillations\nof the relative population in $m_F=2$ (solid line), $m_F=1$ (dashed line)\nand $m_F=0$ (dotted line).}\n\\label{popo}\n\\end{figure}\n\n\nThe theoretical curves, calculated for a Rabi frequency of 26~kHz\ncorresponding to the amplitude of our oscillating rf magnetic field\nB$_{rf}$=3.6~$\\mu$T, are shown together with the experimental data.\nThe population of each Zeeman state is calculated by solving the set\nof the Bloch equation in the presence of an external rf coupling\nfield. These results clearly show that we can control in a\nreproducible way the relative populations of the multiple condensates.\nIt is worth noting that the use of a static magnetic trap allows a\nstraightforward explanation of the phenomenon; similar investigations\nrecently reported for a time-dependent TOP trap \\cite{15} show that\nthe theory is more complicated in presence of a time varying magnetic\nfield.\n\n\nThe $m_F=-1,-2$ sublevels are also populated by rf induced multistep\ntransitions. However, these condensates are quickly expelled by the\nmagnetic potential and the effect can be observed for shorter times\nafter the rf pulse. This is evident in the image in Fig.\\ref{expell} which is\ntaken under the same conditions of Fig.\\ref{pulse} but with a shorter time\n($\\Delta$t=1.5~ms) in the magnetic trap. As expected, in addition to the\nfree--falling $m_F=0$ condensate atoms coupled--out simply by gravity, an\nelongated cloud appears, corresponding to atoms in the high--field\nseeking states that are repelled from the\ntrap.\n\n\n\\begin{figure}[h,t,b]\n\\begin{displaymath}\n\\epsfxsize=2cm\n\\epsfbox{image5.ps}\n\\end{displaymath}\n\\caption{Absorption imaging of the multicomponent condensate produced\nby a rf pulse. The magnetic trap was on for 1.5~ms after the rf pulse\nand the picture was taken after 20~ms of free expansion.}\n\\label{expell}\n\\end{figure}\n\n\n\\section{cw atom laser}\n\n\nContinuously coupling atoms out of a Bose condensate with resonant rf\nradiation was first proposed by W.~Ketterle et al. \\cite{1ketterle}.\nIn their paper on the rf output coupler they discuss this scheme and\npoint out the necessity to have a very stable magnetic field. I.~Bloch\net al. \\cite{2bloch} realized a cw atom laser based on rf output\ncoupling using an apparatus with a very well controlled magnetic\nfield. They placed a $\\mu$--metal shield around the cell where the\ncondensate forms, achieving residual fluctuations below $10^{-8}$~T.\n\n\nWe explored the regime of continuous coupling by leaving the rf field\non for at least 10~ms. In this case we observed a stream of atoms\nescaping from the trap (Fig.\\ref{alas}). The experimental\nconfiguration is similar to the one described in \\cite{2bloch}, except\nfor the fact that our apparatus is not optimized to minimize magnetic\nfield fluctuations, that are at the level of $10^{-6}$~T.\nNevertheless, our observation demonstrate that these fluctuations do\nnot prevent the operation of a cw atom laser.\n\n\nFig.\\ref{alas} shows an absorption image taken after an rf pulse 10~ms\nlong with an amplitude B$_{rf}$=0.36~$\\mu$T. The first picture\ncorresponds to the temperature of the rubidium atoms being above the\ncritical temperature (T$_c$) for condensation. In this case a very\nweak tail of atoms escaping from the magnetically trapped cloud is\nobserved. Decreasing the temperature below T$_c$ (second picture of\nFig.\\ref{alas}) the beam of atoms leaving the trap becomes sharper and\nmore collimated.\n\n\n\\begin{figure}[h,t,b]\n\\begin{displaymath}\n\\epsfxsize=3cm\n\\epsfbox{image6.ps}\n\\end{displaymath}\n\\caption{Absorption imaging of the atom laser obtained applying a rf\n field for 10~ms and illuminating the atoms 5~ms after switching off\n the\n trap.}\n\\label{alas}\n\\end{figure}\n\n\nWe have demonstrated a cw atom laser. However, the fluctuations in our\nmagnetic field strongly influences not only the reproducibility but\nalso the quality of the extracted beam. The high magnetic field\nstability achieved by the M\\\"{u}nich group allowed the measurement of\nthe spatial coherence of a trapped Bose gas \\cite{16,17} by observing\nthe interference pattern of two matter waves out--coupled from the\ncondensate using a rf field composed of two frequencies.\n\n\n\\section{conclusions}\n\n\nWe have illustrated the rich phenomenology arising from the\ninteraction of an rf field with a $^{87}$Rb condensate originally in\nthe $F=2$, $m_F=2$ state. We have shown that condensates can be\nproduced in each of the five Zeeman sublevels and that the relative\npopulations can be controlled by varying the duration and amplitude of\nthe rf pulse. We investigated the behaviour of both trapped and\nuntrapped condensates as a function of the time in the magnetic trap.\nAt short times we recorded the different behaviour between the atoms\noutput--coupled under gravity only ($m_F=0$), and those with an\nadditional impulse due to the magnetic field ($m_F=-1,-2$). We have\nalso produced a cw atom laser by simply increasing the time duration\nof the rf pulse. In our apparatus no particular care is devoted to\nthe shielding of unwanted magnetic field. The stability and\nhomogeneity requirements seem to be less stringent than those\npredicted in the pioneering work of \\cite{1ketterle} and of those\nof the magnetic field implemented by the original cw atom laser\napparatus \\cite{2bloch}. This could make the cw atom laser based on rf\nout--coupling more generally accessible. Most of the observed\nphenomena may be understood using a simple theoretical model; a more\ndetailed and complete description of the multicomponent condensate\nshould take into account also the mean field potential and interaction\nbetween different condensates.\n\nFuture applications of the experimental set--up we are currently\noperating can be foreseen, for instance for the study of collective\nexcitations induced by the sudden change in the atom number. The\ninteraction between condensates in different internal states may\npossibly be investigated as well as time--domain matter--wave\ninterferometers using a sequence of rf\npulses.\n\n\n\n\\begin{acknowledgments}\n\n We would like to thank M. Prevedelli for his contribution in setting\n up the experiment. This work was supported by the INFM \"Progetto di\n Ricerca Avanzata\" and by the CNR \"Progetto Integrato\". We would like\n to thank also D. Lau for careful reading of the\n manuscript.\n\\end{acknowledgments}\n\n% Bibliography made with BibTeX:\n\n%% apalike is preferred if you have used \\kluwerbib, above.\n\n%% Otherwise you may use any .bst style your editor approves.\n\n\n\n%\\bibliographystyle{apalike}\n\n%\\chapbblname{<name of .bbl file>}\n\n%\\chapbibliography{<name of .bib file>}\n\n\n\\begin{chapthebibliography}{99}\n\n\\bibitem{1ketterle}\nMewes, M. -O., Andrews, M. R., Kurn, D. M., Durfee, D. S., Towsend,\nC. G., and Ketterle, W. (1997)\n\\newblock An output coupler for Bose condensed atoms\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 78}: 582\n\n\n\\bibitem{2bloch}\nBloch, I., H\\\"{a}nsch, T. W. and Esslinger, T. (1999)\n\\newblock An Atom Laser with a cw Output Coupler\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 82}: 3008.\n\n\\bibitem{3anderson} Anderson, B. P., and Kasevich, M. A. (1998)\n\\newblock Macroscopic Quantum Interference from Atomic Tunnel Arrays\n\\newblock {\\em Science} {\\bf 282}: 1686.\n\n\\bibitem{4phillips}Hagley, E. W., Dung, L., Kozuma, M., Wen, J.,\n Helmerson, K., Rolston, S. L., and Phillips, W. D. (1999) \n\\newblock A well Collimated Quasi-Continous Atom Laser\n\\newblock {\\em Science} {\\bf 283}: 1706.\n\n\\bibitem{5} Naraschewski, M., Schenzle, A., and Wallis, H. (1997) \n\\newblock Phase diffusion and the output properties of a cw atom-laser\n\\newblock {\\em Phys. Rev. A} {\\bf 56}: 603.\n\n\\bibitem{6} Ballagh, R. J., Burnett, K., and Scott, T. F. (1997)\n\\newblock Theory of an Output Coupler for Bose-Einstein Condensed Atoms\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 78}: 1607.\n\n\\bibitem{7} Steck, H., Naraschewski, M., and Wallis, H. (1998)\n\\newblock Output of a pulsed Atom Laser\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 80}: 1.\n\n\n\\bibitem{8} Band, Y. B., Julienne, P. S., and Trippenbach, M. (1999)\n \\newblock Radio--frequency output coupling of the Bose-Einstein\n condensate for atom lasers \n\\newblock {\\em Phys. Rev. A} {\\bf 59}: 3823.\n\n\n\\bibitem{9} Edwards, M., Griggs, D. A., Holman, P.L., Clark, C. W.,\n Rolston, S. L., and Phillips, W. D. (1999)\n\\newblock Properties of a Raman atom--laser output coupler\n\\newblock {\\em J. Phys. B} {\\bf 32}: 2935.\n\n\n\\bibitem{10} Fort, C., Prevedelli, M., Minardi, F., Cataliotti, F. S.,\n Ricci, L., Tino, G. M., and Inguscio, M. (2000) \\newblock Collective\n excitations of a $^{87}$Rb Bose condensate in the Thomas Fermi\n regime\n\\newblock {\\em Eur. Phys. Lett.} {\\bf 49}: 8.\n\n\n\\bibitem{11} Fort, C.\n\\newblock Experiments with potassium isotopes\n\\newblock in this Volume.\n\n\n\\bibitem{12} Esslinger, T., Bloch, I., and H\\\"{a}nsch, T. W. (1998)\n\\newblock Bose--Einstein condensation in a quadrupole--Ioffe--configuration trap\n\\newblock {\\em Phys. Rev. A} {\\bf 58}: R2664.\n\n\n\\bibitem{13} Petrich, W., Anderson, M. H., Ensher, J. R., and Cornell,\n E. A. (1994) \n\\newblock Behavior of atoms in a compressed magneto-optical trap\n\\newblock {\\em J. Opt. Soc. Am. B} {\\bf 11}: 1332.\n\n\n\\bibitem{14} Castin, Y., and Dum, R. (1996) \n\\newblock Bose--Einstein Condensates in Time--Dependent Traps\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 77}: 5315.\n\n\\bibitem{15} Martin, J. L., McKenzie, C. R., Thomas, N. R.,\n Warrington, D. M., and Wilson, A. C. \\newblock Production of two\n simultaneously trapped Bose--Einstein condensates by RF coupling in\n a TOP trap\n\\newblock cond-mat/9912045.\n\n\\bibitem{16} Esslinger, T., Bloch, I., Greiner, M., and H\\\"{a}nsch, T. W.\n\\newblock Generating and manipulating Atom Lasers Beams\n\\newblock in this Volume.\n\n\\bibitem{17} Bloch, I., H\\\"{a}nsch, T. W., and Esslinger, T. (2000)\n \\newblock Measurement of the spatial coherence of a trapped Bose gas\n at the phase transition \n\\newblock {\\em Nature} {\\bf 403}: 166.\n\n\\end{chapthebibliography}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n"
}
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[
{
"name": "cond-mat0002171.extracted_bib",
"string": "\\bibitem{1ketterle}\nMewes, M. -O., Andrews, M. R., Kurn, D. M., Durfee, D. S., Towsend,\nC. G., and Ketterle, W. (1997)\n\\newblock An output coupler for Bose condensed atoms\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 78}: 582\n\n\n\n\\bibitem{2bloch}\nBloch, I., H\\\"{a}nsch, T. W. and Esslinger, T. (1999)\n\\newblock An Atom Laser with a cw Output Coupler\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 82}: 3008.\n\n\n\\bibitem{3anderson} Anderson, B. P., and Kasevich, M. A. (1998)\n\\newblock Macroscopic Quantum Interference from Atomic Tunnel Arrays\n\\newblock {\\em Science} {\\bf 282}: 1686.\n\n\n\\bibitem{4phillips}Hagley, E. W., Dung, L., Kozuma, M., Wen, J.,\n Helmerson, K., Rolston, S. L., and Phillips, W. D. (1999) \n\\newblock A well Collimated Quasi-Continous Atom Laser\n\\newblock {\\em Science} {\\bf 283}: 1706.\n\n\n\\bibitem{5} Naraschewski, M., Schenzle, A., and Wallis, H. (1997) \n\\newblock Phase diffusion and the output properties of a cw atom-laser\n\\newblock {\\em Phys. Rev. A} {\\bf 56}: 603.\n\n\n\\bibitem{6} Ballagh, R. J., Burnett, K., and Scott, T. F. (1997)\n\\newblock Theory of an Output Coupler for Bose-Einstein Condensed Atoms\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 78}: 1607.\n\n\n\\bibitem{7} Steck, H., Naraschewski, M., and Wallis, H. (1998)\n\\newblock Output of a pulsed Atom Laser\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 80}: 1.\n\n\n\n\\bibitem{8} Band, Y. B., Julienne, P. S., and Trippenbach, M. (1999)\n \\newblock Radio--frequency output coupling of the Bose-Einstein\n condensate for atom lasers \n\\newblock {\\em Phys. Rev. A} {\\bf 59}: 3823.\n\n\n\n\\bibitem{9} Edwards, M., Griggs, D. A., Holman, P.L., Clark, C. W.,\n Rolston, S. L., and Phillips, W. D. (1999)\n\\newblock Properties of a Raman atom--laser output coupler\n\\newblock {\\em J. Phys. B} {\\bf 32}: 2935.\n\n\n\n\\bibitem{10} Fort, C., Prevedelli, M., Minardi, F., Cataliotti, F. S.,\n Ricci, L., Tino, G. M., and Inguscio, M. (2000) \\newblock Collective\n excitations of a $^{87}$Rb Bose condensate in the Thomas Fermi\n regime\n\\newblock {\\em Eur. Phys. Lett.} {\\bf 49}: 8.\n\n\n\n\\bibitem{11} Fort, C.\n\\newblock Experiments with potassium isotopes\n\\newblock in this Volume.\n\n\n\n\\bibitem{12} Esslinger, T., Bloch, I., and H\\\"{a}nsch, T. W. (1998)\n\\newblock Bose--Einstein condensation in a quadrupole--Ioffe--configuration trap\n\\newblock {\\em Phys. Rev. A} {\\bf 58}: R2664.\n\n\n\n\\bibitem{13} Petrich, W., Anderson, M. H., Ensher, J. R., and Cornell,\n E. A. (1994) \n\\newblock Behavior of atoms in a compressed magneto-optical trap\n\\newblock {\\em J. Opt. Soc. Am. B} {\\bf 11}: 1332.\n\n\n\n\\bibitem{14} Castin, Y., and Dum, R. (1996) \n\\newblock Bose--Einstein Condensates in Time--Dependent Traps\n\\newblock {\\em Phys. Rev. Lett.} {\\bf 77}: 5315.\n\n\n\\bibitem{15} Martin, J. L., McKenzie, C. R., Thomas, N. R.,\n Warrington, D. M., and Wilson, A. C. \\newblock Production of two\n simultaneously trapped Bose--Einstein condensates by RF coupling in\n a TOP trap\n\\newblock cond-mat/9912045.\n\n\n\\bibitem{16} Esslinger, T., Bloch, I., Greiner, M., and H\\\"{a}nsch, T. W.\n\\newblock Generating and manipulating Atom Lasers Beams\n\\newblock in this Volume.\n\n\n\\bibitem{17} Bloch, I., H\\\"{a}nsch, T. W., and Esslinger, T. (2000)\n \\newblock Measurement of the spatial coherence of a trapped Bose gas\n at the phase transition \n\\newblock {\\em Nature} {\\bf 403}: 166.\n\n"
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cond-mat0002172
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Evidence for coexistence of the superconducting gap and the pseudo - gap in Bi-2212 from intrinsic tunneling spectroscopy
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[
{
"author": "V.M.Krasnov$^{a}$"
},
{
"author": "A.Yurgens$^{b}$"
},
{
"author": "D.Winkler$^{c}$"
},
{
"author": "P.Delsing and T.Claeson"
}
] |
We present intrinsic tunneling spectroscopy measurements on small Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ mesas. The tunnel conductance curves show both sharp peaks at the superconducting gap voltage and broad humps representing the $c$-axis pseudo-gap. The superconducting gap vanishes at $T_c$, while the pseudo-gap exists both above and below $T_c$. Our observation implies that the superconducting and pseudo-gaps represent different coexisting phenomena. {PACS numbers: 74.25.-q, 74.50.+r, 74.72.Hs, 74.80.Dm}
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"name": "Pseudogap3.tex",
"string": "\\documentstyle[aps,prl,multicol,epsf]{revtex}\n\n\\begin{document}\n\n\\title{Evidence for coexistence of the superconducting gap and the pseudo -\ngap in Bi-2212 from intrinsic tunneling spectroscopy}\n\n\\author{ V.M.Krasnov$^{a}$, A.Yurgens$^{b}$, D.Winkler$^{c}$,\nP.Delsing and T.Claeson}\n\n\\address{Department of Microelectronics and Nanoscience,\\\\\n Chalmers University of Technology, S-41296 G\\\"oteborg, Sweden}\n\n\\date{\\today }\n\\maketitle\n\n\\begin{abstract}\n\nWe present intrinsic tunneling spectroscopy measurements on small\nBi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ mesas. The tunnel conductance curves\nshow both sharp peaks at the superconducting gap voltage and broad\nhumps representing the $c$-axis pseudo-gap. The superconducting gap\nvanishes at $T_c$, while the pseudo-gap exists both above and\nbelow $T_c$. Our observation implies that the superconducting and\npseudo-gaps represent different coexisting phenomena.\n\n{PACS numbers: 74.25.-q, 74.50.+r, 74.72.Hs, 74.80.Dm}\n\n\\end{abstract}\n\n\\begin{multicols}{2}\n\nThe existence of a pseudo-gap (PG) in the quasiparticle density of\nstates (DOS) in the normal state of high-$T_c$ superconductors\n(HTSC) has been revealed by different experimental techniques\n\\cite{Renner,Miyak,Suzuki,ARPES,Bianc,Oleg,Inter,Puchkov}. For a\nreview, see Refs.~\\cite{Puchkov,Randeira,CDW,Halbritt}.\nUp-to-date, there is no consensus about the origin of the PG, the\ncorrelation between the superconducting gap (SG) and PG, or the\ndependencies of both gaps on material and experimental parameters.\nA clarification of these issues is certainly important for\nunderstanding HTSC.\n\nFrom surface tunneling experiments, it was concluded that the SG\nis almost temperature independent~\\cite{Renner,ARPES}. At $T>T_c$,\nit continuously evolves into the PG, which can persist up to room\ntemperature. Furthermore, it was observed that such a\nsuperconducting gap has no correlation with $T_c$ and continues to\nincrease in underdoped samples despite a reduction of\n$T_c$~\\cite{Renner,Miyak,ARPES}. This was the basis for a\nsuggestion that the PG-state at $T>T_c$ is a precursor of\nsuperconductivity~\\cite{Randeira}. On the other hand, surface\ntunneling into HTSC has several drawbacks\\cite{Mallet}, e.g. it is\nsensitive to surface deterioration. The growing controversy\nrequires further studies with alternative techniques.\n\nIntrinsic tunneling spectroscopy has become a powerful tool in\nstudying the quasiparticle DOS {\\it inside} bulk single crystals\nof layered HTSC~\\cite{Suzuki,Inter,Schlenga,Gough,Winkler,Meso}\nand thus avoiding the sensitivity to surface deterioration. First\nexperiments were recently attempted~\\cite{Suzuki,Gough,Winkler} to\nstudy the PG in mesas fabricated on surface of\nBi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (Bi-2212) single crystals.\nUnfortunately, intrinsic tunneling experiments also have several\nproblems, such as internal heating and stacking faults (defects)\nin the mesas. To reduce overheating, a pulse technique was applied\nby Suzuki {\\it et al},~\\cite{Suzuki}. The result at $T\\sim T_c$\nwas essentially similar to the surface measurements~\\cite{Renner},\nleaving the obscure relationship between SG and\nPG~\\cite{Suzuki,Gough} unresolved.\n\nIn this paper we present results of intrinsic tunneling\nspectroscopy for Bi-2212 mesas with considerably smaller areas,\ncompared to previous studies~\\cite{Suzuki,Gough}. Smaller areas\nallowed us to avoid stacking faults in the mesas and to avoid\nmixing between the $c$- axis and $ab$- plane transport. As a\nresult, clean and clear tunnel-type current-voltage (I-V)\ncharacteristics were observed, which allowed us to distinguish\nsuperconducting and pseudo - gaps in a wide range of temperatures.\nIn contrast to surface and earlier intrinsic tunneling\nexperiments~\\cite{Renner,Suzuki,Gough}, we have clearly traced\ndifferent behaviors of the SG and PG. Thorough studies of I-V\ncurves close to $T_c$ revealed that the superconducting gap does\nvanish, while the PG does not change at $T=T_c$. All this speaks\nin favor of different origins of the two coexisting phenomena and\nagainst the precursor-superconductivity scenario of the PG.\nFinally, we discuss interplay between Coulomb interaction and low\ndimensionality as a possible mechanism for the c-axis PG in an\ninherent two-dimensional (2D) system, such as the Bi-2212 single\ncrystals.\n\nMesas with different dimensions from 2 to 20 $\\mu $m were\nfabricated simultaneously on top of Bi-2212 single crystals. To\nreduce the mesa area we adopted a self-alignment technique, see\nRef. ~\\cite{Meso} for details of sample fabrication. The {\\it\nc}-axis I-V characteristics were measured in a three-probe\nconfiguration. The contact resistance was small, about two orders\nof magnitude less than the total resistance of the mesa at the\ncorresponding current. All the leads to a mesa were filtered from\nhigh-frequency electrical noise. Altogether, more than 50 mesas\nmade on different crystals were investigated. Parameters of the\nmesas are listed in Table 1. The three figures in the mesa number\nrepresent the batch number, the crystal number and the number of\nthe mesa on the crystal, respectively. Letters \"Ar\" or \"Ch\"\nindicate whether the mesa was made by Ar-ion or chemical etching.\nHere we present results for slightly overdoped ($T_c=89$~K) and\noptimally doped ($T_c=93-94$~K) samples. The pristine crystals\nwere slightly overdoped. Overdoped mesas were obtained by wet\nchemical etching, which does not significantly change the oxygen\ncontent. Optimally doped mesas were made by Ar-ion etching, during\nwhich mesas partly loose oxygen. Such mesas had larger $T_c$,\n$c$-axis resistivity, $\\rho_c$ see Table 1, and pseudo-gap, see\nFig. 4.\n\nOur fabrication procedure provides samples with highly\nreproducible properties. This is illustrated in the\n\n\\begin{figure}\n\\noindent\n\\begin{minipage}{0.48\\textwidth}\n%\\epsfysize=3.0in\n\\epsfxsize=0.9\\hsize \\centerline{ \\epsfbox{fig1s.eps} }\n\\vspace*{6pt} \\caption{I-V curves per junction for the 423Ar mesa\nat different $T$. Top inset shows normalized I-V curves at\n$T$=4.2K for three different mesas. Bottom inset shows the\ntemperature dependence of the zero bias resistance, $R_0$ (open\ncircles), large bias resistance, $R_n$ (solid squares) and the\ntotal sub-gap resistance, $R_0^N$ (solid triangles).}\n\\end{minipage}\n\\end{figure}\n\n\\noindent top inset of Fig.~1, in which current density, $j=I/S$,\nvs. voltage per junction, $v=V/N$, curves at $T=4.2$~K are shown\nfor three mesas with different areas from different batches and\ncrystals. Here $S$ is the area and $N$ is the number of intrinsic\nJosephson junctions (IJJ's) in the mesa. We note, that all\nnormalized I-V curves collapse into a single curve. In the\nfollowing, we will denote quantities corresponding to the whole\nmesa by capital letters, and those related to an individual IJJ,\nby small letters. The subscripts \"s\", \"pg\" and \"n\" will correspond\nto the {\\underline s}uperconducting, {\\underline\np}seudo-{\\underline g}ap and {\\underline n}ormal state properties,\nrespectively.\n\nIn Fig. 1, $I-v$ curves and in Fig. 2 the voltage dependence of\nthe dynamic conductance $\\sigma(v)= {\\rm d}I/{\\rm d}v(v)$ are\nshown for the optimally doped mesa 423Ar at different\ntemperatures. Figs. 1 and 2 exhibit a typical tunnel-junction\nbehavior. At large bias current, there is a well defined\nnormal-state part of tunneling I-V curves with tunnel resistance\n$R_n$. $R_n$ decreases by merely $\\sim$ 15\\% from 300 K to 4.2 K\nand has no feature at $T=T_c$, as shown in the bottom inset of\nFig. 1. This is in accordance with the pure tunnel junction\nbehavior, for which $R_n$ is expected to be temperature\nindependent. The weak $T$-dependence of $R_n$ indicates an absence\nof mixing between $c$-axis and $ab$-plane transport in our mesas.\nPreviously, however, a strong change of $R_n$ at $T=T_c$ has been\nreported for larger mesas~\\cite{Suzuki,Gough}. On the other hand,\nthe zero bias resistance, $R_0$, has a strong temperature\ndependence, see bottom inset in Fig. 1. Below $T_c$, $R_0$ is\ndetermined by the sub-gap resistance of the first IJJ, $R_0^1$, At\n$T<$40K, a small critical current in the first IJJ appears, see\nFig. 3 a), and $R_0$ drops to the contact\n\n\\begin{figure}\n\\noindent\n\\begin{minipage}{0.48\\textwidth}\n%\\epsfysize=3.0in\n\\epsfxsize=0.9\\hsize \\centerline{ \\epsfbox{fig2s.eps} }\n\\vspace*{6pt} \\caption{Dynamic conductance, $\\sigma(v)$, at\ndifferent temperatures for 423Ar mesa. Inset shows detailed curves\nfor high $T$. Coexistence of the superconducting peak, $v_s$, and\nthe pseudo-gap hump, $v_{pg}$, is clearly visible at $T=77.7$ K. }\n\\end{minipage}\n\\end{figure}\n\n\\noindent resistance. Such a two stage decrease of $R_0$ is due to\na deterioration of IJJ's at the surface of the mesa \\cite{Lee}.\n\nAt low $T$, there is a sharp peak in $\\sigma(v)$, which we\nattribute to the superconducting gap voltage, $v_s=2\\Delta_s /e$.\nWith increasing $T$, the peak at $v_s$ reduces in amplitude and\nshifts to lower voltages, reflecting the decrease in\n$\\Delta_s(T)$. At $T\\sim 83\\ {\\rm K}$ $(< T_c \\simeq 93$~K), the\nsuperconducting peak is smeared out completely and only a smooth\ndepletion of $\\sigma(0)$ (a dip) plus a hump in conductance at\n$v=v_{pg}\\simeq 70$~mV remain. The dip and the hump are correlated\nto each other and both flatten simultaneously with increasing $T$,\nsee inset in Fig. 2. Therefore, both reflect the existence of the\npseudo-gap in the tunneling DOS. The $\\sigma(0)$ gradually\nincreases with temperature but the I-V curves remain non-linear\nnearly up to room temperature. At $T>T_c$, the zero-bias\nresistance, $R_0$, is fairly well described by the\nthermal-activation formula,\n\n\\begin{equation}\nR_0 \\propto exp( T^* /T), \\ T^* \\simeq 150 \\pm 20 K, \\label{Eq1}\n\\end{equation}\n\n\\noindent as shown by the dashed line in bottom inset of Fig. 1.\n\nIn agreement with surface tunneling\nexperiments\\cite{Renner,Miyak}, there are no sharp changes at\n$T_c$. As shown in bottom inset of Fig. 1, at $T<T_c$, $R_0$\nevolves continuously into the total (all $N$ junctions in the\nresistive state) sub-gap resistance, $R_0^N$. This implies that\nthe PG persists also in the superconducting state. The gradual\nevolution of the PG hump upon cooling through $T_c$ is most\nclearly shown in inset of Fig. 2. It is seen that the PG dip/hump\nfeature does not change qualitatively upon cooling through $T_c$.\nMoreover, the I-V curve at $T=$77.7 K shows that the\nsuperconducting peak at $v_s$ emerges on top of the PG - features\nwhich demonstrates a coexistence of both SG and PG features. From\nFig. 2 it is seen that by further decreasing the temperature, the\nsuperconducting peak\n\n\\begin{figure}\n\\noindent\n\\begin{minipage}{0.47\\textwidth}\n%\\epsfysize=3.0in\n\\epsfxsize=0.9\\hsize \\centerline{ \\epsfbox{fig3.eps} }\n\\vspace*{6pt} \\caption{Detailed views of QP branches for the 423Ar\nmesa at different temperatures. Thin lines in Fig. a) represent\npolynomial fits and indicate good scaling of QP branches. The\ninset in b) shows the first five QP branches at $T$ close to\n$T_c$, in an expanded scale. The arrow demonstrates the tendency\nfor vanishing $\\delta v_s(T)$ at $T\\rightarrow T_c$ even when\nmeasured at one and the same current. }\n\\end{minipage}\n\\end{figure}\n\n\\noindent shifts to higher voltages, increases in amplitude and\neventually the PG hump is washed out by the much stronger\nsuperconducting peak. For optimally doped mesas, the PG hump can\nbe resolved at $T>$60 K, i.e. well below $T_c$. The gradual\nopening of $\\Delta_s$ at $T<T_c$, in addition to the PG, can also\nbe seen from a steeper growth of the total sub-gap resistance,\n$R_0^N$, at $T<T_c$, as compared to the thermal-activation\nbehavior of $R_0$ at $T \\ge T_c$, as shown in bottom inset in\nFig.1.\n\nAt low bias and $T<T_c$, multiple quasiparticle (QP) branches are\nseen in the I-V curves, representing a one-by-one switching of the\nIJJ's into the resistive state~\\cite{Suzuki,Schlenga}. A detailed\nview of multiple QP branches is shown in Fig. 3 for different $T$.\nDots and thin lines in Fig. 3 a) represent the experimental points\nand a polynomial fit, correspondingly. Only the last branch,\nhaving many data-points, was actually fitted, all the other thin\nlines were obtained by dividing the voltages of this fit, $V_{\\rm\nfit}(I)$, by the integer number $N^*=N-n+1$, where $n$ is the\nnumber of IJJ's in the resistive state. A good scaling of QP\nbranches is seen, which implies that there is no significant\noverheating of the mesa at the operational current. If there were\noverheating, $V_{\\rm fit}(I)/N^*$ would not go\n\n\\begin{figure}\n\\noindent\n\\begin{minipage}{0.48\\textwidth}\n%\\epsfysize=3.0in\n\\epsfxsize=0.9\\hsize \\centerline{ \\epsfbox{fig4.eps} }\n\\vspace*{6pt} \\caption{Temperature dependence of parameters of the\noptimally doped (solid symbols) and overdoped (open symbols)\nsamples: the superconducting peak voltage, $v_s=2\\Delta_s/e$, the\nspacing between QP branches, $\\delta v_s$, and the pseudo-gap hump\nvoltage, $v_{pg}$. It is seen that the superconducting gap\nvanishes at $T_c$, while the pseudo-gap exists both above and\nbelow $T_c$.}\n\\end{minipage}\n\\end{figure}\n\n\\noindent through the data points because switching of additional\nIJJ's would cause a progressive increase of the internal\ntemperature and the branches with increasing count numbers would\nhave lower voltages due to the strong temperature dependence of\n$R_0^N$ and $\\Delta_s$.\n\nThe separation between QP branches, $\\delta v_s$, is the\nadditional quantity, provided by intrinsic tunneling spectroscopy,\nwhich can be used to estimate $\\Delta_s$ in a wider range of\ntemperatures. From Fig.~3 b) it is seen that multiple QP branches\nare clearly distinguishable up to $T\\sim T_c-2$~K. From Table~1 it\nis seen that $\\delta v_s$ scales with $\\Delta_s$. The $\\delta v_s$\nis less than $V_s/N$ simply because the critical current, $I_c$,\nis less than $V_s/R_n$ and all IJJ's switch to the resistive state\nbefore they reach the gap voltage, see Fig.~1. The $\\delta v_s\n(I=I_c)$ continuously decreases with $T$ and vanishes at $T_c$. In\nprinciple, the temperature dependence of $I_c$ is also involved in\n$\\delta v_s(T)$, since we measure $\\delta v_s$ at $I\\approx I_c$.\nHowever, the inset in Fig.3 b) reveals that $\\delta v_s(T)$ still\ntends to vanish at $T\\rightarrow T_c$ even if we evaluate $\\delta\nv_s$ at one and the same current for all $T$.\n\nIn Fig.~4, the temperature dependencies of the superconducting\npeaks, $v_s$ (squares), $\\delta v_s (I=I_c)$ (circles), and\npseudo-gap humps, $v_{pg}$ (triangles), are shown for optimally\ndoped (solid) and overdoped (open symbols) samples. Small solid\nsymbols represent $v_s$ for the rest of the mesas listed in\nTable~1, and the lines are guides for the eye. In agreement with\nprevious studies, both $v_s$ and $v_{pg}$ increase upon going from\noverdoped to optimally doped samples\\cite{Renner,Miyak,ARPES}. The\nsuperconducting gap deduced from the sum-gap voltage\n$V_s=2N\\Delta_s/e$ is $\\Delta_s(4.2K)\\simeq 33$~meV for the\noptimally doped sample, and $\\simeq 26$~meV for the overdoped one.\n%The values $\\Delta/T_c$ are considerably larger\\cite{Renner,Miyak,ARPES}\n%than in low-$T_c$ superconductors, and range from $\\sim 3.4$ for\n%overdoped to $\\sim 4.1$ in optimally doped mesas.\nIn contrast to\nsurface tunneling experiments, we observe that $\\Delta_s$\ndecreases considerably with temperature. The robust decrease of\n$\\Delta_s(T)$ from 4.2K to $T_c$ is more than 80 \\% for the\noverdoped mesas. Moreover, we can measure $\\delta v_s (I=I_c)$ in\na wider range of $T$ and observe that it vanishes at $T\\rightarrow\nT_c$.\n\nAll this brings us to the conclusion that the superconducting gap\ndoes close at $T_c$, in agreement with the previous observations\nof vanishing of the superfluid density (divergence of the magnetic\npenetration depth)~\\cite{Lambda} and the Josephson plasma\nfrequency~\\cite{Jplasma}. On the contrary, the PG is almost\ntemperature independent and exists both above and below $T_c$.\nTherefore, the SG is not developing from the PG, and these two gaps\nrepresent different coexisting phenomena. The recently observed\nindependence of the PG on magnetic field~\\cite{Oleg} supports our conclusion\nand also casts doubts about the precursor-superconductivity origin of the PG~\\cite{Randeira}.\n\nOne possible \"non-superconducting\" PG-scenario is the formation of\ncharge or spin density waves (CDW or SDW)~\\cite{Bianc,CDW}. HTSC's\nare composed of quasi-two dimensional electronic systems with a\ncertain degree of Fermi-surface nesting~\\cite{Bianc}, which can\nmake the system unstable with respect to CDW or SDW\nformation~\\cite{Gruner,Nesting}. A CDW or SDW is accompanied by a\nPG in DOS, detectable by a surface-tunneling\nspectroscopy~\\cite{Inger}. Many similarities exist between the PG\nin CDW or SDW (including ARPES~\\cite{Nesting}, optical\nconductivity and NMR~\\cite{Gruner,CDW}) and the PG in HTSC. On the\nother hand, an opening of the gap due to CDW or SDW is typically\naccompanied by a metal-insulator transition~\\cite{Gruner}, while\nthe $ab$-plane resistivity in Bi-2212 shows the opposite tendency\n~\\cite{Watanabe}.\n\nWe would also like to emphasize a similarity between the PG\nfeatures of $c$-axis tunneling in HTSC and Coulomb PG for\ntunneling into a two-dimensional electron system (2DES). The\nCoulomb PG in 2DES is well studied in connection with\nsemiconducting heterostructures~\\cite{Chan,Efros,Levitov}.\nExperimental $\\sigma (v)$ curves from the inset in Fig.~2 are\nstrikingly similar to \"V-shaped\" tunneling characteristics of\n2DES~\\cite{Chan,Efros}. Certainly, the electron system in Bi-2212\nis highly two-dimensional. Moreover, a Coulomb origin of the HTSC\npseudo-gap would naturally explain the increase of PG with\ndecreasing O-doping and carrier concentration. A large Coulomb PG\nin low conducting 2DES is due to unscreened long-range Coulomb\ninteraction~\\cite{Efros} and/or slow charge accommodation\n\\cite{Levitov}. Large PG could also appear if tunneling occurs via\nintermediate low conducting BiO layers \\cite{Halbritt}.\n\nAn attractive feature of both CDW/SDW and Coulomb PG scenaria is\nthat the PG can persist in the superconducting state. Below $T_c$,\nSG and PG are combined into a larger overall gap\\cite{CDW}. This\nis in agreement with a definite trend for the increase of $v_{pg}$\nat $T<T_c$, see Fig. 4. This might also help in understanding of\nlarge \"superconducting\" gaps seen in underdoped HTSC\n\\cite{Renner,Miyak}. Whether the CDW/SDW or Coulomb PG scenaria\ncan explain all PG features in HTSC remains to be clarified.\n\nIn Conclusion, small mesa structures were used for intrinsic\ntunneling spectroscopy of Bi-2212. We were able to distinguish and\nsimultaneously observe both superconducting and pseudo gaps in a\nwide range of temperatures. The superconducting gap has a strong\ntemperature dependence and vanishes at $T_c$, while the pseudo-gap\nis almost temperature independent and exists both above and below\n$T_c$. This suggests that the pseudo-gap is not directly related\nto superconductivity.\n\n\\begin{references}\n\n\\bibitem[a]{1} also Institute of Solid State Physics,\n142432 Chernogolovka, Russia\n\n\\bibitem[b]{2} also P.L.Kapitza Institute,\n117334 Moscow, Russia\n\n\\bibitem[c]{3} also IMEGO Institute, Aschebergsgatan 46,\nS41133, G\\\"oteborg, Sweden\n\n\\bibitem{Renner} Ch.Renner, et.al,\nPhys.Rev.Lett. \\textbf{80} (1998) 149\n%;ibid.\\textbf{80} (1998) 3606\n\n\\bibitem{Miyak} N.Miyakawa, et.al,\nPhys.Rev.Lett. \\textbf{83} (1999) 1018\n%; ibid. \\textbf{80} (1998) 157\n\n\\bibitem{Suzuki} M.Suzuki, et.al, Phys.Rev.Lett. \\textbf{82}(1999)\n5361\n%; Physica C \\textbf{293} (1997) 124\n\n\\bibitem{ARPES} M.R.Norman, et.al., Phys.Rev.B \\textbf{57} (1998) R11093\n%; F.Ronning, et.al., Science \\textbf{282} (1998) 2067;\n\n\\bibitem{Bianc} N.L.Saini, et.al,\nPhys.Rev.Lett. \\textbf{79} (1997) 3467\n\n\\bibitem{Oleg} K.Gorny, et.al, Phys.Rev.Lett. \\textbf{82} (1999)\n177\n%; G.Q.Zheng, et.al., Phys.Rev.B \\textbf{60} (1999) R9947\n\n\\bibitem{Inter} A.Yurgens, et.al., Cond-Mat/9907159\n\n\\bibitem{Puchkov} A.V.Puchkov, et.al,\nJ.Phys.Cond.Mat. \\textbf{8} (1996) 10049\n\n\\bibitem{Randeira} M.Randeira, Cond-Mat/9710223\n\n\\bibitem{CDW}R.S.Markiewicz, et.al., Cond-mat/9807068\n\n\\bibitem{Halbritt} J.Halbritter,\nPhysica C \\textbf{302} (1998) 221\n\n\\bibitem{Mallet} P.Mallet, et.al,\nPhys.Rev.B \\textbf{54} (1996) 13324\n\n\\bibitem{Schlenga} K.Schlenga, et.al.,\nPhys.Rev.B \\textbf{57}(1998) 14518\n\n\\bibitem{Gough} I.F.G.Parker, et.al., Proc. SPIE \\textbf{3480}(1998) 11\n%in {\\it Superconducting Superlattices II:\n%Native and Artificial}(eds. I.Bozovic, and D.Pavuna)\n\n\\bibitem{Winkler} D.Winkler, et.al, Supercond.Sc.Techn. \\textbf{12} (1999) 1013\n\n%\\bibitem{Yurgens} A.Yurgens, et.al, Appl.Phys.Lett. \\textbf{70} (1997) 1760\n\n\\bibitem{Meso} V.M.Krasnov, et.al, Cond-Mat/0002094\n%Proc. Sat. Conf. LT22, {\\it Electron Transport in\n%Mesoscopic Systems} (August 1999, G\\\"oteborg, Sweden)\n\n\\bibitem{Lee} N.Kim, et.al, Phys.Rev.B \\textbf{59} (1999) 14639\n\n\\bibitem{Lambda}\n%P.Zimmermann, et.al, Phys.Rev.B \\textbf{52} (1995) 541;\nS.F.Lee, et.al, Phys.Rev.Lett. \\textbf{77} (1996) 735\n\n\\bibitem{Jplasma} T.Shibauchi, et.al,\nPhys.Rev.Lett. \\textbf{83} (1999) 1010\n\n\\bibitem{Gruner} G.Gr\\\"uner,\n{\\it Density Waves in Solids}, Addison-Wesley\nPublishing Company (1994)\n\n%\\bibitem{Friend} R.H. Friend and A.D. Yoffe,\n%Adv. Phys. \\textbf{36} (1987) 1\n\n\\bibitem{Nesting} G.H.Gweon, et.al. J. Phys., Cond.Mat. \\textbf{8} (1996) 9923\n%K.Breuer, et.al,Phys.Rev.Lett. \\textbf{76} (1996) 3172;\n\n\\bibitem{Inger} J.J.Kim, I.Ekvall and H.Olin, Phys.Rev.B \\textbf{54} (1996) 2244\n\n\\bibitem{Watanabe} T.Watanabe, et.al, Phys.Rev.Lett. \\textbf{79} (1997) 2113\n\n\\bibitem{Chan} H.B.Chan, et.al.,\nPhys.Rev.Lett. \\textbf{79} (1997) 2867; V.T.Dolgopolov, et.al.,\nibid. \\textbf{79} (1997) 729\n\n\\bibitem{Efros} F.G.Pikus and A.L.Efros,\nPhys.Rev.B \\textbf{51} (1995) 16871\n\n\\bibitem{Levitov} L.S.Levitov and A.V.Shitov,\nJETP Letters \\textbf{66} (1997) 215\n\n\\end{references}\n\n\\begin{table}\n\\noindent\n\\begin{minipage}{0.47\\textwidth}\n\\caption{Parameters of Bi2212 mesas, where $\\rho_c$ is the\n$c$-axis normal-state resistivity at large bias current.}\n\\begin{tabular}{ccccccc}\nmesa & $S$ & $N$ & $T_c$ & $\\Delta_s$(0) & $\\delta v_s$(0) &\n$\\rho_c$(4.2K) \\\\ \\space &($\\mu$m$^2$) & \\space & (K) & (meV) &\n(mV) & ($\\Omega$cm)\\\\ \\hline\n423Ar & $3.5\\times 7.5$ & 10 & 93 & 33.3 & 38.5 & 44.9\\\\\n255Ar & $5.5\\times 6$ & 12 & 92.5 & 32.5 & 35.5 & 45.4\\\\\n251Ar & $6\\times 6$ & 12 & 92.5 & 32.5 & 35.5 & 44.5\\\\\n211Ar & $4\\times 7.5$ & 12 & 94 & 33.0 & 37 & 44.0\\\\\n216Ar & $4\\times 20$ & 10 & 94 & 32.3 & 38.5 & 44.9\\\\\n015Ch & $12\\times 15$ & 9 & 89 & 25.8 & 28.5 & 32.3\n\\end{tabular}\n\\end{minipage}\n\\end{table}\n\n\\end{multicols}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002172.extracted_bib",
"string": "\\bibitem[a]{1} also Institute of Solid State Physics,\n142432 Chernogolovka, Russia\n\n\n\\bibitem[b]{2} also P.L.Kapitza Institute,\n117334 Moscow, Russia\n\n\n\\bibitem[c]{3} also IMEGO Institute, Aschebergsgatan 46,\nS41133, G\\\"oteborg, Sweden\n\n\n\\bibitem{Renner} Ch.Renner, et.al,\nPhys.Rev.Lett. \\textbf{80} (1998) 149\n%;ibid.\\textbf{80} (1998) 3606\n\n\n\\bibitem{Miyak} N.Miyakawa, et.al,\nPhys.Rev.Lett. \\textbf{83} (1999) 1018\n%; ibid. \\textbf{80} (1998) 157\n\n\n\\bibitem{Suzuki} M.Suzuki, et.al, Phys.Rev.Lett. \\textbf{82}(1999)\n5361\n%; Physica C \\textbf{293} (1997) 124\n\n\n\\bibitem{ARPES} M.R.Norman, et.al., Phys.Rev.B \\textbf{57} (1998) R11093\n%; F.Ronning, et.al., Science \\textbf{282} (1998) 2067;\n\n\n\\bibitem{Bianc} N.L.Saini, et.al,\nPhys.Rev.Lett. \\textbf{79} (1997) 3467\n\n\n\\bibitem{Oleg} K.Gorny, et.al, Phys.Rev.Lett. \\textbf{82} (1999)\n177\n%; G.Q.Zheng, et.al., Phys.Rev.B \\textbf{60} (1999) R9947\n\n\n\\bibitem{Inter} A.Yurgens, et.al., Cond-Mat/9907159\n\n\n\\bibitem{Puchkov} A.V.Puchkov, et.al,\nJ.Phys.Cond.Mat. \\textbf{8} (1996) 10049\n\n\n\\bibitem{Randeira} M.Randeira, Cond-Mat/9710223\n\n\n\\bibitem{CDW}R.S.Markiewicz, et.al., Cond-mat/9807068\n\n\n\\bibitem{Halbritt} J.Halbritter,\nPhysica C \\textbf{302} (1998) 221\n\n\n\\bibitem{Mallet} P.Mallet, et.al,\nPhys.Rev.B \\textbf{54} (1996) 13324\n\n\n\\bibitem{Schlenga} K.Schlenga, et.al.,\nPhys.Rev.B \\textbf{57}(1998) 14518\n\n\n\\bibitem{Gough} I.F.G.Parker, et.al., Proc. SPIE \\textbf{3480}(1998) 11\n%in {\\it Superconducting Superlattices II:\n%Native and Artificial}(eds. I.Bozovic, and D.Pavuna)\n\n\n\\bibitem{Winkler} D.Winkler, et.al, Supercond.Sc.Techn. \\textbf{12} (1999) 1013\n\n%\n\\bibitem{Yurgens} A.Yurgens, et.al, Appl.Phys.Lett. \\textbf{70} (1997) 1760\n\n\n\\bibitem{Meso} V.M.Krasnov, et.al, Cond-Mat/0002094\n%Proc. Sat. Conf. LT22, {\\it Electron Transport in\n%Mesoscopic Systems} (August 1999, G\\\"oteborg, Sweden)\n\n\n\\bibitem{Lee} N.Kim, et.al, Phys.Rev.B \\textbf{59} (1999) 14639\n\n\n\\bibitem{Lambda}\n%P.Zimmermann, et.al, Phys.Rev.B \\textbf{52} (1995) 541;\nS.F.Lee, et.al, Phys.Rev.Lett. \\textbf{77} (1996) 735\n\n\n\\bibitem{Jplasma} T.Shibauchi, et.al,\nPhys.Rev.Lett. \\textbf{83} (1999) 1010\n\n\n\\bibitem{Gruner} G.Gr\\\"uner,\n{\\it Density Waves in Solids}, Addison-Wesley\nPublishing Company (1994)\n\n%\n\\bibitem{Friend} R.H. Friend and A.D. Yoffe,\n%Adv. Phys. \\textbf{36} (1987) 1\n\n\n\\bibitem{Nesting} G.H.Gweon, et.al. J. Phys., Cond.Mat. \\textbf{8} (1996) 9923\n%K.Breuer, et.al,Phys.Rev.Lett. \\textbf{76} (1996) 3172;\n\n\n\\bibitem{Inger} J.J.Kim, I.Ekvall and H.Olin, Phys.Rev.B \\textbf{54} (1996) 2244\n\n\n\\bibitem{Watanabe} T.Watanabe, et.al, Phys.Rev.Lett. \\textbf{79} (1997) 2113\n\n\n\\bibitem{Chan} H.B.Chan, et.al.,\nPhys.Rev.Lett. \\textbf{79} (1997) 2867; V.T.Dolgopolov, et.al.,\nibid. \\textbf{79} (1997) 729\n\n\n\\bibitem{Efros} F.G.Pikus and A.L.Efros,\nPhys.Rev.B \\textbf{51} (1995) 16871\n\n\n\\bibitem{Levitov} L.S.Levitov and A.V.Shitov,\nJETP Letters \\textbf{66} (1997) 215\n\n"
}
] |
cond-mat0002173
|
Disorder Induced Diffusive Transport In Ratchets
|
[
{
"author": "M.N. Popescu$^1$"
},
{
"author": "C.M. Arizmendi$^{1,2}$"
},
{
"author": "A. L. Salas- Brito$^{1,3}$"
},
{
"author": "and F. Family$^1$"
}
] |
The effects of quenched disorder on the overdamped motion of a driven particle on a periodic, asymmetric potential is studied. While for the unperturbed potential the transport is due to a regular drift, the quenched disorder induces a significant additional chaotic ``diffusive'' motion. The spatio-temporal evolution of the statistical ensemble is well described by a Gaussian distribution, implying a chaotic transport in the presence of quenched disorder.
|
[
{
"name": "ratchets5.tex",
"string": "\\documentstyle[prl,aps,preprint,amssymb]{revtex}\n\\begin{document}\n\\draft\n\\title{Disorder Induced Diffusive Transport In Ratchets}\n\\author{ M.N. Popescu$^1$, C.M. Arizmendi$^{1,2}$, A. L.\nSalas- Brito$^{1,3}$, and F. Family$^1$}\n\\address {$^1$Department of Physics, Emory University,\nAtlanta, GA 30322, USA\\\\\n$^2$Depto. de F\\'{\\i}sica, Facultad de Ingenier\\'{\\i}a,\nUniversidad Nacional de Mar del Plata,\\\\ Av. J.B. Justo\n4302, 7600 Mar del Plata, Argentina\\\\\n$^3$Laboratorio de Sistemas Din\\'amicos, Departamento de\nCiencias B\\'asicas,\\\\ Universidad Aut\\'onoma\nMetropolitana-Azcapotzalco, Apartado Postal 21-726,\\\\ \nCoyoac\\'an 04000 D.\\ F., M\\'exico}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract} \nThe effects of quenched disorder on the overdamped motion \nof a driven particle on a periodic, asymmetric potential is \nstudied. While for the unperturbed potential the transport \nis due to a regular drift, the quenched disorder induces a \nsignificant additional chaotic ``diffusive'' motion. \nThe spatio-temporal evolution of the statistical ensemble is \nwell described by a Gaussian distribution, implying a chaotic \ntransport in the presence of quenched disorder.\n\\end{abstract}\n\\pacs{87.15.Aa, 87.15.Vv, 05.60.Cd, 05.45.Ac} \n\nStochastic models known as {\\it thermal ratchets} or \n{\\it correlation ratchets} \\cite{general}, in which a non-zero \nnet drift velocity may be obtained from time correlated \nfluctuations interacting with broken symmetry structures \n\\cite{magnasco}, have recently received much attention. This\ninterest is due to the wide range of possible applications\nof these models for understanding such systems as molecular \nmotors \\cite{motors}, nano-scale friction \\cite{friction}, \nsurface smoothening \\cite{barabasi}, coupled Josephson \njunctions \\cite{josephson}, optical ratchets and directed motion \nof laser cooled atoms \\cite{optics}, mass separation and \ntrapping schemes at the microscale \\cite{separation}. Recently, \nspatial disorder in thermal ratchets has been shown to reduce \nthe characteristic drift speed \\cite{Marchesoni,Harms}. \nLittle is known, however, about the effects of quenched \nspatial disorder on the regular or diffusive motion in \notherwise periodic potentials \n\\cite{Marchesoni,Harms,Garcia,Radons,Igloi}. \n\nDiffusion-like motion is observed in many types of\ndeterministic systems. In particular, it has been shown that\nin deterministic chaotic systems, diffusion can be normal\n\\cite{normal}, with the mean-square displacement\n$\\langle x^2 \\rangle$ proportional to time $t$ ($\\langle x^2\n\\rangle \\sim t$), or it can be anomalous \\cite{anomalous}, \nwith $\\langle x^2 \\rangle \\sim t^\\gamma$, (enhanced for \n$\\gamma > 2$, dispersive for $1 < \\gamma < 2$), or have a \nlogarithmic time dependence ($\\gamma = 0$) \\cite{log}.\n\nIn the present work we report on an unusual behavior \nthat occurs in the case of an overdamped ratchet subject to \nan external oscillatory drive: quenched disorder induces a \nnormal diffusive transport in addition to the drift due to the \nexternal drive. For the parameter range considered, this \nprocess is observed even for very small perturbations. \nMoreover, this diffusive motion is enhanced by higher values \nof the quenched disorder. In fact, for high enough disorder \nthe diffusive motion is of the same order of magnitude as the \nregular drift. The possibility of having large fluctuations, \nof the same order of magnitude as the average velocity, can \nbe of great importance for a correct interpretation of \nexperimental results. This may be of particular importance \nin studies of friction, in understanding the motion of \nnanoclusters or monolayers sliding on surfaces, as well as \nfor designing particles separation techniques.\n\nWe consider the one-dimensional, overdamped motion of a\nparticle (in dimensionless units) on a disordered ratchet:\n\\begin{equation}\n\\gamma {{d x} \\over {d t}} = \\cos (x) + \\mu \\cos (2x) + \\Gamma\n\\sin(\\omega t) + \\alpha~\\xi(x). \n\\label{motion} \n\\end{equation}\nHere, $\\gamma$ is the damping coefficient, $\\Gamma$ and\n$\\omega$ are, respectively, the amplitude and frequency of\nan external oscillatory forcing, and $\\alpha~\\xi(x)$ is the\nforce due to the quenched disorder. For the present study,\n$\\xi(x) \\in [-1,1]$ are independent, uniformly distributed\nrandom variables with no spatial correlations, and $\\alpha\n\\geq 0$ is the amount of quenched disorder. This corresponds\nto a piecewise constant force on the interval \n$\\biglb[ 2 k \\pi, 2 (k + 1) \\pi \\bigrb)$, \n$k \\in \\Bbb Z$. The unperturbed ratchet potential, \n\\begin{equation} \nU(x) = - \\sin(x) - \\mu \\sin(2x)\n\\label{potential} \n\\end{equation} \nhas been the subject of extensive recent studies\n\\cite{Bartussek,Jung,Lindner}. The quenched disorder term\n($\\alpha \\neq 0$) is expected to give either a more realistic\nrepresentation of a real substrate or potential landscape, \nor to model fluctuations in DC current amplitude, as for arrays \nof Josephson junctions.\n\nIt is well known \\cite{Bartussek,Jung} that in the absence\nof quenched disorder ($\\alpha = 0$) there are unbounded\nsolutions of Eq.\\ (\\ref{motion}), provided that the driving\namplitude $\\Gamma$ is large enough. These solutions tend\nasymptotically to a constant average velocity independent of\nthe initial conditions \\cite {Jung}. We have identified a\nset of parameters where, in the absence of disorder, the\nsystem shows non-zero current (regular transport).\nSpecifically, we have selected $\\gamma = 1.0$, $\\mu=0.25$,\n$\\omega = 0.1$, and we have studied the behavior for several\nvalues of $\\Gamma$ with $\\Gamma \\geq 1.4$ (see below).\n\nFor $\\alpha > 0$ the periodicity of the unperturbed potential\nis destroyed as a result of the spatial randomness, and \nsolutions of Eq.\\ (\\ref{motion}) begin to show a very\ncomplex behavior, including chaotic motion. The chaotic behavior\nis characterized by the rate of divergence of trajectories\nstarting from very close initial conditions, in other words by\nthe leading positive Lyapunov exponent. For a given, fixed \nrealization of the quenched disorder, and for several values of\n$\\Gamma \\geq 1.4$, we have computed Lyapunov exponents over \ntrajectories starting from origin. We have found positive \nLyapunov exponents $\\Lambda$ ranging from 2.55, for \n$\\Gamma = 1.4$, to 3.22 for $\\Gamma = 1.76$, which shows a \nvery strong chaotic behavior. As a consequence of this chaotic \nbehavior, the motion of the particle in the perturbed \npotential should be characterized by ensemble averages \nperformed not only over realizations of disorder, but also \nover the spatial distribution of the positions of the \nparticle in a given realization of the quenched disorder. \n\nNumerical solutions of Eq.\\ (\\ref{motion}) were obtained\nusing a variable step Runge-Kutta-Fehlberg method\n\\cite{NumRec}. Averages were performed over ensembles of\n5000 trajectories starting from different initial conditions \nvery close to the origin $x = 0$. The ensemble described above \nwas left to evolve for 1000 external drive periods $T$, and \nevery 10 periods the positions $x(t)$ were stored for further \nanalysis.\n\nWe have first analyzed the motion in a given realization of \ndisorder. In Fig.\\ \\ref{fig1} we show results for the second \nmoment, \n$C_2(t) = \\langle {(x(t)-\\langle x(t) \\rangle)}^2 \\rangle$, \nwhere $\\langle~\\dots~\\rangle$ means average over the\nensemble, as a function of the time $t$, for two different,\nfixed realizations of quenched disorder, and for two \nvalues of the disorder parameter $\\alpha$, $\\alpha = 0.05$ \n(panel (a)), respectively $\\alpha = 0.10$ (panel (b)). \nThe most striking feature is the fact that the second moment \nwhich is zero in the absence of disorder (corresponding to a \npurely deterministic motion), becomes non-zero in the presence \nof the perturbation. The non-zero second moment confirms the\nchaotic behavior mentioned above, showing a {\\it\ndisorder-induced} sensitive dependence on the initial \nconditions. It can be seen also that the time-dependence of\nthe second moment is very complicated, and it is dependent\non both the realization of quenched disorder and the\namplitude $\\alpha$ of disorder. \n\nIn order to perform averages over the realizations of\ndisorder, we have used for each trajectory a different\nrandom sequence $\\xi(x)$. In this way, the averages over the\nensemble of trajectories include also averages over\nrealizations of disorder. Figure\\ \\ref{fig2} shows results \nfor the first two moments, $C_1(t) = \\langle x(t) \\rangle$\nand \n$C_2(t) = \\langle {(x(t)-\\langle x(t) \\rangle)}^2 \\rangle$,\nas a function of the time $t$ for $\\Gamma=1.5$ for several\nvalues of disorder parameter $\\alpha$. In contrast to\naverages over a given \"landscape\", in this case both first\nand second moment show an asymptotic linear dependence on\ntime $t$, $C_1(t) \\simeq v(\\alpha)~t$, \n$C_2(t) \\simeq D(\\alpha)~t$. We have considered several other \nvalues $1.40 \\leq \\Gamma \\leq 1.76$, and we have observed the\nsame linear behavior for all $\\alpha$ values below a\nthreshold value which depends on $\\Gamma$. The quenched\ndisorder induces fluctuations in the spatial position \naround the average value in our system, and the dynamics is \nno longer regular, but rather consists in a superposition of \nregular drift and diffusion-like chaotic motion. Moreover, \neven for reasonably small amounts of disorder, for example \n$\\alpha = 0.1$, it can be seen that these spatial \nfluctuations are of the same order of magnitude as the first \nmoment, so the knowledge of a particular $x(t)$ no longer \ngives relevant information about the position of the center \nof mass of the distribution, an observation that can be of \nimportance in studies of friction, particularly the sliding \nmotion of clusters on surfaces \\cite{friction}.\n\nThe fluctuations in the position are characterics of chaotic \nbehavior in deterministic systems. The description of the \ninitial ensemble is then given by a probability distribution \nfunction $p_t(x)$, whose first two moments are linear in time \nas we have shown above. We have also calculated the higher \norder cumulants $C_n(t)$, for $n \\leq 6$, and we have found \nthat they increase slower than $t^{n/2}$. Therefore, $p_t(x)$ \nis asymptotically a Gaussian, and it is determined by the first \ntwo moments \\cite{Jung}. In Figure\\ \\ref{fig3} we show the \ndistributions $P(z)$, where $z = x-\\langle x \\rangle$, for \n$\\Gamma=1.50$ and two values of disorder parameter \n$\\alpha = 0.05$ (panel (a)), and $\\alpha = 0.10$ (panel (b)), \nat several times $t$, and the scaled distributions \n$f(y) = P(z) \\times \\sqrt(t)$, where $y = z/\\sqrt {t}$; one \ncan see that the distribution is indeed well aproximated by a \nGaussian. This asymptotic Gaussian behavior also supports the conclusion that the motion is chaotic, as it was shown \nby Jung {\\it et al} \\cite{Jung}. The reason for this chaotic \nbehavior is the existence of discontinuities in the velocity at \n$x_k = 2 \\pi k$, where $k$ is an integer, introduced by the \nquenched disorder perturbation. These random kicks keep into a transitory regime the trajectories that in the absence of \ndisorder would have asymptotically converged to the asymptotic \nconstant speed state mentioned above. This ``mixing'' of \ntransitory regimes causes the chaotic behavior, and we emphasize \nagain that it is an effect due solely to the perturbation \ninduced by quenched spatial disorder.\n\nFor several values of the external drive amplitude $\\Gamma$,\nwe have computed from the slopes of the first two moments,\nthe drift velocity $v(\\alpha)$, and the diffusion\ncoefficient $D(\\alpha)$, as functions of the amount of \ndisorder $\\alpha$. The results shown in Fig.\\ \\ref{fig4}\nindicate that below a ($\\Gamma$ dependent) threshold value of\n$\\alpha$ the drift is slightly decreasing with increased\nquenched disorder, while the diffusion coefficient is steadily\nincreasing and tends to saturate at high amounts of disorder.\nThe fact that disorder has little effect on the drift motion\nis explained by the fact that the drift is a consequence of\nthe initial asymmetry in the potential, and this asymmetry is\nonly weakly influenced by small perturbations. We note\nhere that there is no decrease in the diffusion coefficient\nover the range of disorder considered in this study. This\nis in contrast to the decrease of the diffusion coefficient \nobserved in other systems \\cite{Marchesoni,Radons}. \nThe ``divergence'' of $D(\\alpha)$ above the threshold can be \nunderstood if we consider the fact that $v(\\alpha)$ decreases \nto zero. For large enough $\\alpha$, some of the trajectories \nin the ensemble become bounded, and their contribution to the \nsecond moment is proportional to the displacement of \nthe center of mass, thus with $t^2$. The number of bounded \ntrajectories increases with time, as shown by the steady \ndecrease of the drift velocity toward zero. The contribution \nto the second moment (fluctuations) of the $t^2$ term thus \nincreases in time, and becomes dominant at late time, \nleading to the above mentioned ``divergence'' of $D(\\alpha)$.\n\nThere is a number of experimental situations where small \nperturbations of a ratchet potential are relevant, including \nsuch systems as surface electromigration \\cite{barabasi}, \ndielectrophoretic trapping, and particle separation \ntechniques \\cite{separation}. \nOur preliminary results for the case of a non-negligible \ninertial term in Eq.\\ (\\ref{motion}) are qualitatively similar \nto the ones for the over-damped case, showing disorder \ninduced chaotic diffusion. In this case, however, both the \n``diffusion coefficient'' $D$ and the drift velocity $v$ depend \non the mass of the particle. Based on the similarities \nmentioned, our results may be relevant for experiments where \nthe mass dependence of the drift velocity or diffusion \ncoefficient is essential. The efficiency of a nano-scale surface \nsmoothening by an ac field suggested by Der\\'enyi {\\it et al} \n\\cite{barabasi} could be actually significantly smaller than \ntheoretically predicted because of the chaotic diffusion, induced \nby the inherent \"disorder\" of a real surface, superimposed on \nthe net downhill current. On the other hand, the rough, imperfect \nsurface of the electrodes in the dielectrophoretic separation \ntechnique suggested by Gorre-Talini {\\it et al} \\cite{separation} \ncan actually lead to a better efficiency of the process by \nsuperimposing the chaotic diffusion and drift on top of the \nthermal, Brownian motion. Moreover, the ac-separation techniques \nusing a two-dimensional sieve discussed by Der\\'enyi and Astumian \n\\cite{separation} can be modified in a very natural way to take \nadvantage of the inherent imperfection of the two-dimensional \nstructure. This can be done by replacing the Brownian diffusion \nalong the drift direction with an additional ac-field along that \ndirection. Also, in this way one can fine tune both the drift \nvelocity and the diffusion coefficient along the separation \ndirection by a convenient choice of the ac-field parameters. \nThe temperature can then be used for an independent tuning of the electrophoretic mobility, thus for the transverse displacement.\n\nIn summary we have shown that the addition of small amounts \nof quenched disorder in the equation of motion of a continuous \ntime system induces a strong diffusive motion. In addition, we \nhave found that the presence of small amounts of disorder \nslightly decreases the regular current (drift motion), but \nsignificantly increases the transport by chaotic diffusion. \nWe have shown also that in the presence of disorder the spatial \ndistribution of positions, averaged over the realizations of \ndisorder, is described by a time-dependent Gaussian \ndistribution, which is a signature of chaotic motion. These \nunexpected results may help in the interpretation of \nexperimental results in studies of friction, particularly at \nthe nanoscale, as well as in understanding transport processes \nin molecular motors or designing particle separation techniques.\n\n\\bigskip \n\\noindent {\\bf Acknowledgments}\n\nThis work was supported by grants from the Office of Naval \nResearch, and from the Universidad Nacional de Mar del Plata. \nA. L. Salas-Brito wants to thank M.\\ Mina and C. Ch. Ujaya \nfor their friendly support and acknowledges the partial support \nof CONACyT through grant 1343P-E9607. \n\n\\begin{references}\n\n\\bibitem {general} \nC.R. Doering, {\\it Il Nuovo Cimento} {\\bf 17}, 685 (1995).\n\n\\bibitem {magnasco} \nA. Ajdari and J. Prost, \n{\\it C. R. Acad. Sci. Paris} {\\bf 315}, 1635 (1992); \nM. O. Magnasco, {\\it Phys. Rev. Lett.} {\\bf 71}, 1477 (1993).\n\n\\bibitem {motors}\nJ. Maddox, {\\it Nature} {\\bf 365}, 203 (1993); \n{\\bf 368}, 287 (1994); {\\bf 369}, 181 (1994); \nS. Leibler, {\\it ibid.} {\\bf 370}, 412 (1994); \nR. D. Astumian and M. Bier, \n{\\it Phys. Rev. Lett.} {\\bf 72}, 1766 (1994); \nC. Doering, B. Ermentrout, and G. Oster, \n{\\it Biophysical Journal} {\\bf 69}, 2256 (1995); \nR. D. Astumian and I. Der\\'enyi, \n{\\it Eur. Biophys. J.} {\\bf 27}, 474 (1998).\n\n\\bibitem {friction} \nJ. Krim, D. H. Solina, and R. Chiarello, \n{\\it Phys. Rev. Lett.} {\\bf 66}, 181 (1991); \nJ. B. Sokoloff, J. Krim, and A. Widom, \n{\\it Phys. Rev. B} {\\bf 48}, 9134 (1993);\nL. Daikhin and M. Urbakh, \n{\\it Phys. Rev. E} {\\bf 49}, 1424 (1994);\nC. Daly and J. Krim, \n{\\it Phys. Rev. Lett.} {\\bf 76}, 803 (1996);\nM. R. S$\\o$rensen, K. W. Jacobsen, and P. Stoltze, \n{\\it Phys. Rev. B} {\\bf 53}, 2101 (1996). \n\n\\bibitem {barabasi}\nI. Der\\'enyi, Choongseop Lee, and Albert-L\\'aszl\\'o Barab\\'asi,\n{\\it Phys. Rev. Lett.} {\\bf 80}, 851 (1998).\n\n\\bibitem {josephson}\nI. Zapata, R. Bartussek, F. Sols, and P. H\\\"anggi, \n{\\it Phys. Rev. Lett.} {\\bf 77}, 2292 (1996). \n\n\\bibitem {optics}\nL.P. Faucheux, L.S. Bourdieu, P.D. Kaplan, and A.J. Libchaber, \n{\\it Phys. Rev. Lett.} {\\bf 74}, 1504 (1995);\nC. Mennerat-Robilliard, D. Lucas, S. Guibal, J. Tabosa, \nC. Jurczak, J.-Y. Courtois, and G. Grynberg, \n{\\it Phys. Rev. Lett.} {\\bf 82}, 851 (1999).\n\n\\bibitem {separation}\nA.Adjari, D. Mukamel, L. Peliti, and J. Prost,\n{\\it J. Phys. I France} {\\bf 4}, 1551 (1994);\nL. Gorre-Talini, J.P. Spatz, and P. Silberzan, \n{\\it Chaos} {\\bf 8}, 650 (1998);\nI. Der\\'enyi and R. Dean Astunian, \n{\\it Phys. Rev. E} {\\bf 58}, 7781 (1998).\n\n\\bibitem {Marchesoni}\nF. Marchesoni, {\\it Phys. Rev. E} {\\bf 56}, 2492 (1997).\n\n\\bibitem {Harms}\nT. Harms and R. Lipowsky, \n{\\it Phys. Rev. Lett.} {\\bf 79}, 2895 (1997).\n\n\\bibitem {Garcia} \nE. Hern\\'andez-Garc\\'{\\i}a, M.A. Rodr\\'{\\i}guez, L. Pesquera, \nand M. San Miguel, {\\it Phys. Rev. B} {\\bf 42}, 10653 (1990).\n\n\\bibitem {Radons} \nG. Radons, {\\it Phys. Rev. Lett.} {\\bf 77}, 4748 (1996).\n\n\\bibitem {Igloi} \nF. Igl\\'oi, L. Turban, and H. Rieger, \n{\\it Phys. Rev. E} {\\bf 59}, 1465 (1999).\n\n\\bibitem {normal}\nT. Geisel and J. Nierwertberg, \n{\\it Phys. Rev. Lett.} {\\bf 48}, 7 (1982); \nJ. A. Blackburn and N. Gr$\\o$nbech-Jensen, \n{\\it Phys. Rev. E} {\\bf 53}, 3068 (1996). \n\n\\bibitem {anomalous}\nM. F. Shlesinger, G. Zaslavsky, and J. Klafter,\n{\\it Nature} {\\bf 363}, 31 (1993); \nJ. Klafter, M. F. Shlesinger, G. Zumofen, \n{\\it Physics Today} {\\bf 49}, 33 (1996);\nE. Barkai and J. Klafter, \n{\\it ibid.} {\\bf 79}, 2245 (1997).\n\n\\bibitem {log}\nE. Marinari, G. Parisi, D. Ruelle and P. Windey, \n{\\it Phys. Rev. Lett.} {\\bf 50}, 1223 (1983);\nJ. Krug and H. T. Dobbs, \n{\\it Phys. Rev. Lett.} {\\bf 76}, 4096 (1996).\n\n\\bibitem {Bartussek} \nR. Bartussek, P. H\\\"anggi, and J.G. Kissner, \n{\\it Europhys. Lett.}, {\\bf 28}, 459 (1994).\n\n\\bibitem {Jung} \nP. Jung, J.G. Kissner, and P. H\\\"anggi, \n{\\it Phys. Rev. Lett.} {\\bf 76}, 3436 (1996).\n\n\\bibitem {Lindner} \nB. Lindner, L. Schimansky-Geier, P. Reimann, P. H\\\"anggi, \nand M. Nagaoka, {\\it Phys. Rev. E} {\\bf 59}, 1417 (1999). \n\n\\bibitem {NumRec} \nW. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, \n{\\it Numerical Receipes in FORTRAN}, (Cambridge University Press, \nCambridge, 1995), pp. 704 - 716\n\n\\end{references}\n\n\n\\begin{figure}\n\\caption{Second moment $C_2(t)$ as a function of time for two\ndifferent, fixed realizations of quenched disorder (solid,\nrespectively dashed lines). The parameters used in Eq.\\\n(\\ref{motion}) are $\\gamma = 1$, $\\mu=0.25$, \n$\\Gamma = 1.76$, $\\omega = 0.1$, and $\\alpha = 0.05$ (panel\n(a)), respectively $\\alpha = 0.10$ (panel (b)).}\n\\label{fig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\caption{(a) First moment $C_1(t)$ as a function of time for\nseveral values of the amount of quenched disorder parameter\n$\\alpha$. From top to bottom, $\\alpha =\n0,~0.05,~0.10,~0.15,~0.20$. (b) Second moment $C_2(t)$ as a\nfunction of time for several values of the amount of\nquenched disorder parameter $\\alpha$. From bottom to top,\n$\\alpha = 0,~0.05,~0.10,~0.15,~0.20$. The parameters used in\nEq.\\ (\\ref{motion}) are $\\gamma = 1$, $\\mu=0.25$, \n$\\Gamma = 1.50$, and $\\omega = 0.1$.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Time evolution of the unscaled spatial distributions \n$P(z)$, where $z = x-\\langle x \\rangle$; from top to bottom, \ntime $t = 20~T, 40~T,\\dots, 100~T$. The inset shows the scaled distributions $P(z) \\times \\sqrt{t}$ {\\it vs} $z/\\sqrt{t}$. \nThe solid line in the insets shows the theoretical, asymptotic \nGaussian form. The parameters used in Eq.\\ (\\ref{motion}) are \n$\\gamma = 1$, $\\mu=0.25$, $\\Gamma = 1.50$, $\\omega = 0.1$, and \n$\\alpha = 0.05$ (panel (a)), respectively $\\alpha = 0.10$ \n(panel (b)).}\n\\label{fig3}\n\n\\end{figure}\n\n\\begin{figure}\n\\caption{(a) Diffusion coefficient $D(\\alpha)$ as a function\nof the amount of quenched disorder for\n$\\Gamma=1.40,~1.50,~1.55,~1.65,~1.76$. (b) Drift velocity\n$v(\\alpha)$ as a function of the amount of quenched disorder\nfor $\\Gamma=1.40,~1.50,~1.55,~1.65,~1.76$. The parameters are\n$\\gamma = 1$, $\\mu=0.25$, $\\omega = 0.1$.}\n\\label{fig4}\n\\end{figure}\n\n\\end{document}\n\n\n"
}
] |
[
{
"name": "cond-mat0002173.extracted_bib",
"string": "\\bibitem {general} \nC.R. Doering, {\\it Il Nuovo Cimento} {\\bf 17}, 685 (1995).\n\n\n\\bibitem {magnasco} \nA. Ajdari and J. Prost, \n{\\it C. R. Acad. Sci. Paris} {\\bf 315}, 1635 (1992); \nM. O. Magnasco, {\\it Phys. Rev. Lett.} {\\bf 71}, 1477 (1993).\n\n\n\\bibitem {motors}\nJ. Maddox, {\\it Nature} {\\bf 365}, 203 (1993); \n{\\bf 368}, 287 (1994); {\\bf 369}, 181 (1994); \nS. Leibler, {\\it ibid.} {\\bf 370}, 412 (1994); \nR. D. Astumian and M. Bier, \n{\\it Phys. Rev. Lett.} {\\bf 72}, 1766 (1994); \nC. Doering, B. Ermentrout, and G. Oster, \n{\\it Biophysical Journal} {\\bf 69}, 2256 (1995); \nR. D. Astumian and I. Der\\'enyi, \n{\\it Eur. Biophys. J.} {\\bf 27}, 474 (1998).\n\n\n\\bibitem {friction} \nJ. Krim, D. H. Solina, and R. Chiarello, \n{\\it Phys. Rev. Lett.} {\\bf 66}, 181 (1991); \nJ. B. Sokoloff, J. Krim, and A. Widom, \n{\\it Phys. Rev. B} {\\bf 48}, 9134 (1993);\nL. Daikhin and M. Urbakh, \n{\\it Phys. Rev. E} {\\bf 49}, 1424 (1994);\nC. Daly and J. Krim, \n{\\it Phys. Rev. Lett.} {\\bf 76}, 803 (1996);\nM. R. S$\\o$rensen, K. W. Jacobsen, and P. Stoltze, \n{\\it Phys. Rev. B} {\\bf 53}, 2101 (1996). \n\n\n\\bibitem {barabasi}\nI. Der\\'enyi, Choongseop Lee, and Albert-L\\'aszl\\'o Barab\\'asi,\n{\\it Phys. Rev. Lett.} {\\bf 80}, 851 (1998).\n\n\n\\bibitem {josephson}\nI. Zapata, R. Bartussek, F. Sols, and P. H\\\"anggi, \n{\\it Phys. Rev. Lett.} {\\bf 77}, 2292 (1996). \n\n\n\\bibitem {optics}\nL.P. Faucheux, L.S. Bourdieu, P.D. Kaplan, and A.J. Libchaber, \n{\\it Phys. Rev. Lett.} {\\bf 74}, 1504 (1995);\nC. Mennerat-Robilliard, D. Lucas, S. Guibal, J. Tabosa, \nC. Jurczak, J.-Y. Courtois, and G. Grynberg, \n{\\it Phys. Rev. Lett.} {\\bf 82}, 851 (1999).\n\n\n\\bibitem {separation}\nA.Adjari, D. Mukamel, L. Peliti, and J. Prost,\n{\\it J. Phys. I France} {\\bf 4}, 1551 (1994);\nL. Gorre-Talini, J.P. Spatz, and P. Silberzan, \n{\\it Chaos} {\\bf 8}, 650 (1998);\nI. Der\\'enyi and R. Dean Astunian, \n{\\it Phys. Rev. E} {\\bf 58}, 7781 (1998).\n\n\n\\bibitem {Marchesoni}\nF. Marchesoni, {\\it Phys. Rev. E} {\\bf 56}, 2492 (1997).\n\n\n\\bibitem {Harms}\nT. Harms and R. Lipowsky, \n{\\it Phys. Rev. Lett.} {\\bf 79}, 2895 (1997).\n\n\n\\bibitem {Garcia} \nE. Hern\\'andez-Garc\\'{\\i}a, M.A. Rodr\\'{\\i}guez, L. Pesquera, \nand M. San Miguel, {\\it Phys. Rev. B} {\\bf 42}, 10653 (1990).\n\n\n\\bibitem {Radons} \nG. Radons, {\\it Phys. Rev. Lett.} {\\bf 77}, 4748 (1996).\n\n\n\\bibitem {Igloi} \nF. Igl\\'oi, L. Turban, and H. Rieger, \n{\\it Phys. Rev. E} {\\bf 59}, 1465 (1999).\n\n\n\\bibitem {normal}\nT. Geisel and J. Nierwertberg, \n{\\it Phys. Rev. Lett.} {\\bf 48}, 7 (1982); \nJ. A. Blackburn and N. Gr$\\o$nbech-Jensen, \n{\\it Phys. Rev. E} {\\bf 53}, 3068 (1996). \n\n\n\\bibitem {anomalous}\nM. F. Shlesinger, G. Zaslavsky, and J. Klafter,\n{\\it Nature} {\\bf 363}, 31 (1993); \nJ. Klafter, M. F. Shlesinger, G. Zumofen, \n{\\it Physics Today} {\\bf 49}, 33 (1996);\nE. Barkai and J. Klafter, \n{\\it ibid.} {\\bf 79}, 2245 (1997).\n\n\n\\bibitem {log}\nE. Marinari, G. Parisi, D. Ruelle and P. Windey, \n{\\it Phys. Rev. Lett.} {\\bf 50}, 1223 (1983);\nJ. Krug and H. T. Dobbs, \n{\\it Phys. Rev. Lett.} {\\bf 76}, 4096 (1996).\n\n\n\\bibitem {Bartussek} \nR. Bartussek, P. H\\\"anggi, and J.G. Kissner, \n{\\it Europhys. Lett.}, {\\bf 28}, 459 (1994).\n\n\n\\bibitem {Jung} \nP. Jung, J.G. Kissner, and P. H\\\"anggi, \n{\\it Phys. Rev. Lett.} {\\bf 76}, 3436 (1996).\n\n\n\\bibitem {Lindner} \nB. Lindner, L. Schimansky-Geier, P. Reimann, P. H\\\"anggi, \nand M. Nagaoka, {\\it Phys. Rev. E} {\\bf 59}, 1417 (1999). \n\n\n\\bibitem {NumRec} \nW. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, \n{\\it Numerical Receipes in FORTRAN}, (Cambridge University Press, \nCambridge, 1995), pp. 704 - 716\n\n"
}
] |
cond-mat0002174
|
Foundations of Dissipative Particle Dynamics
|
[
{
"author": "Eirik G. Flekk\\o y$^1$"
},
{
"author": "Peter V. Coveney$^2$ and Gianni De Fabritiis$^2$"
}
] |
%NB We derive a mesoscopic modeling and simulation technique that is very close to the technique known as dissipative particle dynamics. The model is derived from molecular dynamics by means of a systematic coarse-graining procedure. This procedure links the forces between the dissipative particles to a hydrodynamic description of the underlying molecular dynamics (MD) particles. In particular, the dissipative particle forces are given directly in terms of the viscosity emergent from MD, while the interparticle energy transfer is similarly given by the heat conductivity derived from MD. In linking the microscopic and mesoscopic descriptions we thus rely on the macroscopic description emergent from MD. Thus the rules governing our new form of dissipative particle dynamics reflect the underlying molecular dynamics; in particular all the underlying conservation laws carry over from the microscopic to the mesoscopic descriptions. We obtain the forces experienced by the dissipative particles together with an approximate form of the associated equilibrium distribution. Whereas previously the dissipative particles were spheres of fixed size and mass, now they are defined as cells on a Voronoi lattice with variable masses and sizes. This Voronoi lattice arises naturally from the coarse-graining procedure which may be applied iteratively and thus represents a form of renormalisation-group mapping. It enables us to select any desired local scale for the mesoscopic description of a given problem. Indeed, the method may be used to deal with situations in which several different length scales are simultaneously present. We compare and contrast this new particulate model with existing continuum fluid dynamics techniques, which rely on a purely macroscopic and phenomenological approach. Simulations carried out with the present scheme show good agreement with theoretical predictions for the equilibrium behavior.
|
[
{
"name": "all5.tex",
"string": "%\\documentstyle[aps,prl,manuscript]{revtex} \n\\documentstyle[aps,prl,multicol,psfig]{revtex}\n\\newcommand{\\be}{\\begin{equation}} \n\\newcommand{\\ee}{\\end{equation}} \n\\newcommand{\\bea}{\\begin{eqnarray}} \n\\newcommand{\\eea}{\\end{eqnarray}} \n\\newcommand{\\eq}{Eq.~(\\ref} \n\\def\\dm{\\dot{\\tilde{M}}_{kl}}\n\\def\\bi{{\\bf i}} %\n\\def\\bs{{\\bf s}} %\n\\def\\bu{{\\bf u}} %\n\\def\\bU{{\\bf U}} %\n\\def\\bP{{\\bf P}} %\n\\def\\bQ{{\\bf Q}} %\n\\def\\bom{\\mbox{\\boldmath$ \\omega $}}\n\\def\\bv{{\\bf v}} %\n\\def\\bx{{\\bf x}} %\n\\def\\br{{\\bf r}} %\n\\def\\bj{{\\bf j}} %\n\\def\\bee{{\\bf e}} %\n\\def\\bg{{\\bf g}} %\n\\def\\bn{{\\bf n}} %\n\\def\\bS{{\\bf S}} %\n\\def\\bF{{\\bf F}} %\n\\def\\bG{{\\bf G}} %\n\\def\\bL{{\\mathcal{L} }} %\n\\def\\bJ{{\\bf J}} %\n\\def\\la{\\langle} %\n\\def\\ra{\\rangle} %\n\\def\\dd{\\text{d}} %\n\\begin{document}\n\\draft\n\\title{Foundations of Dissipative Particle \nDynamics}\n\\author{Eirik G. Flekk\\o y$^1$, Peter V. Coveney$^2$ \nand Gianni De Fabritiis$^2$}\n\\address{%ad---------------------------------------------------------\n$^1$Department of Physics, University of Oslo\\\\\n P.O. Box 1048 Blindern, 0316 Oslo 3, Norway\\\\\n$^2$Centre for Computational Science, Queen Mary and Westfield College,\\\\\n University of London, London E1 4NS, United Kingdom}\n\\date{\\today}\n\\maketitle \n\\begin{abstract}\n%NB\nWe derive a mesoscopic modeling and simulation\ntechnique that is very close to the\ntechnique known as dissipative particle dynamics.\nThe model is derived from molecular dynamics \nby means of a systematic coarse-graining procedure. This procedure\nlinks the forces between the dissipative particles\nto a hydrodynamic description of the underlying molecular dynamics (MD)\nparticles.\nIn particular, the dissipative particle forces are given \ndirectly in terms of the viscosity emergent from MD, while the \ninterparticle energy transfer is similarly given by the heat \nconductivity derived from MD.\nIn linking the microscopic and mesoscopic descriptions we thus\nrely on the macroscopic description emergent from MD.\nThus the rules governing our new\nform of dissipative particle dynamics reflect the \nunderlying molecular dynamics; in particular all the \nunderlying conservation laws carry over from the \nmicroscopic to the mesoscopic descriptions.\nWe obtain the forces experienced \nby the dissipative particles together with an approximate \nform of the associated equilibrium distribution.\n Whereas previously the dissipative particles were spheres of \nfixed size and mass, now they\nare defined as cells on a Voronoi lattice with variable masses and sizes.\nThis Voronoi lattice arises naturally from the coarse-graining \nprocedure which may be applied iteratively\nand thus represents a form of renormalisation-group mapping.\nIt enables us to select any desired local scale\nfor the mesoscopic description of a given problem. Indeed, the method \nmay be used to deal with situations in which several \ndifferent length scales are simultaneously present. We compare and \ncontrast this new particulate model with existing continuum\nfluid dynamics techniques, which rely on a purely macroscopic and \nphenomenological approach.\nSimulations carried out with the present scheme show \ngood agreement with theoretical predictions for the equilibrium behavior.\n\\end{abstract}\n\\pacs{Pacs numbers:\n47.11.+j % Computational methods in fluid dynamics\n47.10.+g % General theory -- fluid dynamics\n05.40.+j % Fluctuation phenomena, Random processes and Brownian motion\n} \n\n%==================================================================\n\\begin{multicols}{2}\n\n\\section{Introduction}\n\nThe non-equilibrium behavior of fluids continues to present \na major challenge for both theory and numerical simulation. In recent times, \nthere has\nbeen growing interest in the study of so-called `mesoscale' modeling and \nsimulation \nmethods, particularly for the description of the complex dynamical behavior of \nmany kinds\nof soft condensed matter, whose properties have thwarted more \nconventional approaches. \nAs an example, consider the case of complex fluids with many\ncoexisting length and time scales, for which hydrodynamic descriptions are \nunknown and may not even exist. These kinds of fluids \ninclude multi-phase flows, particulate and colloidal suspensions, polymers,\nand amphiphilic fluids, including emulsions and microemulsions. \nFluctuations and\nBrownian motion are often key features controlling their behavior.\n\n{}From the standpoint of traditional fluid dynamics, a general problem\nin describing \nsuch fluids is the lack of adequate continuum models. \nSuch descriptions, which are usually based on simple\nconservation laws, approach the physical description from the \nmacroscopic side, that is in a `top down' manner, \nand have certainly proved successful \nfor simple Newtonian fluids \\cite{landau59}.\nFor complex fluids, however, equivalent phenomenological representations \nare usually unavailable\nand instead it is necessary to base the modeling approach\non a microscopic (that is on a particulate) description of the system, \nthus working from the bottom upwards, along the general lines \nof the program for statistical mechanics pioneered by \nBoltzmann \\cite{boltzmann1872}.\nMolecular dynamics (MD) presents itself as the most accurate \nand fundamental method~\\cite{koplik95} but it is far too\ncomputationally intensive to provide a practical option for \nmost hydrodynamic problems involving complex fluids.\nOver the last decade several alternative `bottom up' strategies have\ntherefore been introduced. Hydrodynamic lattice gases \\cite{frisch86},\nwhich model the fluid as a discrete set of particles, represent\na computationally efficient spatial and temporal discretization of the \nmore conventional molecular dynamics. The\nlattice-Boltzmann method \\cite{mcnamara88}, originally derived from \nthe lattice-gas paradigm by invoking Boltzmann's {\\em Stosszahlansatz}, \nrepresents an \nintermediate (fluctuationless) approach \nbetween the top-down (continuum) and bottom-up (particulate)\nstrategies, insofar as the basic\nentity in such models is a single particle distribution function; but\nfor interacting systems even these lattice-Boltzmann methods can be \nsubdivided into bottom-up~\\cite{chan93} \nand top-down models~\\cite{swift95}.\n\nA recent contribution to the family of bottom-up approaches\nis the dissipative particle dynamics (DPD) method introduced\nby Hoogerbrugge and Koelman in 1992~\\cite{hoogerbrugge92}.\nAlthough in the original formulation\nof DPD time was discrete and space continuous, a more recent re-interpretation \nasserts that\nthis model is in fact a finite-difference approximation to the `true'\nDPD, which is defined by a set of continuous time \nLangevin equations with momentum\nconservation between the dissipative particles~\\cite{espanol95b}.\nSuccessful applications of the technique have been made to colloidal\nsuspensions~\\cite{boek97}, polymer solutions~\\cite{schlijper95} and\nbinary immiscible fluids~\\cite{coveney96}.\nFor specific applications where comparison is possible,\nthis algorithm is orders of magnitude faster than MD~\\cite{groot97}.\nThe basic elements of the DPD scheme are particles that \nrepresent rather ill-defined `mesoscopic' quantities of the \nunderlying molecular fluid. \nThese dissipative particles are stipulated to evolve in the \nsame way that MD \nparticles do, but with different inter-particle forces: since \nthe DPD particles are pictured to have \ninternal degrees of freedom, the forces between them have \nboth a fluctuating and a dissipative component in addition\nto the conservative forces that are present at the MD level.\nNewton's third law is still satisfied, however, and consequently \nmomentum conservation together with mass conservation produce \nhydrodynamic behavior at the macroscopic level.\n\nDissipative particle dynamics has been shown to produce the\ncorrect macroscopic (continuum) theory; that is, for a one-component DPD\nfluid, the Navier-Stokes equations emerge in the \nlarge scale limit, and the fluid viscosity \ncan be computed~\\cite{espanol95,MBE1}.\n%NB\nHowever, even though dissipative particles have\ngenerally been viewed as clusters of molecules, \nno attempt has been made to link DPD to the underlying \nmicroscopic dynamics, and DPD thus remains a foundationless\nalgorithm, as is that of the hydrodynamic lattice gas and {\\em a\nfortiori} the lattice-Boltzmann\nmethod. It is the principal purpose of the present paper\nto provide an atomistic foundation for dissipative\nparticle dynamics. Among the numerous benefits gained by achieving\nthis, \nwe are then able to provide a precise definition of the term `mesoscale', \nto relate the hitherto purely\nphenomenological parameters in the algorithm to underlying molecular \ninteractions, and thereby to formulate DPD simulations for {\\em specific} \nphysicochemical systems, defined in terms of their molecular constituents.\nThe DPD that we derive is a representation of the underlying MD.\nConsequently, to the extent that the \napproximations made are valid, the DPD and MD will have the same hydrodynamic \ndescriptions,\nand no separate kinetic theory for, say, the DPD viscosity\nwill be needed once it is known for the MD system.\nSince the MD degrees of freedom will be integrated out in our\napproach the MD viscosity will appear in the DPD model as \na parameter that may be tuned freely.\n\nIn our approach, the `dissipative particles' (DP) are defined in terms\nof appropriate weight functions that sample portions of the underlying \nconservative MD particles, and the \nforces between the dissipative particles are obtained from the \nhydrodynamic \ndescription of the MD system: the microscopic conservation laws carry \nover directly to the \nDPD, and the hydrodynamic behavior of MD is thus reproduced by the \nDPD, albeit at a coarser scale. The mesoscopic (coarse-grained) scale\nof the DPD can be precisely\nspecified in terms of the MD interactions.\nThe size of the dissipative particles, as specified by \nthe number of MD particles within them, furnishes the \nmeaning of the term `mesoscopic' in the present context.\nSince this size is a freely tunable parameter of the model, the\nresulting DPD\nintroduces a general procedure for simulating \nmicroscopic systems at \nany convenient scale of coarse graining, provided that\nthe forces between the dissipative particles are known.\nWhen a hydrodynamic description of the underlying particles \ncan be found, these forces follow directly; in\ncases where this is not possible, the forces between dissipative\nparticles must be supplemented \nwith the additional components of the physical description \nthat enter on the mesoscopic level.\n\nThe DPD model which we derive from molecular dynamics\nis formally similar to conventional, albeit foundationless, \nDPD~\\cite{espanol95}. \nThe interactions are pairwise and conserve mass and momentum, as\nwell as energy \\cite{avalos97,espanol97}.\nJust as the forces conventionally used to define DPD have\nconservative, dissipative and fluctuating \ncomponents, so too do the forces in the present case. \nIn the present model, the role of the conservative force is played \nby the pressure forces.\nHowever, while conventional dissipative particles possess spherical\nsymmetry and experience interactions mediated by purely central\nforces, our dissipative particles \nare defined as space-filling cells on a Voronoi lattice whose forces\nhave both central and tangential components.\nThese features are shared with a model studied by Espa\\~{n}ol\n\\cite{espanol98b}. This model links DPD to smoothed particle \nhydrodynamics \\cite{monaghan92} and defines the DPD \nforces by hydrodynamic considerations in a way analogous to earlier\nDPD models.\nEspa\\~{n}ol {\\it et al.} \\cite{espanol97b} have also carried out MD\nsimulations with a superposed Voronoi mesh \nin order to measure the coarse grained inter-DP forces.\n\nWhile conventional DPD defines dissipative particle masses\nto be constant, this feature is not preserved in our new model. \nIn our first publication on this theory \\cite{flekkoy99}, we \nstated that, while the dissipative particle masses fluctuate due to the\nmotion of MD particles across their boundaries, the average masses\nshould be constant. In fact, the DP-masses\nvary due to distortions of the Voronoi cells, \nand this feature is now properly incorporated\nin the model.\n\n\nWe follow two distinct routes to obtain the\nfluctuation-dissipation relations that \ngive the magnitude of the thermal forces.\nThe first route follows the conventional path which makes\nuse of a Fokker-Planck equation~\\cite{espanol95b}.\n We show that the DPD system is described in an approximate \nsense by the isothermal-isobaric ensemble.\nThe second route is based on the theory of\nfluctuating hydrodynamics and it is argued that this\napproach corresponds to the statistical mechanics of\nthe grand canonical ensemble. \nBoth routes lead to the same result for the fluctuating \nforces and simulations confirm that, \nwith the use of these forces, the measured \nDP temperature is equal to the MD temperature which is provided as input.\nThis is an important finding in the present context as\nthe most significant \napproximations we have made \nunderlie the derivation of the thermal forces.\n\n\n\\section{Coarse-graining molecular dynamics: from micro to mesoscale}\n\nThe essential idea motivating our definition of mesoscopic \ndissipative particles is to specify them as clusters of MD particles\nin such a way that the MD particles themselves remain unaffected while\n{\\em all} being represented by the dissipative particles. \nThe independence of the molecular dynamics from the superimposed\ncoarse-grained dissipative particle dynamics implies that \nthe MD particles are able to move between the dissipative particles.\nThe stipulation that all MD particles must be fully represented by the\nDP's implies that while the mass, momentum and energy\nof a single MD particle may be shared between DP's, the sum of the\nshared components must always equal the mass and momentum of the MD particle.\n\n\\subsection{Definitions}\nFull representation of all the MD particles can be achieved in a\ngeneral way by introducing a sampling function\n\\be\nf_k(\\bx )= \\frac{s(\\bx - \\br_k )}{\\sum_l s(\\bx - \\br_l )}\n\\label{sampling} \\; .\n\\ee\nwhere the positions $\\br_k$ and $\\br_l$ define the DP \ncenters, $\\bx$ is an arbitrary position and $s(\\bx )$\nis some localized function. \nIt will prove convenient to choose it as a Gaussian \n\\be \ns(\\bx ) = \\exp{( - x^2/a^2)}\n\\ee\nwhere the distance $a$ sets the scale of the sampling function,\nalthough this choice is not necessary.\nThe mass, momentum and internal energy $E$ of the $k$th DP are then \ndefined as\n\\bea\nM_k &=& \\sum_i f_k(\\bx_i ) m , \\nonumber \\\\\n\\bP_k &=& \\sum_i f_k(\\bx_i )m \\bv_i , \\nonumber \\\\\n\\frac{1}{2} M_k U_k^2 + E_k &=& \\sum_i f_k(\\bx_i )\n \\left( \\frac{1}{2} m v_i^2 + \\frac{1}{2} \\sum_{j\\neq i} \nV_{MD}(r_{ij}) \\right) \\nonumber \\\\\n& \\equiv & \\sum_i f_k(\\bx_i ) \\epsilon_i , \\label{DPdef}\n\\eea\nwhere $\\bx_i$ and $\\bv_i$ are the position and velocity \nof the $i$th MD particle, which are all assumed to have identical masses $m$, \n$\\bP_k$ is the momentum \nof the $k$th DP and $V_{MD}(r_{ij})$ is the potential energy \nof the MD particle pair $ij$, separated a distance $r_{ij}$.\nThe particle energy $\\epsilon_i$ thus contains both the kinetic\nand a potential term.\nThe kinematic condition\n\\be\n\\dot{\\br}_k =\\bU_k \\equiv \\bP_k/M_k \n\\ee\ncompletes the definition of our dissipative particle dynamics.\n\nIt is generally true that mass and momentum conservation\nsuffice to produce hydrodynamic behavior. However, \nthe equations expressing these conservation laws contain the\nfluid pressure. In order to get the fluid pressure a \nthermodynamic description of the system is needed.\nThis produces an equation of state, which\ncloses the system of hydrodynamic equations.\nAny thermodynamic potential may be used to obtain\nthe equation of state.\nIn the present case we shall take this potential to be \nthe internal energy $E_k$ of the dissipative particles,\nand we shall obtain the equations of motion\nfor the DP mass, momentum and energy.\nNote that the internal energy would also have to be computed\nif a free energy had been chosen for the thermodynamic\ndescription. For this reason it is not possible to complete\nthe hydrodynamic description without taking the\nenergy flow into account. As a byproduct of this \nthe present DPD also contains a description\nof the heat flow and corresponds to the recently\nintroduced DPD with energy conservation~\\cite{avalos97,espanol97}.\nEspa\\~{n}ol previously introduced an angular momentum variable\ndescribing the dynamics of extended particles~\\cite{espanol98b}: this\nis needed\nwhen forces are non-central \nin order to avoid dissipation of energy in a rigid rotation of the fluid.\nAngular momentum could be included on the same footing as momentum \nin the following developments.\nHowever for reasons both of space and \nconceptual economy we shall omit it in the present context, even\nthough it is probably important in applications\nwhere hydrodynamic precision is important.\n% be seen by considering a vortex enclosed in angular momentum \n% conserving boundaries. Within such boundaries both a system \n% of DP's with and without an internal angular momentum variable will\n% produce an angular momentum conserving flow, the difference\n% being that the flow without the angular momentum variables \n% will not relax to a rigid body rotation but rather a vortex \n% with some internal shear (though without dissipation).\nIn the following sections, we shall \nuse the notation $\\br$, $M$, $\\bP$ and $E$ with the \nindices $k\\; , l\\; , m$ and $ n$ to denote DP's\nwhile we shall use $\\bx$, $m$, $\\bv$ and $\\epsilon$ \nwith the indices $i$ and $j$ to denote MD particles.\n\n\\subsection{Equations of motion for the dissipative particles\nbased on a microscopic description} \nThe fact that all the MD particles are represented at all instants in the\ncoarse-grained scheme is guaranteed by the normalization condition \n$\\sum_k f_k (\\bx ) = 1$. This implies directly that \n\\bea\n\\sum_k M_k &=& \\sum_i m \\nonumber \\\\\n\\sum_k \\bP_k &=& \\sum_i m \\bv_i \\nonumber \\\\\n\\sum_k E_k^{\\text{tot}} &=&\n\\sum_k \\left( \\frac{1}{2} M_k \\bU_k^2 + E_k \\right) = \\sum_i \\epsilon_i \\;;\n\\eea\nthus with mass, momentum and energy conserved at the MD \nlevel, these quantities are also conserved at the DP level.\nIn order to derive the equations of motion for\ndissipative particle dynamics we now take the time derivatives \nof Eqs.~(\\ref{DPdef}).\nThis gives\n\\bea\n\\frac{\\dd {M_k}}{\\dd t} &=& \\sum_i \\dot{f}_k (\\bx_i )m \\label{mass}\\\\\n\\frac{\\dd \\bP_k }{\\dd t} &=& \\sum_i \\left( \\dot{f}_k (\\bx_i ) m \\bv_i\n+{f}_k (\\bx_i ) \\bF_i \\right) \\label{momentum} \\\\\n\\frac{\\dd E_k^{\\text{tot}}}{\\dd t} &=&\\sum_i \\left( \\dot{f}_k (\\bx_i ) \\epsilon_i\n+{f}_k (\\bx_i ) \\dot{\\epsilon}_i \\right) \\label{energy} \n\\eea\nwhere $\\dd /\\dd t$ is the substantial derivative and $\\bF_i = m\\dot{\\bv}_i$\nis the force on particle $i$.\n\nThe Gaussian form of $s$ implies that \\\\ $\\dot{s} (\\bx ) = -(2/a^2)\n\\dot{\\bx} \\cdot \\bx s(\\bx )$. This makes it possible to write\n\\be\n\\dot{f}_k (\\bx_i ) = f_{kl}(\\bx_i ) (\\bv'_i \\cdot \\br_{kl}\n+ \\bx'_i \\cdot \\bU_{kl} )\n\\label{dotf}\n\\ee\nwhere the overlap function $f_{kl}$ is defined as\n$f_{kl} (\\bx )\\equiv (2/a^2 )f_{k} (\\bx ) f_{l} (\\bx ) $,\n$\\br_{kl} \\equiv (\\br_k - \\br_l)$ and $\\bU_{kl} \\equiv (\\bU_k - \\bU_l)$,\nand we have rearranged terms so as to get them in terms of the \ncentered variables\n\\bea \n\\bv'_i &=& \\bv_i - \\frac{(\\bU_k +\\bU_l)}{2} \\nonumber \\\\\n\\bx'_i &=& \\bx_i - \\frac{(\\br_k +\\br_l)}{2} \\; .\n\\label{primes}\n\\eea\n%****** VORONOI STORY*******YY\n\nBefore we proceed with the derivation of the \nequations of motion it is instructive \nto work out the actual forms of $f_k(\\bx )$\nand $f_{kl}(\\bx )$\nin the case of only two particles $k$ and $l$.\n Using the Gaussian choice of $s$ we immediately get \n\\be\nf_k (\\bx ) = \\frac{1}{1+ \\left[ \\exp{((\\bx - (\\br_k + \\br_l)/2)\n\\cdot (\\br_{kl})/(a^2) )}\\right]^2} \\; .\n\\ee \nThe overlap function similarly follows:\n\\be \nf_{kl} (\\bx ) =\n\\frac{1}{2a^2} \\cosh^{-2}{ \\left( \\left( \\bx - \\frac{\\br_k + \\br_l}{2} \\right) \n\\cdot \\left( \\frac{\\br_{kl}}{a^2} \\right) \\right) }\n\\label{overlap} \\; .\n\\ee\n\\begin{figure} \n\\centerline{\\hbox{\\psfig{figure=overlap.eps,width=8cm}}}\n\\caption{\\label{fig2}\n\\protect \\narrowtext \nThe overlap region between two Voronoi \ncells is shown in grey. The sampling function $f_k (\\br )$ is shown in the\ntop graph and the overlap function $f_{kl} ( \\br ) = \n(2/a^2) f_{k} (\\br ) f_{l}(\\br )$ in the bottom graph. The width of the overlap\nregion is $ a^2 / |\\br_k - \\br_l |$ and its length is denoted by $l$.}\n\\end{figure}\nThese two functions are shown in Fig.\\ref{fig2}.\nNote that the scale of the overlap region is not $a$ but $a^2/|\\br_k - \\br_l|$.\nDissipative particle interactions only\ntake place where the overlap function is non-zero.\nThis happens along the dividing line which is equally far \nfrom the two particles.\nThe contours of non-zero $f_{kl}$\nthus define a Voronoi lattice with lattice segments of length \n$l_{kl}$.\nThis Voronoi construction is shown in Fig.~\\ref{fig1}\nin which MD particles in the overlap region defined by $f_{kl} >0.1$,\nare shown, though presently not actually simulated as dynamic\nentities. The volume of the Voronoi cells will in general \nvary under the dynamics. \nHowever, even with arbitrary dissipative particle\nmotion the cell volumes will approach zero only exceptionally, \nand even then the identities\nof the DP particles will be preserved so that they subsequently re-emerge.\n\n\\subsubsection{Mass equation}\n\nThe mass equation (\\ref{mass}) \ntakes the form \n\\be\n\\frac{\\dd {M_k}}{\\dd t} \\equiv \\sum_l \\dot{M}_{kl}\n\\nonumber \n\\ee\nwhere\n\\be \\dot{M}_{kl} = \\sum_i {f}_{kl} (\\bx_i )m (\n \\bv_i'\\cdot \\br_{kl} + \\bx'_i \\cdot \\bU_{kl} )\n\\label{mass2} \\; .\n\\ee\nThe $\\bv'_i$ term will be shown to be negligible\nwithin our approximations. The $ \\bx'_i \\cdot \\bU_{kl}$-term \nhowever describes the geometric\neffect that the Voronoi cells do not conserve their \nvolume: The relative motion of the DP centers causes the cell boundaries\nto change their orientation. We will return to give this `boundary \ntwisting' term a quantitative content when the equations of motion\nare averaged--an effect which was overlooked in our \nfirst publication of this \ntheory \\cite{flekkoy99} where it was stated that $\\la \\dot{M}_{kl} \\ra = 0$.\n\n\\begin{figure}\n\\centerline{\\hbox{\\psfig{figure=particle_voronoi.eps,width=8cm}}}\n\\caption{\\label{fig1}\n\\protect \\narrowtext The Voronoi lattice defined by the \ndissipative particle positions $\\br_k$. The grey dots \nwhich represent the underlying MD particles are drawn \nonly in the overlap region.}\n\\end{figure}\n\n\\subsubsection{Momentum equation}\n\nThe momentum equation (\\ref{momentum}) takes the form \n\\bea\n\\frac{\\dd \\bP_k }{\\dd t} &=& \n \\sum_{li} {f}_{kl} (\\bx_i )m \\bv_i (\n \\bv_i'\\cdot \\br_{kl} + \\bx'_i \\cdot \\bU_{kl} )\n \\nonumber \\\\ \n& +& \\sum_{li} f_k (\\bx_i ) \\bF_i \n \\label{momentum2}\n\\eea\nWe can write the force as $\\bF_i = m{\\bf g} + \\sum_j \\bF_{ij}$\nwhere the first term is an external force and the second term\nis the internal force caused by all the other particles.\nNewton's third law then takes the form $\\bF_{ij} = - \\bF_{ji}$.\nThe last term in \\eq{momentum2}) may then be rewritten as\n\\be\n\\sum_i f_k (\\bx_i ) \\bF_i =\nM_k \\bg + \\sum_{ij} f_k (\\bx_i ) \\bF_{ij}\n\\label{gravity}\n\\ee\nwhere \n\\bea\n\\sum_{ij} f_k (\\bx_i ) \\bF_{ij} &=& - \\sum_{ij} f_k (\\bx_i ) \\bF_{ji} \n\\nonumber \\\\\n&=& - \\sum_{ij} f_k (\\bx_j + \\Delta \\bx_{ij} ) \\bF_{ji} \\nonumber \\\\\n&\\approx & - \\sum_{ij} f_k (\\bx_j ) \\bF_{ji} - \n \\sum_{ij} \\left(\\Delta \\bx_{ij} \\cdot \\nabla \nf_k (\\bx_i ) \\right) \\bF_{ji} \\nonumber \\\\\n&=& -\\frac{1}{2} \\sum_{ij} \\left(\\Delta \\bx_{ij} \\cdot \\nabla \nf_k (\\bx_i ) \\right) \\bF_{ji} \\nonumber \\\\\n&=& \\sum_l \\left\\{ \\sum_{ij} \\frac{1}{2} \nf_{kl} (\\bx_i ) \\bF_{ij} \\Delta \\bx_{ij} \\right\\} \\cdot \\br_{kl}\n\\label{virial}\n\\eea\nwhere $\\Delta \\bx_{ij} = \\bx_i - \\bx_j$,\nwe have Taylor expanded $f_k (\\bx )$ around $\\bx_j $ and used a result\nsimilar to \\eq{dotf}) to evaluate $\\nabla f_k (\\bx )$.\nIn passing from the third to the fourth line in the above\nequations we have moved the first term on the right hand side\nto the left hand side and divided by two.\nNow, if we group the last term above with the $\\br_{kl}$ term in \n\\eq{momentum2}), make use of \\eq{primes}), and do some rearranging \nof terms we get \n\\bea\n\\frac{\\dd \\bP_k }{\\dd t} &=& M_k \\bg +\n\\sum_l \\dot{M}_{kl} \\frac{ \\bU_{k}+\\bU_{l}}{2} \\nonumber \\\\\n&+& \\sum_{li} f_{kl}(\\bx_i ){\\bf \\Pi}_i' \n \\cdot \\br_{kl} \\nonumber \\\\\n&+& \\sum_{li} f_{kl} (\\bx_i ) m \\bv_i' \\bx_i' \\cdot \\bU_{kl} \n\\label{momentum3}\n\\eea\nwhere we have used the relation \n$\\dot{M}_k = \\sum_l \\dot{M}_{kl}$ and defined\nthe general momentum-flux tensor\n\\be\n{\\bf \\Pi}_i = m \\bv_i \\bv_i + \\frac{1}{2} \\sum_{j}\n\\bF_{ij} \\Delta \\bx_{ij} \\label{momentum_flux} \\;.\n\\ee\nThis tensor is the momentum analogue of the mass-flux vector\n$m\\bv_i$. The prime indicates that the velocities on the right hand \nside are those defined in \\eq{primes}).\nThe tensor ${\\bf \\Pi}_i$ describes both the momentum that the particle\ncarries around through its own motion and the momentum exchanged\nby inter-particle forces. It may be arrived at by \nconsidering the momentum transport \nacross imaginary cross sections of the volume in which the particle is \nlocated. \n\n%*****ENERGY EQUATION: ****************\n\n\\subsubsection{Energy equation}\nIn order to get the microscopic energy equation of motion \nwe proceed as with the mass and momentum equations\nand the two terms that appear on the right hand side \nof \\eq{energy}).\n\nTaking $V_{MD}$ to be a central potential and using the relations \n$\\nabla V_{MD}(r_{ij}) = V_{MD}'(r_{ij}) \\bee_{ij} = -\\bF_{ij}$ and\n$\\dot{V}_{MD}(r_{ij}) = V_{MD}'(r_{ij}) \\bee_{ij}\\cdot \\bv_{ij} \n= -\\bF_{ij}\\cdot \\bv_{ij}$ where $\\bv_{ij} = \\bv_i - \\bv_j$ we get the \ntime rate of change of the particle energy\n\\be\n\\dot{\\epsilon_i } = m\\bg \\cdot \\bv_i + \\frac{1}{2} \\sum_{j \\neq i}\n\\bF_{ij} \\cdot (\\bv_i + \\bv_j ) \\; .\n\\ee\nThis gives the \nfirst term of \\eq{energy}) in the form \n\\be\n\\sum_i f_k (\\bx_i ) \\dot{\\epsilon} = \\bP_k \\cdot \\bg\n+ \\frac{1}{2} \\sum_{i\\neq j} f_k (\\bx_i ) \\bF_{ij} \\cdot (\\bv_i + \\bv_j ) \\; .\n\\ee\nThe last term of this equation is odd under the exchange $i \\leftrightarrow j$\nand exactly the same manipulations as in \n\\eq{virial}) may be used to give\n\\bea\n\\sum_i f_k (\\bx_i ) \\dot{\\epsilon} &=& \\bP_k \\cdot \\bg \\nonumber \\\\ \n& +& \\sum_{l,i\\neq j} f_{kl} (\\bx_i ) \\frac{1}{4} \n\\bF_{ij} \\cdot (\\bv_i + \\bv_j ) \\Delta \\bx_{ij} \\cdot \\br_{kl} \\nonumber \\\\ \n&=& \\bP_k \\cdot \\bg\n+ \\sum_{l,i\\neq j} f_{kl} (\\bx_i ) \n\\left( \\frac{1}{4} \n\\bF_{ij} \\cdot (\\bv'_i + \\bv'_j )\n \\right. \\nonumber \\\\\n &+& \\left. \\frac{1}{2} \n\\bF_{ij} \\cdot \\frac{\\bU_k + \\bU_l}{2} \\right) \n\\Delta \\bx_{ij} \\cdot \\br_{kl} \n\\label{virial2}\n\\eea\nwhere for later purposes we have used Eqs.~(\\ref{primes}) \nto get the last equation.\nThe last term of \\eq{energy}) is easily written down using\n\\eq{dotf}). This gives\n\\be\n\\sum_i \\dot{f}_k(\\bx_i) \\epsilon_i = \n\\sum_{li} f_{kl} (\\bx_i ) (\\bv'_i\\cdot \\br_{kl} + \\bx'_i \\cdot \n\\bU_{kl} )\\epsilon_i \\; . \\label{energyA}\n\\ee\nAs previously we write the particle velocities in terms \nof $\\bv_i'$. The corresponding expression for the \nparticle energy is $\\epsilon_i = \\epsilon'_i + m\\bv_i' \\cdot\n(\\bU_k + \\bU_l)/2 + (1/2) m ((\\bU_k + \\bU_l)/2 )^2 $ where \nthe prime in $\\epsilon'_i$ denotes that the particle velocity\nis $\\bv'_i$ rather than $\\bv_i$.\nEquation (\\ref{energyA}) may then be written \n\\bea\n\\sum_i \\dot{f}_k(\\bx_i) \\epsilon_i &= &\n \\sum_l \\frac{1}{2}\\dot{M}_{kl} \\left( \\frac{\\bU_k +\\bU_l}{2} \\right)^2 \n\\nonumber \\\\ \n&+& \\sum_{li} f_{kl} (\\bx_i )\n\\left( \\epsilon'_i \\bv_i' + m \\bv_i' \\bv_i' \\cdot \\frac{\\bU_k +\\bU_l}{2}\\right)\n\\cdot \\br_{kl} \\nonumber \\\\\n&+& \\sum_{li} f_{kl} (\\bx_i ) \\epsilon_i \\bx'_i \\cdot \\bU_{kl} \\; .\n\\eea\nCombining this equation with \\eq{virial2}) we obtain\n\\bea\n\\dot{E}_k^{\\text{tot}} &=& \\sum_{li}\n f_{kl} (\\bx_i)\\left( \\bJ_{\\epsilon i}'\n+ \\Pi'_i \\cdot \\frac{\\bU_k +\\bU_l}{2} \\right)\n\\cdot \\br_{kl} \\nonumber \\\\\n&+& M_k \\bU_k \\cdot \\bg +\n\\sum_l \\frac{1}{2} \\dot{M}_{kl}\\left( \\frac{\\bU_k +\\bU_l}{2} \\right)^2\n\\nonumber \\\\\n&+& \\sum_{li} f_{kl} (\\bx_i) \\left( \\epsilon'_i \n+ m \\bv'_i \\cdot \\left( \\frac{\\bU_k + \\bU_l}{2} \\right)\n \\right) \\bx'_i \\cdot \\bU_{kl} \\; .\n\\label{aa}\n\\eea\nwhere the momentum-flux tensor is\ndefined in \\eq{momentum_flux}) and \nwe have identified the energy-flux vector associated \nwith a particle $i$ \n\\be\n\\bJ_{\\epsilon i} = \\epsilon_i \\bv_i + \\frac{1}{4} \\sum_{i\\neq j}\n\\bF_{ij} \\cdot (\\bv_i + \\bv_j) \\Delta \\bx_{ij} \\; . \n\\label{enrgy_flux}\n\\ee\nAgain the prime denotes that the velocities are \n$\\bv'_i$ rather than $\\bv_i$.\nTo get the internal energy $\\dot{E}_k$ \ninstead of $\\dot{E}_k^{\\text{tot}}$\nwe note that $\\dd (\\bP^2_k/2M_k)/\\dd t = \\bU_k \\cdot \\dot{\\bP}_k \n- (1/2) \\dot{M}_k \\bU^2_k$. \nUsing this relation, the momentum equation\n\\eq{momentum3}), as well as the \nsubstitution $(\\bU_k + \\bU_l)/2 = \\bU_k - \\bU_{kl}/2$\nin \\eq{aa}), followed by some rearrangement of the $\\dot{M}_{kl}$\nterms we find that\n\\bea\n\\dot{E}_k^{\\text{tot}} &=&\n \\frac{\\dd }{\\dd t} \\left( \\frac{1}{2} M_k \\bU_k^2 \\right) \\nonumber \\\\\n&+& \\sum_l \\frac{1}{2} \\dot{M}_{kl} \\left( \\frac{\\bU_{kl}}{2} \\right)^2\n+ \\sum_{li} f_{kl} (\\bx_i)\\left( \\bJ_{\\epsilon i}'\n- \\Pi'_i \\cdot \\frac{\\bU_{kl}}{2} \\right)\n\\cdot \\br_{kl} \\nonumber \\\\\n&+& \\sum_{li} f_{kl} (\\bx_i)\\left( \\epsilon'_i - m\\bv'_i \\cdot \n\\frac{\\bU_{kl}}{2} \\right) \\bx'_i \\cdot \\bU_{kl} \\; .\n\\label{energy_micro}\\eea\n\nThis equation has a natural physical interpretation. The first \nterm represents the translational kinetic energy of the DP as \na whole. The remaining terms represent the internal energy $E_k$.\nThis is a purely thermodynamic quantity which cannot depend\non the overall velocity of the DP, i.e. it must be Galilean\ninvariant. This is easily checked as the relevant terms\nall depend on velocity differences only.\n\nThe $\\dot{M}_{kl}$ term represents\nthe kinetic energy received through mass exchange with\nneighboring DPs. As will become evident when \nwe turn to the averaged description, \nthe term involving the momentum and energy \nfluxes represents the work done on the DP by its neighbors\nand the heat conducted from them.\nThe $\\epsilon'_i$-term represents the energy\nreceived by the DP due to the same `boundary twisting'\neffect that was found in the mass equation.\nUpon averaging, the last term proportional \nto $\\bv_i'$ will be shown to be relatively small since $\\langle\n\\bv'_i \\rangle = 0$ in our approximations. This is true also\nin the mass and momentum equations.\nEquations (\\ref{mass2}), (\\ref{momentum3}) and (\\ref{energy_micro}) \nhave the coarse grained form that will remain in the final \nDPD equations. Note, however, that they retain the full microscopic\ninformation about the MD system, and for that reason they are \ntime-reversible. Equation (\\ref{momentum3}) for instance\ncontains only terms of even order in the velocity. In the next section\nterms of odd order will appear when this equation is averaged.\n\n\nIt can be seen that the rate of change of momentum in \\eq{momentum3})\nis given as a sum of separate pairwise contributions from the other\nparticles, and that these terms are all odd under the \nexchange $l\\leftrightarrow k$. Thus \nthe particles interact in a pairwise fashion and\nindividually fulfill Newton's third law; in other words, \nmomentum conservation is again explicitly upheld.\nThe same symmetries hold for the mass conservation \nequation (\\ref{mass2}) and energy equation (\\ref{aa}).\n\n\n\\section{Derivation of dissipative particle \ndynamics: average and fluctuating forces}\n\nWe can now investigate the average \nand fluctuating parts of Eqs.~(\\ref{energy_micro}), (\\ref{momentum3}) and \n(\\ref{mass2}).\nIn so doing we shall need to draw on a hydrodynamic\ndescription of the underlying molecular dynamics and \n construct a statistical mechanical\ndescription of our dissipative particle dynamics. \nFor concreteness we shall take the hydrodynamic description\nof the MD system in question to be that of a simple \nNewtonian fluid \\cite{landau59}.\nThis is known to be a good description for MD\nfluids based on Lennard-Jones or hard sphere potentials,\nparticularly in three dimensions~\\cite{koplik95}.\nHere we shall carry out the analysis for systems in two spatial \ndimensions; the generalization to three dimensions is straight forward,\nthe main difference being of a practical nature as the Voronoi \nconstruction becomes more involved.\n\nWe shall begin by specifying a scale separation\nbetween the dissipative particles and the molecular dynamics \nparticles by assuming that\n\\be\n{|\\bx_i - \\bx_j|} << {|\\br_k - \\br_l|} \\; ,\n\\label{scale}\n\\ee\nwhere $\\bx_i$ and $\\bx_j$ denote the positions of neighbouring MD particles. Such a scale separation is in\ngeneral necessary in order for the coarse-graining procedure to be\nphysically meaningful. Although for the most part in this paper we are\nthinking of the molecular interactions as being mediated by short-range\nforces such as those of Lennard-Jones type,\na local description of the interactions will still be\nvalid for the case of long-range Coulomb interactions in an\nelectrostatically neutral system, provided that\nthe screening length is shorter than the width of the overlap region between\nthe dissipative particles. Indeed, as we shall show here,\nthe result of doing a local averaging is that the original Newtonian equations \nof\nmotion for the MD system become a set of Langevin equations \nfor the dissipative particle dynamics. These Langevin equations admit\nan associated Fokker-Planck equation.\n An associated fluctuation-dissipation relation \nrelates the amplitude of the Langevin force\nto the temperature and damping in the system.\n\n\\subsection{Definition of ensemble averages}\n\nWith the mesoscopic variables now available, \nwe need to define the correct average corresponding to \na dynamical state of the system.\nMany MD configurations are consistent\nwith a given value of the set $\\{ \\br_k, M_k, \\bU_k, E_k \\}$, and \naverages are computed by means of an ensemble\nof systems with common {\\it instantaneous} values of the set $\\{ \\br_k, M_k, \n\\bU_k, E_k \\}$.\nThis means that only the time derivatives of the set $\\{ \\br_k, M_k, \\bU_k, E_k \n\\}$,\ni.e. the forces, have a fluctuating part. \nIn the end of our development \napproximate distributions for $\\bU_k$'s and $E_k$'s will\nfollow from the derived Fokker-Planck equations. These \ndistributions refer to the larger equilibrium ensemble that contains all\nfluctuations in $\\{ \\br_k, M_k, \\bU_k, E_k \\}$.\n\nIt is necessary, to compute the average MD particle velocity \n$\\langle \\bv \\rangle $ {\\it between} dissipative particle\ncenters, given $\\{ \\br_k, M_k, \\bU_k, E_k \\}$. This velocity depends\non all neighboring dissipative particle velocities. However, for \nsimplicity we shall only employ\na ``nearest neighbor'' approximation, which consists in assuming \nthat $\\langle \\bv \\rangle $ interpolates linearly between the \ntwo nearest dissipative particles.\nThis approximation is of the same nature as the approximation\nused in the Newtonian fluid stress--strain relation which is linear\nin the velocity gradient.\nThis implies that in the overlap region between dissipative particles \n$k$ and $l$\n\\be\n\\langle \\bv' \\rangle = \\langle \\bv' \\rangle (\\bx ) = \\frac{\\bx' \\cdot\n\\br_{kl}}{r_{kl}^{2}} \\bU_{kl} \\; ,\n\\label{average}\n\\ee\nwhere the primes are defined in Eqs.~(\\ref{primes})\nand $r_{kl}= |\\br_k - \\br_l|$.\n\n\nA preliminary mathematical observation is useful in splitting\nthe equations of motion into average and fluctuating parts.\nLet $r(\\bx )$ be an arbitrary, slowly varying function on the \n$a^2/r_{kl}$ scale. Then \nwe shall employ the approximation corresponding to a linear \ninterpolation between DP centers, that \n$r(\\bx ) = (1/2) (r_k + r_l) $ where $\\bx$ is a position\nin the overlap region between DP k and l and $r_k$ and $r_l$\nare values of the function $r$ associated with the DP centers k and l\nrespectively.\n\\begin{figure}\n\\centerline{\\hbox{\\psfig{figure=boundary_twist.eps,width=5cm}}}\n\\caption{\\label{twist}\n\\protect \\narrowtext \nTwo interacting Voronoi cells. The length of the intersection\nbetween DP's k and l is $l_{kl}$, the shift from the center\nof the intersection between\n$\\br_{kl}$ and $l_{kl}$ is $L_{kl}$ ($L_{kl}=0$ when \n$\\br_{kl}$ intersects $l_{kl}$ in the middle)\n and the unit vector $\\bi_{kl}$ is normal to $\\bee_{kl}$.\nThe coordinate system x-y used for the integration has its origin on\nthe intersection.}\n\\end{figure}\nThen\n \\bea \n& \\sum_i & f_{kl}(\\bx_i ) r(\\bx ) \\approx \\int dx \\; dy \\frac{\\rho_k + \n\\rho_l}{2} f_{kl}(\\bx )\n\\frac{r_k + r_l}{2} \\nonumber \\\\\n&\\approx & \\frac{l_{kl}}{2a^2} \\frac{\\rho_k + \\rho_l}{2}\n\\frac{r_k + r_l}{2} \\int_{-\\infty}^\\infty dx'\\; \\cosh^{-2}{(x' r_{kl}/a^2 )}\n\\nonumber \\\\\n&=& \\frac{l_{kl} }{r_{kl}} \\frac{\\rho_k + \\rho_l}{2}\\frac{r_k + r_l}{2},\n\\label{prefactor}\n\\eea\nwhere $\\frac{\\rho_k + \\rho_l}{2}$ is the MD particle number density \nand we have used the identity $\\tanh'(x) \n= \\cosh^{-2}(x)$.\nWe will also need the first moment in $\\bx'$ \n\\bea\n\\sum_i & f_{kl} (\\bx_i ) & \\bx'_i r(\\bx_i ) \\approx \n\\int dx dy \\frac{\\rho_k + \\rho_l}{2} f_{kl}(\\bx ) \\bx'\n\\frac{r_k + r_l}{2}\\nonumber \\\\\n& \\approx & \n\\frac{1}{2 a^2} \\frac{\\rho_k + \\rho_l}{2} \\frac{r_k + r_l}{2} \n\\int dx \\; dy \\cosh^{-2} \n\\left(\\frac{x r_{kl}}{ a^2} \\right) y \\bi_{kl} \\nonumber \\\\\n&= & \\frac{l_{kl}}{2 r_{kl}} L_{kl} \n \\frac{\\rho_k + \\rho_l}{2} \\frac{r_k + r_l}{2} \\bi_{kl}\n\\label{small}\n\\eea \nwhere the unit vectors $\\bee_{kl} = \\br_{kl}/r_{kl}$ and $\\bi_{kl}$\nare shown in Fig.~\\ref{twist}, we have used the fact that the \nintegral over $x \\bee_{kl} \\cosh^{-2} ...$ vanishes\nsince the integrand is odd, and the last \nequation follows by the substitution $x \\rightarrow (a^2/r_{kl}) x$.\nIn contrast to the vector $\\bee_{kl}$\nthe vector $\\bi_{kl}$ is even under the exchange $k \\leftrightarrow l$,\nas is $L_{kl}$. This is a matter of definition only as it would be equally\npermissible to let $\\bi_{kl}$ and $L_{kl}$ be odd under this exchange.\nHowever, it is important for the symmetry properties of the \nfluxes that $\\bi_{kl}$ and $L_{kl}$ have the same symmetry \nunder $k \\leftrightarrow l$.\n\n\\subsection{The mass conservation equation}\n\nTaking the average of \\eq{mass2}), we observe that \nthe first term vanishes if \\eq{average}) is used,\nand the second term follows directly from \\eq{small}).\nWe thus obtain \n\\be\n\\dot{M}_k = \n\\sum_l (\\la \\dot{M}_{kl} \\ra + \\dm) \n\\label{mass_dpd1}\n\\ee\nwhere\n\\be \n \\langle \\dot{M}_{kl} \\rangle = \n\\sum_{li} f_{kl} m (\\bx_i ) \\la \\bx'_i \\ra \\cdot \\bU_{kl} =\n\\frac{l_{kl}}{2 r_{kl}} L_{kl} \\frac{\\rho_k + \\rho_l}{2}\n\\bi_{kl} \\cdot \\bU_{kl}\n\\label{mass0} \\; ,\n\\ee\nand $\\dm = \\dot{M}_{kl} - \\la \\dot{M}_{kl} \\ra $. \nThe finite value of $\\langle \\dot{M}_{kl} \\rangle $ is caused\nby the relative DP motion {\\em perpendicular} to $\\bee_{kl}$.\nThis is a geometric effect intrinsic to the Voronoi lattice.\nWhen particles move the Voronoi boundaries change\ntheir orientation, and this boundary twisting causes\nmass to be transferred between DP's. This mass \nvariation will be visible in the energy flux, though not\nin the momentum flux.\n It will later be shown that the effect of\nmass fluctuations in the momentum and energy equations\nmay be absorbed in the force and heat flux fluctuations.\n% \n% Thus $\\dd M_k / \\dd t $ has a fluctuating component but its\n% average is identically zero. If the fluctuating component\n% is assumed to have no correlations, $M_k$ will perform a \n% random walk with time $t$, and the variance will grow\n% linearly with $t$. This implies that there will\n% be no equilibrium average value of $M_k$, except \n% possibly one set by the size of the system.\n% However, it is probable that the fluctuations\n% are indeed correlated in time as a mass fluctuation \n% will result in a pressure response within the dissipative particles.\n% This is consistent with the standard fluctuation dissipation \n% result that relates the mass fluctuations in an \n% open system to the compressibility \\cite{landau59c}.\n% Hence the correlation time may increase with the\n% size of the dissipative particles owing to the increased time it \n% takes to propagate the information of a pressure variation.\n% Such correlations in conserved quantities are well\n% known in theories of the `long-time tails' that persist in the\n% velocity autocorrelation functions of Brownian particles \\cite{hauge73}.\n\n% However, since mass and momentum \n% fluctuations are generally uncorrelated~\\cite{landau59c},\n% they may be studied separately, and so here \n% we shall simply neglect the mass\n% fluctuations of the dissipative particles, and focus \n% on the momentum fluctuations. That mass and momentum fluctuations are\n% indeed uncorrelated follows \n% from isotropy, since the probability of a mass fluctuation, $\\delta M$\n% $P(\\delta M)$, cannot depend on the direction of a corresponding \n% momentum fluctuation $\\delta {\\bf P}$; that is, \n% $\\langle \\delta M \\delta {\\bf P}\\rangle = {\\bf 0 }$.\n% This means that an MD simulation where \\eq{DPdef}) is used \n% to define the dissipative particles will correspond to the\n% theoretical description elaborated herein only for times shorter\n% than the time it takes for $M_k$ to depart significantly \n% from its initial value. \n% However, due to the absence of any mass--momentum correlations,\n% such a simulation may still be used to measure the inter\n% dissipative-particle forces.\n\n\\subsection{The momentum conservation equation}\n\nUsing \\eq{mass0}) we may split \\eq{momentum3}) into \naverage and fluctuating parts to get \n\\bea\n \\frac{\\dd \\bP_k }{\\dd t} &=& M_k \\bg \\nonumber \\\\\n &+& \\sum_l \\langle \\dot{M}_{kl} \\rangle \\frac{\\bU_{k} + \\bU_l}{2} \n+ \\sum_{li} f_{kl}(\\bx_i )\n\\langle {\\bf \\Pi}_i \\rangle \\cdot \\br_{kl} \\nonumber \\\\\n&+& \\sum_i f_{kl} (\\bx_i ) m \\langle \\bv_i' \\bx_i' \\rangle\n \\cdot \\bU_{kl} + \\sum_l \\tilde{\\bF}_{kl}\n\\label{momentum4} \\; ,\n\\eea\nwhere the fluctuating force or, equivalently, the momentum flux is\n\\bea\n\\tilde{\\bF}_{kl}&=& \\sum_i f_{kl}(\\bx_i ) [({\\bf \\Pi}_i -\\langle {\\bf\n \\Pi}_i \\rangle ) \\cdot \\br_{kl} \\nonumber \\\\\n&+& m (\\bv'_i \\bx'_i - \\la \\bv'_i \\bx'_i \\ra )\\cdot \\bU_{kl} ] \\nonumber \\\\\n& + &\n\\dm \\frac{\\bU_{k} + \\bU_l}{2} \\; .\n\\eea\nNote that by definition $\\tilde{\\bF}_{lk} = -\\tilde{\\bF}_{kl}$.\nThe fact that we have absorbed \nmass fluctuations with the fluctuations in $\\tilde{\\bF}_{kl}$\ndeserves a comment. \n In general force fluctuations will\ncause mass fluctuations, which in turn will couple back to \ncause momentum fluctuations. The time scale over which this will\nhappen is $t_{\\eta}= r_{kl}^2/\\eta$, where \n$\\eta$ is the dynamic viscosity of the MD system.\nThis is the time it takes for \na velocity perturbation to decay over a distance of $r_{kl}$.\nPerturbations mediated by the pressure, i.e. sound waves, will\nhave a shorter time. In the sequel we shall need to make the\nassumption that the forces are Markovian, and it is clear that \nthis assumption may only be valid on time scales larger than $t_{\\eta}$.\nSince the time scale of a hydrodynamic perturbation of size $l$, say, is \nalso given as $l^2/\\eta$ this restriction implies the scale separation\nrequirement $r_{kl}^2 << l^2$, consistent with the scale $r_{kl}$ being\nmesoscopic.\n\nSince $\\langle {\\bf \\Pi}_i \\rangle$ is in general \ndissipative in nature, Eq.~(\\ref{momentum4}) \nand its mass- and energy analogue will be referred to as DPD1. \nIt is at the point of \ntaking the average in Eq.~(\\ref{momentum4}) that time reversibility is\nlost. Note, however, that we do not claim to treat the\nintroduction of irreversibility into the problem in a mathematically \nrigorous way. This is a very difficult problem\nin general which so far has only been realized by rigorous methods \nin the case of\nsome very simple dynamical systems with well defined ergodic \nproperties~\\cite{pvcop92,oppvc94,pvcrrh90}. We shall\ninstead use the constitutive relation for a Newtonian fluid which, as\nnoted earlier, is an emergent property of Lennard-Jones and hard sphere\nMD systems, to give\nEq.~(\\ref{momentum4}) a concrete content.\nThe momentum-flux tensor then has the following simple form\n\\be\n\\rho \\langle {\\bf \\Pi }_i\\rangle = m \\rho \\bv \\bv + \n{\\bf I} p - \\eta (\\nabla \\bv + (\\nabla \\bv )^T)\n\\label{constitutive}\n\\ee\nwhere $p$ is the pressure and $\\bv$ the average velocity \nof the MD fluid, $^T$ denotes the transpose and $\\bf I$ is the\nidentity tensor~\\cite{landau59}. In the above equation \nwe have for simplicity assumed that the bulk viscosity $\\zeta = (2/d)\\eta$ where \n$d$ is \nthe space dimension 2.\nThe modifications to include an independent $\\zeta$ are \ncompletely straight forward. \n\nUsing the \nassumption of linear interpolation (\\eq{average})), \nthe advective term $\\rho \\bv \\bv$\nvanishes in the frame of reference of the overlap region since \nthere $\\bv' \\approx 0$.\nThe velocity gradients in \\eq{constitutive}) may be evaluated \nusing \\eq{average}); the result is \n\\be\n\\nabla \\bv + (\\nabla \\bv )^T = \\frac{1}{r_{kl}} \n\\left( \\bee_{kl} \\bU_{kl} + \\bU_{kl} \\bee_{kl} \\right)\n\\label{gradient} \\; .\n\\ee\n\n%************ SURFACE INTEGRAL ********\nNote further that $\\sum_l l_{kl} $ is in fact a surface integral\nover the DP surface. Consequently \n\\be \n\\sum_l l_{kl} \\bee_{kl} g_k = 0\n\\label{surface_integral}\n\\ee\nfor any function $g_k$ that does not depend on $l$.\nIn particular we have $\\sum_l l_{kl}\\bee_{kl} (p_k + p_l)/2 = - \\sum_l \nl_{kl}\\bee_{kl}\n p_{kl}/2 $, where $p_{kl} = p_k - p_l$.\n%********\nCombining Eqs.~(\\ref{constitutive}), (\\ref{prefactor})\nand (\\ref{gradient}), \\eq{momentum4})\nthen takes the form\n\\bea\n && \n\\frac{\\dd \\bP_k }{\\dd t} = \n M_k \\bg + \\sum_l \\langle \\dot{M}_{kl} \\rangle \\frac{\\bU_{k} + \\bU_l}{2}\n\\nonumber \\\\\n &-& \\sum_l l_{kl} \\left( \n\\frac{p_{kl}}{2} \\bee_{kl} +\n\\frac{\\eta}{r_{kl}} \\left( \\bU_{kl} + (\\bU_{kl}\n \\cdot \\bee_{kl}) \\bee_{kl} \\right)\n\\right)\\nonumber \\\\\n&+& \\sum_l \\tilde{\\bF}_{kl}\n\\label{momentum5} \\; ,\n\\eea\nwhere we have assumed that the pressure $p$, as \nwell as the average velocity, interpolates \nlinearly between DP centers, and we have omitted the \n$ \\la \\bv'_i \\bx'_i \\ra \\approx 0 $ term.\nNote that all terms except the gravity term \non the right hand side of \\eq{momentum5}) are odd when\n$k \\leftrightarrow l$. This shows that Newton's third law\nis unaffected by the approximations made and that momentum\nconservation holds exactly. The same statements\ncan be made for the mass equation and the energy equation.\nThe pressure will eventually follow from an equation of state\nof the form $p_k= p(E_k,V_k,M_k)$ where $V_k$ is the volume \nand $M_k$ is the mass of DP $k$.\n\n\\subsection{The energy conservation equation}\n\n% For the present purposes we shall limit our attention to the \n% ensemble averaged value of $E_k$. This is sufficient\n% to obtain $p_k$, which itself is an averaged quantity.\n% In the same spirit of maximum simplicity we shall use\n% the ideal gas equation of state\n% \\be\n% p_k V_k = \\frac{M_k}{m} k_BT = \\frac{2}{3} E_k\n% \\label{eq_state} \\; . \n% \\ee\n% This, however is not a necessary choice. Any \n% equation of state written in terms of the energy,\n% volume and pressure would do.\n% A smaller compressibility than that of the ideal gas may be\n% desirable in order to approach the \n% limit of incompressible fluid flow more quickly.\nSplitting Eq.~(\\ref{energy_micro}) into an average and a fluctuating part gives \n\\bea\n\\dot{E}_k &=& \\sum_{li} f_{kl} (\\bx_i)\\left( \\langle \\bJ_{\\epsilon i}' \\rangle\n- \\langle \\Pi'_i \\rangle \\cdot \\frac{\\bU_{kl}}{2} \\right)\n\\cdot \\br_{kl} \\nonumber \\\\\n&+& \\sum_{li} f_{kl} (\\bx_i) \\langle \\epsilon'_i \\bx'_i \n\\rangle \\cdot \\bU_{kl} \\nonumber \\\\\n&+& \\sum_l \\frac{1}{2} \n\\langle \\dot{M}_{kl} \\rangle \\left( \\frac{\\bU_{kl}}{2} \\right)^2 \\nonumber \\\\\n&-& \\sum_l \\tilde{\\bF}_{kl} \\cdot \\frac{\\bU_{kl}}{2} + \\tilde{q}_{kl}\\; . \n\\label{energy_dpd1} \n\\eea\nwhere we have defined\n\\bea\n\\tilde{q}_{kl} &=& \\sum_i f_{kl}(\\bx_i) \n(\\bJ'_{\\epsilon i} - \\langle \\bJ'_{\\epsilon i} \\rangle )\\cdot \\br_{kl}\n+ \\frac{\\dot{\\tilde{M}}_{kl}}{2} \n\\left( \\frac{\\bU_{kl}}{2} \\right)^2 \\nonumber \\\\\n&+& \\sum_i f_{kl}(\\bx_i) [ ( \\epsilon'_i \\bx'_i- \n\\la \\epsilon'_i \\bx'_i \\ra )\n\\nonumber \\\\\n&-& m \\frac{\\bU_{kl}}{2} \\cdot \\bv'_i \\bx'_i ] \\cdot \\bU_{kl} \n\\eea\ni.e. the fluctuations in the heat flux also contains\nthe energy fluctuations caused by mass fluctuations.\nThis is like the momentum case.\n\nNote that in taking the average in \\eq{energy_dpd1}) the\n${\\bf \\Pi} \\cdot \\bU_{kl}$ product presents no problem as\n$\\bU_{kl}$ is kept fixed under this average. If we had \naveraged over different values of $\\bU_{kl}$ the\nproduct of velocities in ${\\bf \\Pi} \\cdot \\bU_{kl}$ would \nhave caused difficulties.\nEquation (\\ref{energy_dpd1}) is the third component in the description at the\nDPD1 level.\n\nThe average of the energy flux vector $\\bJ_{\\epsilon}$\nis taken to have the general form \\cite{landau59}\n\\be\n\\rho \\langle \\bJ_{\\epsilon} \\rangle\n = \\epsilon \\bv + \\sigma \\cdot \\bv - \\lambda \\nabla T\n\\label{constitutive2}\n\\ee\nwhere $\\sigma = {\\bf \\Pi} - \\rho \\bv \\bv$ is the stress tensor, and $\\lambda$ \nthe thermal conductivity\nand $T$ the local temperature.\nNote that in \\eq{energy_micro}) only $\\bJ'_{\\epsilon}$ appears.\nSince $\\bv' \\approx {\\bf 0}$ we have $\\langle \\bJ'_{\\epsilon} \\rangle =\n\\lambda \\nabla T$.\nAveraging of \\eq{energy_dpd1}) gives\n\\bea\n&&\\dot{E}_{k} = -\\sum_l l_{lk} \\lambda \\frac{T_{kl} }{r_{kl}} \\nonumber \\\\\n&-& \\sum_l l_{lk} \\left( \\frac{p_{k}+p_l}{2} \\bee_{kl} - \\frac{\\eta}{r_{kl}}\n(\\bU_{kl} + (\\bU_{kl}\\cdot \\bee_{kl})\\bee_{kl}) \\right) \\cdot\n\\frac{\\bU_{kl}}{2} \\nonumber \\\\\n &+& \\sum_l\n\\frac{1}{2} \\langle \\dot{M}_{kl} \\rangle \\left(\\frac{\\bU_{kl}}{2} \\right)^2\n+ \\frac{l_{kl}}{4 r_{kl}} L_{kl} \n\\bi_{kl} \\cdot \\bU_{kl}\n\\left( \\frac{E_k}{V_k} + \\frac{E_l}{V_l} \\right) \n\\nonumber \\\\\n&-& \\sum_l \\tilde{\\bF}_{kl} \\cdot \\frac{\\bU_{kl}}{2} + \\tilde{q}_{kl}\\; .\n\\label{energy_dpd}\n\\eea\nwhere $T_{kl} = T_k - T_l$ is the temperature \ndifference between DP's $k$ and $l$, and we have \nused linear interpolation to write \n$\\la \\epsilon'_1 \\ra = (1/2)(E_k/V_k + E_l/V_l)$.\nThe first term above describes the heat flux according to Fourier's law.\nThe next non-fluctuating terms, which are multiplied by $\\bU_{kl}/2$\nrepresent the (rate of) work done by the interparticle forces, and the \n$\\tilde{\\bF}_{kl}$ term represents the work done by the fluctuating force.\n\nAs has been pointed out by Avalos et al and Espanol\\cite{avalos97,espanol97}\nthe work done by $\\tilde{\\bF}_{kl}$ has the effect that it increases the thermal \nmotion of the DP's at the expense of a reduction in $E_k$. This\nis the case here as well since the above $ \\tilde{\\bF}_{kl} \\cdot \\bU_{kl}$\nterm always has a positive average due to the positive correlation between the \nforce\nand the velocity increments.\n\nEquation (\\ref{energy_dpd}) is identical in form to the energy equation \npostulated \nby Avalos and Mackie \\cite{avalos97}, save for the fact that here the \nconservative \nforce $ (({p_{k}+p_{l}})/{2} )\\bee_{kl} \\cdot {\\bU_{kl}}/{2} $ (which sums\nto zero under $\\sum_k$) is present. \nThe pressure forces in the present case \ncorrespond to the conservative forces in conventional DPD--it will be \nobserved that they are both derived from a potential. \nHowever, while the conservative force in conventional DPD must be thought to be \ncarried by some field external to the particles, the pressure \nforce in our model has its origin within the particles themselves.\nThere is also a small difference between the present form of Fourier's law\nand the description of thermal conduction employed by Avalos and Mackie.\nWhile the heat flux here is taken to be linear in differences in $T$, \nAvalos and Mackie use a flux linear in differences in $ (1/T)$. As both \ntransport laws are approximations valid to lowest order in differences in $T$, \nthey should be considered equivalent.\n\nWith the internal energy variable at hand it is possible\nto update the pressure and temperature $T$ of the DP's\nprovided an equation of state for the underlying MD system is assumed,\nand written in the form $P=P(E,V,m)$ and $T=T(E,V,m)$.\nFor an ideal gas these are the well known relations $PV=(2/d)E$ and\n$k_BT=(2/d)mE$.\n%Please check. Gianni\n\nNote that we only need the average evolution of the pressure and temperature.\nThe fluctuations of $p$ are already contained in \n$\\tilde{\\bF}_{kl}$ and the effect of temperature fluctuations is\ncontained within $\\tilde{q}_{kl}$.\n% The Fokker-Planck equation \n% with the energy treated as a fluctuating variable as\n% well as the fluctuation-dissipation relation that \n% relates the magnitude of the heat flow fluctuations\n% to the thermal conductivity has been worked out in \n% Refs.~\\cite{avalos97,espanol97}.\n\nAt this point we may compare the forces arising in the present model to \nthose used in conventional DPD. In conventional DPD\nthe forces are pairwise and act in a direction \nparallel to $\\bee_{kl}$, with a conservative part \nthat depends only on $r_{kl}$\nand a dissipative part proportional to \n$(\\bU_{kl}\\cdot \\bee_{kl})\\bee_{kl}$~\\cite{hoogerbrugge92,espanol95b,marsh98c}. \nThe forces in our new version of\nDPD are pairwise too. \nThe analog of the conservative force, $ l_{kl} (p_{kl}/2) \\bee_{kl}$, \nis central and its $\\br$ dependence is given by the Voronoi \nlattice. When there is no overlap $l_{kl}$ between dissipative particles\ntheir forces vanish. (A cut--off distance,\nbeyond which no physical interactions are permitted, \nwas also present in the earlier versions of DPD--see, for example, \nRef.~\\cite{hoogerbrugge92}--where it was introduced \nto simplify the numerical treatment.)\nDue to the existence of an overlap region in our model, the \ndissipative force has both a component parallel to $\\bee_{kl}$\nand a component parallel to the relative velocity $\\bU_{kl}$.\nHowever, due to the linear nature\nof the stress--strain relation in the Newtonian MD\nfluid studied here, \n this force has the same simple linear \nvelocity dependence that \nhas been postulated in the literature. \n\nThe friction coefficient\nis simply the viscosity $\\eta$ of the underlying fluid times\nthe geometric ratio $l_{kl}/r_{kl}$.\nAs has been pointed out both in the context of DPD \\cite{espanol95}\nand elsewhere, the viscosity is generally {\\em not} \nproportional to a friction coefficient between the particles.\nAfter all, conservative systems like MD \nare also described by a viscosity.\nGenerally the viscosity will be caused by the combined effect\nof particle interaction (dissipation, if any) and the momentum\ntransfer caused by particle motion. The latter \ncontribution is proportional to the mean free path.\nThe fact that the MD viscosity $\\eta$, the DPD viscosity\nand the friction coefficient are one and the same\ntherefore implies that the mean free path effectively vanishes.\nThis is consistent with the space filling nature of the particles.\nSee Sec.~\\ref{low_visc} for a further discussion of the zero viscosity limit.\n\nNote that constitutive relations like Eqs.~(\\ref{constitutive}) \nand (\\ref{constitutive2})\nare usually regarded as components\nof a top-down or macroscopic description of a fluid.\nHowever, any bottom-up mesoscopic description\nnecessarily relies on the use of some kind of averaging procedure;\nin the present context, these relations represent a \nnatural and convenient\nalthough by no means a necessary choice of average. \nThe derivation of emergent constitutive relations is itself part of the\nprogramme of non-equilibrium statistical mechanics (kinetic theory), \nwhich provides a link \nbetween the microscopic and the macroscopic levels. However, as noted\nabove, no general and rigorous procedure for deriving such relations has \nhitherto been realised; in the present theoretical treatment, such \nassumed constitutive relations are therefore a necessary \ninput in the linking of the microscopic and mesoscopic levels.\n\n\\section{Statistical mechanics of dissipative particle dynamics}\n \n\nIn this section we discuss the statistical properties of the DP's\nwith the particular aim of obtaining the magnitudes of $\\tilde{\\bF}_{kl}$ and\n$\\tilde{q}_{kl}$.\nWe shall follow two distinct routes that lead to the same \nresult for these \nquantities, one based on the conventional Fokker-Planck description\nof DPD\\cite{avalos97}, and one based on Landau's and Lifshitz's\nfluctuating hydrodynamics \\cite{landau59}.\n\nIt is not straightforward to obtain a general statistical mechanical\ndescription of the DP-system. The reason is that the DP's,\nwhich exchange mass, momentum, energy and volume,\nare not captured by any standard statistical ensemble.\nFor the grand canonical ensemble, the system in question\nis defined as the matter within a fixed volume, and\nin the case of a the isobaric ensemble the \nparticle number is fixed.\nNeither of these requirements hold for a DP in general.\n\nA system which exchanges mass, momentum, energy and volume\nwithout any further restrictions will generally \nbe ill-defined as it will lose its identity in the course of time.\nThe DP's of course remain well-defined by virtue of the coupling \nbetween the momentum and volume variables: The DP volumes are defined\nby the positions of the DP-centers and the DP-momenta govern\nthe motion of the DP-centers. Hence the quantities that are exchanged\nwith the surroundings are not independent and the ensemble must be \nconstructed accordingly.\n\nHowever, for present purposes we shall leave aside the\ninteresting challenge of designing the statistical mechanical\nproperties of such an ensemble, and derive the magnitude\nof $\\tilde{\\bF}_{kl}$ and $\\tilde{q}_{kl}$ \nfrom two different approximations.\nThe approximations are both justifiable from the assumption\nthat $\\tilde{\\bF}_{kl}$ and $\\tilde{q}_{kl}$ have a negligible\ncorrelation time. It follows that their properties\nmay be obtained from the DP behavior on such short time\nscales that the DP-centers may be assumed fixed in space.\nAs a result, we may take \neither the DP volume or the system of MD-particles fixed for\nthe relevant duration of time.\nHence for the purpose of getting $\\tilde{\\bF}_{kl}$ and\n$\\tilde{q}_{kl}$ we may use either the isobaric ensemble, applied to\nthe DP system, or the grand canonical ensemble, applied to the MD system.\nWe shall find the same results from either route.\nThe analysis of the DP system using the isobaric ensemble follows the\nstandard procedure using the Fokker-Planck equation,\nand the result for the equilibrium distribution is only valid\nin the short time limit.\nThe analysis of the MD system \ncorresponding to the grand canonical ensemble could\nbe conducted along the similar lines. However, it is also possible \nto obtain the magnitude of $\\tilde{\\bF}_{kl}$ and\n$\\tilde{q}_{kl}$ directly\nfrom the theory of fluctuating hydrodynamics since \nthis theory is derived from coarse-graining\nthe fluid onto a grid. The pertinent fluid velocity and stress\nfields thus result from averages over {\\em fixed volumes} associated\nwith the grid points: Since mass flows freely between these volumes\nthe appropriate ensemble is thus the grand canonical one.\n\n\n\\subsection{The isobaric ensemble}\n\nWe consider the system of $N_k \\gg 1$ \nMD particles inside a given DP$_k$ at \na given time, say all the MD particles with positions \nthat satisfy $f_k(\\bx_i ) \n>1/2$\nat time $t_0$. At later times it will be possible to \nassociate a certain volume \nper\nparticle with these particles, and by definition the system they form will\nexchange volume and energy but not mass. \nWe consider all the remaining DP's as a thermodynamic bath with which \nDP$_k$ is in equilibrium.\nThe system defined in this way will be described\nby the Gibbs free energy and the isobaric ensemble.\nDue to the diffusive spreading\nof MD-particles, this system will only initially coincide\nwith the DP; during this transient time interval, however, we may treat\nthe DP's as systems of fixed mass and describe them by the\napproximation $\\la \\dot{M}_{kl} \\ra =0$.\nThe magnitudes of $\\tilde{q}$ and $\\tilde{\\bF}$\nfollow in the form of fluctuation-dissipation relations\nfrom the Fokker-Planck equivalent of our Langevin equations.\nThe mathematics involved in obtaining fluctuation-dissipation relations\nis essentially well-known from the literature \\cite{espanol95b}, and\nour analysis parallels that of Avalos and Mackie~\\cite{avalos97}.\nHowever, the fact that the conservative part of the conventional DP forces\nis here replaced by the pressure and that the present DP's\nhave a variable volume makes a separate treatment enlightening.\n\nThe probability $\\rho (V_k,\\bP_k,E_k)$ of finding DP$_k$ with a volume $V_k$, \nmomentum $\\bP_k$ and \ninternal energy $E_k$ is then proportional to $\\exp (S_T/k_B)$\nwhere $S_T$ is the entropy of all DP's given that \nthe values $(V_k,\\bP_k,E_k)$ are known for DP$_k$\\cite{reif65}.\nIf $S'$ denotes the entropy of the bath we can write $S_T$ as\n\\bea\nS_T &=& S'(V_T-V_k, \\bP_T - \\bP_k, E_T - \\frac{P_k^2}{2M_k} - E_k) + S_k \n\\nonumber \\\\\n&\\approx & S'(V_T, \\bP_T, E_T ) - \\frac{\\partial S'}{\\partial E} \\left( \nE_k + \\frac{P_k^2}{2M_k} \\right) -\\frac{\\partial S'}{\\partial V} V_k \\nonumber \n\\\\\n&-& \\frac{\\partial S'}{\\partial \\bP} \\bP_k + S_k \n\\eea\nwhere the derivatives are evaluated at $(V_T,\\bP_T,E_T)$ \nand thus characterize the bath only. \nAssuming that $\\bP_T$ vanishes there is nothing in the system \nto give the vector ${\\partial S'}/{\\partial \\bP}$ a direction, and it must \ntherefore vanish as well \\cite{landau59c}.\nThe other derivatives give the pressure $p_0$ and temperature $T_0$\nof the bath and we obtain\n\\be\nS_T = S'(V_T, \\bP_T, E_T ) - \\frac{1}{T_0} \\left( G_k + \\frac{P_k^2}{2M_k} \n\\right)\n\\ee\nwhere the Gibbs free energy has the standard form $G_k = E_k + p_0 V_k - T_0 \nS_k$.\nSince there is nothing special about DP$_k$ it immediately follows that the \nthe full equilibrium distribution has the form\n\\be\n\\rho^{\\text{eq}} = Z^{-1}(T_0,p_0)\\exp \\left( -\\beta_0 \\sum_k \\frac{P_k^2}{2M_k} \n+ G_k \\right) \\; ,\n\\label{distribution}\n\\ee\nwhere $\\beta_0 = 1/(k_BT_0)$.\nThe temperature $T_k = (\\partial S_k /\\partial E_k)^{-1} $ and pressure \n$p_k = T_k (\\partial S_k /\\partial V_k) $ will fluctuate around the\nequilibrium values $T_0$ and $p_0$.\nThe above distribution is analyzed by Landau and Lifshitz \\cite{landau59c} who\nshow that the fluctuations have the magnitude\n\\be\n\\langle \\Delta P_k^2\\rangle = \\frac{k_BT_0}{V_k \\kappa_S }, \n\\; \\; \\langle \\Delta T^2_k \\rangle = \\frac{k_BT_0^2}{V c_v } \n\\label{fluctuations}\n\\ee\nwhere the isentropic compressibility $\\kappa_S = -(1/V) (\\partial V/\\partial \nP)_S$ and\nthe specific heat capacity $c_v$ are both intensive quantities.\nComparing our expression \nwith the distribution postulated by Avalos and Mackie, we have \nreplaced the Helmholtz by the Gibbs free energy in \\eq{distribution}).\nThis is due to the fact that our DP's exchange volume as well as \nenergy.\n\nWe write the fluctuating force as\n\\be\n\\tilde{\\bF}_{kl} = \\bom_{kl \\parallel} W_{kl\\parallel} + \\bom_{kl \\perp} \nW_{kl\\perp}\n\\label{fluc_force}\n\\ee \nwhere, for reasons soon to become apparent, we have chosen\nto decompose $\\tilde{\\bF}_{kl}$ into components parallel\nand perpendicular to $\\bee_{kl}$.\nThe $W$'s are defined as Gaussian random variables with the\n correlation function \n\\bea\n\\langle W_{kl \\alpha} (t) W_{nm \\beta}(t') \\rangle \n&=& \\delta_{\\alpha \\beta} \\delta (t-t') (\\delta_{kn}\\delta_{lm}\n+\\delta_{km}\\delta_{ln})\n \\label{correlations}\n\\eea\nwhere $\\alpha$ and $\\beta$ denote either $\\perp$ or $\\parallel$.\nThe product of $\\delta$ factors ensures that only \nequal vectorial components of the forces between a pair\nof DP's are correlated, while Newton's third law \nguarantees that $\\bom_{kl} = - \\bom_{lk}$. \nLikewise the fluctuating heat flux takes the form\n\\be\n\\tilde{q}_{kl} = \\Lambda_{kl} W_{kl}\n\\ee\nwhere $W_{kl}$ satisfies \\eq{correlations}) without the \n$\\delta_{\\alpha \\beta}$ factor and energy conservation implies\n$\\Lambda_{kl} = -\\Lambda_{lk} $.\n\nThe force correlation function then takes the form \n\\bea \n\\langle \\tilde{\\bF}_{kn}(t) \\tilde{\\bF}_{lm}(t') \\rangle &=& \n(\\bom_{kn\\perp} \\bom_{lm\\perp} + \\bom_{kn\\parallel} \\bom_{lm\\parallel})\n\\nonumber \\\\ \n&& ( \\delta_{kl} \\delta_{nm}\n+ \\delta_{km} \\delta_{ln} ) \\delta (t-t') \\nonumber \\\\\n&\\equiv & \\bom_{klnm} ( \\delta_{kl} \\delta_{nm}\n+ \\delta_{km} \\delta_{ln} ) \\delta (t-t') \\;\n\\label{force_correlations}\n\\eea\nwhere we have introduced the second order tensor $ \\bom_{knlm}$.\n\nIt is a standard result in non-equilibrium statistical mechanics \nthat a Langevin\ndescription of a dynamical variable $\\bf y$\n\\be \n\\dot{\\bf y} = {\\bf a} ({\\bf y}) + \\tilde{\\bf G}\n\\label{langevin}\n\\ee\nwhere $\\tilde{\\bG}$ is a delta-correlated force\nhas an equivalent probabilistic representation \nin terms of the Fokker-Planck equation\n\\be \n\\frac{\\partial \\rho({\\bf y},t)}{\\partial t} = -\\nabla \\cdot \n ({\\bf a} ({\\bf y}) \\rho({\\bf y})) + \\frac{1}{2} \\nabla \\nabla \n\\colon ({\\bf A}({\\bf y})\n \\rho({\\bf y}))\n\\ee\nwhere $\\nabla $ denotes derivatives with respect to ${\\bf y}$\nand $\\rho({\\bf y},t)$ is the probability\ndistribution for the variable ${\\bf y}$ at time $t$, \n $ \\langle \n \\tilde{\\bG}({\\bf y},t) \\tilde{\\bG}({\\bf y},t')\\rangle = {\\bf A} \\delta (t-\nt')$ and\n$\\bf A$ is a symmetric tensor of rank two~\\cite{gardiner85}.\n\nIn the preceding paragraph, $\\bG$ denotes all the fluctuating terms in \nEqs.~(\\ref{momentum5})\nand (\\ref{energy_dpd}). \nUsing the above definitions and $\\la \\dot{M}_{kl} \\ra = 0$\nit is a standard matter \\cite{espanol95b} to obtain the Fokker-Planck\nequation\n\\be\n\\frac{\\partial \\rho}{\\partial t} = (L_0 + L_{\\text{DIS}} +L_{\\text{DIF}}), \\rho\n\\label{fokker_planck}\n\\ee\nwhere \n\\bea\nL_0 &=& -\\sum_k \\frac{\\partial }{\\partial \\br_k} \\cdot \\bU_k +\n\\sum_{k\\neq l} l_{kl} \\left( \\frac{\\partial }{\\partial \\bP_k} \\cdot \\bee_{kl} \n\\frac{p_{kl}}{2} \\right. \\nonumber \\\\\n&+& \\left. \\frac{\\partial }{\\partial E_k } \\bee_{kl} \\cdot \\bU_{kl} \\frac{p_k \n+ p_l}{4} \\right) \\nonumber \\\\\n L_{\\text{DIS}} &=& \\sum_{k\\neq l} l_{kl} \\left( \\frac{\\partial }{\\partial \n\\bP_k } \\cdot\n\\bF^D_{kl} - \\frac{\\partial }{\\partial E_k }\\left( \\frac{\\bU_{kl}}{2} \\cdot \n\\bF^D_{kl} -\n\\lambda \\frac{T_{kl}}{r_{kl}} \\right) \\right) \\nonumber \\\\\n L_{\\text{DIF}} &=& \\frac{1}{2} \\sum_{k\\neq l} \\left( \\bom_{klkl} \\cdot \n\\frac{\\partial }{\\partial \\bP_k } \\cdot\n\\bL_{kl} - \\frac{\\partial }{\\partial E_k } \\left( \\bom_{klkl} \\cdot \n\\frac{\\bU_{kl}}{2} \\cdot \\bL_{kl} \\right. \\right. \\nonumber \\\\\n&-& \\left. \\left. \\Lambda^2_{kl} \\left( \\frac{\\partial }{\\partial E_k } - \n\\frac{\\partial }{\\partial E_l }\n\\right) \\right) \\right),\n\\label{operators}\n\\eea\n$\\bF^D_{kl} = (\\eta/r_{kl}) (\\bU_{kl} + (\\bU_{kl}\\cdot \\bee_{kl} ) \n\\bee_{kl} )$, and\nthe sum $\\sum_{k\\neq l}$ runs over both $k$ and $l$. The operator $\\bL_{kl}$\nis defined as in Ref.~\\cite{avalos97}:\n\\be\n\\bL_{kl} = \\left( \\frac{\\partial }{\\partial \\bP_k } \n- \\frac{\\partial }{\\partial \\bP_l }\\right) - \\frac{\\bU_{kl}}{2} \n\\left( \\frac{\\partial }{\\partial E_k } - \\frac{\\partial }{\\partial E_l } \n\\right) \\; .\n\\ee\n\nThe steady-state solution of \\eq{fokker_planck}) is already given by \n\\eq{distribution}); following conventional procedures we can\nobtain the fluctuation-dissipation relations\nfor $\\bom$ and $\\Lambda$ by inserting $\\rho^{\\text{eq}}$ in \n\\eq{fokker_planck}).\n\n\nApart from the tensorial nature of $\\bom_{klkl}$ the operators $ \nL_{\\text{DIS}}$\nand $ L_{\\text{DIF}}$ are essentially identical to those published\nearlier in conventional DPD~\\cite{avalos97,espanol97}.\nHowever, the `Liouville' operator $L_0$ plays a somewhat different role \nas it contains the $\\partial /\\partial E_k$ term, corresponding to the fact that \nthe pressure forces do work on the DP's to change their internal energy.\n\nWhile $L_0 \\rho^{\\text{eq}}$ conventionally vanishes exactly\nby construction of the inter-DP forces, here it vanishes \nonly to order $1/N_k$. \nIn order to evaluate $L_0 \\rho^{\\text{eq}}$ we need the following \nrelationship\n\\be \n \\frac{\\partial }{\\partial \\br_k } = \\frac{1}{2} \\sum_{k\\neq l} l_{kl} \\bee_{kl}\n \\left( \\frac{\\partial }{\\partial V_l } - \\frac{\\partial }{\\partial V_k } \n\\right) \\; ,\n\\ee\nwhich is derived by direct geometrical consideration of the\n Voronoi construction. By repeated use of \n\\eq{surface_integral}) it is then a straightforward algebraic task \nto obtain \n\\be\nL_0 \\rho^{\\text{eq}} = \\frac{\\rho^{\\text{eq}} }{4} \\sum_{k\\neq l} l_{kl} \n\\bee_{kl}\\cdot \\bU_k \\left[ \\frac{\\partial p_l}{\\partial E_l} -\\frac{ p_{kl} \nT_{kl}}{k_BT_kT_l} \\right] \\; ,\n\\label{L0}\n\\ee\nwhich does not vanish identically.\nHowever, note that if we estimate $E_l \\approx N_l k_BT$ we obtain ${\\partial \np_l}/{\\partial E_l} \\approx (1/N_k) (p_l/k_BT)$.\nSimilarly we may estimate $p_{kl} $ and $T_{kl}$ from \\eq{fluctuations}) to \nobtain\n\\be\n\\frac{p_{kl}T_{kl}}{k_BT_kT_l} \\approx \\frac{\\sqrt{ \\Delta P^2 \\Delta \nT^2}}{k_BT_kT_l} = \\frac{1}{N_k}\n\\sqrt{\\frac{N_k/V_k}{\\kappa_S c_v T_0^2}} \\; .\n\\ee\nThe last square root is an intensive quantity of the order $p_0/(k_BT_0)$, as \nmay be easily demonstrated for the case\nof an ideal gas.\nSince each separate quantity that is contained in the differences in the square \nbrackets of \\eq{L0}) is of the \norder $p_0/T_0$ we have shown that they cancel up to relative order $1/N_k\\ll 1$.\nIn fact, it is not surprising that Langevin equations which approximate \nlocal gradients to first order only in the corresponding differences, \nlike $T_{kl}$, give rise to a Fokker-Planck description \nthat contains higher order correction terms.\n\nHaving shown that $L_0 \\rho^{\\text{eq}}$ vanishes to a good approximation \nwe may proceed to obtain the fluctuation-dissipation relations from\nthe equation $ ( L_{\\text{DIS}} + L_{\\text{DIF}}) \\rho^{\\text{eq}} = 0$.\nIt may be noted from \\eq{operators}) that this equation is satisfied if \n\\bea\n(l_{kl} \\bF^D_{kl} + \\frac{1}{2} \\bom_{klkl} \\bL_{kl} ) \\rho^{\\text{eq}} &=& 0 \n\\nonumber \\\\\n\\left( l_{kl} \\lambda \\frac{T_{kl}}{r_{kl}} + \\frac{1}{2} \\Lambda^2_{kl} \\left( \n\\frac{\\partial }{\\partial E_k } - \\frac{\\partial }{\\partial E_l }\\right)\\right) \n\\rho^{\\text{eq}} &=& 0\\; .\n\\label{fd0}\n\\eea\nUsing the identity\n\\be\n\\bee_{kl}\\bee_{kl} + \\bi_{kl}\\bi_{kl} = {\\bf I}\n\\ee\nwhere $\\bi_{kl}$ a \nvector normal to $\\bee_{kl}$, \nwe may show that \\eq{fd0}) implies that \n\\bea\n\\omega_{kl\\parallel}^2 &=& 2 \\omega_{kl\\perp}^2 \n=4 \\eta k_B \\Theta_{kl} \\frac{l_{kl} }{ r_{kl} } \\nonumber \\\\\n\\Lambda_{kl}^2 &=& 2 k_B T_k T_l \\lambda \\frac{l_{kl}}{r_{kl}} \\; ,\n\\label{fd} \n\\eea\nwhere $\\Theta^{-1}_{kl} = (1/2) ( T_k^{-1} + T_l^{-1} ) $.\n\n%******END PAD\n\n\\subsection{$\\tilde{\\bF}$ from fluctuating hydrodynamics}\n\nHaving derived the fluctuation-dissipation relations from the \napproximation of the isobaric ensemble we now derive the \nsame result from fluctuating hydrodynamics, which corresponds \nto the grand canonical ensemble. We shall only derive the \nmagnitude of $\\tilde{\\bF}_{kl}$ since $\\tilde{q}$ follows \non the basis of the same\nreasoning.\n\nFluctuating hydrodynamics \\cite{landau59} is based on the conservation equations\nfor mass, momentum and energy with the modification that the momentum \nand energy fluxes contain an additional fluctuating term.\nSpecifically, the momentum flux tensor takes the form\n$-\\nabla P + \\rho \\bv \\bv + \\sigma'$, where $P$ is the pressure,\n$\\bv$ is the velocity field and the viscous stress tensor is given as\n\\be \n\\sigma' = \\eta \\left( \\nabla \\bv + \\nabla \\bv^T - \\frac{2}{d}\\nabla \\cdot \\bv \n\\right)\n+ \\zeta \\nabla \\cdot \\bv + \\bs ,\n\\ee\nwhere $\\bs$ is the fluctuating component of the momentum flux.\n{}From the same approximations as we used in deriving \\eq{fd}), i.e.\na negligible correlation time for the fluctuating forces,\nLandau and Lifshitz derive\n\\bea\n&& \\la \\bs (\\bx , t) \\cdot \\bn \\bs (\\bx' , 0)\\cdot \\bn \\ra = 2 k_BT \\left( \n\\eta (1+ \\bn \\bn )+ (\\zeta - \\frac{2}{d} \\eta ) \\bn \\bn \\right) \\nonumber \\\\\n&& \\delta (t) \\left\\{ \\begin{array}{cc}\n\\frac{1}{\\Delta V_n} & \\mbox{if } \\bx , \\bx' \\varepsilon \\Delta V_n \\\\\n0 & \\mbox{otherwise}\n\\end{array} \\right . \\; \n\\eea\nwhere $\\bn$ is an arbitrary unit vector and $n$ labels the volume element \n$\\Delta V_n$.\nBy following the derivations presented by Landau and Lifshitz, \nit may be noted that nowhere is it\nassumed that\n the $\\Delta V_n$'s are cubic or stationary.\n\\begin{figure}\n\\centerline{\\hbox{\\psfig{figure=voronoi_fh.eps,width=8cm}}}\n\\caption{\\label{figV}\n\\protect \\narrowtext \nPairwise Voronoi-cell interactions. Dark gray: The volume $V_{kl}$ associated with the interaction \nbetween a single DP-pair. The light gray region shows the \nvolume of the neighboring interaction.}\n\\end{figure}\nBy making the identifications $\\zeta = (2/d) \\eta $, $\\bn \\rightarrow \\bee_{kl} \n$ $\\tilde{\\bF}_{kl} = \nl_{kl} \\bs \\cdot \\bee_{kl} $, $\\Delta V_n \\rightarrow V_{kl}$,\n(shown in Fig.~\\ref{figV}), and \n$T = \\Theta_{kl}$ we may immediately write down \n\\bea\n\\la \\tilde{\\bF}_{kl} (t) \\tilde{\\bF}_{nm} (0) \\ra &= &\n2 \\frac{k_B \\Theta_{kl}l_{kl}^2}{V_{kl}} \\eta\n(1 + \\bee_{kl} \\bee_{nm} ) \\delta (t) \\nonumber \\\\\n& (& \\delta_{kn}\\delta_{lm} + \n\\delta_{km}\\delta_{ln} )\n\\label{grunch}\n\\eea \nwhere again the last \nsum of $\\delta$-factors ensures that $kl$ and $nm$ denote the same DP \npair.\nObserving from Fig.~\\ref{figV} that \n$V_{kl}= l_{kl} r_{kl}$, \nit now follows directly from \\eq{grunch}) that \n\\bea\n\\la \\tilde{\\bF}_{kl} (t) \\cdot \\bee_{kl} \\tilde{\\bF}_{nm} (0) \\cdot \\bee_{nm} \n\\ra\n&=& 2 \\la \\tilde{\\bF}_{kl} (t) \\cdot \\bi_{kl} \\tilde{\\bF}_{nm} (0) \\cdot \n\\bi_{nm} \\ra \\nonumber \\\\\n&=& 4 k_B \\Theta_{kl} \\frac{l_{kl}}{r_{kl}}\n \\eta \\delta (t) \\nonumber \\\\\n&(& \\delta_{kn}\\delta_{lm} + \\delta_{km}\\delta_{ln} )\n\\label{fd2}\n\\eea\nwhich is nothing but the momentum part of \\eq{fd}). That the\nfluctuating heat flux\n$\\tilde{q}$ produces the form of \nfluctuation-dissipation relations given in \\eq{fd}) \nfollows from a similar analysis.\nThus the approximation of fixed DP volume \n$V_k$ produces the\nsame result as the approximation of fixed number of MD particles\n$N_k$. This is due to the fact that both approximations \nare based on the assumption\nthat the DP's are only considered within a time interval \nwhich is longer than the \ncorrelation time of the fluctuations but shorter than the time \nneeded for the DP's to move significantly.\n\nThe result given in \\eq{fd2}) was derived from the \nsomewhat arbitrary choice of discretizaton volume $V_{kl}$;\nthis is the volume which corresponds to the segment $l_{kl}$\nover which all forces have been taken as constant. It is thus \nthe smallest discretization volume we may consistently choose.\nIt is reassuring that \\eq{fd2}) also follows from different\nchoices of $\\Delta V_n$. For example, one may readily \ncheck that \\eq{fd2}) is obtained\nif we split $V_{kl}$ in two along $r_{kl}$ and \nconsider $\\tilde{\\bF}_{kl}$ to be the sum of two independent\nforces acting on the two parts of $l_{kl}$.\n\n\nWe are now in a position to quantify the average component \n$\\langle \\dot{\\tilde{E}}_k \\rangle \\equiv \\sum_{l\\neq k} \\langle \n\\tilde{\\bF}_{kl} \\cdot \\bU_{kl}/2\\rangle $ \nof the fluctuations in the internal energy \ngiven in \\eq{energy_dpd}). \nWriting the velocity in response to $\\tilde{\\bF}_{kl}$ as \n$\\tilde{\\bU}_k = \\sum_{l\\neq k} \\int_{-\\infty}^t \\dd t' \\tilde{F}_{kl}(t')/M_k$,\nwe get that $ \\langle \\dot{\\tilde{E}}_k \\rangle = \\sum \\int_{-\\infty}^t \\dd t' \n\\langle \n\\tilde{F}_{kl}(t') \\tilde{F}_{kl}(t) \\rangle$ which by \nEqs.~(\\ref{fd}) and (\\ref{force_correlations}) becomes\n$\\langle \\dot{\\tilde{E}}_k \\rangle = (1/M_k) \\sum 3 l_{kl} \\eta k_B \n\\Theta_{kl}/r_{kl} $.\nThis result is the same as one would have obtained applying the rules of\nIt\\^{o} calculus to $\\tilde{\\bU}_k^2/(2M_k)$. \nIt yields the modified, though equivalent, energy equation\n\\bea\n&&\\dot{E}_{k} = -\\sum_l l_{lk} \\lambda \\frac{T_{kl} }{r_{kl}} \\nonumber \\\\\n&-& \\sum_l l_{lk} \\left( \\frac{p_{k}+p_l}{2} \\bee_{kl} - \\frac{\\eta}{r_{kl}}\n(\\bU_{kl} + (\\bU_{kl}\\cdot \\bee_{kl})\\bee_{kl}) \\right) \\cdot\n\\frac{\\bU_{kl}}{2} \\nonumber \\\\\n&-& \\sum_l \\tilde{\\bF'}_{kl} \\cdot \\frac{\\bU_{kl}}{2} - \n3 \\frac{l_{kl}}{r_{kl}} \\eta k_B \\Theta_{kl} + \\tilde{q}_{kl}\\; .\n\\label{energy_dpd2}\n\\eea\nwhere we have written $\\tilde{\\bF'}_{kl}$ with a prime to denote\nthat it is uncorrelated with $\\bU_{kl}$. In a numerical implementation \nthis implies that $\\tilde{\\bF'}_{kl}$ must be \ngenerated from a different random variable than $\\tilde{\\bF}_{kl}$,\nwhich was used to update $\\bU_{kl}$.\n\nThe fluctuation-dissipation relations Eqs.~(\\ref{fd})\n complete our theoretical description of dissipative\nparticle dynamics, which has been derived by a coarse-graining of\nmolecular dynamics. All the parameters and properties of this new\nversion of DPD are related directly to the underlying molecular\n dynamics, and properties such as the viscosity which are \nemergent from it.\n\n\\section{Simulations}\n\nWhile the present paper primarily deals with theoretical\ndevelopments we have carried out simulations to test the \nequilibrium behavior of the model in the case of the isothermal model.\n This is a crucial test \nas the derivation of the fluctuating forces relies on the\nmost significant approximations.\nThe simulations are carried out using a periodic Voronoi tesselation\ndescribed in detail elsewhere~\\cite{defabritiis99}.\n\n\n\\begin{figure}\n\\centerline{\\hbox{\\psfig{figure=temp3.eps,width=8cm}}}\n\\caption{\\label{figkT}\n\\protect \\narrowtext \nThe DPD temperature (energy units) averaged over 5000 dissipative\nparticles as a function of time \n(iteration number in the integration scheme), \nshowing good convergence to the underlying\nmolecular dynamics temperature which was set at one. This simulation\nprovides strong support for the approximations used to derive the\nfluctuation-dissipation relations \nin our DPD model from molecular dynamics.}\n\\end{figure}\n\nFigure \\ref{figkT} shows the relaxation process towards equilibrium of an \ninitially\nmotionless system. The DP temperature is measured as\n$\\langle \\bP_k^2/(2 M_k) \\rangle$ for a system of DPs \nwith internal energy equal to unity.\nThe simulations were run for 4000 iterations of 5000 dissipative\nparticles and a timestep\n$\\dd t =0.0005$ using an initial molecular density $\\rho =5$ for each DP.\nThe molecular mass was taken to be $m=1$, \nthe viscosity was set at $\\eta=1$, \nthe expected mean free path is 0.79, \nand the Reynolds number (See Sec.~\\ref{low_visc}) is Re=2.23.\nIt is seen that the convergence of the DP system \ntowards the MD temperature is good, a result that provides\nstrong support for the \nfluctuation-dissipation relations of \\eq{fd}).\n\n\\section{Possible applications}\n\\subsection{Multiscale phenomena}\n \nFor most practical applications involving complex fluids, additional \ninteractions and boundary conditions need to be specified.\nThese too must be \ndeduced from the microscopic dynamics, just as we have done for \nthe interparticle forces.\nThis may be achieved by considering a \nparticulate\ndescription of the boundary itself and including molecular interactions\nbetween the fluid MD particles and other objects, such as \nparticles or walls. Appropriate modifications can then be made\non the basis of the\nmomentum-flux tensor of \\eq{momentum_flux}), which \nis generally valid.\n\nConsider for example the case of a colloidal suspension, which is\nshown in Fig.~\\ref{fig3}. Beginning with \nthe hydrodynamic momentum-flux tensor \\eq{momentum_flux}) and \n\\eq{momentum5}), it is evident that we also need to define \nan interaction region where the DP--colloid forces act:\nthe DP--colloid interaction may be \nobtained in the same form as the DP--DP interaction \nof \\eq{momentum5}) by making the replacement $l_{kl}\\rightarrow L_{kI}$,\nwhere $L_{kI}$ is the length (or area in 3D) of the arc \nsegment where the dissipative particle\nmeets the colloid (see Fig.~\\ref{fig3}) and the velocity \ngradient \n$r_{kl}^{-1}( ({\\bf U}_{kl}\\cdot {\\bf e}_{kl}) {\\bf e}_{kl} \n + {\\bf U}_{kl})$ \nis that between the dissipative particle \nand the colloid surface. The latter may be computed \nusing ${\\bf U}_k $ and the velocity of the colloid surface \ntogether with a no-slip boundary condition on this surface.\nIn \\eq{fd}) the replacement $l_{kl} \\rightarrow L_{KI}$\nmust also be made. \n\n\\begin{figure}\n\\centerline{\\hbox{\\psfig{figure=colloid.eps,width=8cm}}}\n\\caption{\\label{fig3}\n\\protect \\narrowtext {\\bf Multiscale modeling of colloidal fluids.}\nAs usual, the dissipative particles are defined as cells\nin the Voronoi lattice. Note that there are four relevant length\nscales in this problem: the scale of the large, gray colloid particles,\nthe two distinct scales of the dissipative particles in between and\naway from the colloids and finally the molecular scale of the \nMD particles. These mediate the mesoscopic \ninteractions and are shown as dots on the \nboundaries between dissipative and \ncolloidal particles.}\n\\end{figure}\n\nAlthough previous DPD simulations of colloidal fluids have proved\nrather successful~\\cite{boek97} at low to intermediate solids \nvolume fractions, they break down for dense systems\nwhose solids volume fraction exceeds a value of about \n40\\% because the existing\nmethod is unable to handle multiple lengthscale phenomena. However,\nour new version of the algorithm provides the freedom to define\ndissipative particle sizes according to the local resolution\nrequirements as illustrated in Fig.~\\ref{fig3}. \nIn order to increase the spatial resolution where \ncolloidal particles are within close proximity \nit is necessary and perfectly admissible to introduce a \nhigher density of dissipative particles\nthere; this ensures that fluid lubrication and hydrodynamic \neffects are properly maintained.\nAfter these dissipative particles have moved it may be \nnecessary to re-tile\nthe DP system; this is easily achieved by distributing \nthe mass and momentum of the old dissipative particles on \nthe new ones according to \ntheir area (or volume in 3D). Considerations of space prevent us from\ndiscussing this problem further in the present paper, but \nwe plan to report in detail on such\ndense colloidal particle simulations using our method in \nfuture publications. We note in passing that a wide variety of other\ncomplex systems exist where modeling and simulation are challenged by\nthe presence of several simultaneous length scales, for example in\npolymeric and amphiphilic fluids, particularly in confined geometries\nsuch as porous media~\\cite{coveney98}.\n\n\\subsection{The low viscosity limit and high Reynolds numbers}\n\\label{low_visc}\nIn the kinetic theory derived by Marsh, Backx and Ernst [15]\n% \\cite{MBE1}\nthe viscosity is explicitly shown to have a kinetic\ncontribution $\\eta_K = \\rho D/2$ where $D$ is the DP self diffusion \ncoefficient \nand $\\rho$ the mass density.\nThe kinetic contribution to the viscosity was measured by Masters and Warren \n\\cite{masters99}\nwithin the context of an improved theory.\nHow then can the viscosity $\\eta$ used in our model \nbe decreased to zero while kinetic \ntheory\nputs the lower limit $\\eta_K$ to it?\n\n%\\subsubsection{Non-dimensionalizing the DPD equations}\n\nTo answer this question we must define a physical way of decreasing the MD \nviscosity\nwhile keeping other quantities fixed, or, alternatively rescale the system in a \nway that \nhas the equivalent effect.\nThe latter method is preferable as it allows the underlying microscopic system \nto \nremain fixed.\n In order to do this we non-dimensionalize the DP\nmomentum equation \\eq{momentum5}).\n\nFor this purpose we introduce the characteristic equilibrium velocity, $U_0 = \n\\sqrt{k_BT/M}$,\nthe characteristic distance $r_0$ as the typical DP size. Then the \ncharacteristic\ntime $t' = r_0/U_0$ follows. \n% \\eq{momentum5}) \n\nNeglecting gravity for the time being \\eq{momentum5}) \n takes the form\n\\bea\n && \\frac{\\dd \\bP_k' }{\\dd t'} = \n - \\sum_l l'_{kl} \\left( \n\\frac{p'_{kl}}{2} \\bee_{kl} +\\frac{1}{\\text{Re}} \\left( \\bU'_{kl} + (\\bU'_{kl}\n \\cdot \\bee_{kl}) \\bee_{kl} \\right)\n\\right)\\nonumber \\\\\n&+& \n\\sum_l \\frac{l'_{kl}L'_{kl}}{2 r'_{kl}} \\frac{\\rho'_k + \\rho'_l }{2}\n\\bi_{kl}\\cdot \\bU'_{kl}\\frac{\\bU'_k + \\bU'_l}{2}\n+ \\sum_l \\tilde{\\bF}'_{kl}\n\\label{non_dim} \\; ,\n\\eea\nwhere $\\bP_k'= \\bP_k/(MU_0)$, $p'_{kl}= p_{kl}r_0^2/(MU_0^2)$,\n$M = \\rho r_0^2$ in 2d, the Reynolds number $\\text{Re} = U_0 r_0\\rho /\\eta$\nand $\\tilde{\\bF}'_{kl}= (r_0/MU_0^2)\\tilde{\\bF}_{kl}$ \nwhere $\\tilde{\\bF}_{kl}$ is given by Eqs.~(\\ref{fluc_force}) and (\\ref{fd}).\nA small calculations then shows that if $\\tilde{\\bF}'_{kl}$ is \nrelated to $ \\omega_{kl}'$ and $t'$ like $\\tilde{\\bF}_{kl}$ related to \n$ \\omega_{kl}$ and $t$, then\n\\be\n\\omega_{kl}^{'2} \\approx \\frac{1}{\\text{Re}} \\frac{k_BT}{M U_0^2} \\approx \n\\frac{1}{\\text{Re}} \n\\ee\nwhere we have neglected dimensionless geometric prefactors like \n$l_{kl}/r_{kl}$ and used\nthe fact that the ratio of the thermal to kinetic energy by definition of $U_0$ \nis one.\n\nThe above results imply that when the DPD system \nis measured in non-dimensionalized \nunits \neverything is determined by the value of the mesoscopic Reynolds number Re.\nThere is thus no observable difference in this system between increasing \n$r_0$ and decreasing $\\eta$.\n\nReturning to dimensional units again the DP diffusivity may be obtained from \nthe \nStokes-Einstein relation \\cite{einstein05} as \n\\be\nD= \\frac{k_BT}{a r_0 \\eta}\n\\label{einstein}\n\\ee\nwhere $a$ is some geometric factor ($a=6\\pi $ for a sphere) and \nall quantities on the right hand side except $r_0$ refer directly to the\nunderlying MD. \nAs we are keeping the MD system fixed and \nincreasing Re by increasing $r_0$, \nit is seen that $D$ and hence $\\eta_K$ vanish in the process.\n\nWe note in passing that if $D$ is written in terms of the mean free path \n$\\lambda$:\n$D=\\lambda \\sqrt{k_BT/(\\rho r_0^2)}$ and this result is compared with \n\\eq{einstein})\nwe get $\\lambda' = \\lambda /r_0 \\sim 1/r_0 $ in 2d, i.e. the mean free path, \nmeasured\nin units of the particle size decreases as the inverse particle size.\nThis is consistent with the decay of $\\eta_K$.\nThe above argument shows that decreasing $\\eta$ is equivalent to \nkeeping the microscopic MD system fixed while increasing the DP size,\nin which case the mean free path effects on viscosity is decreased to zero\nas the DP size is increased to infinity. It is in this limit that \nhigh Re values may be achieved.\n\nNote that in this limit the thermal forces $\\tilde{\\bF}_{kl} \\sim \\text{Re}^{-\n1/2}$\nwill vanish, and that we are effectively left with a macroscopic, \nfluctuationless\ndescription.\n This is no problem when using the present Voronoi construction.\nHowever, the effectively spherical particles of conventional DPD will \nfreeze into a colloidal crystal, i.e. into a \nlattice configuration [8,9] in this \nlimit.\nAlso while conventional DPD has usually \nrequired calibration simulations\nto determine the viscosity, due to discrepancies between theory\nand measurements, \nthe viscosity in this new form of DPD is simply an input parameter.\nHowever, there may still be \ndiscrepancies due to the approximations made in going from \nMD to DPD. These approximations include the linearization of the \ninter-DP velocity fields,\nthe Markovian assumption in the force correlations \nand the neglect of a DP angular momentum variable.\n\nNone of the conclusions from the above arguments would change if \nwe had worked in three dimensions in stead of two.\n\n\\section{Conclusions}\n\nWe have introduced a systematic procedure \nfor deriving the mesoscopic modeling and simulation method known as \ndissipative particle dynamics \nfrom the underlying description in terms of molecular dynamics. \n\\begin{figure} \n\\centerline{\\hbox{\\psfig{figure=logic_chart.eps,width=8cm}}}\n\\caption{\\label{fig5}\n\\protect \\narrowtext Outline of the derivation of dissipative particle\ndynamics from molecular dynamics as presented in the present paper.\nThe MD viscosity is denoted by $\\eta$ and $\\omega$ is the amplitude\nof the fluctuating force $\\tilde{\\bf F}$ as defined \nin \\protect Eq.~(\\ref{fluc_force})}\n\\end{figure}\nFigure \\ref{fig5} illustrates the structure of the theoretical development\nof DPD equations from MD as presented in this paper.\nThe initial coarse graining leads to equations of essentially \nthe same structure\nas the final DPD equations. However, they are still invariant under time-\nreversal.\nThe label DPD1 refers to Eqs.~(\\ref{mass_dpd1}), \n(\\ref{momentum4}) and (\\ref{energy_dpd1}), whereas\nthe DPD2 equations have been supplemented with specific constitutive relations \nboth for the non-equilibrium fluxes (momentum and heat) and an equilibrium \ndescription of the thermodynamics. These equations are \nEqs.~(\\ref{momentum5}) and (\\ref{energy_dpd}) along with Eqs.~(\\ref{fd}).\nThe development we have made which is shown in Fig.~\\ref{fig5} does\nnot claim\nto derive the irreversible DPD equations from the reversible ones of\nmolecular dynamics in a rigorous manner, although it \ndoes illustrate where the transition takes place with the \nintroduction of molecular averages. The kinetic equations of this new\nDPD satisfy an \n$H$-theorem, guaranteeing an\nirreversible approach to the equilibrium state.\nNote that in passing to the time-asymmetric description\nby the introduction of the averaged description of \\eq{constitutive}),\na time asymmetric non-equilibrium ensemble is required~\\cite{oppvc94}.\n\nThis is the first time that\nany of the various existing \nmesoscale methods have been put on a firm `bottom up'\ntheoretical foundation, a development which brings\nwith it numerous new insights as well as practical advantages.\nOne of the main virtues of this procedure \nis the capability it provides to choose one or more coarse-graining \nlengthscales to suit the particular modeling problem at hand.\nThe relative scale between molecular dynamics and the chosen dissipative\nparticle dynamics, which may be defined \nas the ratio of their number densities \n$\\rho_{\\text{DPD}}/\\rho_{\\text{MD}}$,\nis a free parameter within the theory.\nIndeed, this rescaling may be viewed as a renormalisation group procedure\nunder which the fluid viscosity remains constant:\nsince the conservation laws hold exactly\nat every level of coarse graining, the result of doing two rescalings,\nsay from MD to DPD$\\alpha$ and from DPD$\\alpha$ to DPD$\\beta$, \nis the same as doing just one with a larger ratio, i.e.\n$\\rho_{\\text{DPD}\\beta}/\\rho_{\\text{MD}} = (\\rho_{\\text{DPD}\\beta}/\n\\rho_{\\text{DPD}\\alpha})(\\rho_{\\text{DPD}\\alpha}/\\rho_{\\text{MD}})$. \n% Indeed, the asymptotic limit of infinitely large (or\n% indefinitely iterated) coarse-grained rescaling applied to our \n% single-component\n% DPD fluid picks out the macroscopic\n% Navier-Stokes equations of continuum fluid dynamics as a fixed\n% point of this renormalisation group.\n% Provided the system at hand as well as its boundaries \n% exhibit scale invariance this procedure should give rise \n% to corresponding scaling laws.\n\nThe present coarse graining scheme is not limited to hydrodynamics.\nIt could in principle be used to rescale the local description\nof any quantity of interest. However, only for \n%NB In view of Giannis findings with Saoro: 'locally' is new\nlocally\nconserved quantities will\nthe DP particle interactions take the form of surface terms as here,\nand so it is unlikely that the scheme will produce a useful\ndescription of non-conserved quantities. \n\nIn this context, we note that the bottom-up approach to fluid\nmechanics presented here may throw new light on\naspects of the problem of homogeneous and inhomogeneous\nturbulence. Top-down multiscale methods and, to a more limited extent,\nideas taken from renormalisation group theory have been applied \nquite widely in recent years to provide insight into the nature of \nturbulence~\\cite{frisch95,bensoussan78}; one might expect an\nalternative perspective \nto emerge from a fluid dynamical theory originating at\nthe microscopic level, in which the central \nrelationship between conservative\nand dissipative processes is specified in a more fundamental manner.\nFrom a practical point of view it is noted that, \nsince the DPD viscosity is the same as the viscosity emergent from the\nunderlying MD level, it\nmay be treated as a free parameter in the DPD model, and thus high \nReynolds numbers may be reached. In the $\\eta \\rightarrow 0$\nlimit the model thus represents \na potential tool for hydrodynamic simulations of turbulence. \nHowever, we have not investigated the potential numerical \ncomplications of this limit.\n\nThe dissipative particle dynamics which we have derived is formally\nsimilar to the conventional version,\nincorporating as it does \nconservative, dissipative and fluctuating forces. The\ninteractions are pairwise, and conserve mass and momentum as well as\nenergy.\nHowever, now all these forces have been derived from the underlying \nmolecular dynamics. The conservative and dissipative forces arise\ndirectly from the hydrodynamic \ndescription of the molecular dynamics and \nthe properties of the fluctuating forces \nare determined via a fluctuation--dissipation relation.\n\nThe simple hydrodynamic description of the molecules chosen \nhere is not a necessary requirement.\nOther choices for the average of the general momentum and energy \nflux tensors\nEqs.~(\\ref{enrgy_flux}) and (\\ref{momentum_flux}) \nmay be made and we hope these will be explored in\nfuture work. \nMore significant is the fact that our \nanalysis permits the introduction of specific\nphysicochemical interactions at the mesoscopic level, together with\na well-defined scale for this mesoscopic description.\n\nWhile the Gaussian basis we used for the sampling functions is an\narbitrary albeit convenient choice, the Voronoi geometry itself emerged \nnaturally from the requirement that all the MD particles be fully\naccounted for. Well defined procedures already \nexist in the literature for the computation of\nVoronoi tesselations~\\cite{guibas92} and so algorithms \nbased on our model are not computationally\ndifficult to implement. Nevertheless, it should be appreciated that \nthe Voronoi construction represents a significant\ncomputational overhead. This overhead is of order\n$N\\log N$, a factor $\\log N$ larger than the \nmost efficient multipole \nmethods in principle \navailable for handling the particle interactions in \nmolecular dynamics. \nHowever, the prefactors are likely to be much larger in the \nparticle interaction case.\n \nFinally we note the formal similarity of the present\nparticulate description \nto existing continuum fluid dynamics methods \nincorporating adaptive meshes, \nwhich start out from a top-down or \nmacroscopic description. These top-down approaches include in particular \nsmoothed particle hydrodynamics~\\cite{monaghan92} and finite-element \nsimulations.\nIn these descriptions too the computational method is based \non tracing the motion of elements of the fluid on the basis of the \nforces acting between them~\\cite{boghosian98}. \nHowever, while such top-down computational \nstrategies depend on a \nmacroscopic and purely phenomenological fluid description,\nthe present approach rests on a {\\em molecular} basis.\n\n\\acknowledgments \nIt is a pleasure to thank Frank Alexander, Bruce Boghosian and Jens\nFeder for many helpful and stimulating discussions. We are grateful\nto the Department of Physics at the University of Oslo and\nSchlumberger Cambridge Research for financial support which enabled\nPVC to make several visits to Norway in the course of 1998; and to\nNATO and the Centre for Computational Science at Queen Mary and\nWestfield College for funding visits by EGF to London in 1999 and\n2000.\n\n\\begin{thebibliography}{10}\n\n\\bibitem{landau59}\nL.~D. Landau and E.~M. Lifshitz, {\\em Fluid Mechanics} (Pergamon Press, New\n York, 1959).\n\n\\bibitem{boltzmann1872}\nL. Boltzmann., {\\em Vorlesungen \\\"{u}ber Gastheorie} (Leipzig,\n 1872).\n\n\\bibitem{koplik95}\nJ. Koplik and J.~R. Banavar, Ann. Rev. Fluid Mech. {\\bf 27}, 257 (1995).\n\n\\bibitem{frisch86}\nU. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. {\\bf 56}, 1505\n (1986).\n\n\\bibitem{mcnamara88}\nG. McNamara and G. Zanetti, Phys. Rev. Lett. {\\bf 61}, 2332 (1988).\n\n\\bibitem{chan93}\nX.~W. Chan and H. Chen, Phys. Rev E {\\bf 49}, 2941 (1993).\n\n\\bibitem{swift95}\nM.~R. Swift, W.~R. Osborne, and J. Yeomans, Phys. Rev. Lett. {\\bf 75}, 830\n (1995).\n\n\\bibitem{hoogerbrugge92}\nP.~J. Hoogerbrugge and J.~M.~V.~A. Koelman, Europhys. Lett. {\\bf 19}, 155\n (1992).\n\n\\bibitem{espanol95b}\nP. Espa{\\~{n}}ol and P. Warren, Europhys. Lett. {\\bf 30}, 191 (1995).\n\n\\bibitem{boek97}\nE.~S. Boek, P.~V. Coveney, H.~N.~W. Lekkerkerker, and P. van~der Schoot, Phys.\n Rev. E {\\bf 54}, 5143 (1997).\n\n\\bibitem{schlijper95}\nA.~G. Schlijper, P.~J. Hoogerbrugge, and C.~W. Manke, J. Rheol. {\\bf 39}, 567\n (1995).\n\n\\bibitem{coveney96}\nP.~V. Coveney and K.~E. Novik, Phys. Rev. E {\\bf 54}, 5143 (1996).\n\n\\bibitem{groot97}\nR.~D. Groot and P.~B. Warren., J. Chem. Phys. {\\bf 107}, 4423 (1997).\n\n\\bibitem{espanol95}\nP. Espa{\\~{n}}ol, Phys. Rev. E {\\bf 52}, 1734 (1995).\n\n\\bibitem{MBE1}\nC.~A. Marsh, G. Backx, and M. Ernst, Phys. Rev. E {\\bf 56}, 1676 (1997).\n\n\\bibitem{avalos97}\nJ.~B. Avalos and A.~D. Mackie, Europhys. Lett. {\\bf 40}, 141 (1997).\n\n\\bibitem{espanol97}\nP. Espa{\\~{n}}ol, Europhys. Lett {\\bf 40}, 631 (1997).\n\n\\bibitem{espanol98b}\nP. Espa{\\~{n}}ol, Phys. Rev. E {\\bf 57}, 2930 (1998).\n\n\\bibitem{monaghan92}\nJ.~J. Monaghan, Ann. Rev. Astron. Astophys. {\\bf 30}, 543 (1992).\n\n\\bibitem{espanol97b}\nP. Espa{\\~{n}}ol, M. Serrano, and I.~Zu{\\~{n}}iga, Int. J. Mod. Phys. C\n{\\bf 8}, 899 (1997).\n\n\\bibitem{flekkoy99}\nE.~G. Flekk{\\o}y and P.~V. Coveney, Phys. Rev. Lett. {\\bf 83}, 1775 (1999).\n\n\n\\bibitem{pvcop92}\nP.~V. Coveney and O. Penrose, J. Phys A: Math. Gen. {\\bf 25}, 4947 (1992).\n\n\\bibitem{oppvc94}\nO. Penrose and P.~V. Coveney, Proc. R. Soc. {\\bf 447}, 631 (1994).\n\n\\bibitem{pvcrrh90}\nP.~V. Coveney and R. Highfield, {\\em The Arrow of Time} (W.~H.Allen, London,\n 1990).\n\n\\bibitem{marsh98c}\nC.~A. Marsh and P.~V. Coveney, J. Phys. A: Math. Gen. {\\bf 31}, 6561 (1998).\n\n\\bibitem{reif65}\nF. Reif, {\\em Fundamentals of statistical and thermal physics} (Mc Graw-Hill,\n Singapore, 1965).\n\n\\bibitem{landau59c}\nL.~D. Landau and E.~M. Lifshitz, {\\em Statistical Physics} (Pergamon Press, New\n York, 1959).\n\n\\bibitem{gardiner85}\nC.~W. Gardiner, {\\em Handbook of stochastic methods} (Springer Verlag, Berlin\n Heidelberg, 1985).\n\n\\bibitem{defabritiis99}\nG.~D. Fabritiis, P.~V. Coveney, and E.~G. Flekk{\\o}y, in {\\em Proc. 5th\n European SGI/Cray MPP Workshop, Bologna, Italy 1999}.\n%(PUBLISHER, ADDRESS, 1999).\n\n\\bibitem{coveney98}\nP.~V. Coveney, J.~B. Maillet, J.~L. Wilson, P.~W. Fowler, O. Al-Mushadani and\nB.~M. Boghosian, Int. J. Mod. Phys. C {\\bf 9}, 1479 (1998).\n\n\\bibitem{masters99}\nA.~J. Masters and P.~B. Warren, {\\it preprint} cond-mat/9903293\n (http://xxx.lanl.gov/) (unpublished).\n\n\\bibitem{einstein05}\nA. Einstein, Ann. Phys. {\\bf 17}, 549 (1905).\n\n\\bibitem{frisch95}\nU. Frisch, {\\em Turbulence} (Cambridge University Press, Cambridge, 1995).\n\n\\bibitem{bensoussan78}\nA. Bensoussan, J.~L. Lions, and G. Papanicolaou, {\\em Asymptotic Analysis for\n Periodic Structures} (North-Holland, Amsterdam, 1978).\n\n\\bibitem{guibas92}\nD.~E.~K. L.~J.~Guibas and M. Sharir, Algorithmica {\\bf 7}, 381 (1992).\n\n\\bibitem{boghosian98}\nB.~M. Boghosian, Encyclopedia of Applied Physics {\\bf 23}, 151 (1998).\n\n\\end{thebibliography}\n\n% \\bibliographystyle{../../bibliography/prsty}\n% \\bibliography{../../bibliography/all}\n\n%\\appendix\n%\\section{Mass fluctuations}\n%\\label{appendix1}\n\\end{multicols}\n%\\newpage\n\n\\end{document} \n"
}
] |
[
{
"name": "cond-mat0002174.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{landau59}\nL.~D. Landau and E.~M. Lifshitz, {\\em Fluid Mechanics} (Pergamon Press, New\n York, 1959).\n\n\\bibitem{boltzmann1872}\nL. Boltzmann., {\\em Vorlesungen \\\"{u}ber Gastheorie} (Leipzig,\n 1872).\n\n\\bibitem{koplik95}\nJ. Koplik and J.~R. Banavar, Ann. Rev. Fluid Mech. {\\bf 27}, 257 (1995).\n\n\\bibitem{frisch86}\nU. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. {\\bf 56}, 1505\n (1986).\n\n\\bibitem{mcnamara88}\nG. McNamara and G. Zanetti, Phys. Rev. Lett. {\\bf 61}, 2332 (1988).\n\n\\bibitem{chan93}\nX.~W. Chan and H. Chen, Phys. Rev E {\\bf 49}, 2941 (1993).\n\n\\bibitem{swift95}\nM.~R. Swift, W.~R. Osborne, and J. Yeomans, Phys. Rev. Lett. {\\bf 75}, 830\n (1995).\n\n\\bibitem{hoogerbrugge92}\nP.~J. Hoogerbrugge and J.~M.~V.~A. Koelman, Europhys. Lett. {\\bf 19}, 155\n (1992).\n\n\\bibitem{espanol95b}\nP. Espa{\\~{n}}ol and P. Warren, Europhys. Lett. {\\bf 30}, 191 (1995).\n\n\\bibitem{boek97}\nE.~S. Boek, P.~V. Coveney, H.~N.~W. Lekkerkerker, and P. van~der Schoot, Phys.\n Rev. E {\\bf 54}, 5143 (1997).\n\n\\bibitem{schlijper95}\nA.~G. Schlijper, P.~J. Hoogerbrugge, and C.~W. Manke, J. Rheol. {\\bf 39}, 567\n (1995).\n\n\\bibitem{coveney96}\nP.~V. Coveney and K.~E. Novik, Phys. Rev. E {\\bf 54}, 5143 (1996).\n\n\\bibitem{groot97}\nR.~D. Groot and P.~B. Warren., J. Chem. Phys. {\\bf 107}, 4423 (1997).\n\n\\bibitem{espanol95}\nP. Espa{\\~{n}}ol, Phys. Rev. E {\\bf 52}, 1734 (1995).\n\n\\bibitem{MBE1}\nC.~A. Marsh, G. Backx, and M. Ernst, Phys. Rev. E {\\bf 56}, 1676 (1997).\n\n\\bibitem{avalos97}\nJ.~B. Avalos and A.~D. Mackie, Europhys. Lett. {\\bf 40}, 141 (1997).\n\n\\bibitem{espanol97}\nP. Espa{\\~{n}}ol, Europhys. Lett {\\bf 40}, 631 (1997).\n\n\\bibitem{espanol98b}\nP. Espa{\\~{n}}ol, Phys. Rev. E {\\bf 57}, 2930 (1998).\n\n\\bibitem{monaghan92}\nJ.~J. Monaghan, Ann. Rev. Astron. Astophys. {\\bf 30}, 543 (1992).\n\n\\bibitem{espanol97b}\nP. Espa{\\~{n}}ol, M. Serrano, and I.~Zu{\\~{n}}iga, Int. J. Mod. Phys. C\n{\\bf 8}, 899 (1997).\n\n\\bibitem{flekkoy99}\nE.~G. Flekk{\\o}y and P.~V. Coveney, Phys. Rev. Lett. {\\bf 83}, 1775 (1999).\n\n\n\\bibitem{pvcop92}\nP.~V. Coveney and O. Penrose, J. Phys A: Math. Gen. {\\bf 25}, 4947 (1992).\n\n\\bibitem{oppvc94}\nO. Penrose and P.~V. Coveney, Proc. R. Soc. {\\bf 447}, 631 (1994).\n\n\\bibitem{pvcrrh90}\nP.~V. Coveney and R. Highfield, {\\em The Arrow of Time} (W.~H.Allen, London,\n 1990).\n\n\\bibitem{marsh98c}\nC.~A. Marsh and P.~V. Coveney, J. Phys. A: Math. Gen. {\\bf 31}, 6561 (1998).\n\n\\bibitem{reif65}\nF. Reif, {\\em Fundamentals of statistical and thermal physics} (Mc Graw-Hill,\n Singapore, 1965).\n\n\\bibitem{landau59c}\nL.~D. Landau and E.~M. Lifshitz, {\\em Statistical Physics} (Pergamon Press, New\n York, 1959).\n\n\\bibitem{gardiner85}\nC.~W. Gardiner, {\\em Handbook of stochastic methods} (Springer Verlag, Berlin\n Heidelberg, 1985).\n\n\\bibitem{defabritiis99}\nG.~D. Fabritiis, P.~V. Coveney, and E.~G. Flekk{\\o}y, in {\\em Proc. 5th\n European SGI/Cray MPP Workshop, Bologna, Italy 1999}.\n%(PUBLISHER, ADDRESS, 1999).\n\n\\bibitem{coveney98}\nP.~V. Coveney, J.~B. Maillet, J.~L. Wilson, P.~W. Fowler, O. Al-Mushadani and\nB.~M. Boghosian, Int. J. Mod. Phys. C {\\bf 9}, 1479 (1998).\n\n\\bibitem{masters99}\nA.~J. Masters and P.~B. Warren, {\\it preprint} cond-mat/9903293\n (http://xxx.lanl.gov/) (unpublished).\n\n\\bibitem{einstein05}\nA. Einstein, Ann. Phys. {\\bf 17}, 549 (1905).\n\n\\bibitem{frisch95}\nU. Frisch, {\\em Turbulence} (Cambridge University Press, Cambridge, 1995).\n\n\\bibitem{bensoussan78}\nA. Bensoussan, J.~L. Lions, and G. Papanicolaou, {\\em Asymptotic Analysis for\n Periodic Structures} (North-Holland, Amsterdam, 1978).\n\n\\bibitem{guibas92}\nD.~E.~K. L.~J.~Guibas and M. Sharir, Algorithmica {\\bf 7}, 381 (1992).\n\n\\bibitem{boghosian98}\nB.~M. Boghosian, Encyclopedia of Applied Physics {\\bf 23}, 151 (1998).\n\n\\end{thebibliography}"
}
] |
cond-mat0002175
|
Temporal correlations versus noise in the correlation matrix formalism: an example of the brain auditory response
|
[
{
"author": "J. Kwapie\\'n$^{1}$"
},
{
"author": "S. Dro\\.zd\\.z$^{1,2}$ and A.A. Ioannides$^{3}$"
}
] |
We adopt the concept of the correlation matrix to study correlations among sequences of time-extended events occuring repeatedly at consecutive time-intervals. As an application we analyse the magnetoencephalography recordings obtained from human auditory cortex in epoch mode during delivery of sound stimuli to the left or right ear. We look into statistical properties and the eigenvalue spectrum of the correlation matrix $C$ calculated for signals corresponding to different trials and originating from the same or opposite hemispheres. The spectrum of $C$ largely agrees with the universal properties of the Gaussian orthogonal ensemble of random matrices, with deviations characterised by eigenvectors with high eigenvalues. The properties of these eigenvectors and eigenvalues provide an elegant and powerful way of quantifying the degree of the underlying collectivity during well defined latency intervals with respect to stimulus onset. We also extend this analysis to study the time-lagged interhemispheric correlations, as a computationally less demanding alternative to other methods such as mutual information.
|
[
{
"name": "tcvn.tex",
"string": "\\documentstyle[aps,preprint]{revtex}\n\\begin{document}\n\n\\title{Temporal correlations versus noise in the correlation matrix formalism:\nan example of the brain auditory response}\n\n\\author{J. Kwapie\\'n$^{1}$, S. Dro\\.zd\\.z$^{1,2}$ and A.A. Ioannides$^{3}$}\n\n\\address{\n $^{1}$ Institute of Nuclear Physics, PL--31-342 Krak\\'ow, Poland,\\\\\n $^{2}$ Institut f\\\"ur Kernphysik, Forschungszentrum J\\\"ulich,\n D--52425 J\\\"ulich, Germany,\\\\\n $^{3}$ Laboratory for Human Brain Dynamics, Brain Science Institute, RIKEN, Wako-shi,351-0198, Japan.}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\n\n We adopt the concept of the correlation matrix to study correlations\n among sequences of time-extended events occuring repeatedly at consecutive \n time-intervals. As an application we analyse the magnetoencephalography \n recordings obtained from human auditory cortex in epoch\n mode during delivery of sound stimuli to the left or right ear. We look\n into statistical properties and the eigenvalue spectrum of the\n correlation matrix $\\bf C$ calculated for signals corresponding to\n different trials and originating from the same or opposite hemispheres.\n The spectrum of $\\bf C$ largely agrees with the universal properties of the\n Gaussian orthogonal ensemble of random matrices, with deviations \n characterised by eigenvectors with high eigenvalues. The properties of \n these eigenvectors and eigenvalues provide an elegant and powerful way of \n quantifying the degree of the underlying collectivity during well defined \n latency intervals with respect to stimulus onset. We also extend this \n analysis to study the time-lagged interhemispheric correlations, as a \n computationally less demanding alternative to other methods such as mutual \n information.\n\n\\end{abstract}\n\n\\newpage\n\n\\section{Introduction}\n\n Studying complex systems is typically based on analyzing large,\n multivariate data. Since, in general terms, complexity is primarily\n connected with coexistence of collectivity and chaos or even noise,\n it is of crucial importance to find an appropriate low dimensional \n representation of an underlying high dimensional dynamical system.\n In many cases this aims at denoising and compressing dynamic imaging \n data. Such a problem is particularly frequent in the area of the brain \n research where a complex but relatively sparse connectivity prevails. \n Understanding brain function requires a characterisation and quantification \n of the correlations in the signals generated at different areas. \n\n Direct pathways connect the sensory organs with the corresponding primary \n cortical areas. In the auditory system of interest here, delivery \n of a stimulus to either the left or the right ear is relayed to both \n primary auditory cortices, with stronger and earlier response on \n the contralateral side. The first cortical response arrives very early, \n well within 20 milliseconds, but it is too weak to be mapped non-invasively \n from outside. Successive waves of cortical activation follow with the \n strongest around 80-100 ms. For a simple stimulus and no cognitive task \n required the response as seen in the average is effectively over within \n the first 200-300 milliseconds. More elaborate analysis shows that the \n \"echoic memory\" last for a few seconds~\\cite{NR1,NR2}. Furthermore \n the activity in each area of the cortex, including the auditory cortex and \n its subdivisions, is determined by a plethora of interactions with \n other areas and not just the direct pathway from the cochlea. The \n variability of the evoked response possibly reflects the many ways a given \n input in the periphery can be modulated before the strong cortical \n activations emerge~\\cite{Liu98}. Our treatment of the activity from each \n auditory cortex as an independent signal bypasses this complexity by \n lumping many effects into information theoretic measures. The advantage \n of this approach is that it leads to quantitative analysis of stochastic \n and collective aspects of the complex phenomena in the auditory cortex and \n the brain at large.\n\n In our previous work~\\cite{Kwapien98} we have established the existence of \n correlations between activity in the two auditory cortices, using mutual\n information~\\cite{Fraser86} as a measure of statistical dependence.\n The analysis showed that collectivity and noise were present in the \n data~\\cite{Drozdz99a}.\n\n Usually, one analyzes a set of simultaneously recorded signals which emerge \n from the activity of sub-components of the system.\n Consequently, the presence of correlations in\n such signals is to be interpreted as a certain sort of \n cooperation among several or all of these sub-components.\n Though closely related, our present approach is somewhat different.\n Instead of studying many subsystems at the same time, we deal with two\n brain areas only and aim at identifying repetitive structures and their\n time-relations in consecutive independent trials of delivery of the stimulus.\n We thus construct the correlation matrix (which is a normalized version\n of the covariance matrix~\\cite{Broomhead86,Liu96}) whose entries express\n correlations among all the trials that are delivered by experiment.\n The difference relative to a conventional use of the correlation matrix\n is that now the indices of this matrix are labeling different presentations \n of the stimulus and not different subsystems. The resulting eigenspectrum\n is then expected to carry information about deterministic,\n non-random properties, separated out from the noisy background whose\n nature can also be quantified. \n \n\n\n\\section{Experiment and Data}\n\n The details of the experiment can be found in our earlier\n articles~\\cite{Liu96,Liu98,Kwapien98}. Here, for completeness, we\n sketch briefly only the most important facts. Five healthy male \n volunteers participated in the auditory experiment. We used\n 2x37-channel, two-dewar MEG apparatus (each dewar covered the temporal\n area in one hemisphere) to measure magnetic field generated by the\n cortical electric activity~\\cite{Hamalainen93}. The stimuli were 1 kHz\n tones lasting 50 ms each delivered in three runs to the left, right or\n both ears in 1 second intervals. The single trial of delivery of\n stimulus was repeated 120 times for each kind of stimulation. The\n cortical signals were sampled with 1042 Hz frequency.\n Pilot runs were used to place each dewar in turn so that both the positive \n and negative magnetic field extrema were captured by the 37 channel array. \n With such a coverage it is feasible to construct linear combinations of the \n signals which act like virtual electrodes \"sensing\" the activity in the \n auditory cortex~\\cite{Liu98}. This computation can be done at each\n timeslice of each single trial independently, thus building the\n timeseries for each auditory cortex for further analysis~\\cite{Kwapien98}.\n\n Delivery of a sound stimulus or any change in the continuous stimulus\n causes a characteristic activity in the auditory cortex which is best \n illustrated by averaging many such events~\\cite{Creutzfeldt95}. \n The (averaged) evoked potential, appears in both hemispheres and has a \n form of several positive and negative deflections of the magnetic field. \n The most prominent feature of the average is a high amplitude deflection \n at about 80-100 ms after the onset of the stimulus (so called M100). The \n details of the average evoked response are hardly visible in each single \n trial, partly because of strong background activity, which is not related \n to the stimulus and partly because of the latency jitter introduced by the \n many feed-forward and feed-back interactions that occur intermittently \n between the periphery and the cortex. If as signal we consider what is \n fairly time-locked to the stimulus onset then signal-to-noise ratio is much \n improved by averaging the signal over all single trials. \n\n\n We will consider two runs, corresponding to stimuli delivered to the left \n and right ear. Each run comprises $N=120$ \n single trials, thus we have 120 signals for each hemisphere and each\n kind of stimulation. The signals are represented by the time series \n $x^{L,R}_{\\alpha} (t_i)$ of length of $T=1042$ time slices \n $(i = 1,..., 1042, \\alpha = 1,..., 120)$\n each evenly covering 1 second time interval. Since all the stimuli were\n provided in precisely specified equidistant instants of time, \n all the series can be adjusted so \n that the onset of each stimulus corresponds to the same time slice $i =\n 230$. Each signal starts 220 ms before and ends 780 ms after the onset.\n A band pass filter was applied in the 1-100 Hz range.\n\n\n For a simple auditory stimulus and no cognitive task associated with it, the \n average evoked response lasts for 200-300 ms; this is also reflected in \n our earlier mutual information study of the signals~\\cite{Kwapien98}. \n Since other parts of each series are associated with activity which is not \n time-locked to the stimulus, the appearence of similar events in both \n hemispheres and across trials results in correlations that are much stronger\n in the first few hundred millisecond. The presence of correlations \n and collectivity can not be excluded {\\em a priori} from other periods and \n it is therefore of considerable interest to compare two such intervals. \n We have settle on two such intervals, each with 250 timeslices: the first \n we call the Evoked Potential (EP) interval and it covers the first 250 \n timeslices after stimulus onset, i.e. 250 time slices $(i=231,480)$ \n (2-241 ms); this is the period where the average signal is strong. \n The second interval we consider as baseline or background (B) and for\n this we choose the interval from 501 ms and ending 740 ms after the\n onset of the stimulus $(i=751,1000)$. Since the time between stimuli\n is one second our choice avoids the time just before stimulus onset,\n when anticipation and expectation is high while being as far as\n possible from the stimulus onset. \n\n\n\\section{Correlation matrix analysis}\n\n For the two time-series \n $x_{\\alpha}(t_i)$ and $x_{\\beta}(t_i)$ of the same length, $(i = 1,...,T)$\n one defines the correlation function by the relation\n \n\\begin{equation}\n C_{\\alpha,\\beta} = {\\sum_{i} (x_{\\alpha}(t_i)-\\bar{x}_{\\alpha})\n (x_{\\beta}(t_i)-\\bar{x}_{\\beta})\n \\over {\\sqrt{\\sum_{i}{(x_{\\alpha}(t_i) - \\bar{x}_{\\alpha})^2} \n \\sum_{j}{(x_{\\beta}(t_j)- \\bar{x}_{\\beta})^2}}}},\n\\label{eq:cab}\n\\end{equation}\n where $\\bar{x}$ denotes a time average over the period studied.\n For two sets of $N$ time-series $x_{\\alpha}(t_i)$ each \n $(\\alpha, \\beta=1,...,N)$\n all combinations of the elements $C_{\\alpha,\\beta}$ can be used \n as entries of the $N \\times N$ correlation matrix $\\bf C$.\n By diagonalizing $\\bf C$ \n \\begin{equation}\n {\\bf C}{\\bf v}^k = {\\lambda}_k {\\bf v}^k,\n \\label{eq:diag}\n \\end{equation}\n one obtains the eigenvalues $\\lambda_k$ $(k=1,...,N)$ \n and the corresponding eigenvectors ${\\bf v}^k = \\{v^k_{\\alpha}\\}$.\n\n In the limiting case of entirely random correlations the distribution\n $\\rho_C(\\lambda)$ is known analytically~\\cite{Edelman98} and reads:\n\n \\begin{equation}\n \\rho_C(\\lambda) = {Q \\over{2 \\pi \\sigma^2}} \n {\\sqrt{(\\lambda_{max} - \\lambda)(\\lambda - \\lambda_{min})} \\over \\lambda}\n \\label{eq:rho}\n \\end{equation} \n where \n \\begin{equation}\n \\lambda_{min}^{max} = \\sigma^2 (1 + 1/Q \\pm 2 \\sqrt{1/Q})\n \\label{eq:lambda}\n \\end{equation}\n with $\\lambda_{min} \\le \\lambda \\le \\lambda_{max}$, $Q=T/N \\ge 1$, and\n where $\\sigma^2$ is equal to the variance of the time series \n (unity in our case).\n\n For our present detailed numerical analysis we select two characteristic\n subjects (DB and FB) out of all five subjects who participated in\n the experiment. The background activity in both subjects does not\n reveal any dominant rhythm which, if present in two signals, may\n introduce additional, spontaneous correlations not related to the stimulus. \n The signals of DB reveal a relatively strong EPs and a good signal-to-noise\n ratio. FB is somehow on the other side of the spectrum of subjects,\n as its EPs are small and hardly visible and the signals are dominated\n by a high-frequency noise which results in a poor SNR. \n The signals forming pairs in eq.~(\\ref{eq:cab}) may come either from the\n same or from the opposite hemispheres. The first possibility we term\n the {\\it one-hemisphere} correlation matrix and the latter one the {\\it\n cross-hemisphere} correlation matrix. The first matrix is, by\n definition, real symmetric and the second one must be real but, in\n general, it is not symmetric.\n \n An interesting global characteristics of the dynamics encoded in $\\bf C$\n is provided by the distribution of its elements. An example for such\n a distribution is shown in Fig.~1 for the one-hemipshere correlation \n matrix. As one can see in the background region (solid lines) the \n distributions are Gaussian-like centered at zero. This implies that the\n corresponding signals are statistically independent to a large extent.\n A significantly different situation is associated with the evoked potential\n part of the signal. \n The most obvious effect is that the centre of mass of the distribution is\n shifted towards the positive values. In this respect \n there is also a difference between the subjects:\n the average value of elements for DB (approx. 0.35) is considerably\n higher than for FB (0.1). This indicates that the signals in FB are on\n average less correlated even in the EP region than the signals recorded\n from DB. This may originate from either a smaller amplitude of the\n collective response of FB's cortex or from a much smaller\n signal-to-noise ratio. For the cross-hemisphere correlation matrix\n the relevant characteristics are similar. The only difference is that\n the shifts (in both subjects) are slightly smaller.\n\n More specific properties of the\n correlation matrix can be analysed after diagonalazing $\\bf C$. \n The one-hemisphere correlation matrix \n is real and symmetric and consequently all its eigenvalues are real. \n The structure of their distribution is displayed in Fig.~2. \n The eigenvalues are shown for several characteristic cases: \n two subjects, the left and right hemispheres and two regions (EP and B).\n\n The structure of the eigenvalue spectra depends on the\n subject but first of all on the region of the signal. \n There is a clear separation of the\n largest eigenvalue from the rest of the spectrum in the EP region in DB.\n This effect is much less pronounced for FB and considerably reduced in B. \n This can be understood if we compare this result with Fig.~1.\n To a first approximation the distribution of elements in EP can be \n described as a shifted Gaussian~\\cite{Drozdz99b}: \n\n \\begin{equation}\n {\\bf C} = {\\bf G} + \\gamma {\\bf U},\n \\label{eq:gu}\n \\end{equation} \n where ${\\bf G}$ denotes a Gaussian matrix centered at zero and ${\\bf U}$\n is a matrix whose entries are all unity. $\\gamma$ is a real\n number $0 \\le \\gamma \\le 1$. Of course, the rank of ${\\bf U}$ is one and,\n therefore, the second term alone in eq.~(\\ref{eq:gu}) develops only one\n nonzero eigenvalue of magnitude $\\gamma$. Since the expansion coefficients\n of this particular state are all equal this assigns a maximum of collectivity\n to such a state. If $\\gamma$ is significantly larger than zero the structure\n of $\\bf C$ is predetermined by the second term in eq.~(\\ref{eq:gu}).\n As a result the spectrum of $\\bf C$ comprises one collective state with\n large eigenvalue. Since in this case $\\bf G$ constitutes only a 'noise'\n correction to $\\gamma {\\bf U}$ all the other states are connected \n with significantly smaller eigenvalues. In terms of the signals analysed \n here the first component of (\\ref{eq:gu}) corresponds to uncorrelated\n background activity and noise and the second one originates from the\n synchronous response of the cortex to external stimuli. \n Similar characteristics of collectivity on the level of the correlation \n matrix has recently been identified~\\cite{Drozdz99b} in correlations \n among companies on the stock market. \n\n In relation to eq.~(\\ref{eq:rho}) the presence of a strongly separated \n eigenvalue is one obvious deviation which is consistent with the non-random\n character of the corresponding eigenstate. Further deviations can be \n identified by comparing the boundaries of our calculated spectrum to\n $\\lambda_{min}^{max}$ of eq.~(\\ref{eq:lambda}). For $Q=T/N=250/120$ we\n obtain $\\lambda_{min}=0.944$ and $\\lambda_{max}=2.866$. Clearly, there\n are several eigenvalues more which are larger than $\\lambda_{max}$. This\n may indicate that the corresponding eigenstates absorb a fraction of the \n collectivity. However a closer inspection shows that also on the other\n side of the spectrum there are eigenvalues smaller than $\\lambda_{min}$\n and basically no empty strip between $0$ and $\\lambda_{min}$ can be\n seen. By this our empirical distribution seems to indicate that an\n effective $Q$ which determines this distribution is significantly smaller \n than $Q=T/N$. This, in turn, may signal that the information content in\n the time-series of length $T$ is equivalent to a significantly shorter\n time-series. This conclusion is supported \n by the time-dependence of the autocorrelation function\n calculated~\\cite{Drozdz99a} from our signals. It drops down relatively\n slowly and reaches zero only after 20-30 time-steps between consecutive\n recordings. Memory effects are present and hence neighboring recordings \n are not independent; this of course is not surprising because neural \n activity in the brain has a finite duration (and 25-30 ms is an important \n time scale) and there are plenty of time-delayed processes and interactions \n which will produce activity in neighbouring times with shared information. \n One could explicitly test whether this is a reason our calculated \n $\\rho_C(\\lambda)$ deviates from the prediction of eq.~(\\ref{eq:rho}) by \n recomputing $\\bf C$ with appropriately sparser time-series. Unfortunately, \n the number of recordings covering the EP is too small for this. \n Instead we perform the following analysis: we generate the new time-series\n $d_{\\alpha} (t_i)$ such that \n $d_{\\alpha} (t_i) = x_{\\alpha} (t_{i+1}) - x_{\\alpha}(t_i)$, i.e., the\n time-series of differences. These destroy the memory effects and now the\n autocorrelation function drops down very fast. \n Fig.~3 shows the density of eigenvalues of the correlation matrix \n generated from $d_{\\alpha} (t_i)$. Now the agreement with eq.~(\\ref{eq:rho})\n improves and becomes relatively good already when every second \n time-point $i$ from $d_{\\alpha} (t_i)$ is taken, such that the total number \n of them remains the same $(T=250)$.\n Taking more distant points, leaving out intermediate ones, drastically \n reduces the correlation between the remaining successive points. \n The above thus illustrates the subtleties \n connected with the correlation matrix analysis of time-series. \n Replacing our original time-series $x_{\\alpha} (t_i)$ by $d_{\\alpha} (t_i)$\n improves the agreement with eq.~(\\ref{eq:rho}) but at the same time the\n collective state connected with EP dissolves. This is due to disappearance\n in $d_{\\alpha} (t_i)$ of the memory effects present in \n $x_{\\alpha} (t_i)$. Therefore, in the following we return to our\n original time-series.\n\n Another statistical measure of spectral fluctuations is provided \n by the nearest-neighbor spacing distribution $P(s)$. The corresponding\n spacings $s = \\lambda_{i+1}-\\lambda_i$ \n are computed after renormalizing the eigenvalues in such a way \n that the average distance between the neighbors equals unity. \n A related procedure is known as unfolding~\\cite{Brody81,Mehta91,Drozdz91}.\n Two characteristic and typical examples of such distributions \n corresponding to EP and B regions are shown in Fig.~4 (for DB).\n While in both cases these distributions agree well with the \n Wigner distribution which corresponds to the Gaussian orthogonal ensemble\n (GOE) of random matrices, some deviations\n on the level of larger distances between neighboring states are more visible\n in the EP than in the B region. This in fact is consistent with the presence \n of larger eigenvalues in the EP case as shown in Fig.~2. \n Interestingly, the bulk of $P(s)$ even here agrees well with GOE. \n In order to further quantify the observed deviations\n we also fitted the histograms with the so-called Brody distribution \n \\begin{equation}\n P_r(s) = (1+r) a s^r \\exp (-as^{(1+r)})\n \\label{eq:brody}\n \\end{equation}\n where\n $a = [\\Gamma((2+r)/(1+r))]^{1+r}$.\n Depending on a value of the repulsion parameter $r$, this distribution\n describes the intermediate situations between the Poisson \n (no repulsion, $r=0$) and the standard Wigner $(r=1)$ distribution (GOE).\n The best fit in terms of eq.~(\\ref{eq:brody}) gives $r=0.95$ in the EP\n and $r=0.93$ in the B case, respectively. Thus we clearly see that the \n measurements share the universal properties of GOE. A departure\n betraying some collectivity is nevertheless present in both B and EP\n intervals, but even in the EP interval the effect of the stimulus does\n not change this picture significantly: it results in one or at most few\n remote distinct states in the sea of low eigenvalues of the GOE type.\n \n In order to further explore this effect we look at the distribution \n of the eigenvector components $v^k_{\\alpha}$ for the same cases as in \n Fig.~4. Fig.~5 displays such a distribution generated from eigenvectors\n associated to one hundred lowest eigenvalues (main panels of the\n Figure) calculated both for the EP (upper part) and B (lower part)\n regions. The result is a perfectly Gaussian distribution in both cases. \n However, in EP a completely different distribution (upper inset)\n corresponds to the state with the largest eigenvalue. The charactersitic \n peak located at around 0.1 documents that majority of the trials \n contribute to this eigenvector with similar strength. This eigenvector\n is thus associated with a typical behavior of many single-trial signals. \n The component values in the largest eigenvalue in B also deviate from a \n Gaussian distribution (inset in the lower part of Fig.~5) although in \n this case their distribution is largely symmetric with respect to zero. \n This makes the two $k=120$ eigenvectors in B and EP regions approximately \n orthogonal which indicates a different mechanism generating collectivity \n in these two regions. \n\n A more explicit way to visualise the differences among the eigenvectors \n is to look at the superposed signals \n \\begin{equation}\n x_{\\lambda_k} (t_i) = \\sum_{\\alpha=1}^{120} v^{k}_{\\alpha}\n x_{\\alpha} (t_i). \n \\label{eq:sup}\n \\end{equation}\n For $k=120$, 119 and 75 these are shown in Fig.~6 using the eigenvectors\n calculated for the EP (middle panel) and for B (lower panel) regions.\n The signals corresponding to the largest eigenvalues $(k=120)$ develop\n the largest amplitudes in both cases. In the first case (EP) it very closely \n resembles a simple average (upper panel) over all the trials. In the second \n case (B) long range correlations are clearly present, demonstrating that \n there is more in the signal than the short latency correlations in EP. \n The large eigenvalues in B also show a \n degree of collectivity. When signals weighted by the eigenvectors with\n the highest eigenvalue in EP and B are compared we see that there is\n essentially no amplification in the other region (i.e. in the EP\n interval when the B-weighted signals are used). This provides another\n indication that different mechanisms are responsible for the\n collectivity at these two different latency ranges. Analogous effects of\n collectivity for $k=119$ are already much weaker and disappear\n completely as an example of $k=75$ shows.\n\n We now turn to the cross-hemisphere correlation function, obtained by \n forming pairs in eq.~(\\ref{eq:cab}) from the time-series representing \n opposite hemispheres ($x_{\\alpha}^L(t_i)$ with $x_{\\beta}^R(t_i)$). \n Introducing in addition a time-lag $\\tau$ between such \n signals~\\cite{Kwapien98}, and dropping the rather obvious superscripts for \nthe left and right hemisphere, we define a delayed correlation matrix\n\n\\begin{equation}\n\\label{cf_lagged}\n C_{\\alpha,\\beta}(\\tau) = {\\sum_{i} (x_{\\alpha}(t_i)-\\bar{x}_{\\alpha})\n (x_{\\beta}(t_i+\\tau))-\\bar{x}_{\\beta})\n \\over {\\sqrt{\\sum_{i}{(x_{\\alpha}(t_i) - \\bar{x}_{\\alpha})^2}\n \\sum_{j}{(x_{\\beta}(t_j+\\tau))- \\bar{x}_{\\beta})^2}}}},\n\\ \\ \\ \\ \\ \\alpha,\\beta = 1,...,N.\n\\end{equation}\n\nA similar cross-correlation time-lag function has been employed in the past \nto investigate across trials correlations, but because of the high \ncomputational load of an exhaustive comparison across different delays \nthe analysis was restricted to the computation of the time-lagged \ncross-correlation between the average and individual single \ntrials~\\cite{Liu96}.\nThe spectral decomposition of the cross-correlation matrix provides a more \nelegant approach, requiring the solution of the $\\tau$-dependent eigenvalue \nproblem\n\n\\begin{equation}\n {\\bf C}(\\tau){\\bf v}^k(\\tau) = {\\lambda}_k(\\tau) {\\bf v}^k(\\tau),\n \\ \\ \\ \\ \\ k = 1,...,N.\n\\label{cf_lag_diag}\n\\end{equation}\nSince $\\bf C$ can now be asymmetric its eigenvalues $\\lambda_k$ can be complex \n(but forming pairs of complex conjugate values since $\\bf C$ remains real) \nand in our case they generically are complex indeed. \nOne anticipated exception may occur when similarity of the\nsignals in both hemispheres takes place for a certain value of $\\tau$.\nIn this case $\\bf C$ is dominated by its symmetric component and the effect,\nif present, is thus expected to be visible predominantly on the largest \neigenvalue. It is more likely to see this effect in the EP\nregion of the time-series. We thus calculate the cross-hemisphere\ncorrelation matrix from the $T=250$-long subintervals of\n$x_{\\alpha}^L(t_i)$ and $x_{\\beta}^R(t_i)$ covering the EPs. \nFig.~7 presents the resulting real and imaginary parts of the largest \neigenvalue as a function of $\\tau$ for \ntwo subjects and two kinds of stimulation (left and right ear).\nAs it is clearly seen the large real parts are accompanied by\nvanishing imaginary parts.\nBased on this figure several other interesting observations are to be made.\nFirst of all $\\lambda_{max}(\\tau)$ strongly depends on $\\tau$ and reaches\nits maximum for a significantly nonzero value of $\\tau$. This reflects the\nalready known fact~\\cite{Kwapien98} that the contralateral (opposite to the\nside the stimulus is delivered) hemisphere responds first and thus the \nmaximum of synchronization occurs when the signals from the opposite \nhemispheres are shifted in time relative to each other. \n(Here $\\tau > 0$ means that the signal from the right hemisphere \nis retarded relative to the left hemisphere and the opposite applies \nto $\\tau < 0$). Furthermore, the magnitude $(\\tau \\sim 10$ms) of the time-delay\nestimated from locations of the maxima agrees with an independent estimate\nbased on the mutual information~\\cite{Kwapien98}. \nEven a stronger degree of synchronization for DB relative to FB, as can be\nconcluded from a significantly larger value of $\\lambda_{max}$ in the former\ncase, agrees with this previous study. \n\nFinally, Fig.~8 shows some examples of the eigenvalue distribution on the\ncomplex plane. In the EP region the specific value of the time-delay \n$(\\tau=7$ms, upper panel) corresponds to maximum synchronization between \nthe two hemispheres for this particular subject. Here we see one strongly\nrepelled eigenvalue with a large real part $(\\sim 36.5)$ and vanishing \nimaginary part. An interesting sort of collectivity can be inferred from\nan example shown in the middle panel $(\\tau=-40$ms) of Fig.~8. \nHere the largest eigenvalue is about a factor of 3 repelled more in the \nimaginary axis direction than in the real direction. \nThis indicates that the antisymmetric part of $\\bf C$ is dominating it \nwhich expresses certain effects of antisynchronization\n(synchronization between the signals opposite in phase). \nIn the B region, on the other hand, there are basically no such effects \nof synchronization between the two hemispheres and, consequently,\nthe complex eigenvalues are distributed more or less uniformly around (0,0)\nas an example in the lowest panel of Fig.~8 shows. \n\n\\section{Conclusions}\n \n The standard application of the correlation matrix formalism is to study\n correlations among (nearly) coincident events in different \n parts of a given system. A typical principal aim of the related analysis \n is to extract a low-dimensional, non-random component which carries some \n system specific information from the whole multi-dimensional background\n activity. The advantage of the correlation matrix formalism is that\n it allows to directly relate the results to universal predictions of \n the theory of random matrices. \n The present study shows that the correlation matrix provides a \n useful tool for studying the underlying mechanism which gives rise to \n collectivity from a collection of events or signals sampled in different \n regions. The brain auditory experiment considered here is one example \n where there is a need for such an analysis. In this way we were thus \n able to quantify the nature of the background brain activity in two \n distinct periods which turns out to be \n largely consistent with the Gaussian orthogonal ensemble of random matrices, \n both in absence as well as in presence of the evoked potentials. \n The analysis also allows to compare the degree of collectivity from the \n properties of the eigenvectors with the highest eigenvalues. Crucially \n the same analysis allows also a quantification of the degree of collectivity. \n The beginnings of how the method can be extended to study correlations \n between the two sources of signals was also outlined. In this case the correlation matrix is asymmetric and results\n in complex eigenvalues. An immediate application of such an extension \n is to look at correlations among signals recorded in our experiment from\n the opposite hemispheres. Introducing in addition the time-lag between \n the signals one can study the effects of delayed synchronization between\n the two hemispheres. The quantitative \n characteristics of such synchronization remain in agreement with those\n found by other means~\\cite{Kwapien98}.\n\n\\begin{references}\n\n\\bibitem{NR1} Z.L.~L\\\"{u}, S.J.~Williamson and L.~Kaufman, Science {\\bf\n 258}, 1668(1992).\n\\bibitem{NR2} L.C.~Liu, A.A.~Ioannides and J.G.~Taylor (1998), \n NeuroReport {\\bf 9}, 2679(1998)\n\\bibitem{Liu98} L.C.~Liu, A.A.~Ioannides and H.W.~M\\\"{u}ller-G\\\"{a}rtner (1998) \n Electroenceph. Clin. Neurophysiol. {\\bf 106}, 64(1998)\n\\bibitem{Kwapien98} J.~Kwapie\\'n, S.~Dro\\.zd\\.z, L.C.~Liu and\n A.A.~Ioannides, Phys. Rev. {\\bf E58}, 6359 (1998)\n\\bibitem{Fraser86} A.M.~Fraser, and H.L.~Swinney, Phys. Rev. {\\bf A33},\n 1134 (1986)\n\\bibitem{Drozdz99a} S.~Dro\\.zd\\.z, J.~Kwapie\\'n, A.A.~Ioannides and\n L.C.~Liu, in {\\it Collective excitations in Fermi and Bose systems},\n edited by C.A.~Bertulani, L.P.~Canto and M.S.~Hussein (World Scientific,\n Singapore, 1999), pp. 62-77\n\\bibitem{Broomhead86} D.S.~Broomhead and G.P.~King, Physica {\\bf 20D},\n 217(1986)\n\\bibitem{Liu96} L.C.~Liu and A.A.~Ioannides, Brain Topogr. {\\bf 8b(4)},\n 385 (1996)\n\\bibitem{Hamalainen93} M.~H\\\"am\\\"al\\\"ainen, R.~Hari, R.J.~Ilmoniemi,\n J.~Knuutila and O.~Lounasmaa, Rev. Mod. Phys. {\\bf 65}, 413(1993)\n\\bibitem{Creutzfeldt95} O.D.~Creutzfeldt, {\\it Cortex Cerebri}, (Oxford\n University Press, Oxford, 1995)\n\\bibitem{Edelman98} A.~Edelman, SIAM J. Matrix Anal. Appl. {\\bf 9}, \n 543(1988); \\\\\n A.M. Sengupta and P.P.~Mitra, Phys. Rev. {\\bf E60}, 3389(1999)\n\\bibitem{Drozdz99b} S.~Dro\\.zd\\.z, F.~Gr\\\"ummer, F.~Ruf and J.~Speth, \n {\\it Dynamics of competition between collectivity and noise \n in the stock market}, LANL preprint , cond-mat/9911168\n\\bibitem{Brody81} T.A.~Brody, J.~Flores, J.B.~French, P.A.~Mello, A.~Panday,\n and S.S.M.~Wong, Rev. Mod. Phys. {\\bf 53}, 385 (1981)\n\\bibitem{Mehta91} M.L.~Mehta, {\\it Random Matrices} (Academic Press,\n Boston,1991)\n\\bibitem{Drozdz91} S.~Dro\\.zd\\.z and J.~Speth, Phys. Rev. Lett. {\\bf 67},\n 529(1991)\n\n\\end{references}\n\n\\newpage\n\\begin{center}\n{\\bf FIGURE CAPTIONS}\n\\end{center}\n%------------------------------------------------------------------------\n{\\bf Fig.~1.} Distributions of $C_{\\alpha,\\beta}$ for the one-hemisphere \n correlation matrix. The upper panel corresponds to DB and the lower one\n to FB. The solid lines display such distributions evaluated in the\n regions beyond evoked activity (B) and the dashed lines in the EP\n region. \\\\\n%--------------------------------------------------------------------------\n{\\bf Fig.~2.} Structure of the eigenvalue spectra of the correlation matrices\n (one-hemisphere correlations) for the two discussed regions of the\n signals (evoked potential - EP, background activity - B) for DB (upper\n part) and FB (lower part). In each panel there are two spectra of\n eigenvalues, corresponding to the right hemisphere (circles) and the\n left one (triangles). The eigenvalues are ordered from the smallest to\n the largest. \\\\\n%--------------------------------------------------------------------------\n{\\bf Fig.~3} Density of eigenvalues of the correlation matrix calculated\n from the $T=250$ points of the time-series $d_{\\alpha}(t_i)$ of\n increments of the original time-series $x_{\\alpha}(t_i)$, i.e., \n $d_{\\alpha}(t_i) = x_{\\alpha}(t_{i+1}) - x_{\\alpha}(t_i)$.\n In the lower panel every second point of $d_{\\alpha}(t_i)$ is taken but the \n number of such points is still 250. \n The dashed line corresponds to the distribution prescribed by\n eq.~(\\ref{eq:rho}). \\\\ \n%--------------------------------------------------------------------------\n{\\bf Fig.~4.} Nearest-neighbor $(s)$ spacing distribution (histogram) of the \n eigenvalues of $\\bf C$ for subject DB. The upper panel corresponds to \n the evoked potential (EP) region of the time-series and the lower panel\n to the background (B) activity part. The distributions have been created\n after unfolding the eigenvalues. The smooth solid curves illustrate the\n Wigner distribution and the dashed curves the best fit in terms of the\n Brody distribution. \\\\\n%--------------------------------------------------------------------------\n{\\bf Fig.~5.} Distribution of the eigenvector components ($v_{\\alpha}^k$) \n for EP (upper part) and B (lower part) regions (subject DB). The main\n panels correspond to one hundred lowest eigenvalues, while the insets\n show plots of the same quantity for the eigenvector corresponding to \n $\\lambda_{max}$ ($k=120$). For comparison, Gaussian best fits are\n also presented (dotted lines). (Note different scales in the Figure.) \\\\\n%--------------------------------------------------------------------------\n{\\bf Fig.~6.} The comparison of the signal obtained by simple average over\n all 120 trials (upper panel) and the signals obtained from \n eq.~(\\ref{eq:sup}) for both regions, EP (middle part) and B (lower\n part) for subject DB. Signals in the middle and lower panels denote\n superpositions for $k=120$ (solid line), $k=119$ (dashed line) and\n $k=75$ (dotted line). \\\\\n%--------------------------------------------------------------------------\n{\\bf Fig.~7.} $\\lambda_{max}(\\tau)$ calculated from the cross-hemisphere \n correlation matrix. The upper part corresponds to DB and the lower part\n to FB. Both panels illustrate two kinds of stimulation: left ear (LE) \n and right ear (RE). The solid lines denote the real part of\n $\\lambda_{max}$ while the dashed and dotted ones its imaginary part.\n The sign of $\\tau$ denotes retardation of a signal from the\n right hemisphere ($\\tau > 0$) or the left one ($\\tau < 0$). \\\\\n%--------------------------------------------------------------------------\n{\\bf Fig.~8.} Examples of the eigenvalue distribution of the\n cross-hemisphere correlation matrix for the right ear stimulation for \n DB obtained from the EP region (upper and middle panels) and the B\n region (lower panel). All parts present the distributions on the\n complex plane. The eigenvalues for $\\tau = 7$, which corresponds to the\n maximum of $\\lambda_{max}(\\tau)$ in Fig.~7, are shown in the upper panel \n and the eigenvalues for $\\tau = -40$ (corresponding to strong\n antisymmetry of {\\bf C}) are presented in the middle one. A typical\n distribution of the eigenvalues in the B region is illustrated in the\n lower part. (Note different scale in the middle panel.)\n%--------------------------------------------------------------------------\n\n\\end{document}\n\n"
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{
"name": "cond-mat0002175.extracted_bib",
"string": "\\bibitem{NR1} Z.L.~L\\\"{u}, S.J.~Williamson and L.~Kaufman, Science {\\bf\n 258}, 1668(1992).\n\n\\bibitem{NR2} L.C.~Liu, A.A.~Ioannides and J.G.~Taylor (1998), \n NeuroReport {\\bf 9}, 2679(1998)\n\n\\bibitem{Liu98} L.C.~Liu, A.A.~Ioannides and H.W.~M\\\"{u}ller-G\\\"{a}rtner (1998) \n Electroenceph. Clin. Neurophysiol. {\\bf 106}, 64(1998)\n\n\\bibitem{Kwapien98} J.~Kwapie\\'n, S.~Dro\\.zd\\.z, L.C.~Liu and\n A.A.~Ioannides, Phys. Rev. {\\bf E58}, 6359 (1998)\n\n\\bibitem{Fraser86} A.M.~Fraser, and H.L.~Swinney, Phys. Rev. {\\bf A33},\n 1134 (1986)\n\n\\bibitem{Drozdz99a} S.~Dro\\.zd\\.z, J.~Kwapie\\'n, A.A.~Ioannides and\n L.C.~Liu, in {\\it Collective excitations in Fermi and Bose systems},\n edited by C.A.~Bertulani, L.P.~Canto and M.S.~Hussein (World Scientific,\n Singapore, 1999), pp. 62-77\n\n\\bibitem{Broomhead86} D.S.~Broomhead and G.P.~King, Physica {\\bf 20D},\n 217(1986)\n\n\\bibitem{Liu96} L.C.~Liu and A.A.~Ioannides, Brain Topogr. {\\bf 8b(4)},\n 385 (1996)\n\n\\bibitem{Hamalainen93} M.~H\\\"am\\\"al\\\"ainen, R.~Hari, R.J.~Ilmoniemi,\n J.~Knuutila and O.~Lounasmaa, Rev. Mod. Phys. {\\bf 65}, 413(1993)\n\n\\bibitem{Creutzfeldt95} O.D.~Creutzfeldt, {\\it Cortex Cerebri}, (Oxford\n University Press, Oxford, 1995)\n\n\\bibitem{Edelman98} A.~Edelman, SIAM J. Matrix Anal. Appl. {\\bf 9}, \n 543(1988); \\\\\n A.M. Sengupta and P.P.~Mitra, Phys. Rev. {\\bf E60}, 3389(1999)\n\n\\bibitem{Drozdz99b} S.~Dro\\.zd\\.z, F.~Gr\\\"ummer, F.~Ruf and J.~Speth, \n {\\it Dynamics of competition between collectivity and noise \n in the stock market}, LANL preprint , cond-mat/9911168\n\n\\bibitem{Brody81} T.A.~Brody, J.~Flores, J.B.~French, P.A.~Mello, A.~Panday,\n and S.S.M.~Wong, Rev. Mod. Phys. {\\bf 53}, 385 (1981)\n\n\\bibitem{Mehta91} M.L.~Mehta, {\\it Random Matrices} (Academic Press,\n Boston,1991)\n\n\\bibitem{Drozdz91} S.~Dro\\.zd\\.z and J.~Speth, Phys. Rev. Lett. {\\bf 67},\n 529(1991)\n\n"
}
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cond-mat0002176
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A comparison between broad histogram and multicanonical methods
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[
{
"author": "A. R. Lima\\cite{arlima}"
},
{
"author": "P. M. C. de Oliveira and T. J. P. Penna"
}
] |
We discuss the conceptual differences between the Broad Histogram (BHM) and reweighting methods in general, and particularly the so-called Multicanonical (MUCA) approaches. The main difference is that BHM is based on microcanonical, fixed-energy averages which depends only on the good statistics taken {inside} each energy level. The detailed distribution of visits among different energy levels, determined by the particular dynamic rule one adopts, is irrelevant. Contrary to MUCA, where the results are extracted from the dynamic rule itself, within BHM any microcanonical dynamics could be adopted. As a numerical test, we have used both BHM and MUCA in order to obtain the spectral energy degeneracy of the Ising model in $4 \times 4 \times 4$ and $32 \times 32$ lattices, for which exact results are known. We discuss why BHM gives more accurate results than MUCA, even using {the same} Markovian sequence of states. In addition, such advantage increases for larger systems. {Key Words}: Monte Carlo Methods, Ising Model, Computational Physics
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{
"name": "compara.tex",
"string": "%\\documentstyle[12pt]{article}\n\\documentstyle[epsfig, rotate, aps, pre, preprint]{revtex}\n%\n%\n%\n\\begin{document}\n%\n\\widetext\n%\n%\n\n\\title{A comparison between broad histogram and multicanonical methods}\n\n\\author{A. R. Lima\\cite{arlima}, P. M. C. de Oliveira and T. J. P. Penna}\n\\address{Instituto de F\\'{\\i}sica, Universidade Federal Fluminense \\\\\n Av. Litor\\^anea, s/n$^o$ - 24210-340 Niter\\'oi, RJ, Brazil}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\n We discuss the conceptual differences between the Broad Histogram\n (BHM) and reweighting methods in general, and particularly the\n so-called Multicanonical (MUCA) approaches. The main difference is\n that BHM is based on microcanonical, fixed-energy averages which\n depends only on the good statistics taken {\\bf inside} each energy\n level. The detailed distribution of visits among different energy\n levels, determined by the particular dynamic rule one adopts, is\n irrelevant. Contrary to MUCA, where the results are extracted from\n the dynamic rule itself, within BHM any microcanonical dynamics could\n be adopted. As a numerical test, we have used both BHM and\n MUCA in order to obtain the spectral energy degeneracy of the\n Ising model in $4 \\times 4 \\times 4$ and $32 \\times 32$ lattices,\n for which exact results are known. We discuss why BHM gives more\n accurate results than MUCA, even using {\\bf the same} Markovian\n sequence of states. In addition, such advantage increases for larger\n systems.\n \n {\\bf Key Words}: Monte Carlo Methods, Ising Model, Computational\n Physics\n\\end{abstract}\n\n\\section{Introduction}\n%\nDevelopment of tools for optimization of computer simulations is a\nfield of great interest and activity. Cluster updating algorithms\n\\cite{coniglio,swendsen92,wolff89}, probability reweighting procedures\n\\cite{salzburg59,dickman84,swendsen93} and, more recently, methods\n\\cite{berg91,lee93,hesselbo95} that obtain directly the spectral\ndegeneracy $g(E)$ are a few examples of very successful approaches\n(for reviews of these methods see, for instance,\n\\cite{marinari96,defelicio96,newman99} and references therein). The\nMulticanonical (MUCA) and Broad Histogram (BHM) methods belong to the\nformer category. The Entropic Sampling Method (ESM) \\cite{lee93} was\nproven to be an equivalent formulation of MUCA \\cite{berg95}. From\nthe knowledge of $g(E)$, these methods allow us to obtain any\nthermodynamical quantity of interest for the system under study, as\nthe canonical average\n%\n\\begin{equation}\n\\label{canonical}\n\\langle Q\\rangle_T = {\\sum_E g(E) \\langle Q(E)\\rangle \\exp(-E/T)\n\\over \\sum_E g(E) \\exp(-E/T)}\n\\end{equation}\n%\nof some macroscopic quantity $Q$ (magnetization, density,\ncorrelations, etc). Both sums run over all allowed energies (for\ncontinuous spectra, they must be replaced by integrals), $T$ is the\nfixed temperature, and the Boltzmann constant was set to unity. The\ndegeneracy function $g(E)$ simply counts the number of states with\nenergy $E$ (which, must be interpreted as a density within a narrow\nwindow ${\\rm d}E$, for continuous spectra). Also,\n%\n\\begin{equation}\n\\label{microcanonical}\n\\langle Q(E)\\rangle = {\\sum_{S[E]} Q_S \\over g(E)}\n\\end{equation}\n%\nis the microcanonical, fixed-$E$ average of $Q$. The sum runs {\\bf\nuniformly} over all states $S$ with energy $E$ (or, again, within the\nsmall window ${\\rm d}E$, for continuous spectra). Note that neither\n$g(E)$ nor $\\langle Q(E)\\rangle$ depend on the particular environment\nthe system is actually interacting with, for instance the canonical\nheat bath represented by the exponential Boltzmann factors in eq.\n(\\ref{canonical}). Thus, these methods go far beyond the canonical\nensemble: once $g(E)$ and $\\langle Q(E)\\rangle$ were already\ndetermined for a given system, one can study its behavior under\ndifferent environments, or different ensembles, using the same $g(E)$\nand $\\langle Q(E)\\rangle$. Accordingly, as an additional advantage,\nonly one computer run is enough to evaluate the quantities of\ninterest in a large range of temperatures and other parameters.\n\nMUCA was introduced in 1991 by Berg and Neuhaus\\cite{berg91}. The\nbasic idea of the method is to sample microconfigurations performing\na biased random walk (RW) in the configuration space leading to\nanother unbiased random walk (i.e. uniform distribution) along the\nenergy axis. Thus, the visiting probability for each energy level $E$\nis inversely proportional to $g(E)$. By tuning the acceptance\nprobability of movements in order to get a uniform distribution of\nvisits along the energy axis, one is able to get $g(E)$ at the end.\nMUCA has been proven to be very useful and efficient to obtain\nresults in many different problems, such as first order phase\ntransitions, confinement/deconfinement phase transition in SU(3)\ngauge theory, relaxation paths \\cite{shteto97}, conformal studied of\npeptides, helix-coil transition and protein folding, evolutionary\nproblems \\cite{choi97} and to study phase equilibrium in binary lipid\nbilayer \\cite{besold99} (for reviews of the method see\n\\cite{mucareviews}).\n\nBHM was introduced three years ago by de Oliveira {\\em et al}\n\\cite{pmco96}. It is based on an exact relation between the spectral\ndegeneracy, $g(E)$, and the microcanonical averages of some special\nmacroscopic quantities. A remarkable feature of BHM is its\ngenerality, since these macroscopic quantities can be averaged by\ndifferent procedures\n\\cite{pmco96,pmco98a,pmco98b,pmco98c,pmco98d,pmco99,wang98,munoz98,lima99,wang99,promb,j1j2}.\nBecause it is not restricted to a rule like a biased random walk,\nmore adequate dynamics can be adopted for each different application.\nBHM has been applied to a variety of magnetic systems such as the 2D\nand 3D Ising Models (also with external fields, next nearest neighbor\ninteractions), 2D and 3D XY and Heisenberg Models, Ising Spin Glass\n\\cite{pmco96,pmco98a,pmco98b,pmco98c,pmco98d,pmco99,wang98,munoz98,lima99,wang99,promb,j1j2}\nwith accurate results in a very efficient way. Theoretically, the\nmethod can be applied to any statistical system \\cite{pmco98c}.\nAnother distinguishing feature of BHM is that its numerical results\ndo not rely on the number of visits $H(E)$ to each energy level, a\nquantity which is updated by one unit for each new visited state. On\nthe contrary, BHM is based on microcanonical averages of macroscopic\nquantities. Each visited state contributes with a macroscopic upgrade\nfor the measured quantities. Thus the numerical accuracy is much\nbetter than that obtained within all other methods, better yet for\nlarger and larger systems, considering the same computer effort.\n\nBesides the practical points described above, the most important feature of\nBHM is the following conceptual one. All reweighting methods\n\\cite{salzburg59,dickman84,swendsen93,berg91,lee93,hesselbo95} depend on the\nfinal distribution of visits $H(E)$ along the energy axis. Histogram methods\n\\cite{salzburg59,dickman84,swendsen93} adopt a canonical dynamics, getting\n$H_{T_0}(E)$ for some fixed temperature $T_0$; a new distribution $H_T(E)$ is\nthen analytically inferred for another (not simulated) temperature $T$.\nFollowing the same reasoning, one can also obtain $g(E)$ \\cite{dickman84}.\nMulticanonical approaches \\cite{berg91,lee93,hesselbo95}, on the other hand,\ntune appropriate dynamics in order to obtain a flat distribution $H(E)$. In\nboth cases, the actually implemented transition probabilities from energy\nlevel $E$ to another value $E'$ are crucial. In other words, in both cases the\nresults depend on the comparison of $H(E)$ with the neighboring $H(E')$. All\nthose reweighting methods are, thus, extremely sensitive to the particular\ndynamic rule adopted during the computer run, i.e. to the prescribed\ntransition probabilities from $E$ to $E'$.\n\nBHM is not a reweighting method. It does not perform any reweighting on the\ndistribution of visits $H(E)$. It needs only the knowledge of the\nmicrocanonical, fixed-$E$ averages of some particular macroscopic quantities.\nThe possible transitions from the current energy $E$ to other values are {\\bf\n exactly} taken into account within these quantities (see section\n\\ref{bh:section}), instead of performing a numerical measurement of the\ncorresponding probabilities during the computer run. Thus, the only important\nrole played by the actually implemented dynamic rule is to provide a good\nstatistics {\\bf within each energy level, separately}: the relative weight of\n$H(E)$ as compared to $H(E')$, i.e. the relative visitation frequency for\ndifferent energy levels, is completely irrelevant. One can even decide to\nsample more states inside a particularly important region of the energy axis\n(near the critical point, for instance) \\cite{pmco99}, instead of a flat\ndistribution. In short, any dynamic rule can be adopted within BHM, the only\nconstraint is to sample with uniform probability the various states belonging\nto the same energy level, not the relative probabilities concerning different\nenergies.\n\nIn this paper we have used the formulation of MUCA given by Lee \\cite{lee93}\ncalled Entropic Sampling, which, from now on, we call ESM. We present a\ncomparison between ESM/MUCA and BHM, focusing on both accuracy and the use of\nCPU time. We choose to start our study with the same example used in the\noriginal ESM paper by Lee \\cite{lee93}: the $4 \\times 4 \\times 4$ simple cubic\nIsing model, for which the exact energy spectrum is known \\cite{pearson82}.\nOur results show that BHM gives more accurate results than ESM/MUCA with the\nsame number of Monte Carlo steps. Despite the fact that one Monte Carlo step\nin BHM takes more CPU time, in measuring further macroscopic averages, the\noverall CPU time is smaller for the same accuracy, at least for this model.\nAlso, BHM can be applied to larger lattices without the problems faced by\nESM/MUCA, as we show in our simulations of the same model in a $32 \\times 32$\nlattice.\n\nThis paper is structured as follows: in section \\ref{es:section} and\n\\ref{bh:section} we review the implementation of ESM/MUCA and BHM (including a\ndetailed description of the distinct dynamics adopted in this work). In\nsection \\ref{nt:section} our numerical tests are presented and discussed.\nConclusions are in section \\ref{conc:section}.\n\n\\section{The Multicanonical Method}\n\\label{es:section}\n\nThe idea of the Multicanonical method is to obtain the spectral\ndegeneracy of a given system using a biased RW in the configuration\nspace \\cite{berg91,lee93}. The transition probability between two\nstates $X_{\\rm old}$ and $X_{\\rm new}$ is given by\n%\n\\begin{equation}\n\\label{estransition}\n\\tau(X_{\\rm old}, X_{\\rm new}) = e^{-[S(E(X_{\\rm new}))-S(E(X_{\\rm\nold}))]} = \\frac{g(E_{\\rm old})}{g(E_{\\rm new})}\n\\end{equation}\n%\nwhere $S(E(X))=\\ln g(E)$ is the entropy and $E(X)$ is the energy of\nstate $X$. The transitional probability (\\ref{estransition})\nsatisfies a detailed balance equation and leads to a distribution of\nprobabilities where a state is sampled with probability $\\propto\n1/g(E)$. The successive visitations along the energy axis follow a\nuniform distribution. However, $g(E)$ is not known, {\\em a priori}.\nIn order to obtain $g(E)$, Lee \\cite{lee93} proposes the following\nalgorithm:\n\n{\\em Step 1:} Start with $S(E)=0$ for all states;\n\n{\\em Step 2:} Perform a few unbiased RW steps in the configuration space and\nstore $T(E)$, the number of tossed movements to each energy $E$\n(in this stage, $T(E) = H(E)$ because all movements are accepted);\n\n{\\em Step 3:} Update $S(E)$ according to\n\n\\begin{equation}\n\\label{esentropy}\nS(E) = \\left\\{ \\begin{array}{ll}\n S(E) + \\ln T(E) &\\mbox{, if $T(E) \\neq 0$} \\\\\n S(E) &\\mbox{, otherwise.}\n \\end{array}\n \\right. \n\\end{equation}\n\n{\\em Step 4:} Perform a much longer MC run using the transitional probability\ngiven by eq. (\\ref{estransition}), storing $T(E)$.\n\n{\\em Step 5:} Repeat 3 and 4. This is considered one iteration.\n\nThis implementation is known to be quite sensitive to the lengths of\nthe MC runs in steps 2 and 4. In section \\ref{nt:section} we study, in two\nexamples, how the accuracy depends on the total number and size of each\niteration.\n\n\\section{The Broad Histogram Method}\n\\label{bh:section}\n\nBHM \\cite{pmco96} enables us to directly calculate the energy\nspectrum $g(E)$, without any need for a particular choice of the\ndynamics to be used \\cite{pmco98a}. Many distinct dynamic rules could\nbe used, and indeed some were already tested\n\\cite{pmco96,pmco98a,pmco98b,pmco98c,pmco98d,pmco99,wang98,munoz98,lima99,wang99,promb,j1j2}.\n\nWithin BHM, the energy degeneracy is calculated through the following steps\n(alternatively, other quantities could replace $E$):\n\n{\\em Step 1}: Choice of a reversible protocol of allowed movements in the\nstate space. Reversible means simply that for each allowed movement $X_{\\rm\n old}\\rightarrow X_{\\rm new}$ the back movement $X_{\\rm new} \\rightarrow\nX_{\\rm old}$ is also allowed. It is important to note that these movements are\nvirtual, since they are not actually performed. In this work we take the\nflips of one single spin as the protocol of movements;\n\n{\\em Step 2}: For a configuration $X$, to compute $N(X,\\Delta E)$,\nthe number of possible movements that change the energy $E(X)$ by a\ngiven amount $\\Delta E$. Therefore $g(E) \\langle N(E, \\Delta\nE)\\rangle$ is the total number of movements between energy levels $E$\nand $E+\\Delta E$, according to the definition (\\ref{microcanonical})\nof microcanonical averages;\n\n{\\em Step 3}: Since the total number of possible movements from level\n$E+\\Delta E$ to level $E$ is equal to the total number of possible movements\nfrom level $E$ to level $E+\\Delta E$ (step 1, above), we can\nwrite down the equation \\cite{pmco96}\n%\n\\begin{equation}\n\\label{bhrel}\ng(E)\\langle N(E, \\Delta E)\\rangle = g(E+\\Delta E)\\langle\nN(E+\\Delta E, -\\Delta E)\\rangle .\n\\end{equation}\n%\nThe relation above is exact for any statistical model and energy spectrum\n\\cite{pmco98c}. It can be rewritten as\n\\begin{equation}\n\\ln g(E+\\Delta E) - \\ln g(E) = \\ln \\frac{\\langle N(E, \\Delta\n E)\\rangle}{\\langle N(E+\\Delta E, -\\Delta E)\\rangle}\n\\end{equation}\nThis equation can be easily solved for all values of $E$, after $\\langle\nN(E,\\Delta E) \\rangle$ is obtained by any procedure, determining $g(E)$ along\nthe whole energy axis. In cases where $\\Delta E$ can assume more than one\nvalue, eq.~(\\ref{bhrel}) becomes an overdetermined system of equations.\nHowever, the spectral degeneracy can be obtained without need of solving all\nequations simultaneously, since the spectral degeneracy is the same for all\nvalues of $\\Delta E$.\n\nThe exact Broad Histogram relation (\\ref{bhrel}) is independent of the\nprocedure by which $\\langle N(E, \\Delta E)\\rangle$ is obtained\n\\cite{pmco98a,pmco98b,pmco98c,pmco98d,pmco99,wang98}. Therefore, virtually\nany procedure can be adopted in this task, for instance, an unbiased energy RW\n\\cite{pmco96}, a microcanonical simulation \\cite{pmco98d}, or a mixture of\nboth \\cite{pmco98c,pmco99}. Even the juxtaposition of histograms obtained\nthrough canonical simulations at different temperatures, a completely\nunphysical procedure, could be used, as in some of the results presented in\n\\cite{pmco98a}, and explicitly used in \\cite{wang99} where BHM is\nre-formulated under a transition matrix \\cite{smith} approach. Here, we are\ngoing to introduce an alternative procedure, referred as Entropic\nSampling-based Dynamics for BHM (ESDYN, hereafter). First, one implements ESM,\nas described in section II, in order to perform the visitation in the\nconfiguration space. Additionally, for each visited state $X$, we store the\nvalues of $N(X, \\Delta E)$ cumulatively into $E-$histograms. Therefore, at the\nend we have two choices for the determination of the spectral degeneracy,\neither by using the entropy accumulated through $T(E)$ (that is the\ntraditional ESM/MUCA) or by using the accumulated $\\langle N(E, \\Delta E)\n\\rangle$ and the BHM relation, eq.(\\ref{bhrel}). Because of this special\nimplementation, we can guarantee that exactly the same states are visited for\nboth methods. Hence, the eventual difference in the performances reported in\nthis work must be credited to the methods themselves and not to purely\nstatistical factors.\n\nThe dynamic rule originally used in order to test BHM \\cite{pmco96}\nprescribes an acceptance probability $p = \\langle N(E+\\Delta\nE,-\\Delta E)\\rangle / \\langle N(E,\\Delta E)\\rangle$. Both the\nnumerator and the denominator are read from the currently accumulated\nhistograms, and thus $p$ varies during the simulation. Wang\n\\cite{wang98} has proposed a new approach: instead of using the\ndynamically updated values of $\\langle N(E, \\Delta E) \\rangle$ as in\n\\cite{pmco96}, the transitional probabilities follow a previously\nobtained (from a canonical simulation, for example) distribution\n$\\langle N_{\\rm fixed}(E,\\Delta E )\\rangle$, kept fixed during the\nsimulation. An alternative and simpler derivation of Wang's dynamics\ncan be done by using the BHM relation (\\ref{bhrel}) itself.\n>From this, we readily obtain that Wang's dynamics is the same as\nusing the transitional probability $p = g_{\\rm fixed}(E)/g_{\\rm\nfixed}(E+\\Delta E)$ with approximated values $g_{\\rm fixed}(E)$\nkept fixed during the simulation. We refer to this dynamics as\napproximated Wang's dynamics, since $\\langle N_{\\rm fixed}(E,\\Delta\nE)\\rangle$ (or $g_{\\rm fixed}(E)$) is actually only an approximation\nof the real $\\langle N(E,\\Delta E)\\rangle$ (or $g(E)$). We will use\nthe results obtained by ESM or BHM with ESDYN as inputs to the\napproximate Wang's dynamics. These dynamics will be called AWANG1 and\nAWANG2, respectively.\n\nFor comparison, we also implemented a dynamics that uses the ESM\nprobabilities taken from the exact values of $g(E)$. It is worth\nnoticing that, as pointed out in the previous paragraph, this is\nequivalent to Wang's proposal with exact values for $\\langle N_{\\rm\nfixed}(E,\\Delta E)\\rangle$. We refer to this dynamics as WANG. Its\npurpose is only to test the accuracies of the other approaches, once\none does not know the exact $g(E)$ a priori, in real implementations.\n\n\\section{Numerical Tests}\n\\label{nt:section}\n\nWe start our comparison with the smallest system, since it is also\npresent in the ESM original paper by Lee \\cite{lee93}. The partition\nfunction for the $4 \\times 4 \\times 4$ simple cubic Ising model is\nexactly known \\cite{pearson82}. It is given by the polynomial\nfunction\n\\begin{equation}\nZ(\\beta)=\\sum_{n=0}^{96}{C(n)u^n}\n\\end{equation}\n%\nwhere $u=\\exp(-4 \\beta )$, $\\beta = 1/T$, $T$ is the temperature. The\nenergy spectrum is written in terms of the coefficients $C(n)$ as\n$g(E)=g(2n)=C(n)$ for $n=0$ to $n=96$. For this model, only the first\n$49$ coefficients are necessary by symmetry. The other $48$\ncoefficients are mirror images of the first ones. Our results will\nbe expressed in terms of $S(E)=\\ln g(E)$.\n\nIn order to compare the entropies as obtained by both ESM and\nBHM, we normalize the entropy such that $g(96)=1$, i.e. $S(96)=0$.\nThis point corresponds to the center of the energy\nspectrum, or, alternatively, to infinite temperature. Of course, the\nerror relative to the exact value vanishes for $E=96$. In fig. (1), we\ncompare the normalized (with respect to its exact value) entropy as\nfunction of $E$ obtained by ESM and BHM (with four different\ndynamics). In AWANG1 and AWANG2 dynamics we use as $\\langle N_{\\rm\nfixed}(E,\\Delta E )\\rangle$ the results obtained by ESM and BHM with\nESDYN, respectively. BHM with the entropic\nsampling dynamics or using the exact relation proposed by Wang present\nerrors within the same order of magnitude, while pure ESM gives the\nworst results, as clearly seen in the inset. It is also clear that\nAWANG1 and AWANG2 results are worse than BHM with ESDYN.\n\n The methods can be better compared by the ratio of their\nrelative errors rather than their absolute values. We define the\nrelative errors in the entropy, for a given energy $E$, as\n\\begin{equation}\n\\epsilon(E) = \\left\\vert\n \\frac{ S(E)-S(E)_{\\rm exact} }{S(E)_{\\rm exact}}\n \\right\\vert\n\\end{equation}\n\nIn fig. (2) we show the ratio between the BHM relative errors (obtained by\nESDYN, AWANG1 and AWANG2 dynamics) and the ESM ones. The inset shows the\nratio between the errors from BHM with WANG and the ESM ones. In its worst\nperformance, the error obtained by BHM with ESDYN is roughly one third of\nthe one obtained by ESM. BHM with Wang's exact dynamics gives slightly\nbetter results once it uses the exact $g(E)$ as input in order to get $g(E)$\nas output. BHM with ESDYN gives on average results $11$ times more\naccurate than ESM, for\nthis number of iterations. However, BHM with the AWANG1 and AWANG2\ndynamics give relatively poorer results. Therefore, we have shown that the\nES dynamics is a powerful approach (among other possibilities\n\\cite{pmco96,pmco98a,pmco98b,pmco98c,pmco98d,pmco99,wang98,munoz98,lima99,wang99})\nto obtain with great accuracy the microcanonical averages\n$\\langle N(E,\\Delta E )\\rangle$ needed by BHM. Let us stress that the\nresults for\n$g(E)$ obtained with AWANG2 are worse than the input they use, namely the\nvalues of $g(E)$ obtained as output of BHM with ESDYN. In order to obtain good\nresults with Wang's dynamics, we need a pretty good estimative of $g(E)$ as\ninput, and not only some crude estimation as claimed in \\cite{wang98}.\n\n In fig. (3), we show the time evolution of the mean error as\na function of the number of Monte Carlo steps (MCS). One iteration in\nESM corresponds to perform many RW steps, according to\neq.~(\\ref{estransition}), storing the number of visits at each energy\nand, after this fixed number of RW steps, to update the entropy,\naccording to eq.~(\\ref{esentropy}). As we pointed out before, the ESM\nperformance is quite sensitive to the choice of the number of RW\nsteps before each entropy update. For a fixed number of MCS ($10^6$), we\nplot the time evolution for both ESM and BHM, but with different\nnumber of iterations. As one can see in fig.(3), the best results\nfor ESM correspond to the smaller number of iterations since, in\nthis case, more RW steps are performed and, consequently, we have\nbetter statistics for the determination of the entropy.\n\nIt is worth noticing that the ESM/MUCA errors seem to stabilize after a\nfew iterations. We believe that this effect occurs because the number of\nvisits to each energy is not sufficient to provide a good statistics in\ndetermining the entropy. The only way to improve the accuracy in the\nspectral degeneracy is to perform more RW steps between each entropy\nupdate (it does not increase the CPU time if the total number of MCS is\nkept constant). Conversely, for BHM the error decreases monotonically\nbecause macroscopic quantities are stored, leading to a good statistics\neven if some states are not frequently visited. After the very first\nsteps, the errors decay as $t^{-1/2}$, as expected. It is also remarkable\nthe accuracy when using BHM: we reach the same accuracy of the best\nperformance of ESM/MUCA, by performing roughly 30 times less MCS.\n\nOf course, accuracy is not the only important factor concerning the efficiency\nof a computational method. BHM with ES dynamics has one additional step\ncompared to the traditional implementation of ESM, namely the storage of\nthe macroscopic quantities $N(X,\\Delta E)$. So, we need to know the cost\nof this additional step. In table I, we show the CPU time (in seconds)\nspent in both implementations on a 433 MHz DEC Alpha, in order to obtain\nthe results shown in fig.(1). For direct comparison, we also present the\nCPU time relative to the ESM CPU time. All dynamics tested within BHM\ntakes roughly the same CPU time. As one can see, BHM uses twice more CPU\ntime than ESM. However, for the same number of steps, and the best\nstrategy for ESM, BHM is at least 10 times more accurate. If we consider\naccuracy and CPU use, we argue that BHM is more efficient than ESM.\n\nUp to now, we have tested both approaches in a very small lattice.\nNevertheless, we can show that BHM is even more efficient for larger\nsystems, as expected due to the macroscopic character of the\nquantities $N(X,\\Delta E)$. For $L^d$ Ising spins on a lattice, for\ninstance, even restricting the allowed movements only to single-spin\nflips, the total number of movements starting from $X$ is just $L^d$.\nThus, being a finite fraction of them, $N(X,\\Delta E)$ is a\nmacroscopic quantity (this is true along the whole energy axis,\nexcept at the ground state where $g(E)$ presents a macroscopic jump\nrelative to the neighboring energy levels). In fig. (4) we present\nresults for the time evolution of the mean error for a $32 \\times 32$\nsquare lattice Ising model (a lattice that is 16 times larger than\nthe one in the previous results). The exact solution for this system\nis also known \\cite{beale96}. Here, the accuracy of BHM is two orders\nof magnitude higher than that of ESM. Again the errors decrease as\n$t^{-1/2}$ for BHM with ESDYN, while the ESM mean errors seem to\nstabilize. In summary, BHM with ESDYN can obtain accurate results for\nlattices much larger than the ones considered as limit for ESM.\n\n\\section{Conclusions}\n\\label{conc:section}\n\n The Multicanonical \\cite{berg91,lee93} and the Broad\nHistogram \\cite{pmco96} methods are completely distinct from canonical\nMonte Carlo methods, once they focus on the determination of the\nenergy spectrum degeneracy $g(E)$. This quantity is independent of\nthermodynamic concepts and depends only on the particular system under\nstudy. It does not depend on the interactions of the system with the\nenvironment. Thus, once one has determined $g(E)$, the effects of\ndifferent environments can be studied using always the same data for\n$g(E)$. Different temperatures, for instance, can be studied without\nneed of a new computer run for each $T$. \n\nThe goal of this paper is to discuss the conceptual differences\nbetween Multicanonical and Broad Histogram frameworks, and to compare\nboth methods concerning accuracy and speed. We obtained the energy\nspectrum of the Ising model in $4 \\times 4 \\times 4$ and $32 \\times\n32$ lattices, for which the exact results are known and, therefore,\nprovide a good basis for comparison. Our findings show that a\ncombination of the Broad Histogram method and the Entropic Sampling\nrandom walk dynamics (BHM with ESDYN) gives very accurate results\nand, in addition, it needs much less Monte Carlo steps to obtain the\nsame accuracy as the pure Entropic Sampling method. This advantage of\nthe Broad Histogram method grows with the system size, and it does\nnot present the limitations of the Multicanonical or Entropic\nSampling methods concerning large systems.\n\n The reason for the better performance is that the BHM\n\\cite{pmco96,pmco98a,pmco98b,pmco98c,pmco98d,pmco99,wang98,munoz98,lima99,wang99,promb,j1j2}\nuses the microcanonical averages $\\langle N(E,\\Delta E) \\rangle$\n\\cite{pmco96} of the macroscopic quantity $N(X,\\Delta E)$ - the number of\npotential movements which could be done starting from the current\nstate $X$, leading to an energy variation of $\\Delta E$. In\nthis way, each new visited state contributes with a macroscopic value\nfor the averages one measures during the computer simulation. Being\nmacroscopic quantities, the larger the system, the more accurate are\nthe results for these averages. Conversely, Histogram\n\\cite{salzburg59,dickman84,swendsen93} and Multicanonical\n\\cite{berg91,lee93,hesselbo95} approaches rely exclusively on $H(E)$,\nthe number of visits to each energy. Therefore, each new averaging\nstate contributes with only one more count to the averages being\nmeasured, i.e., $H(E)\\rightarrow H(E)+1$, independent of the system\nsize.\n\nUnder a conceptual point of view, BHM is also completely distinct from\nthe other methods which are based on the final distribution of visits\n$H(E)$. Alternatively, it is based on the determination of microcanonical,\nfixed-$E$ averages $\\langle N(E,\\Delta E )\\rangle$ \\cite{pmco96},\nconcerning each energy level separately. Thus, the relative\nfrequency of visitation between distinct energy levels, which is\nsensitive to the particular dynamic rule one adopts, i.e. the comparison\nbetween $H(E)$ and $H(E')$, does not matter. The only requirement for the\ndynamics is to provide a uniform sampling probability for the states\nbelonging to the same energy level. The transition probabilities from one\nlevel to the others are irrelevant.\n\n\\section{Acknowledgments}\n\n This work has been partially supported by Brazilian agencies\nCAPES, CNPq and FAPERJ. The authors acknowledge D. C.~Marcucci and J.\nS. S\\'a Martins for suggestions, discussions and critical readings of\nthe manuscript.\n\n\\begin{thebibliography}{0}\n\\bibitem[*]{arlima} E-mail: arlima@if.uff.br\n\n\\bibitem{coniglio} Coniglio A. and Klein W., \\emph{J. Phys.} \\textbf{A13},\n 2775 (1980).\n \n\\bibitem{swendsen92} Swendsen R. H., J. -S. Wang and Ferrenberg A. M., in {\\em\n The Monte Carlo Method in Condensed Matter Physics}, ed. K. Binder,\n (Springer, Berlin), Topics in Applied Physics Vol 71 p. 75 (1992), and\n references therein.\n \n\\bibitem{wolff89} Wolff U., {\\em Phys. Rev. Lett.} {\\bf 62}, 361 (1989).\n \n\\bibitem{salzburg59} Salzburg Z.W., Jacobson J.D. Fickett W. and Wood W.W.,\n {\\em J. Chem. Phys.}, {\\bf 30}, 65 (1959).\n \n\\bibitem{dickman84} Dickman R. and Schieve W.C., {\\em J. de Physique}\n {\\bf 45}, 1727 (1984).\n \n\\bibitem{swendsen93} Swendsen R.W., {\\em Physica} {\\bf A194}, 53 (1993),\nand references therein.\n\n\\bibitem{berg91} Berg B. A. and Neuhaus T., {\\em Phys. Lett.}\n{\\bf B267}, 249 (1991); \\\\\nBerg B. A. {\\em Int. J. Mod. Phys.} {\\bf C3}, 1083 (1992).\n\n\\bibitem{lee93} Lee J., {\\em Phys. Rev. Lett.} {\\bf 71}, 211 (1993).\n\n\\bibitem{hesselbo95} Hesselbo B. and Stinchcombe R.B., {\\em Phys.\nRev. Lett.} {\\bf 74}, 2151 (1995).\n \n\\bibitem{marinari96} Marinari E. {\\em Optimized Monte Carlo Methods}.\n Lectures given at the 1996 Budapest Summer School on Monte Carlo Methods,\n pag. 50, edited by J. Kert\\'esz and I. Kondor, Springer, Berlin-Heidelberg\n (1998). Available at {\\bf cond-mat/9612010}.\n \n\\bibitem{defelicio96} de Fel\\'{\\i}cio J. R. D. and L\\'{\\i}bero V. L., {\\em Am.\n J. Phys.} {\\bf 64}, 1281 (1996).\n\n\\bibitem{newman99} Newman M.E.J. and Barkema G.T., {\\em Monte Carlo\nMethods in Statistical Physics}, (Oxford, New York) (1999).\n \n\\bibitem{berg95} Berg B.A., Hansmann U.H.E. and Okamoto Y., {\\em\nJ. Phys. Chem.} {\\bf 99}, 2236 (1995).\n\n\\bibitem{shteto97} Shteto I., Linares J. and Varret F., {\\em Phys. Rev.} {\\bf\n E56}, 5128 (1997).\n\n\\bibitem{choi97} Choi M. Y., Lee H. Y. and Park S. H., {\\em J. Phys.} {\\bf\n A30}, L748 (1997).\n\n\\bibitem{besold99} Besold G., Risbo J. and Mouritsen O.G., {\\em\nComput. Mat. Science} {\\bf 15}, 311 (1999).\n\n\\bibitem{mucareviews} Berg B.A., in: Karsch F., Monien B. and Satz\nH. (Eds.), {\\sl Proceedings of the International Conference on Multiscale\nPhenomena and their Simulations} (Bielefeld, October 1996). World\nScientific, Singapore (1997);\\\\ Janke W., {\\em Physica} {\\bf A254},\n164 (1998);\\\\ D\\\"unweg B., in Binder K. and Ciccotti C. (Eds.), {\\sl Monte\nCarlo and Molecular Dynamics of Condensed Matter Physics} (Como, July\n1995). Societ\\'a Italiana di Fisica, Bologna, p. 215 (1996);\\\\ Berg\nB.A, Hansman U.H.E. and Heuhaus T, {\\em Phys. Rev.} {\\bf B47}, 497\n(1993).\n\n\\bibitem{pmco96} de Oliveira P. M. C., Penna T. J. P. and Herrmann H.J.,\n {\\em Braz. J. Phys.} {\\bf 26}, 677 (1996). Available at {\\bf\n cond-mat/9610041}. \n\n\\bibitem{pmco98a} de Oliveira P. M. C., Penna T. J. P. and Herrmann H.J.,\n {\\em Eur. Phys. J.} {\\bf B1}, 205 (1998).\n \n\\bibitem{pmco98b} de Oliveira P. M. C., in {\\sl Computer Simulation\n Studies in Condensed-Matter Physics XI}, edited by D.P. Landau,\n Springer, Berlin, 169 (1998).\n\n\\bibitem{pmco98c} de Oliveira P. M. C., {\\em Eur. Phys. J.} {\\bf B6}, 111\n(1998). Available at {\\bf cond-mat/9807354}.\n\n\\bibitem{pmco98d} de Oliveira P. M. C., {\\em Int. J. Mod. Phys.} {\\bf C9}, 497 \n (1998).\n \n\\bibitem{pmco99} de Oliveira P. M. C., CCP\n 1998 conference proceedings; {\\it Comp. Phys. Comm.} {\\bf 121-122},\n 16 (1999).\n \n\\bibitem{wang98} Wang J.-S., {\\it Eur. Phys. J.} {\\bf B8}, 287-291\n (1999); \\\\\n ``Monte Carlo algorithms based on the number of potential\n moves'' in {\\bf cond-mat/9903224}.\n \n\\bibitem{munoz98} Munoz J. D. and Herrmann H. J., {\\em Int. J. Mod. Phys.}\n {\\bf C10}, 95 (1999); \\\\\n CCP 1998 conference proceedings, {\\it Comp. Phys. Commun.}, also in\n {\\sl Computer Simulation Studies in Condensed-Matter Physics XII},\n edited by\n D.P. Landau (Springer,Berlin, in print) and \\textbf{cond-mat/9810292}.\n \n\\bibitem{lima99} Lima A. R., S\\'a Martins J. S. and Penna T. J. P., {\\em\n Physica} {\\bf A268}, 553 (1999).\n\n\\bibitem{wang99} Wang J. S., Tay T. K. and Swendsen R. H., {\\em Phys. Rev.\n Lett.} {\\bf82}, 476 (1999).\n\n\\bibitem{promb} Kastner M., Promberger M. and Munoz J. D., ``Broad\n Histogram Method: Extension and Efficiency Test'' in {\\bf\n cond-mat/9906097}.\n\n\\bibitem{j1j2} Lima A. R., de Oliveira P. M. C. and Penna T. J. P.,\n {\\em Solid State Comm.} (2000). Available at {\\bf cond-mat/9912152}. \n\n\\bibitem{pearson82} Pearson R. B., {\\em Phys. Rev.} {\\bf B26}, 6285 (1982).\n\n\\bibitem{smith} Smith G. R. and Bruce A. D., {\\em J. Phys.} {\\bf A28},\n 6623 (1995); \\\\\n {\\em Phys. Rev.} {\\bf E53}, 6530 (1996); \\\\\n {\\em Europhysics Lett.} {\\bf 39}, 91 (1996).\n\n\n\\bibitem{beale96} Beale P. D., {\\em Phys. Rev. Lett.} {\\bf 76}, 78 (1996).\n\n\\end{thebibliography}\n\n\n\n\\begin{table*}[!h]\n \\label{table:cputime}\n \\begin{center}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n Method & Dynamics & {\\begin{tabular}{c} CPU \\\\ Time(s)\n \\end{tabular}} & $\\frac{t}{t_{\\rm ESM}}$ &\n $\\frac{1}{\\langle\\epsilon(E)/\\epsilon(E_{\\rm ESM})\\rangle}$\\\\\n \\hline\n ESM & ESDYN & 4181 & 1 & 1 \\\\\n \\hline\n BHM & AWANG1 & 9360 & 2.24 & 2.94 \\\\\n & AWANG2 & 9772 & 2.33 & 4.75 \\\\\n & ESDYN & 8754 & 2.09 & 10.68 \\\\ \n \\hline\n \\end{tabular}\n \\end{center}\n \\caption{CPU time for each method and dynamics. ESM/MUCA is at least\n twice faster than BHM with the various dynamics used in this work.\n However, BHM is more than 10 times more accurate than the\n multicanonical approach. For AWANG1 dynamics, one must add the\n previous ESM time spent in determining $\\langle N_{\\rm\n fixed}(E,\\Delta E )\\rangle$, i.e. 13541s. Analogously, AWANG2 spent a\n total time of 18526s. The simulations were carried out on a 433MHz DEC\n Alpha. }\n\\end{table*}\n\n\\centerline{FIGURE CAPTIONS}\n\n\\baselineskip=22pt\n\n\\vspace{1cm}\n\n\\begin{itemize}\n\n\\item[FIG. 1 - ] Entropies (normalized by its exact values) for the\n $4 \\times 4 \\times 4$ Ising ferromagnet, obtained\n by ESM/MUCA and BHM for $100$ iterations of $10^6$ Monte Carlo steps,\n each. The inset shows a detailed view of the first fourth of the whole\n spectrum. ESM/MUCA gives the largest errors. AWANG1 and AWANG2\n dynamics also give worse results than BHM with both ES or WANG dynamics.\n\n\\item[FIG. 2 - ] Ratio between the relative errors for BHM and\n the ESM/MUCA ones. The horizontal lines are the mean relative errors.\n BHM is on average more than ten times more accurate than\n ESM/MUCA, for this number of iterations. The exact Wang's dynamics\n is, as expected, slightly more accurate than ESDYN (as shown in\n the inset) once it uses the exact $g(E)$ as input in order to get\n $g(E)$ as output. AWANG1 and AWANG2 give the worst results among\n the four distinct dynamics presently used in order to test BHM\n (see text).\n \n\\item[FIG. 3 - ] Time evolution of the mean error for BHM with ESDYN (a)\n and ESM/MUCA (b). In both cases, exactly the same averaging states are\n visited: thus, the differences are due to the methods themselves,\n not to statistics. As described in the text, the entropic sampling\n dynamics is quite sensitive to the number of Monte Carlo steps\n between each iteration, that correspond to an update of the entropy.\n Here, we present the results for $N=10,100,1000$ iterations, that\n means $10^5,10^4$ and $10^3$ Monte Carlo steps, respectively,\n between each iteration. The more Monte Carlo steps, the smaller is\n the error for the entropic sampling. Conversely BHM does not\n seem to depend on the computational strategy adopted. Moreover, the\n errors seem to stabilize after some steps, in the ESM case. On the\n contrary, for BHM, to get a better accuracy is a simple matter of\n increasing the computer time, once the errors decay as $t^{-1/2}$.\n The results are averaged over ten realizations.\n \n\\item[FIG. 4 - ] Time evolution of the mean error for BHM with ESDYN (a)\n and ESM/MUCA (b) for the $32 \\times 32$ square lattice Ising Model. Now\n we consider $10^7$ MC steps. Again we present the results for\n $N=100,1000$ iterations, that means $10^5$ and $10^4$ Monte Carlo\n steps, respectively, between each iteration. The results are\n averaged over ten realizations. The ratio between the accuracies of\n BHM and ESM/MUCA is even higher for large systems.\n\\end{itemize}\n\n\n\\pagestyle{empty}\n\\newpage\n\\begin{figure}\n\\centerline{\\epsfig{file=comp1.epsi,height=20cm}}\n\\end{figure}\n\\newpage\n\\begin{figure}\n\\centerline{\\epsfig{file=comp2.epsi,height=20cm}}\n\\end{figure}\n\\newpage\n\\begin{figure}\n\\centerline{\\epsfig{file=comp3.epsi,height=20cm}}\n\\end{figure}\n\\newpage\n\\begin{figure}\n\\centerline{\\epsfig{file=comp4.epsi,height=20cm}}\n\\end{figure}\n\n\\end{document}\n\n\n"
}
] |
[
{
"name": "cond-mat0002176.extracted_bib",
"string": "\\begin{thebibliography}{0}\n\\bibitem[*]{arlima} E-mail: arlima@if.uff.br\n\n\\bibitem{coniglio} Coniglio A. and Klein W., \\emph{J. Phys.} \\textbf{A13},\n 2775 (1980).\n \n\\bibitem{swendsen92} Swendsen R. H., J. -S. Wang and Ferrenberg A. M., in {\\em\n The Monte Carlo Method in Condensed Matter Physics}, ed. K. Binder,\n (Springer, Berlin), Topics in Applied Physics Vol 71 p. 75 (1992), and\n references therein.\n \n\\bibitem{wolff89} Wolff U., {\\em Phys. Rev. Lett.} {\\bf 62}, 361 (1989).\n \n\\bibitem{salzburg59} Salzburg Z.W., Jacobson J.D. Fickett W. and Wood W.W.,\n {\\em J. Chem. Phys.}, {\\bf 30}, 65 (1959).\n \n\\bibitem{dickman84} Dickman R. and Schieve W.C., {\\em J. de Physique}\n {\\bf 45}, 1727 (1984).\n \n\\bibitem{swendsen93} Swendsen R.W., {\\em Physica} {\\bf A194}, 53 (1993),\nand references therein.\n\n\\bibitem{berg91} Berg B. A. and Neuhaus T., {\\em Phys. Lett.}\n{\\bf B267}, 249 (1991); \\\\\nBerg B. A. {\\em Int. J. Mod. Phys.} {\\bf C3}, 1083 (1992).\n\n\\bibitem{lee93} Lee J., {\\em Phys. Rev. Lett.} {\\bf 71}, 211 (1993).\n\n\\bibitem{hesselbo95} Hesselbo B. and Stinchcombe R.B., {\\em Phys.\nRev. Lett.} {\\bf 74}, 2151 (1995).\n \n\\bibitem{marinari96} Marinari E. {\\em Optimized Monte Carlo Methods}.\n Lectures given at the 1996 Budapest Summer School on Monte Carlo Methods,\n pag. 50, edited by J. Kert\\'esz and I. Kondor, Springer, Berlin-Heidelberg\n (1998). Available at {\\bf cond-mat/9612010}.\n \n\\bibitem{defelicio96} de Fel\\'{\\i}cio J. R. D. and L\\'{\\i}bero V. L., {\\em Am.\n J. Phys.} {\\bf 64}, 1281 (1996).\n\n\\bibitem{newman99} Newman M.E.J. and Barkema G.T., {\\em Monte Carlo\nMethods in Statistical Physics}, (Oxford, New York) (1999).\n \n\\bibitem{berg95} Berg B.A., Hansmann U.H.E. and Okamoto Y., {\\em\nJ. Phys. Chem.} {\\bf 99}, 2236 (1995).\n\n\\bibitem{shteto97} Shteto I., Linares J. and Varret F., {\\em Phys. Rev.} {\\bf\n E56}, 5128 (1997).\n\n\\bibitem{choi97} Choi M. Y., Lee H. Y. and Park S. H., {\\em J. Phys.} {\\bf\n A30}, L748 (1997).\n\n\\bibitem{besold99} Besold G., Risbo J. and Mouritsen O.G., {\\em\nComput. Mat. Science} {\\bf 15}, 311 (1999).\n\n\\bibitem{mucareviews} Berg B.A., in: Karsch F., Monien B. and Satz\nH. (Eds.), {\\sl Proceedings of the International Conference on Multiscale\nPhenomena and their Simulations} (Bielefeld, October 1996). World\nScientific, Singapore (1997);\\\\ Janke W., {\\em Physica} {\\bf A254},\n164 (1998);\\\\ D\\\"unweg B., in Binder K. and Ciccotti C. (Eds.), {\\sl Monte\nCarlo and Molecular Dynamics of Condensed Matter Physics} (Como, July\n1995). Societ\\'a Italiana di Fisica, Bologna, p. 215 (1996);\\\\ Berg\nB.A, Hansman U.H.E. and Heuhaus T, {\\em Phys. Rev.} {\\bf B47}, 497\n(1993).\n\n\\bibitem{pmco96} de Oliveira P. M. C., Penna T. J. P. and Herrmann H.J.,\n {\\em Braz. J. Phys.} {\\bf 26}, 677 (1996). Available at {\\bf\n cond-mat/9610041}. \n\n\\bibitem{pmco98a} de Oliveira P. M. C., Penna T. J. P. and Herrmann H.J.,\n {\\em Eur. Phys. J.} {\\bf B1}, 205 (1998).\n \n\\bibitem{pmco98b} de Oliveira P. M. C., in {\\sl Computer Simulation\n Studies in Condensed-Matter Physics XI}, edited by D.P. Landau,\n Springer, Berlin, 169 (1998).\n\n\\bibitem{pmco98c} de Oliveira P. M. C., {\\em Eur. Phys. J.} {\\bf B6}, 111\n(1998). Available at {\\bf cond-mat/9807354}.\n\n\\bibitem{pmco98d} de Oliveira P. M. C., {\\em Int. J. Mod. Phys.} {\\bf C9}, 497 \n (1998).\n \n\\bibitem{pmco99} de Oliveira P. M. C., CCP\n 1998 conference proceedings; {\\it Comp. Phys. Comm.} {\\bf 121-122},\n 16 (1999).\n \n\\bibitem{wang98} Wang J.-S., {\\it Eur. Phys. J.} {\\bf B8}, 287-291\n (1999); \\\\\n ``Monte Carlo algorithms based on the number of potential\n moves'' in {\\bf cond-mat/9903224}.\n \n\\bibitem{munoz98} Munoz J. D. and Herrmann H. J., {\\em Int. J. Mod. Phys.}\n {\\bf C10}, 95 (1999); \\\\\n CCP 1998 conference proceedings, {\\it Comp. Phys. Commun.}, also in\n {\\sl Computer Simulation Studies in Condensed-Matter Physics XII},\n edited by\n D.P. Landau (Springer,Berlin, in print) and \\textbf{cond-mat/9810292}.\n \n\\bibitem{lima99} Lima A. R., S\\'a Martins J. S. and Penna T. J. P., {\\em\n Physica} {\\bf A268}, 553 (1999).\n\n\\bibitem{wang99} Wang J. S., Tay T. K. and Swendsen R. H., {\\em Phys. Rev.\n Lett.} {\\bf82}, 476 (1999).\n\n\\bibitem{promb} Kastner M., Promberger M. and Munoz J. D., ``Broad\n Histogram Method: Extension and Efficiency Test'' in {\\bf\n cond-mat/9906097}.\n\n\\bibitem{j1j2} Lima A. R., de Oliveira P. M. C. and Penna T. J. P.,\n {\\em Solid State Comm.} (2000). Available at {\\bf cond-mat/9912152}. \n\n\\bibitem{pearson82} Pearson R. B., {\\em Phys. Rev.} {\\bf B26}, 6285 (1982).\n\n\\bibitem{smith} Smith G. R. and Bruce A. D., {\\em J. Phys.} {\\bf A28},\n 6623 (1995); \\\\\n {\\em Phys. Rev.} {\\bf E53}, 6530 (1996); \\\\\n {\\em Europhysics Lett.} {\\bf 39}, 91 (1996).\n\n\n\\bibitem{beale96} Beale P. D., {\\em Phys. Rev. Lett.} {\\bf 76}, 78 (1996).\n\n\\end{thebibliography}"
}
] |
cond-mat0002177
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Congested Traffic States in Empirical Observations and Microscopic Simulations
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[
{
"author": "Martin Treiber"
},
{
"author": "Ansgar Hennecke"
},
{
"author": "and Dirk Helbing"
}
] |
We present data from several German freeways showing different kinds of congested traffic forming near road inhomogeneities, specifically lane closings, intersections, or uphill gradients. The states are localized or extended, homogeneous or oscillating. Combined states are observed as well, like the coexistence of moving localized clusters and clusters pinned at road inhomogeneities, or regions of oscillating congested traffic upstream of nearly homogeneous congested traffic. The experimental findings are consistent with a recently proposed theoretical phase diagram for traffic near on-ramps [D. Helbing, A. Hennecke, and M. Treiber, Phys. Rev. Lett. {82}, 4360 (1999)]. % We simulate these situations with a novel continuous microscopic single-lane model, {the ``intelligent driver model'' (IDM), using the empirical boundary conditions.} All observations, including the coexistence of states, {are qualitatively reproduced by describing inhomogeneities with} local variations of one model parameter. % We show that the results of the microscopic model can be understood by formulating the theoretical phase diagram for {bottlenecks in a more general way.} In particular, a local drop of the road capacity induced by parameter variations has {practically} the same effect as an on-ramp.
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[
{
"name": "empsim.tex",
"string": "\n%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!\n%############ Pfeile fuer die Fahrtrichtung in den 3D Plots !!!!!!!\n%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!\n\n\n\\documentstyle[aps,epsfig,preprint]{revtex}\n%\\documentstyle[aps,prl,epsfig,twocolumn]{revtex}\n\n\\begin{document}\n\n\\newcommand{\\pathdefs}{/home/treiber/tex/definitions}\n\\newcommand{\\pathfigs}{.}\n%\\input{\\pathdefs/defs.tex}\n%\\input{\\pathdefs/defstraffic.tex}\n\n\\newcommand{\\erw}[1]{\\mbox{$\\langle #1 \\rangle$}} %Erwartungswert\n\n\n\\tighten\n\\onecolumn\n%\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname\n%{\\protect\n\n\n\\title{Congested Traffic States in Empirical Observations\n and Microscopic Simulations}\n\\author{Martin Treiber, Ansgar Hennecke, and Dirk Helbing}\n\\address{II. Institute of Theoretical Physics, University of Stuttgart,\n Pfaffenwaldring 57, D-70550 Stuttgart, Germany\\\\\n{\\tt http://www.theo2.physik.uni-stuttgart.de/treiber/, helbing/} \n}\n\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nWe present data from several German freeways showing different\nkinds of congested traffic forming near road inhomogeneities,\nspecifically lane closings,\nintersections, or uphill gradients. \nThe states are localized or extended,\nhomogeneous or oscillating. \nCombined states are observed as well, \nlike the coexistence of moving localized clusters\nand clusters pinned at road inhomogeneities,\nor regions of oscillating congested\ntraffic upstream of nearly homogeneous congested traffic.\nThe experimental findings are consistent with a recently proposed\ntheoretical phase diagram for traffic near on-ramps \n[D. Helbing, A. Hennecke, and M. Treiber, \nPhys. Rev. Lett. {\\bf 82}, 4360 (1999)].\n%\nWe simulate these situations with a novel continuous microscopic\nsingle-lane model, {the ``intelligent driver model'' (IDM), using the\nempirical boundary conditions.} All observations,\nincluding the coexistence of states, {are qualitatively\nreproduced by describing inhomogeneities with} \nlocal variations of one model parameter. \n%\nWe show that the results of the microscopic model\ncan be understood by formulating the\ntheoretical phase diagram for {bottlenecks in a more general way.}\nIn particular, a local drop of the road capacity induced\nby parameter variations has {practically} the same effect\nas an on-ramp.\n\n\n\\end{abstract}\n\n\\pacs{02.60.Cb, 05.70.Fh, 05.65.+b, 89.40.+k}\n\n%} ] %end twocolumn[ .. oder am Schluss??\n\n%###################################################################\n\\section{Introduction}\n%###################################################################\n\nRecently, there is much interest in the \ndynamics of traffic breakdowns \nbehind bottlenecks\n\\cite{Hall1,Daganzo-ramp,Kerner-rehb96-2,Kerner-sync,Nagatani-sync,Persaud,Kerner-wide,sync-Letter,TSG-science,Lee,Lee99,Lee-emp,Daganzo-ST,Phase,Kerner-Transp}.\nMeasurements of traffic breakdowns on various freeways\nin the USA \\cite{Daganzo-ST,Hall1,Daganzo-ramp,Persaud},\nGermany, \\cite{Kerner-rehb96-2,Kerner-sync,Kerner-rehb98,numerics}, \nHolland \\cite{Helb-book,Helb-emp97,GKT-scatter,Smulders1,Vladi-98}, \nand Korea \\cite{Lee-emp}\nsuggest that many dynamic aspects are universal and \ntherefore accessible to\na physical description.\nOne common property is the capacity drop (typically of the order of\n20\\%) associated with a breakdown\n\\cite{Hall1,Daganzo-ST,Persaud}, \nwhich leads to hysteresis effects and is the basis of\napplications like dynamic traffic control with the aim of\navoiding the breakdown.\nIn the majority of cases, traffic breaks down upstream of a\nbottleneck and the congestion has a stationary donstream\nfront at the bottleneck. The type of bottleneck, e.g.,\non-ramps\n\\cite{Daganzo-ramp,Daganzo-ST,Lee-emp,Kerner-sync}, \nlane closings, or uphill gradients \\cite{numerics}, seems not to be \n{of importance}.\nSeveral types of congested traffic have been found, among them\nextended states with a relatively high traffic flow. These states,\nsometimes referred to as ``synchronized traffic'' \\cite{Kerner-sync},\ncan be more or less homogeneously flowing, or show distinct\noscillations in the time series of detector data \n\\cite{Kerner-rehb96-2}.\nVery often, the congested traffic flow is, apart from fluctuations,\nhomogeneous near the\nbottleneck, but oscillations occur further upstream \\cite{Kerner-wide}.\nIn other cases, one finds isolated stop-and-go waves\nthat propagate in the upstream direction with a characteristic\nvelocity of about 15 km/h \\cite{Kerner-rehb96,Kerner-rehb98}. \nFinally, there is also {an} observation of\na traffic breakdown to a pinned localized\ncluster near an on-ramp \\cite{Lee-emp}.\n\nThere are several possibilities to {delineate traffic mathematically,} \namong them macroscopic models describing the dynamics in\nterms of aggregate quantities like density or flow\n\\cite{Lighthill-W,Cremer-93,KK-94,GKT,Lee}, and microsopic\nmodels describing the motion of individual vehicles.\nThe latter include continuous-in-time models (car-following models)\n\\cite{Newell,Gipps81,Wiedemann,Bando,Krauss-traff98,Tilch-GFM,MITSIM,Howe,coexist,TGF99-Treiber},\nand cellular automata \n\\cite{Biham,Cremer-multi,Nagel-S,Barlovic,Helb-sblock,Wolf-Galilei}.\nTraffic breakdowns behind bottlenecks have been simulated with the\nnon-local, gaskinetic-based traffic model\n(GKT model) \\cite{GKT}, the \nK\\\"uhne-Kerner-Konh\\\"auser-Lee model (KKKL model)\n\\cite{Kuehne,KK-94,Lee}, and with \na new car-following model, which will\nbe reported below.\n\nFor a direct comparison with empirical data,\none would prefer car-following\nmodels. {As the} position and velocity of each car is known in such\nmodels, one can reconstruct the way how data are obtained by usual \ninduction-loop detectors.\nTo this end, one introduces ``virtual'' detectors recording\npassage times and velocities of crossing vehicles and compares this\nouput with the empirical data.\nBecause traffic density is not a primary variable,\nthis avoids the problems associated {with} determining the traffic density\nby temporal averages \\cite{Helb-book}.\n\n\n\nThe present study \nrefers to publication \\cite{Phase}, where, based on\na gas-kinetic-based macroscopic simulation model, it has been concluded\nthat there should be five different congested traffic states on freeways\nwith inhomogeneities like on-ramps. The kind of congested state\ndepends essentially on the inflow into the considered freeway section\nand on the ``bottleneck strength'' characterizing the inhomogeneity.\nThis can be summarized by a phase diagram depicting the kind of\ntraffic state as a function of these two parameters.\nA similar phase diagram has been obtained for the KKKL model \\cite{Lee99}.\nThe question is, whether this finding\nis true for some macroscopic models only, or universal for a larger class of \ntraffic models and confirmed by empirical data.\n\nThe relative positions of some of the traffic states in this phase space\nhas been qualitatively\nconfirmed for a Korean freeway \\cite{Lee-emp}, but only one type of \nextended state has been measured there. Furthermore, this state\ndid not have the characteristic properties \nof extended congested traffic on most other freeways,\ncf. e.g., \\cite{Kerner-sync}.\nIt is an open question\nto confirm the relative position of the other states.\nMoreover, to our knowledge, there are no direct simulations of the different\nbreakdowns using empirical data as boundary conditions,\nneither with microscopic nor with macroscopic models.\n\nCareful investigations\nwith a new follow-the-leader model (the IDM model) \n\\cite{coexist,TGF99-Treiber} show that\n(i) the conclusions of Ref \\cite{Phase} are also valid for certain\n microscopic traffic models (at least deterministic models with\n a metastable density range),\n(ii) the results can be systematically transferred to more general\n (in particular flow-conserving) kinds of bottlenecks, and a formula\n allowing to quantify the bottleneck strength is given [see Eq. (15)], \n(iii) the existence of all predicted traffic states is empirically\n supported,\nand finally,\n(iv) all different kinds of breakdowns can be simulated with the IDM\n model with {\\it empirically measured} boundary conditions,\n varying only one parameter (the average time headway T), which\n is used to specify the capacity of the stretch.\n\nThe applied IDM model \nbelongs to the class of deterministic\nfollow-the-leader models like the optimal velocity model by Bando et al.\n\\cite{Bando}, but it has the following advantages:\n(i) it behaves accident-free because of the dependence on the\n relative velocity,\n(ii) for similar reasons and because of metastability, it shows\n the self-organized characteristic traffic constants demanded \n by Kerner et al. \\cite{KK-94} (see Fig. 4), hysteresis effects\n\\cite{Treiterer,Daganzo-ST}, and complex states\n\\cite{Kerner-rehb96-2,Kerner-wide},\n(iii) all model parameters have a reasonable interpretation, are known to \n be relevant, are empirically measurable, and have the expected\n order of magnitude \\cite{TGF99-Treiber},\n(iv) the fundamental diagram and the stability properties of the model\n can be easily (and separately) calibrated to empirical data,\n(v) it allows for a fast numerical simulation,\nand\n(vi) an equivalent macroscopic version of the model is known\n\\cite{TGF99-Hennecke}, which is\n not the case for most other microscopic traffic models.\n\nThese aspects are discussed in Sec.~II, while Sec.~III is not\nmodel-specific at all. Section III presents ways to specify and quantify\nbottlenecks, as well as the traffic states resulting for different traffic\nvolumes and bottleneck strengths. The analytical expressions for the phase\nboundaries of the related phase diagram allow to conclude that similar\nresults will be found for any other traffic model with a stable,\nmetastable, and unstable density range. Even such subtle features like\ntristability first found in macroscopic models \n\\cite{Lee99,TGF99-Treiber} are observed. It would be certainly interesting\nto investigate in the future, whether the same phenomena are also found\nfor CA models or stochastic traffic models like the one by Krauss\n\\cite{Krauss-traff98}.\n\nSection~IV discusses empirical data using\nrepresentative examples out of a sample\nof about 100 investigated breakdowns. \nThanks to a new method for presenting\nthe cross section data (based on a smoothing and interpolation procedure),\nit is possible to present 3d plots of the empirical density or average\nvelocity as a function of time {\\it and} space. This allows a good imagination\nof the traffic patterns and a direct visual comparison with simulation\nresults.\nIn the IDM microsimulations, we used a very\nrestricted data set, namely only the measured flows \nand velocities at the upstream and downstream boundaries omitting the\ndata of the up to eight detectors in between.\nAlthough the simulated sections were up to 13\nkm long and the boundaries were typically outside of congestions, \nthe simulations reproduced \nqualitatively the sometimes very complex\nobserved collective dynamics.\n\nAll in all the study supports the idea of the suggested phase diagram of\ncongested traffic states quite well and suggests ways to simulate real\ntraffic breakdowns at bottlenecks with empirical boundary conditions.\n\n\n\n\n\n\n\n\n%\n%From a physics point of view, a microscopic traffic model\n%should be as simple as possible. The parameters should be intuitive,\n%easy to calibrate, and the corresponding values should be realistic.\n%The collective dynamics should reproduce all observed\n%localized and extended traffic states \\cite{Lee-emp},\n%including synchronized traffic and the wide scattering\n%of congested traffic data \\cite{Kerner-rehb96-2}.\n%Furthermore, the observed\n%hysteresis effects \\cite{Treiterer,Daganzo-ST}, complex states\n%\\cite{Kerner-rehb96-2,Kerner-wide},\n%and the existence of self-organized \n%quantities like the constant propagation velocity of stop-and-go waves\n%or the outflow from a traffic jam \\cite{KK-94} \n%should be reproduced.\n%To be consistent with macroscopic models, a deterministic\n%instability mechanism is favourable.\n%Finally, the dynamics must not lead to vehicle collisions\n%and the model should allow for a fast numerical simulation.\n%\n%Recently, the ``intelligent driver model'' (IDM) has been \n%introduced as a new car-following model\n%\\cite{coexist} to explain a special type \n%of oscillating congested traffic,\n%the ``pinch effect'' \\cite{Kerner-wide}. Simulations with the IDM \n%reproduced the typical extended and localized\n%congested traffic states behind bottlenecks, \n%that were implemented by local speed limits \\cite{TGF99-Treiber}.\n%A macroscopic version of the IDM with nearly identical\n%collective properties has been {also} proposed.\n%This ``micro-macro link'' was applied to implement on-ramps in the \n%microsimulations by simulating the ramp section macroscopically and\n%the rest with the {microscopic} IDM \\cite{TGF99-Hennecke}.\n%\n%In this paper, we test the applicability of the IDM\n%to the realistic simulation of traffic breakdowns and the validity of the\n%phase diagram.\n%We present experimental data of all traffic states\n%predicted by the phase diagram and simulate them with\n%the IDM using open systems with empirical detector data of flow\n%and velocity as boundary conditions. \n%\n%\n%\n%In Sec.~II we describe the IDM in detail,\n%discuss its essential properties and check some of the \n%aforementioned criteria for realistic car-following models.\n%In Sec.~III we describe microscopic simulations of\n%open systems.\n%We propose a microscopic implementation\n%of bottleneck inhomogeneities by locally increasing the safe time headway,\n%which is one of the model parameters.\n%Furthermore, we present \n%the phase diagram of congested traffic states for this type of bottleneck\n%and show that, for identical vehicles,\n%the qualitative features of the phase diagram\n%are the same as for the implementation of bottlenecks with local speed \n%limits. This is also true for hysteresis effects\n%and even for such subtle features like\n%tristability \\cite{Lee99,TGF99-Treiber}.\n%\n%In Sec.~IV we \n%present empirical data from several German freeways which show\n%essentially all traffic states proposed by the phase diagram.\n%We estimate for each situation the control parameters of the phase diagram,\n%{the} traffic flow, and bottleneck stength, and show that\n%the relative positions of the\n%regions of the phase diagram agree qualitatively with the data.\n%The investigated freeway sections contained several types of\n%bottlenecks, including lane closings due to incidents, uphill\n%gradients, and\n%on-ramps, which points to the\n%universality of the phase diagram.\n%We compare the data with IDM microsimulations using measurered flows \n%and velocities\n%as boundary conditions. Although the simulated sections were up to 20\n%km long and the boundaries were typically outside of congestions, \n%the simulations reproduced \n%qualitatively the sometimes very complex\n%observed collective dynamics.\n%\n\n%###################################################################\n\\section{The Microscopic ``Intelligent Driver Model'' (IDM)}\n% subsections werden klein geschrieben in PRX, \n% sections automat alles gross\n%###################################################################\nFor about fifty years now, researchers model\nfreeway traffic by means of continuous-in-time microscopic \nmodels (car-following models) \\cite{Reuschel}.\nSince then, a multitude of car-following models have been\nproposed, both for single-lane and multi-lane traffic including lane\nchanges.\nWe will restrict, here, to phenomena for which lane changes\nare not important and only consider \nsingle-lane models.\nTo motivate our traffic model we first give an overview of the\ndynamical properties of some popular microscopic models.\n\n%##############################################################\n\\subsection{Dynamic Properties of Some Car-Following Models}\n%##############################################################\n%\nContinuous-time single-lane car-following models are defined essentially by \ntheir acceleration function.\nIn many of the earlier models \\cite{Chandler,Herman59,Gazis61,Edie},\nthe acceleration $\\dot{v}_{\\alpha}(t+T_r)$ of vehicle $\\alpha$, delayed\nby a reaction time $T_r$, can be written as\n%\n\\begin{equation}\n\\label{gazis}\n\\dot{v}_{\\alpha}(t+T_r) = \\frac{- \\lambda v_{\\alpha}^m \\Delta v_{\\alpha}}\n {s_{\\alpha}^l}.\n\\end{equation}\n%\nThe deceleration $-\\dot{v}_{\\alpha}(t+T_r)$ is asumed to be\nproportional to the approaching rate\n\\begin{equation}\n \\Delta v_{\\alpha}(t) := v_{\\alpha}(t)-v_{\\alpha-1}(t)\n\\end{equation}\nof vehicle\n$\\alpha$ {with respect} to the leading vehicle\n$(\\alpha-1)$.\nIn addition, the\nacceleration may depend on the own velocity $v_{\\alpha}$\n\\cite{Gazis61} and decrease with some power of\nthe net (bumper-to-bumper) distance\n\\begin{equation}\n s_{\\alpha}=x_{\\alpha-1}-x_{\\alpha}-l_{\\alpha}\n\\end{equation}\nto the leading vehicle (where $l_{\\alpha}$ is the vehicle length)\n\\cite{Gazis61,Edie}.\nSince, according to Eq. (\\ref{gazis}),\n the acceleration\ndepends on {a leading vehicle,}\nthese models are not applicable for very low traffic densities. If \nno leading {vehicle is} present (corresponding\nto $s_{\\alpha}\\to\\infty$), the acceleration is either\nnot determined $(l=0)$ or zero $(l>0)$, regardless of the own velocity.\nHowever, one would expect in this case\nthat drivers accelerate to an individual desired\nvelocity.\nThe car-following behavior in dense traffic is also somewhat\nunrealistic. In particular, the gap $s_{\\alpha}$\nto the respective front vehicle does not {necessarily} relax\nto an equilibrium value. Even small gaps will not\ninduce braking reactions if the velocity difference \n$\\Delta v_{\\alpha}$ is zero.\n\nThese problems are solved by the car-following model \nof Newell \\cite{Newell}. In this model, the velocity\nat time $(t+T_r)$ depends adiabatically on the gap, i.e.,\nthe vehicle adapts exactly to a distance-dependent function $V$\nwithin the reaction time $T_r$,\n%\n\\begin{equation}\n\\label{newell}\nv_{\\alpha}(t+T_r) = V\\Big(s_{\\alpha}(t)\\Big). \n\\end{equation}\n%\nThe ``optimal velocity function'' \n$V(s)=v_0\\{1-\\exp[-(s-s_0)/(v_0 T)]\\}$ includes both, a desired velocity\n$v_0$ for vanishing\ninteractions ($s\\to\\infty$) and a safe time headway $T$ \ncharacterizing the\ncar-following behavior in dense (equilibrium) traffic.\nThe Newell model is collision-free,\nbut the immediate dependence of the velocity\non the density leads to very high accelerations of the order of \n$v_0/T_r$. \nAssuming a typical desired velocity of\n30 m/s and $T_r=1$ s, this would correspond to 30 m/s$^2$, which is clearly\nunrealistic \\cite{Tilch-GFM,Bleile-Diss}. \n\nMore than 30 years later, Bando {\\it et al.} suggested a similar model,\n\\begin{equation}\n\\label{bando}\n\\dot{v}_{\\alpha} = \\frac{V(s_{\\alpha}) - v_{\\alpha}}{\\tau}\n\\end{equation}\nwith a somewhat different optimal velocity function.\nThis ``optimal-velocity mdoel'' \nhas been widely used by physicists because of its simplicity, and because\nsome results could be derived analytically.\nThe dynamical behavior \ndoes not greatly differ from the Newell model,\nsince the reaction time delay $T_r$ of the Newell model can be compared\nwith the velocity relaxation time $\\tau$ of the optimal-velocity model.\nHowever, realistic velocity relaxation\ntimes are of the order of 10 s (city traffic) to 40 s \n(freeway traffic) and\ntherefore much larger than\nreaction delay times (of the order of 1 s).\nFor typical values of the other\nparameters of the optimal-velocity model \\cite{Bando},\ncrashes are only avoided if\n$\\tau < 0.9$ s, i.e., the velocity relaxation time is of the order\nof the reaction time, leading again to\nunrealistically high values \n$v_0/\\tau$ \nof the maximum acceleration.\nThe reason of this unstable behavior is that {effects of}\nvelocity differences are neglected.\nHowever, they play an essential stabilizing\nrole in real traffic, especially when approaching traffic jams. \nMoreover, in models (\\ref{newell}) and (\\ref{bando}),\naccelerations and decelerations are symmetric with respect to the\ndeviation of the actual velocity {from} the equilibrium velocity,\nwhich is unrealistic. The absolute value of\nbraking decelerations is usually stronger than that of\naccelerations.\n\nA relatively simple model with a generalized optimal velocity\nfunction incorporating both, reactions to\nvelocity differences and\ndifferent rules for acceleration and braking has been proposed\nrather recently \\cite{Tilch-GFM}. This ``generalized-force model''\ncould successfully reproduce\nthe time-dependent \ngaps and velocities measured by a sensor-equipped\ncar in congested city traffic. \nHowever, the acceleration and deceleration times in this model are\nstill unrealistically small which requires inefficiently small time\nsteps for the\nnumerical simulation.\n\nBesides these simple models intended for basic investigations, \nthere are also highly complex ``high-fidelity models'' \nlike the Wiedemann model \\cite{Wiedemann} or MITSIM \\cite{MITSIM},\nwhich try to reproduce traffic as realistically as possible, but\nat the cost of a large number of parameters.\n\n{Other} approaches that incorporate ``intelligent'' and\nrealistic braking reactions\nare {the} simple and fast stochastic models\nproposed by Gipps \\cite{Gipps81,Gipps86} and Krauss \n\\cite{Krauss-traff98}.\nDespite their simplicity, these models show a realistic\ndriver behavior, have asymmetric accelerations and decelerations,\nand produce no accidents. Unfortunately,\nthey lose their realistic properties in the\ndeterministic limit. In particular, they show no traffic instabilities \nor hysteresis effects for vanishing fluctuations.\n\n%#####################################################\n\\subsection{Model Equations}\n%#####################################################\n\nThe acceleration assumed in the IDM is a continuous function \nof the velocity $v_{\\alpha}$, the gap $s_{\\alpha}$,\nand the velocity difference (approaching rate)\n$\\Delta v_{\\alpha}$ to the leading vehicle: % kein Doppelpunkt ueblich\n%\n\\begin{equation}\n\\label{IDMv}\n\\dot{v}_{\\alpha} = a^{(\\alpha)}\n \\left[ 1 -\\left( \\frac{v_{\\alpha}}{v_0^{(\\alpha)}} \n \\right)^{\\delta} \n -\\left( \\frac{s^*(v_{\\alpha},\\Delta v_{\\alpha})}\n {s_{\\alpha}} \\right)^2\n \\right].\n\\end{equation}\n%\nThis expression is {an interpolation of the tendency to accelerate \nwith \n$a_f(v_{\\alpha}) := a^{(\\alpha)}[1-(v_{\\alpha}/v_0^{(\\alpha)})^{\\delta}]$ \non a free road and the tendency to brake with deceleration\n$-b_{\\rm int}(s_{\\alpha}, v_{\\alpha}, \\Delta v_{\\alpha})\n:= -a^{(\\alpha)}(s^*/s_{\\alpha})^2$ when vehicle $\\alpha$ comes too\nclose to the vehicle in front.} The deceleration term\n%depends Coulomb-like on the gap. \ndepends on the ratio between the ``desired\nminimum gap'' $s^*$ and the actual gap $s_\\alpha$, where the desired gap\n% \n\\begin{equation}\n\\label{sstar}\ns^*(v, \\Delta v) \n% = s_0 + \\mbox{max} \\left( v T \n% + \\frac{v \\Delta v } {2\\sqrt{a b}}, 0 \\right) \\, .\n = s_0^{(\\alpha)} + s_1^{(\\alpha)} \\sqrt{\\frac{v}{v_0^{(\\alpha)}}}\n + T^{\\alpha} v\n + \\frac{v \\Delta v } {2\\sqrt{a^{(\\alpha)} b^{(\\alpha)}}}\n\\end{equation}\n%\n%!!{\\bf das ``max'' nie gebraucht; nur fuer Multilane Simulationen kann\n%sstar$\\le$s0 auch fuer nicht patholog. AB auftreten: \n%Spurwechsel nach ueberholen}\n%\nis dynamically varying with the velocity and the approaching\nrate.\n\nIn the rest of this paper, \nwe will study the case of identical vehicles whose model\nparameters $v_0^{(\\alpha)}=v_0$, $s_0^{(\\alpha)}=s_0$, \n$T^{(\\alpha)}=T$, $a^{(\\alpha)}=a$,\n$b^{(\\alpha)}=b$, and $\\delta$ are\ngiven\nin Table \\ref{tab:param}.\n%\nHere, our emphasis is on basic investigations with models as simple as\npossible, and therefore\nwe will set $s_1^{\\alpha}=0$ resulting in a model where\nall parameters have an intuitive meaning with {plausible and often\neasily measurable values.}\nWhile the empirical data presented in this paper can be \nnevertheless reproduced, {a distinction of different driver-vehicle types\nand/or} a nonzero $s_1$ \\cite{TGF99-Treiber} is necessary\nfor a more quantitative agreement.\nA nonzero $s_1$ would also be necessary for {features} \nrequiring an inflection point in the equilibrium flow-density\nrelation, e.g., for certain types of\nmulti-scale expansions \\cite{Nagatani-kink}.\n\n\n\n%##################################################################\n\\subsection{Dynamic Single-Vehicle Properties}\n%##################################################################\n%\nSpecial cases of the IDM acceleration (\\ref{IDMv})\nwith $s_1=0$ include the following driving\nmodes:\n\n\n%################\n\\paragraph{Equilibrium traffic:}\nIn equilibrium traffic of arbitrary density\n($\\dot{v}_{\\alpha}=0$, $\\Delta v_{\\alpha}=0$), drivers tend to keep \na velocity-dependent\nequilibrium gap $s_{e}(v_{\\alpha})$ to the front vehicle\ngiven by\n%\n\\begin{equation}\n\\label{seq}\ns_{e}(v) = s^*(v,0) \n \\left[ 1 - \\left(\\frac{v}{v_0}\\right)^{\\delta}\\right]^{-\\frac{1}{2}}\n = (s_0+v T) \n \\left[ 1 - \\left(\\frac{v}{v_0}\\right)^{\\delta}\\right]^{-\\frac{1}{2}}.\n\\end{equation}\n%\nIn particular, the equilibrium gap of homogeneous \n{\\it congested} traffic\n({with} $v_{\\alpha} \\ll v_0$) is essentially equal\nto the desired gap,\n$s_{e}(v) \\approx s_0+v T$, i.e., it is\ncomposed of a bumper-to-bumper space $s_0$ kept in\nstanding traffic and {an additional} velocity-dependent contribution\n$v T$ corresponding to a constant safe time headway $T$.\nThis high-density limit is of\nthe same functional form as\nthat of the Newell model, Eq. (\\ref{newell}).\n%\n%\nSolving Eq. (\\ref{seq}) \nfor $v:=V_e(s)$ leads to simple expressions\nonly for $\\delta=1$, $\\delta=2$, or $\\delta\\to\\infty$.\nIn particular, the equilibium velocity for\n$\\delta=1$ and $s_0=0$ is\n%\n\\begin{equation}\n\\label{veGKT}\nV_e(s)|_{\\delta=1,s_0=0} = \n \\frac{s^2}{2 v_0 T^2} \\left(-1 + \\sqrt{1+\\frac{4 T^2 v_0^2}{s^2}}\n \\right).\n\\end{equation}\n%\nFurther interesting cases are\n%\n\\begin{equation}\n\\label{vedelta2}\nV_e(s)|_{\\delta=2,s_0=0} = \n \\frac{v_0}{\\sqrt{1+\\frac{v_0^2 T^2}{s^2}}},\n\\end{equation}\n%\nand \n%\n\\begin{equation}\n\\label{vedeltainfty}\nV_e(s)|_{\\delta\\to\\infty} = \n\\mbox{min} \\{v_0, (s-s_0)/T \\}.\n\\end{equation}\n%\nFrom a {\\it macroscopic} point of view,\nequilibrium traffic consisting of identical vehicles can\nbe characterized by the \nequilibrium traffic flow $Q_e(\\rho)=\\rho V_e(\\rho)$ \n(vehicles per hour and per lane)\nas a function of the\ntraffic density $\\rho$ (vehicles per km and per lane). \nFor the IDM model, this ``fundamental diagram'' \nfollows from one of the equilibrium relations\n(\\ref{seq}) {to} (\\ref{vedeltainfty}), together with the\nmicro-macro relation between gap and density:\n%\n\\begin{equation}\n\\label{srho}\ns=1/\\rho-l = 1/\\rho-1/\\rho_{\\rm max}.\n\\end{equation} \n%\nHerein, the maximum density $\\rho_{\\rm max}$ is related\nto the vehicle length $l$ by $\\rho_{\\rm max} l=1$.\nFigure \\ref{fig:fund} shows the fundamental diagram and its dependence\non the parameters\n$\\delta$, $v_0$, and $T$.\nIn particular, the fundamental diagram\nfor $s_0=0$ and $\\delta=1$ is \nidentical to the equilibrium relation\nof the macroscopic GKT model, \nif the GKT parameter $\\Delta A$ is set to zero \n(cf. Eq. (23) in Ref. \\cite{GKT}), which is a necessary \ncondition {for} a micro-macro {correspondence} \\cite{TGF99-Hennecke}.\n\n\n%################\n\\paragraph{Acceleration to the desired velocity:}\nIf the traffic density is very low\n($s$ is large), the interaction term is negligible\nand the IDM acceleration reduces to the free-road acceleration\n$a_f(v)=a(1-v/v_0)^{\\delta}$, which is a decreasing\nfunction of the velocity with a maximum value\n$a_f(0)=a$ and $a_f(v_0)=0$.\nIn Fig. \\ref{fig:accbrake}, this regime applies for times\n$t\\le 60$ s.\nThe acceleration exponent $\\delta$ specifies\nhow the\nacceleration decreases when approaching the desired velocity.\nThe limiting case $\\delta\\to\\infty$ corresponds to approaching\n$v_0$ {with} a constant acceleration $a$, while $\\delta=1$\ncorresponds to an exponential relaxation to the desired velocity \nwith the {relaxation time} $\\tau=v_0/a$.\nIn the latter case, the free-traffic acceleration is equivalent to that of\nthe optimal-velocity model (\\ref{bando})\nand also to acceleration functions of many\nmacroscopic models like the KKKL model\n\\cite{KK-94}, or the \nGKT model \\cite{GKT}.\n%The relaxation rate $\\lambda$ of the models \\ref{Bando,Kerner}\n%can be identified with $a/v_0$, and the GKT relaxation time\n%$\\tau$ with $v_0/a$.\nHowever, {the most realistic behavior is expected in between the\ntwo limiting cases of exponential acceleration (for $\\delta = 1$) and constant\nacceleration (for $\\delta \\rightarrow \\infty$), which is confirmed by\nour simulations with the IDM. Throughout this paper we will use\n$\\delta = 4$.}\n\n\n%################\n\\paragraph{Braking as reaction to high approaching rates:} \nWhen\napproaching slower or standing vehicles with \nsufficiently high approaching rates $\\Delta v>0$,\nthe equilibrium part $s_0+v T$ of the dynamical desired\ndistance $s^*$, Eq. (\\ref{sstar}),\ncan be neglected with respect to the\nnonequilibrium % nicht non equilibrium\npart, which is proportional to $v\\Delta v$.\nThen, the interaction part $-a(s^*/s)^2$\nof the acceleration equation (\\ref{IDMv})\nis given by\n%\n\\begin{equation}\n\\label{aapproach}\nb_{\\rm int}(s, v, \\Delta v)\n \\approx \\frac{(v\\Delta v)^2}{4bs^2}.\n\\end{equation}\n%\nThis expression implements anticipative ``intelligent''\nbraking behavior, which we disuss now for the spacial case of\napproaching a standing obstacle\n($\\Delta v=v$).\nAnticipating a constant deceleration during the whole approaching\nprocess,\na minimum kinematic deceleration\n$b_k := v^2/(2 s)$ is\nnecessary to avoid a collision.\nThe situation is assumed to be ``under control'', if $b_k$\nis smaller than the \n``comfortable'' deceleration given by the model parameter $b$,\ni.e., $\\beta := b_k/b \\le 1$. In contrast, an emergency\nsituation is characterized by $\\beta>1$.\nWith these definitions, Eq. (\\ref{aapproach}) becomes\n\\begin{equation}\n\\label{aaproach1}\nb_{\\rm int} (s,v,v) = \\frac{b_k^2}{b} = \\beta b_k. \n%= \\beta^2 b,\n\\end{equation}\nWhile in safe situations the IDM deceleration\nis less than the\nkinematic collision-free deceleration,\ndrivers overreact in\nemergency situations to get the\nsituation again under control. \nIt is easy to show that in both cases \nthe acceleration\napproaches $\\dot{v}=-b$ under the deceleration law (\\ref{aaproach1}).\nNotice that this stabilizing behavior is lost if one replaces in\nEq. (\\ref{IDMv}) the braking term $-a(s^*/s)^2$ by\n$-a'(s^*/s)^{\\delta'}$ with $\\delta' \\le 1$\ncorresponding to $b_{\\rm int}(s,v,v)=\\beta^{\\delta'-1} b_k$.\n%Then, Eq. (\\ref{aaproach1} becomes \n%$b_{\\rm int} (v,v) = b_k^{\\delta'} b^{-\\delta'/2}a^{1-\\delta'/2},\n%For $\\delta'>1$, the braking deceleration approaches\n% $\\dot{v}=\n% a^-((1-\\delta'/2)/(\\delta'-1)) b^(\\delta'/(2\\delta'-2)).\n% For $\\delta'\\le1$, there is no stabilization.\n%\n%Numerical tests\n%show a similar anticipative behavior if $b\\gg a$ is not\n%satisfied \\cite{numerics}, or when approaching moving vehicles.\nThe ``intelligent'' braking behavior of drivers in\nthis regime makes the model collision-free.\nThe right parts of the plots of Fig. \\ref{fig:accbrake} \n($t>70$ s) show the simulated approach of an IDM vehicle to \na standing obstacle.\nAs expected, the maximum deceleration is of the order of $b$.\nFor low velocities, however, the equilibrium term $s_0+vT$ of $s^*$\ncannot be neglected as assumed when deriving Eq. (\\ref{aaproach1}).\nTherefore, the maximum deceleration is somewhat lower than $b$ and the \ndeceleration decreases immediately before the stop while,\nunder the dynamics (\\ref{aaproach1}), one would have $\\dot{v}=-b$.\n\nSimilar braking rules have been implemented in the model\nof Krauss \\cite{Krauss-traff98},\nwhere the model\nis formulated in terms of a time-discretizised update scheme \n(iterated map), where the velocity at timestep $(t+1)$ \nis limited to a\n``safe velocity'' which is calculated on the basis of\nthe kinematic braking distance at a given ``comfortable'' deceleration.\n\n%################\n\\paragraph{Braking in response to small gaps:}\nThe forth driving mode is active when \nthe gap is much smaller than $s^*$ but there are no large velocity\ndifferences.\nThen, the equilibrium part\n$s_0+v T$ of $s^*$ dominates over the dynamic contribution\nproportional to $\\Delta v$. Neglecting the \nfree-road acceleration, Eq. (\\ref{IDMv})\nreduces to \n$\\dot{v} \n\\approx -(s_0+v T)^2/s^2$,\ncorresponding\nto a Coulomb-like repulsion. Such braking\ninteractions are also implemented\nin other models, e.g., in\nthe model of Edie \\cite{Edie}, the GKT model \\cite{GKT},\nor in certain regimes of the Wiedemann\nmodel \\cite{Wiedemann}.\nThe dynamics in this driving regime is not qualitatively different,\nif one replaces $-a(s^*/s)^2$ by\n$-a(s^*/s)^{\\delta'}$ with $\\delta' >0$. This is in contrast to the\napproaching regime, where collisions would be \n{provoked} for $\\delta' \\le 1$.\nFigure \\ref{fig:distance} shows the car-following dynamics in this\nregime.\nFor the standard parameters, one clearly sees an \n{non-oscillatory} relaxation\nto the equilibrium distance {(solid curve), \nwhile for very high values of $b$, the approach to the \nequilibrium distance would occur \nwith damped oscillations (dashed curve).}\nNotice that, for the latter parameter\nset, the {\\it collective} {traffic dynamics}\nwould already be extremely unstable.\n\n%#################################################################\n\\subsection{Collective Behavior and Stability Diagram}\n%#################################################################\n\n\nAlthough we are interested in realistic \n{\\it open} {traffic} systems, \nit turned out that many features can be explained\nin terms of the stability behavior in a {\\it closed} system.\n%{\\bf (on a circular road)}.\nFigure \\ref{fig:stab}(a) shows the stability diagram of \nhomogeneous traffic\non a circular road. The control parameter is the\nhomogeneous density $\\rho_{\\rm h}$. We applied both a very small \nand a large localized perturbation to check for linear and nonlinear\nstability, and plotted the resulting minimum\n($\\rho_{\\rm out}$) and maximum ($\\rho_{\\rm jam}$)\ndensities after a stationary situation was reached. \nThe resulting diagram is very similar to that of the\nmacroscopic KKKL and GKT models \\cite{KK-94,GKT}. \nIn particular, it displays the following realistic features:\n(i) Traffic is stable for very low and high densities, but unstable for\nintermediate densities. \n(ii) There is a density range $\\rho_{\\rm c1}\\le\\rho_{\\rm h}\\le\\rho_{\\rm c2}$\nof metastability, i.e., only perturbations of sufficiently large\namplitudes grow, while smaller perturbations disappear. Note that, \nfor most IDM parameter sets, there is no second metastable range at \nhigher densities, in contrast to the GKT and KKKL models. \nRather, traffic flow becomes stable again for densities exceeding the\ncritical density $\\rho_{\\rm c3}$, or {congested} flows below\n$Q_{\\rm c3}=Q_{\\rm e}(\\rho_{\\rm c3})$.\n(iii) The density $\\rho_{\\rm jam}$ inside of traffic jams \nand the associated flow $Q_{\\rm jam}=Q_{\\rm e}(\\rho_{\\rm jam})$,\ncf. Fig. \\ref{fig:stab}(b),\ndo not depend on\n$\\rho_{\\rm h}$. For the parameter set chosen here, we have\n$\\rho_{\\rm jam}=\\rho_{\\rm c3}=140$ vehicles/km, and $Q_{\\rm jam}=0$,\n{i.e., there is no linearly stable congested traffic with a finite\nflow and velocity.}\nFor other parameters, especially for a nonzero\nIDM parameter $s_1$, both $Q_{\\rm jam}$ and $Q_{\\rm c3}$ \ncan be nonzero and\ndifferent from each other \\cite{TGF99-Treiber}.\n\nAs further ``traffic constants'', at least\nin the density range 20 veh./km $\\le \\rho_{\\rm h} \\le$ 50 veh./km, \nwe observe a constant outflow $Q_{\\rm out}=Q_{\\rm e}(\\rho_{\\rm out})$\nand propagation velocity \n$v_{\\rm g}=(Q_{\\rm out} - Q_{\\rm jam}) / (\\rho_{\\rm out}-\\rho_{\\rm jam})\n\\approx -15$ km/h of {traffic} jams.\nFigure \\ref{fig:stab}(b) shows the stability diagram for the flows.\nIn particular, \nwe have $Q_{\\rm c1}<Q_{\\rm out} \\approx Q_{\\rm c2}$,\nwhere $Q_{{\\rm c}i}=Q_{\\rm e}(\\rho_{{\\rm c}i})$, i.e., the outflow from\ncongested traffic is at the margin of linear stability, which is\nalso the case in the GKT for most parameter sets \\cite{GKT,Phase}.\nFor other IDM parameters, the outflow $Q_{\\rm out}\\in\n[Q_{\\rm c1},Q_{\\rm c2}]$ is metastable \\cite{TGF99-Treiber}, or even at the \nmargin of nonlinear stability \\cite{coexist}.\n\\par\nIn {\\it open} systems, a third type of stability becomes relevant.\nTraffic is {\\it convectively} stable, if, after a sufficiently long time,\nall perturbations are convected out of the system.\nBoth, in the macroscopic models and in the IDM, there is a considerable\ndensity region $\\rho_{\\rm cv}\\le \\rho_{\\rm h}\\le \\rho_{\\rm c3}$,\nwhere traffic is linearly unstable but convectively stable.\n{For the parameters chosen in this paper,} congested traffic is \n{\\it always} linearly unstable, but convectively stable for flows below \n$Q_{\\rm cv}=Q_{\\rm e}(\\rho_{\\rm cv})=1050$ vehicles/h.\nA nonzero jam distance $s_1$ is required for linearly\n{\\it stable} congested traffic with nonzero flows \n\\cite{TGF99-Treiber}, at least, if the model \nshould simultaneously show traffic instabilities.\n%#################################################################\n\\subsection{Calibration}\n%#################################################################\nBesides the vehicle length $l$, the IDM has seven parameters,\ncf. Table \\ref{tab:param}.\nThe {\\it fundamental relations}\nof homogeneous traffic are calibrated\nwith the desired velocity $v_0$ (low density), \n%acceleration exponent $\\delta$ (transition region),\nsafe time headway $T$ (high density), and the jam distances\n$s_0$ and $s_1$ (jammed traffic).\nIn the low-density limit $\\rho\\ll (v_0 T)^{-1}$, \nthe equilibrium flow can be approximated by\n$Q_e\\approx V_0\\rho$. In the high density regime and for $s_1=0$,\none has a linear decrease of the flow \n$Q_e\\approx [1-\\rho(l+s_0)]/T$ which can be used to determine\n$(l+s_0)$ and $T$. \nOnly for nonzero $s_1$, one obtains an inflection point\nin the equilibrium flow-density relation $Q_e(\\rho)$.\nThe acceleration coefficient $\\delta$ \ninfluences the transition region between the free and\ncongested regimes. For $\\delta\\to\\infty$ and $s_1=0$, \nthe fundamental diagram (equilibrium flow-density relation) becomes\ntriangular-shaped: $Q_e(\\rho) = \\mbox{min}(v_0\\rho,\n[1-\\rho(l+s_0)]/T)$. \nFor decreasing $\\delta$, it becomes smoother and \nsmoother, cf. Fig. \\ref{fig:fund}(a).\n\nThe {\\it stability behavior} of traffic in the IDM model\nis determined mainly by\nthe maximum acceleration $a$, desired deceleration $b$, \nand by $T$. Since the accelerations $a$ and $b$ \ndo not influence the fundamental diagram,\nthe model can be calibrated essentially independently with respect to\ntraffic flows and stability.\nAs in the GKT model, traffic becomes more unstable for\ndecreasing $a$ (which corresponds to an increased acceleration time\n$\\tau=v_0/a$), and with decreasing $T$\n(corresponding to reduced safe time headways). \nFurthermore, the instability increases with \nincreasing $b$. This is also plausible, because an increased \ndesired deceleration $b$\ncorresponds to a less anticipative or less defensive braking\nbehavior.\nThe density and flow in jammed traffic\nand the outflow from traffic jams is also influenced by $s_0$ and $s_1$.\nIn particular, for $s_1=0$, the traffic flow $Q_{\\rm jam}$\ninside of traffic jams \nis typically zero after a sufficiently long time [Fig. \\ref{fig:stab}(b)],\nbut nonzero otherwise.\nThe stability of the self-organized outflow $Q_{\\rm out}$ depends\nstrongly on the minimum jam distance $s_0$. It can be unstable\n(small $s_0$), metastable, \nor stable (large $s_0$). In the latter\ncase, traffic instabilities can only lead to single localized\nclusters, not to stop-and go traffic.\n\n\n\n%#################################################################\n%#################################################################\n\\section{\\label{sec:simu} Microscopic Simulation of \nOpen Systems with an Inhomogeneity}\n%#################################################################\n%#################################################################\n\nWe simulated identical vehicles of length $l=5$ m\n{with the typical} IDM model parameters listed in\nTable~\\ref{tab:param}. %We assumed the same values \n%for all simulations shown in this paper.\n{Moreover,} although the various congested states discussed \nin the following were\nobserved on different freeways, {all of them were qualitatively reproduced\nwith very restrictive variations of one single parameter (the safe\ntime headway $T$), while we always used the same values for the other\nparameters (see Table~\\ref{tab:param}).} \nThis indicates that the model is quite realistic and\nrobust. Notice that all parameters have plausible values.\nThe value $T=1.6$ s for the safe time headway \nis slightly lower than suggested by\nGerman authorities (1.8 s).\nThe acceleration parameter $a=0.73$ m/s$^2$ corresponds to\na free-road acceleration from $v=0$ to $v=100$ km/h within 45 s,\ncf. Fig. \\ref{fig:accbrake}(b).\nThis value is obtained by \nintegrating the IDM acceleration \n$\\dot{v}=a[1-(v/v_0)^{\\delta}]$ with \n$v_0=120$ km/h, $\\delta=4$, and the initial value $v(0)=0$. \nWhile this is considerably above\nminimum acceleration times \n(10 s - 20 s for average-powered cars),\nit should be characteristic for {\\it everyday} accelerations. \nThe comfortable deceleration \n$b=1.67$ m/s$^2$ is also consistent with empirical investigations\n\\cite{Bleile-Diss,Tilch-GFM}, \nand with parameters used in more complex models \\cite{MITSIM}.\n\nWith efficient numerical integration schemes,\nwe obtained\na numerical performance of about $10^5$\nvehicles in realtime on a usual workstation \\cite{note-phase2-micperf}.\n\n%############################################\n\\subsection{Modelling of Inhomogeneities}\n%############################################\n%\nRoad inhomogeneities can be classified into flow-conserving\nlocal defects like narrow road sections or gradients, \nand those that do not conserve the average flow per lane,\nlike on-ramps, off-ramps, or lane closings. \n\n{\\it Non-conserving} inhomogeneities can be incorporated \ninto macroscopic models in a natural way by \nadding a source term to the continuity equation \nfor the {vehicle} density \\cite{Kerner-ramp,sync-Letter,Lee}.\nAn explicit microscopic modelling of on-ramps or lane closings,\nhowever, would require\na multi-lane model with an explicit simulation of lane changes. \nAnother possibility {opened by the recently formulated micro-macro link\n\\cite{TGF99-Hennecke} is to} \nsimulate the ramp section {macroscopically with a} source term in\nthe continuity equation \\cite{sync-Letter}, {and to simulate the\nremaining stretch microscopically.}\n\nIn contrast, {\\it flow conserving} inhomogeneities\ncan be implemented easily in both microscopic and macroscopic\nsingle-lane models by locally changing the values of one or more\nmodel parameters or by imposing external decelerations\n\\cite{Nagatani-sync}.\nSuitable parameters for the IDM and the GKT model\nare the desired velocity $v_0$, or the safe time headway $T$.\nRegions with locally decreased desired velocity\ncan be interpreted\neither as sections with speed limits, or as sections with \nuphill \n%{\\bf (and downhill)} \ngradients (limiting the maximum \nvelocity of some vehicles) \\cite{note-phase2-speedlimit}.\nIncreased safe time headways can be\nattributed to more careful driving behavior {along curves,} on\nnarrow, dangerous road sections, {or a reduced range of visibility}. \n\n\nLocal parameter variations act as a bottleneck,\nif the outflow $Q'_{\\rm out}$ from congested traffic \n(dynamic capacity) in the downstream\nsection is reduced with\nrespect to the outflow $Q_{\\rm out}$ in the upstream section.\nThis outflow \ncan be determined from fully developed\nstop-and go waves in a {\\it closed} \nsystem, whose outflow is constant in a rather large range of \naverage densities $\\rho_{\\rm h}\\in$ [20 veh./km, 60 veh./km],\ncf. Fig. \\ref{fig:stab}(b). \nIt will turn out that the outflow $Q'_{\\rm out}$ is the relevant\ncapacity for understanding congested traffic, \nand not the maximum flow $Q_{\\rm max}$ ({\\rm static capacity}),\nwhich can be reached in (spatially homogeneous) equilibrium traffic only.\nThe capacities are decreased, e.g., for \na reduced desired velocity $v'_0<v_0$\nor an increased safe time headway $T'>T$, or both.\nFigure \\ref{fig:Qout} shows $Q_{\\rm out}$ and $Q_{\\rm max}$\nas a function of $T$.\nFor $T'>4$ s,\ntraffic flow is always\nstable, and the outflow from jams is equal to the static capacity.\n\nFor {\\it extended} congested states, \nall types of flow-conserving bottlenecks\nresult in a similar traffic dynamics, if\n$\\delta Q = (Q_{\\rm out} - Q'_{\\rm out})$ is identical,\nwhere $Q'_{\\rm out}$ is the outflow for the changed model parameters\n$v_0'$, $T'$, etc. \nQualitatively the same dynamics is also observed in \n{\\it macroscopic} models including\non-ramps, if the ramp flow satisfies\n$Q_{\\rm rmp}\\approx \\delta Q$ \\cite{TGF99-Treiber}.\n%Q_{\\rm out}(v_0,T)-Q_{\\rm out}(v_0', T')$.\nThis suggests to introduce the following general definition of\nthe ``bottleneck strength'' $\\delta Q$:\n%\n\\begin{equation}\n\\label{deltaQ}\n\\delta Q := Q_{\\rm rmp} + Q_{\\rm out}-Q'_{\\rm out}, \n\\end{equation}\n%\nIn particular, we have $\\delta Q=Q_{\\rm rmp}$ for on-ramp bottlenecks,\nand $\\delta Q=(Q_{\\rm out}-Q'_{\\rm out})$ for flow-conserving\nbottlenecks, {but formula (\\ref{deltaQ}) is also applicable for a\ncombination of both.}\n%##############################################################\n\\subsection{Phase Diagram of Traffic States in Open Systems}\n%##############################################################\n%\nIn contrast to {\\it closed} systems, in which the long-term behavior and\nstability is essentially\ndetermined by the average traffic density $\\rho_h$, the dynamics of \n{\\it open} systems\nis controlled by the inflow $Q_{\\rm in}$ {to the main road (i.e., the\nflow at the upstream boundary).}\nFurthermore, traffic congestions depend on road inhomogeneities and,\nbecause of hysteresis effects, on the history of\nprevious perturbations.\n\nIn this paper, we will implement flow-conserving\ninhomogeneities by a variable safe time headway\n$T(x)$. We chose $T$ as variable model parameter because it\ninfluences the flows {more effectively than} $v_0$,\nwhich has been {varied} in Ref. \\cite{TGF99-Treiber}.\nSpecifically, we increase the\nlocal safe time headway according to\n%\n\\begin{equation}\n\\label{Tx} \nT(x)=\\left\\{ \\begin{array}{ll}\n T & \\ \\ x\\le -L/2 \\\\\n T' & \\ \\ x\\ge L/2 \\\\\n% \\frac{T+T'}{2} + (T'-T)\\frac{x}{L}\n T + (T'-T) \\left(\\frac{x}{L} + \\frac{1}{2} \\right)\n & \\ \\ |x|<L/2,\n\\end{array} \\right.\n\\end{equation}\n%\nwhere the transition region of length\n$L=600$ {is analogous to the ramp length for inhomogeneities that do\nnot conserve the flow.} \n%plays the role of a ramp for flow-conserving inhomogeneities.\nThe bottleneck strength $\\delta Q(T')=[Q_{\\rm out}(T)-Q_{\\rm out}(T')]$\nis an increasing function of $T'$, cf. Fig. \\ref{fig:Qout}(a).\nWe investigated the traffic dynamics for various points\n$(Q_{\\rm in}, \\delta Q)$ or, alternatively,\n$(Q_{\\rm in}, T' - T)$ \nin the control-parameter space.\n\nDue to hysteresis effects and multistability, the phase diagram,\ni.e., the asymptotic traffic state as a function of the control parameters\n$Q_{\\rm in}$ and $\\Delta T$, depends also on the {\\it history}, i.e.,\non initial conditions and on past boundary conditions and\nperturbations.\nSince we cannot explore the whole functional space of initial \nconditions, boundary conditions, and past perturbations, \nwe used the following three representative ``standard'' \nhistories.\n%\n%###################\n\\begin{itemize}\n\\item [A.] Assuming very low values for the initial density and flow,\nwe slowly increased the inflow to the prescribed value\n$Q_{\\rm in}$. \n\\item [B.] \nWe started the simulation with a stable pinned localized cluster\n(PLC) state and a\nconsistent value for the inflow $Q_{\\rm in}$. Then, we \nadiabatically changed the inflow\nto the values prescribed by the point in the phase diagram.\n\\item [C.] After running history A, we \napplied a large perturbation at the downstream boundary.\nIf traffic at the given phase point is metastable, this initiates\nan upstream propagating localized cluster which finally crosses\nthe inhomogeneity, {see} Fig.~\\ref{3dphases}(a)-(c).\nIf traffic is unstable, the breakdown already occurs {during time\nperiod A,} and the additional perturbation has no dynamic influence.\n\\end{itemize}\n%###################\n%\nFor a given history, the resulting phase diagram is unique.\nThe solid lines of Fig. \\ref{phasediag} \nshow the IDM phase diagram for History {C}.\n% Erlaeuterung Dichte \nSpatio-temporal density \nplots of the congested traffic states themselfes are displayed in\nFig.~\\ref{3dphases}. To obtain the spatiotemporal density $\\rho(x,t)$\nfrom the microscopic quantities,\nwe generalize the micro-macro relation (\\ref{srho}) to\ndefine the density at discrete positions \n$x_{\\alpha} + \\frac{l+s_{\\alpha}}{2}$ centered\nbetween vehicle $\\alpha$ and its predecessor,\n%\n\\begin{equation}\n\\label{rhoxt_s}\n\\rho(x_{\\alpha} + \\frac{l+s_{\\alpha}}{2}) = \\frac{1}{l+s_{\\alpha}},\n\\end{equation}\n%\nand interpolate linearly between these positions.\n%\nDepending on $Q_{\\rm in}$ and $\\delta T:=(T'-T)$, the downstream perturbation\n(i) dissipates, resulting in free traffic (FT), \n(ii) travels through the\ninhomogeneity as a moving localized cluster (MLC)\nand neither dissipates nor triggers new breakdowns,\n(iii) triggers a traffic\nbreakdown to a pinned localized cluster (PLC), which remains \nlocalized near the inhomogeneity for \nall times and either is stationary (SPLC),\ncf. Fig.~\\protect\\ref{tristab}(a) for $t<0.2$ h,\nor oscillatory (OPLC).\n(iv) Finally, the initial perturbation can induce \nextended congested traffic (CT), whose downstream boundaries\nare fixed at the inhomogeneity, while the upstream front propagates\nfurther upstream in the course of time. Extended congested traffic \ncan be homogeneous (HCT), oscillatory (OCT), or consist of\ntriggered stop-and-go waves (TSG). We also include in the HCT region a \ncomplex state (HCT/OCT) where traffic is homogeneous only near the bottleneck,\nbut growing oscillations develop further upstream.\nIn contrast to OCT, where there is\npermanently congested traffic at the inhomogeneity (``pinch region''\n\\cite{Kerner-wide,coexist}), the TSG state is characterized by a\nseries of isolated density clusters, each of which triggers a new\ncluster as it passes the inhomogeneity. \n%\nThe maximum perturbation of History C, used also in Ref. \\cite{Phase}, \nseems to select always the\nstable extended congested phase.\nWe also scanned the control-parameter space $(Q_{\\rm in},\\delta Q)$\nwith Histories A and B exploring the maximum phase space of the \n(meta)stable FT and PLC states, respectively, cf. Fig. \\ref{phasediag}.\nIn multistable regions of the control-parameter\nspace, the three histories can be used to select\nthe different traffic states, see below.\n\n\n\\subsection{Multistability}\n%\nIn general, the phase\ntransitions between free traffic,\npinned localized states, and extended congested states are hysteretic. \nIn particular, in all four examples of Fig. \\ref{3dphases},\nfree traffic is possible as a second, metastable state. \nIn the regions between the two dotted lines of the\nphase diagram Fig.~\\ref{phasediag}, both, free and congested traffic is \npossible, depending on the previous history. \nIn particular, \nfor all five indicated phase points, \nfree traffic would persist without the downstream perturbation. In contrast,\nthe transitions PLC-OPLC, and\nHCT-OCT-TSG seem to be non-hysteretic, i.e., the type of pinned\nlocalized cluster or of extended congested traffic, is uniquely\ndetermined by $Q_{\\rm in}$ and $\\delta Q$.\n\nIn a small subset of the metastable region, labelled ``TRI'' in\nFig. \\ref{phasediag}, we even found {\\it tristability} {with the\npossible states} FT, PLC, and OCT. \nFigure \\ref{tristab}(a) shows that\na single moving localized cluster passing the inhomogeneity\ntriggers a transition from PLC to OCT. Starting with free traffic,\nthe same perturbation would trigger OCT as well\n[Fig. \\ref{tristab}(b)], while we never found reverse\ntransitions OCT $\\to$ PLC or OCT $\\to$ FT (without a reduction of the \ninflow). That is, FT and PLC are metastable in the tristable region, \nwhile OCT is stable.\nWe obtained qualitatively the same also for the \nmacroscopic GKT\nmodel with an on-ramp as inhomogeneity [Fig. \\ref{tristab}(c)].\nFurthermore, tristability between FT, OPLC, and OCT\nhas been found for the IDM model with variable $v_0$ \\cite{TGF99-Treiber},\nand for the KKKL model \\cite{Lee99}.\n\n{Such a tristability can only exist if\nthe (self-organized) outflow \n$Q_{\\rm out}^{\\rm OCT}= \\tilde{Q}_{\\rm out}$\nfrom the OCT state is lower than\nthe maximum outflow $Q_{\\rm out}^{\\rm PLC}$ from the PLC state.\nA phenomenological explanation of this condition can be inferred from\nthe positions of the downstream fronts of the OCT and PLC states\nshown in Fig. \\ref{tristab}(a). The downstream front\nof the OCT state ($t>1$ h) is at $x\\approx 300$ m, i.e., at the\n{\\it downstream boundary} of the $L=600$ m wide transition region, \nin which the\nsafe time headway Eq. (\\ref{Tx}) increases from $T$ to $T'>T$.\nTherefore, the local safe time headway\nat the downstream front of OCT is $T'$, or\n$Q_{\\rm out}^{\\rm OCT} \\approx Q'_{\\rm out}$, which was also used to\nderive Eq. (\\ref{Qcong}).\nIn contrast, the PLC state is centered at about $x=0$, so that\nan estimate for the upper boundary $Q_{\\rm out}^{\\rm PLC}$\nof the outflow is given by the self-organized outflow $Q_{\\rm out}$ \ncorresponding to\nthe local value $T(x)=(T+T')/2$ of the safe time headway at $x=0$.\nSince $(T+T')/2 < T'$ this outflow is higher than \n$Q_{\\rm out}^{\\rm OCT}$ (cf. Fig. \\ref{fig:Qout}).\nIt is an open question, however, why the downstream front of the OCT\nstate is further downstream compared to the PLC state.\nPossibly, it can be explained by the close relationship of OCT with the TSG\nstate, for which the newly triggered density clusters even enter the\nregion downstream of the bottleneck\n%where the newly triggered clusters propagate downstream\n%some distance in\n%the bottleneck region, \n[cf. Fig. \\ref{3dphases}(c)]. In accordance with its relative\nlocation in the phase diagram, it is plausible that the OCT state\nhas a ``penetration depth'' into the downstream area that is\nin between the one of the PLC and the TSG states.\n}\n%\n%\n%\n%##################################################################\n\\subsection{Boundaries between and Coexistence of Traffic States}\n%##################################################################\nSimulations show that the outflow $\\tilde{Q}_{\\rm out}$\nfrom the nearly stationary downstream fronts of OCT and HCT\nsatisfies $\\tilde{Q}_{\\rm out}\\le Q'_{\\rm out}$, where\n$Q'_{\\rm out}$ is the outflow from {fully developed density}\nclusters in homogeneous systems for\nthe downstream model parameters. If the bottleneck is not too strong\n(in the phase diagram Fig. \\ref{phasediag}, it must satisfy\n$\\delta Q<350$ vehicles/h),\nwe have $\\tilde{Q}_{\\rm out}\\approx Q'_{\\rm out}$.\nThen, for all types of bottlenecks, the congested traffic flow is\ngiven by\n$Q_{\\rm cong}=\\tilde{Q}_{\\rm out} - Q_{\\rm rmp} \\approx \nQ'_{\\rm out} - Q_{\\rm rmp}$, or\n%\n\\begin{equation}\n\\label{Qcong}\nQ_{\\rm cong} \n% = \\tilde{Q}_{\\rm out} - \\delta Q \n% +Q_{\\rm out} - Q'_{\\rm out}\n\\approx Q_{\\rm out} - \\delta Q.\n\\end{equation}\n%\nExtended congested traffic (CT) \nonly persists, if the inflow $Q_{\\rm in}$ exceeds the congested\ntraffic flow $Q_{\\rm cong}$. Otherwise, it dissolves to PLC.\nThis gives the boundary\n\\begin{equation}\n\\label{CT-PLC}\n\\mbox{CT}\\to\\mbox{PLC}: \\delta Q\\approx Q_{\\rm out}-Q_{\\rm in}.\n\\end{equation}\nIf the traffic flow of CT states is \n{\\it linearly} stable (i.e., \n$Q_{\\rm cong} < Q_{\\rm c3}$), we have HCT. If, for higher flows, it is\nlinearly unstable but\n{\\it convectively} stable,\n$Q_{\\rm cong} \\in [Q_{\\rm c3},Q_{\\rm cv}]$, one has a spatial\n{\\it coexistence} HCT/OCT of states with HCT near the bottleneck and\nOCT further upstream.\nIf, for yet higher flows, congested traffic is also convectively\nunstable, \nthe resulting oscillations lead to TSG or OCT.\nIn summary, the boundaries of the nonhysteretic transitions are given by\n\\begin{equation}\n\\begin{array}{ll}\n\\mbox{HCT}\\leftrightarrow \\mbox{HCT/OCT}:\n & \\delta Q\\approx Q_{\\rm out}-Q_{\\rm c3}, \\\\\n\\mbox{OCT} \\leftrightarrow \\mbox{HCT/OCT}: \n & \\delta Q\\approx Q_{\\rm out}-Q_{\\rm cv}.\n\\end{array}\n\\end{equation}\nCongested traffic of the HCT/OCT type is\nfrequently found in empirical data \\cite{Kerner-wide}.\nIn the IDM, this frequent occurrence is reflected by the \nwide range of flows falling into this regime.\nFor the IDM parameters chosen here, we have\n$Q_{\\rm c3}=0$, and $Q_{\\rm cv}=1050$ vehicles/h, i.e., {\\it all}\ncongested states are linearly unstable and oscillations will develop\nfurther upstream, while $Q_{\\rm c3}$ is nonzero for the parameters of\nRef. \\cite{TGF99-Treiber}.\n\nFree traffic is (meta)stable in the overall system if it is \n(meta)stable in the bottleneck region. This means, a breakdown necessarily\ntakes place if the inflow $Q_{\\rm in}$ exceeds the critical flow \n$[Q'_{\\rm c2}(\\delta Q) - Q_{\\rm rmp}]$, where the linear stability threshold\n$Q'_{\\rm c2}(\\delta Q)$ in the downstream region \nis some function of the bottleneck strength.\nFor the IDM with the parameters\nchosen here, we have $Q'_{\\rm c2}\\approx Q'_{\\rm out}$, {see}\nFig. \\ref{fig:stab}(b). Then, the condition for the maximum inflow \n{allowing} for free traffic simplifies to\n%\n\\begin{equation}\n\\label{FT-CT}\n\\mbox{FT}\\to\\mbox{PLC or FT}\\to\\mbox{CT}: \nQ_{\\rm in} \\approx Q_{\\rm out} - \\delta Q,\n\\end{equation}\n%\ni.e., {it} is equivalent to relation (\\ref{CT-PLC}).\nIn the phase diagram {of} Fig. \\ref{phasediag}, \nthis boundary is given by the dotted line.\nFor bottleneck strengths\n$\\delta Q \\le 350$ vehicles/h, this line coincides with that of the\ntransition CT $\\to$ PLC, in agrement with Eqs (\\ref{FT-CT}) and\n(\\ref{CT-PLC}). For larger bottleneck {strengths,} the\napproximation $\\tilde{Q}_{\\rm out}\\approx Q'_{\\rm out}$ used to derive\nrelation (\\ref{CT-PLC}) is not fulfilled. \n%\n%An open problem are analytic relations for the \n%boundaries of the tristable region. Furthermore, the transitions\n%PLC $\\to$ FT, and PLC $\\to$ CT, could not be explained with \n%considerations {\\bf as the ones above}.\n\n%###################################################################\n\\section{\n\\label{sec:emp}\nEmpirical Data of Congested Traffic States and their \nMicroscopic Simulation}\n%###################################################################\n\nWe analyzed one-minute averages of\ndetector data from the German freeways\nA5-South and A5-North near Frankfurt, A9-South\nnear Munich, and A8-East from Munich to Salzburg.\nTraffic breakdowns occurred \nfrequently on all four freeway sections.\nThe data suggest that the congested states\ndepend not only on the traffic situation but also\non the specific infra\\-structure.\n\nOn the A5-North, we mostly found pinned localized clusters\n(ten {times} during the observation period).\nBesides, we observed \nmoving localized clusters (two {times}), {triggered stop-and-go traffic}\n(three {times}), and oscillating congested traffic (four {times}).\n\nAll eight recorded traffic breakdowns on the A9-South were to\noscillatory congested traffic, and all emerged upstream of \nintersections. The data of the A8 East showed OCT\nwith a more heavily congested HCT/OCT state propagating through it.\nBesides this, we found breakdowns to HCT/OCT\non the\nA5-South (two {times}), one of them caused by lane closing due to\nan external incident. \nIn contrast, HCT states are often found on the Dutch freeway A9 \nfrom Haarlem to Amsterdam behind an on-ramp with a very high inflow\n\\cite{Helb-book,TGF99-Treiber,GKT-scatter,Smulders1,Vladi-98}.\nBefore we present representative data for each traffic state,\nsome remarks about the presentation of the data are in order.\n\n%###################################################################\n\\subsection{Presentation of the Empirical Data}\n%###################################################################\n\nIn all cases, the traffic data were obtained\nfrom several sets of \ndouble-induction-loop detectors recording, separately for each lane,\nthe passage times and velocities of all vehicles. \nOnly aggregate information was stored.\nOn the freeways A8 and A9, the numbers of cars and trucks\nthat crossed a given detector on a given lane in each one-minute\ninterval,\nand the corresponding average\nvelocities was recorded.\n%Every vehicle longer than ??m is considered as a\n%truck.\nOn the freeways A5-South and A5-North, the data\nare available in form of a histogram for the velocity distribution.\nSpecifically, the measured velocities are divided into $n_r$ ranges\n($n_r=15$ for cars and 12 for trucks),\nand the number of cars and trucks driving in each range\nare recorded for every minute.\nThis has the advantage that more ``microscopic'' information \nis given compared to\none-minute averages of the velocity.\nIn particular, the local traffic density \n$\\rho^*(x,t)=Q/V^*$\ncould be estimated using the ``harmonic'' mean\n$V^*= \\erw{1/v_{\\alpha}}^{-1}$ \nof the velocity, instead of the arithmetic mean\n$\\rho=Q/V$ with $V=\\erw{v_{\\alpha}}$.\nHere, $Q$ is the traffic flow (number\nof vehicles per time iterval), and $\\erw{\\cdots}$\ndenotes the {\\it temporal} average \nover all vehicles $\\alpha$ \npassing the detector within the given time interval.\n\n%Especially for high densities and large \n%relative variations of the velocity, the harmonic mean gives a better\n%estimate than using the arithmetic mean \n{The harmonic mean value $V^*$ corrects for the fact\nthat the {\\em spatial} velocity distribution differs from the {\\em\nlocally measured} one} \\cite{Helb-book}. {However,}\nfor better comparison with those freeway data, where this information\nis not available, we\nwill use always the arithmetic velocity average $V$ in this paper.\nUnfortunately, the velocity intervals of the A5 data \nare coarse. In particular, the \nlowest interval ranges from 0 to 20 km/h.\nBecause we used\nthe centers of the intervals as estimates for the velocity, there is\nan artificial\ncutoff in the corrsponding flow-density diagrams \\ref{HCTemptheo}(b),\nand \\ref{PLCMLCemp}(c).\nBelow the line $Q_{\\rm min}(\\rho)=V_{\\rm min} \\rho$ with\n$V_{\\rm min}= 10$ km/h.\nBesides time series of flow and velocity\nand flow-density diagrams, we present the data also\nin form of three-dimensional plots of the\nlocally averaged velocity and traffic density\n as\na function of position and time.\nThis representation is particularly useful to\ndistinguish the different congested states by their\nqualitative spatio-temporal dynamics.\nTwo points are relevant, here.\nFirst, the smallest time scale of the collective effects (i.e., the\nsmallest period of\ndensity oscillations) is of the order of 3 minutes.\nSecond, the spatial resolution of the data is restricted to\ntypical distances between two neighboring detectors \nwhich are of the order of 1 kilometer.\nTo smooth out the small-timescale fluctuations, and\nto obtain a continuous function\n$Y_{\\rm emp}(x,t)$ from the one-minute values \n$Y(x_i,t_j)$ of detector $i$ at time $t_j$ with\n$Y=\\rho$, $V$, or $Q$, we applied\nfor all three-dimensional plots of {an} empirical quantity $Y$\nthe following smoothing and interpolation\nprocedure:\n%\n\\begin{equation}\n\\label{smooth}\nY _{\\rm emp}(x,t) = \\frac{1}{N} \\sum_{x_i} \\sum_{t_j} Y(x_i,t_j)\n\\exp\\left \\{- \\ \\frac{(x-x_i)^2}{2\\sigma_x^2}\n - \\ \\frac{(t-t_j)^2}{2\\sigma_t^2} \n \\right\\}.\n\\end{equation}\n%\nThe quantity\n\\begin{equation}\n\\label{normalization}\nN = \\sum_{x_i} \\sum_{t_j}\n\\exp\\left \\{ - \\ \\frac{(x-x_i)^2}{2\\sigma_x^2} \n - \\ \\frac{(t-t_j)^2}{2\\sigma_t^2} \n \\right\\},\n\\end{equation}\n%\nis a normalization factor.\nWe used smoothing times and length scales of\n$\\sigma_t=1.0$ min and\n$\\sigma_x=0.2$ km, {respectively}.\nFor {consistency,} we applied this smoothing operation \nalso to the simulation results.\nUnless explicitely stated otherwise, we will \nunderstand all empirical data as lane averages.\n\n\n%###########################################################\n\\subsection{\\label{sec:HCT}\nHomogeneous Congested Traffic}\n%###########################################################\n\n\nFigure \\ref{HCTemp} shows data of\na traffic breakdown on the A5-South on\nAugust 6, 1998. \nSketch \\ref{HCTemp}(a) shows the considered section. \nThe flow data at cross section D11 {in} Fig. \\ref{HCTemp}(b)\n{illustrate} that, between 16:20 h and 17:30 h,\nthe traffic flow on the right lane\ndropped to nearly zero.\nFor a short time interval between 17:15 h and 17:25 h \nalso the flow on the\nmiddle lane dropped to nearly zero.\nSimultaneously, there is a sharp drop of the velocity at this cross\nsection on all lanes, cf. Fig. \\ref{HCTemptheo}(d).\nIn contast, the velocities at the downstram\ncross section D12 remained relatively\nhigh during the same time.\nThis suggests\na closing of the right lane at a location\nsomewhere between the detectors\nD11 and D12. \n\nFigures \\ref{HCTemptheo}(d) and (f) show that,\nin most parts\nof the congested region, there were little variations of the velocity.\nThe traffic flow remained relatively high, which is a signature of\nsynchronized traffic \\cite{Kerner-sync}. \nIn the immediate upstream (D11) and downstream (D12) neighbourhood\nof the bottleneck, the amplitude of the fluctuations \nof traffic flow was\nlow as well, in particular, it was lower than in free traffic \n(time series at D11 and D12 for $t<$ 16:20 h, or $t>$ 18:00 h).\nFurther upstream in the congested region (D10), however,\nthe {fluctuation} amplitude increases.\nAfter the bottleneck {was} removed at about 17:35 h, the \npreviously fixed downstream front started moving\nin the {\\it up}stream direction at a characteristic velocity\nof about 15 km/h \\cite{Kerner-rehb98}.\nSimultaneously, the flow\nincreased to about 1600 vehicles/h,\nsee plots \\ref{HCTemptheo}(e) and \\ref{HCTemptheo}(g).\nAfter the congestion dissolved at about 17:50 h, the flow\ndropped to about 900 vehicles/h/lane, which was the inflow at that time.\n\nFigure \\ref{HCTemptheo}(a) shows the flow-density \ndiagram of the lane-averaged one-minute data.\nIn agreement with the absence of large oscillations ({like} stop-and-go\ntraffic), the regions of data points of free and congested traffic were \nclearly separated.\nFurthermore, \nthe transition from the free to the congested state and the reverse\ntransition showed a clear hysteresis.\n\nThe spatio-temporal \nplot of the local velocity in Fig. \\ref{HCT3d}(a)\nshows that \nthe incident induced a breakdown to an extended state\nof essentially homogeneous congested\ntraffic. Only near the upstream boundary, there were {small}\noscillations.\nWhile the upstream front \n(where vehicles {entered} the congested region) propagated upstream,\nthe downstream front (where vehicles {could} accelerate into free traffic)\nremained fixed at the bottleneck at $x\\approx 478$ km.\nIn the spatio-temporal plot of the traffic flow Fig. \\ref{HCT3d}(b),\none clearly can see the flow peak in the region $x>476$ km after the\nbottleneck was removed.\n\nWe estimate now the point in the phase diagram to which \nthis situation {belongs}. The average inflow $Q_{\\rm in}$ ranging from\n1100 vehicles/h at $t=$ 16:00 h to about 900 vehicles/h ($t=$ 18:00 h)\ncan be determined\nfrom an upstream cross section which is not reached by the congestion,\nin our case D6. Because the congestion\nemits no stop-and go waves, we conclude that the free traffic\nin the inflow region is stable,\n$Q_{\\rm in}(t)<Q_{\\rm c1}$.\nWe estimate the bottleneck strength\n$\\delta Q = Q_{\\rm out}-\\tilde{Q}_{\\rm out} \\approx$ 700 vehicles/h\nby identifying the time- and lane-averaged flow \nat D11 during the time of the incident \n(about 900 vehicles/h)\nwith the outflow $\\tilde{Q}_{\\rm out}$ from {the} bottleneck, and\nthe average flow of 1600 vehicles/h during the flow peak \n({when} the congestion dissolved) with\nthe {(universal) dynamical capacity $Q_{\\rm out}$ %of density clusters \non the homogeneous freeway (in the absence of a bottleneck-producing\nincident).} For the short time interval where\ntwo lanes were closed, we even have\n$\\tilde{Q}_{\\rm out} \\approx 500$ vehicles/h corresponding to\n$\\delta Q \\approx 1100$ vehicles/h.\n(Notice, that the lane averages {were always carried out \nover all} three lanes, also if lanes {were} closed.)\nFinally, we conclude from the oscillations near the upstream boundary of \nthe congestion, that the congested traffic flow\n$Q_{\\rm cong}=(Q_{\\rm out}-\\delta Q)$ is linearly unstable, but\nconvectively stable.\nThus, the breakdown corresponds to the HCT/OCT regime.\n\n\n%################## simulation HCT theo ##########################\n\n{We simulated the situation with the IDM parameters from Table\n\\ref{tab:param}, with upstream boundary conditions\ntaken from the data of cross section D6\n[cf. Fig. \\ref{HCTemptheo} (h) and (i)] and homogeneous von Neumann\ndownstream boundary conditions.}\nWe implemented the temporary bottleneck by locally increasing the\nmodel parameter $T$ to some value $T'>T$\nin an 1 km long section centered around the \nlocation of the incident. \nThis section represents the actually closed road section\nand the merging regions upstream and downstream from it.\nDuring the incident, we chose $T'$ such that the outflow \n$\\tilde{Q}'_{\\rm out}$ from the bottleneck agrees roughly with the\ndata of cross-section D11. \nAt the beginning of the\nsimulated incident, we increased $T'$ abruptly from $T=1.6$ s to\n$T'=5 s$, and decreased it linearly to 2.8 s during the time interval\n(70 minutes) of\nthe incident. Afterwards, we assumed again $T'=T=1.6$ s.\n\nThe grey lines of Figs. \\ref{HCTemptheo} (c) to \\ref{HCTemptheo} (j)\nshow time series of the simulated velocity and flow at some\ndetector positions.\nFigures \\ref{HCT3d}(c) and (d) show plots of the smoothed\nspatio-temporal velocity and flow, respectively.\n\nAlthough, in the microscopic picture, the modelled increase of the\nsafe time headway is quite different from lane changes before a\nbottleneck,\nthe qualitative dynamics is essentially\nthe same as that of the data. In particular,\n(i) the breakdown occured immediately after the bottleneck has been\nintroduced. \n(ii) As long as the bottleneck was active,\nthe downstream front of the congested state\nremained stationary and fixed at the bottleneck, while\nthe upstream front propagated further upstream.\n(iii) Most of the congested region consisted of HCT,\nbut oscillations appeared near the upstream front. \nThe typical period of the simulated oscillations\n($\\approx 3$ min), however,\n{was} shorter than that of the measured data ($\\approx $ 8 min).\n(iv) As soon as the bottleneck was removed, the downstream front\npropagated upstream with the well-known characteristic velocity\n$v_g=15$ km/h,\nand there was a flow peak in the downstream\nregions until the congestion {had} dissolved, cf. Figs.\n\\ref{HCTemptheo}(c), \\ref{HCTemptheo}(e), and \\ref{HCT3d}(d).\nDuring this time interval,\nthe velocity increased {\\it gradually} to the value for free traffic.\n\n{Some remarks on the apparently non-identical upstram boundary\nconditions in the empirical and simulated plots \\ref{HCT3d}(a) and\n\\ref{HCT3d}(c) are in order.\nIn the simulation,\nthe {\\em velocity} relaxes quickly from its prescribed value at the upstream\nboundary to\na value corresponding to {\\it free} equilibrium traffic at the given \ninflow. This is a rather generic effect which also\noccurs in macroscopic models \\cite{numerics}.\nThe relaxation takes places within the boundary region \n$3 \\sigma_x=0.6$ km needed \nfor the smoothing procedure (\\ref{smooth}) and is, therefore, \nnot visible in the figures. Consequently, the boundary conditions for\nthe velocity look different, although they have been taken from the data.\n%Although the boundary conditions are taken from the data, this\n%leads to an apparent difference in the\n%spatio-temporal velocity plots. \nIn contrast, the {\\em traffic flow} cannot\nrelax because of the conservation of the number of vehicles\n\\cite{numerics}, \nand the boundary conditions in the corresponding \nempirical and simulated plots look, therefore, consistent \n[see. Figs.~\\ref{HCT3d} (b) and (d)].\nThese remarks apply also to all other simulations below.\n}\n%###########################################################\n\\subsection{Oscillating Congested Traffic}\n%###########################################################\n\nWe now present data from a section of the A9-South near Munich.\nThere are two major intersections I1 and I2 with other \nfreeways, cf. Fig. \\ref{OCTemp}(a). In addition, the number of\nlanes is reduced from three to two downstream of I2.\nThere are three further\nsmall junctions between I1 and I2 which did not appear to be\ndynamically relevant.\n% (Eching km 514.83: 700->300 but 4->3 lanes -> effectively +100/lane,\n% Garching-N km 518.66 +200\n% Garching-S km 520.83 -700 and 3->2 at 522 -> effectively 0)\nThe intersections, however, were major bottleneck inhomogeneities.\nVirtually on each weekday, traffic broke down to \noscillatory congested traffic\nupstream of intersection I2. In addition, we recorded two breakdowns to OCT \n{upstream of} I1 during the observation period of 14 days.\n\nFigure \\ref{OCTemp}(b) shows a spatio-temporal \nplot of the smoothed velocity \nof the OCT state occurring upstream of I2 during the morning rush hour of \nOctober 29, 1998.\nThe oscillations with a period of about 12 min\nare clearly visible in both the time series of the velocity\ndata, plots \\ref{OCTemp}(d)-(f), and the flow,\n\\ref{OCTemp}(g)-(i). In contrast to the observations of\nRef. \\cite{Kerner-wide}, the {density waves apparently} did not merge.\nFurthermore, the velocity in the OCT state rarely exceeded 50 km/h, \ni.e., there was no free traffic between the clusters,\na signature of OCT in comparison with triggered stop-and go waves.\nThe clusters propagated upstream \nat a remarkably constant velocity of 15 km/h, which is \nnearly the same propagation velocity as that of the \ndetached downstream front of the HCT state described above.\n\nFigure \\ref{OCTemp}(c) shows the \nflow-density diagram of this congested state. In contrast to the\ndiagram \\ref{HCTemptheo}(b) for the HCT state, there is no\nseparation between the regions of free and congested\ntraffic. Investigating flow-density diagrams of many other occurrences \nof HCT and OCT, it turned out that this difference can also be used to\nempirically distinguish HCT from OCT states.\n\n\n%################ OCT theo #############################\n\nNow we show that this breakdown to OCT can be qualitatively reproduced \nby a microsimulation with the IDM.\nAs in the previous simulation, we used empirical data for the upstream \nboundary conditions. {(Again, the velocity relaxes quickly to a local\nequilibrium, and only for this reason it looks different from the\ndata.)} We implemented the \nbottleneck by locally increasing the safe time headway in the \ndownstream region.\nIn contrast to the previous simulations, the \nlocal defect causing the breakdown was a permanent\ninhomogeneity of the infrastructure (namely an intersection\nand a reduction from three to two lanes) rather than \na temporary incident. Therefore, we did not assume any time dependence\nof the bottleneck.\nAs upstream boundary conditions, we chose the data of D20, \nthe only cross section where\nthere was free traffic during the whole time interval considered here.\n{Furthermore, we used homogeneous von Neumann boundary conditions at\nthe downstream boundary.}\nWithout assuming a higher-than-observed level of inflow, the\nsimulations showed no traffic breakdowns at all. \nObviously, on the freeway A9 the capacity per lane is lower than\non the freeway A5 (which is several hundret kilometers apart). \nThis lower capacity has been taken into account by a site-specific,\nincreased value of $T=2.2$\\ s in the upstream region $x<-0.2$ km.\n%{We checked, if this difference can be attributed to different\n%truck percentages, but we found about 15\\%\n%trucks on both roads, so the reason for this difference is not yet \n%explained.}\nAn even higher value of $T'=2.5$ s was used in the bottleneck region \n$x>0.2$ km, with a linear increase in the 400 m long transition zone.\nThe corresponding microsimulation is shown in Fig.~\\ref{OCTtheo}.\n%, we show a microsimulation using an increased value \n%$T=2.2$ s \n%in the upstream region $x<-0.2$ km, and a further increased value\n%$T'=2.5$ s in the bottleneck region \n%$x>0.2$ km, with a linear increase in the 400 m long transition zone.\n\n\nIn this way,\nwe obtained a qualitative agreement with the A9 data.\nIn particular, \n(i) traffic broke down at the bottleneck spontaneously, in contrast to\nthe situation on the A5.\n(ii) Similar to the situation on the A5,\nthe downstream front of the resulting OCT state was fixed at the\nbottleneck while the upstream front propagated further upstream.\n(iii) The oscillations showed no mergers and propagated with about 15\nkm/h in upstream direction. Furthermore, their \nperiod (8-10 min) is comparable \nwith that of the data, and the velocity in the OCT region was\nalways much lower than that \nof free traffic.\n(iv) After about 1.5 h, the upstream front reversed\nits propagation direction and eventually dissolved.\nThe downstream front remained always fixed at the permanent inhomogeneity.\n{Since, at no time, there is a clear transition from congested to free traffic\nin the region upstream of the bottleneck (from which \none could determine\n$Q_{\\rm out}$ and compare it with the outflow\n$\\tilde{Q}_{\\rm out}\\approx Q'_{\\rm out}$ from the bottleneck), \nan estimate of the empirical bottleneck strength \n$(Q_{\\rm out} - Q'_{\\rm out})$ is\ndifficult. Only at D26, for times around 10:00 h, \nthere is a region where the vehicles accelerate.\nUsing the corresponding traffic flow\nas coarse estimate for $Q_{\\rm out}$, and the minimum smoothed flow\nat D26 (occurring between $t\\approx$ 8:00 h and 8:30 h) as an estimate for \n$Q'_{\\rm out}$,\nleads to an empirical bottleneck strength\n$\\delta Q$ of 400 vehicles/h. This is consistent with\nthe OCT regime in the phase diagram Fig. \\ref{phasediag}.\nHowever, estimating the theoretical bottleneck strength directly from \nthe difference $(T-T')$, using Fig. \\ref{fig:Qout}, \nwould lead to a smaller value.\nTo obtain a full quantitative agreement, it would probably be\nnecessary to calibrate more than just {\\em one} IDM parameter to the\nsite-specific driver-vehicle behavior, or to\nexplicitely model the bottleneck by on- and off-ramps.}\n\n%###########################################################\n\\subsection{Oscillating Congested Traffic Coexisting with Jammed Traffic}\n%###########################################################\n\nFigure \\ref{flugzeug} shows an example of a more complex\ntraffic breakdown that occurred on the\nfreeway A8 East from Munich to Salzburg during the evening rush hour\n{on} November 2, 1998. \nTwo different kinds of bottlenecks were involved, (i) a relatively\nsteep uphill gradient from $x=38$ km to $x=40$ km\n(``Irschenberg''), and (ii) an\nincident leading to the closing of one of the three lanes between\nthe cross sections D23 and D24 from $t=$\\,17:40 h until $t=$\\,18:10 h.\nThe incident was deduced from the velocity and flow data of the\ncross sections D23 and D24 as described in Section IV B.\n%\\ref{sec:HCT}.\nAs further inhomogeneity, there is a small junction at about\n$x=41.0$ km. {However,} since the involved {ramp} flows were very small,\nwe {assumed}\nthat the junction {had no} dynamical effect.\n\nThe OCT state caused by the uphill gradient had the same qualitative\nproperties as that on the A9 South. In particular, the \nbreakdown was triggered by a short flow peak corresponding \n to a velocity dip in Plot\n\\ref{flugzeug}(b), the downstream\nfront was stationary, while the upstream front moved, \nand all oscillations propagated upstream with a constant velocity.\nThe combined HCT/OCT state caused by the incident had similar\nproperties {as} that on the A5 South. In particular, there was HCT\nnear the location of the incident,\ncorresponding to the downstream boundary of \nthe velocity plot \\ref{flugzeug}(b),\nwhile oscillations developped further upstream. Furthermore, similarly \nto the incident on the A5, the\ndownstream front propagated upstream as soon as the incident was %removed.\n{cleared.} \nThe plot clearly shows, that the HCT/OCT state propagated \nseemingly unperturbed through the\nOCT state upstream of the permanent uphill bottleneck. The \nupstream propagation velocity\n$v_g=15$ km/h of {\\it all} perturbations in the complex state was \nremarkably constant,\nin particular that of (i) the upstream and downstream fronts\nseparating the\nHCT/OCT state from free traffic (for $x>40$ km at $t\\approx$\\,17:40 h and\n18:10 h, respectively), (ii) the fronts separating the HCT/OCT \nfrom the OCT state (35 km $\\le x \\le$ 40 km), and (iii) \n%the propagation velocity of \nthe oscillations within both the HCT and HCT/OCT states.\nIn contrast, the propagation velocity and direction of the\nfront separating the OCT from {\\it free} traffic\nvaried with the inflow.\n\nWe simulated this scenario using empirical (lane averaged) data both\nfor the upstream and downstream boundaries.\nFor the downstream boundary, we used only the velocity information.\nSpecifically, \nwhen, at {some} time $t$, a simulated vehicle $\\alpha$ crosses \nthe downstream boundary of the simulated section $x\\in[0,L]$, \nwe set its\nvelocity to that of the data, $v_{\\alpha}=V_{D15}(t)$, if \n$x_{\\alpha}\\ge L$, and use the velocity and positional information \nof this vehicle to determine the \nacceleration of the vehicle $\\alpha+1$ behind. Vehicle $\\alpha$ is\ntaken out of the simulation as soon as vehicle\n$\\alpha+1$ has crossed the boundary. Then, the velocity of vehicle\n$(\\alpha+1)$ is set to the actual boundary value, and so on.\nThe downstream boundary conditions are only relevant for\nthe time interval around $t=18:00$ h where traffic near this boundary\nis congested. For other time periods,\nthe simulation result is equivalent to using homogeneous\nVon-Neumann downstream boundary conditions.\n\nWe modelled the stationary uphill bottleneck in the usual way by\nincreasing the parameter $T$ to a constant value $T'>T$ in the downstream \nregion. The incident {was already reflected} \nby the downstream boundary conditions.\nFigure \\ref{flugzeug}(c) shows the simulation result in form\nof a spatio-temporal plot of the smoothed velocity.\nNotice that, by using only the\nboundary conditions and a stationary bottleneck as \nspecific information, we obtained a qualitative agreement \nof nearly all dynamical collective\naspects of the {whole complex scenario} described above.\nIn particular, for all times $t<$\\,17:50 h, there was free traffic at both\nupstream and downstream detector positions. Therefore, the boundary \nconditions (the detector data)\ndid not contain any explicit information about the\nbreakdown {to OCT inside the road section,} \nwhich nevertheless was reproduced correctly {as an emergent phenomenon}.\n\n%{\\bf [die Bottleneckstaerken wuerde ich hier weglassen.\n%Es ist ziemlich spekulativ, Qout aus D16 um ungefaehr 19:00h\n%zu bestimmen (einzige Stelle, wo Fahrzeuge in freien Verkehr\n%beschleuigen in der upstream region), da dort die Autobahn viel\n%gerader ist. Man wuerde Qout=1900/h und (von D21) $\\tilde{Q}_{\\rm\n%out}=1400/h$ erhalten, was aber vermutlich die Bottleneckstaerke\n%ueberschaetzt. Ausserdem muesste man dann zusatezliche Bilder von Q(t) \n%von beiden Querschnitten bringen, was wohl wirklich insgesamt zu viel\n%Material waere.\n%Von $T-T'$ erhaelt man etwa $\\delta Q=$ 250/h.]} \n\n\n\n%###########################################################\n\\subsection{Pinned and Moving Localized Clusters} % {\\bf Ueberschrift geaendert}}\n%###########################################################\n%\nFinally, we consider a 30 km long section\nof the A5-North depicted in Fig. \\ref{PLCMLCemp}(a).\nOn this section, we found \none or more traffic breakdowns {on} six out of 21 days, all of them\nThursdays or Fridays. \nOn three out of 20 days,\nwe observed one or more \nstop-and-go waves separated by free traffic. \nThe stop-and go waves were\ntriggered near an intersection and agreed qualitatively with the\nTSG state of the phase diagram. \n{On} one day, two isolated {density} clusters\npropagated through the considered region\nand did not trigger any secondary clusters, which is consistent\nwith moving localized clusters (MLC) and will be discussed below.\nMoving localized clusters were observed quite\nfrequently on this freeway section \\cite{Kerner-rehb96}.\nAgain, they have a constant upstream propagation speed of about 15 km/h,\nand a characteristic outflow \\cite{KK-94}.\nIn addition, we found four breakdowns to OCT, and ten occurrences of\npinned localized clusters (PLC).\nThe PLC states emerged either\nat the intersection I1 (Nordwestkreuz Frankfurt), or\n1.5 km downstream of intersection I2 (Bad Homburg) at cross section \nD13. \nFurthermore, the downstream fronts of all four OCT states\nwere fixed at the latter location. \n\n{On August 6, 1998, we found an interesting transition from an OCT state\nwhose downsteam front was at D13, to a TSG state with a downstram\nfront at intersection I2 (D15).\nConsequently, we conclude\nthat, around detector D13, there is a stationary\nflow-conserving bottleneck with a stronger effect than the\nintersection itself.\nIndeed, there is an uphill section and a relatively sharp \ncurve at this location of the A5-North, which may be the reason for the\nbottleneck.\nThe sudden change of the active bottleneck on August 6 can be explained\nby perturbations and the hysteresis associated with breakdowns.\n}\n\n{The different types of traffic breakdowns are\nconsistent with the relative locations \nof the traffic states in the\n$(Q_{\\rm in},\\delta Q)$ space of the phase\ndiagram in Fig. \\ref{phasediag}.\nThree of the four occurrences of OCT and two of the three TSG states\nwere on Fridays (August 14 and August 21, 1998), on which traffic flows\nwere about 5\\%\n{\\it higher} than on our reference day (Friday, August 7, 1998), which\nwill be discussed in detail below. \nApart from the coexistent PLCs and MLCs\nobserved on the reference day, all PLC states occurred on Thursdays,\nwhere average traffic flows were about 5\\%\n{\\it lower} than on the reference day.\nNo traffic breakdowns were observed on Saturdays to Wednesdays, \nwhere the traffic flows were at least 10\\% \nlower compared to the reference day.\nAs will be shown below, for complex bottlenecks like intersections,\nthe coexistence of MLCs and PLCs\nis only possible for flows just above those triggering pure\nPLCs, but below those triggering OCT states.\nSo, we have, with increasing flows, the sequence FT, PLC, MLC-PLC, and \nOCT or TSG states, in agreement with the theory.\n}\n\nNow, we discuss the traffic breakdowns in August 7, 1998 in detail.\nFigure \\ref{PLCMLCemp}(b) shows the situation \nfrom $t=$ 13:20 h until 17:00 h in form of a spatio-temporal\nplot of the smoothed density. \nDuring the whole time interval, there was a pinned localized cluster \nat cross section D13. \nBefore $t=$\\,14:00 h, the PLC state\nshowed distinct oscillations (OPLC), while it was essentially\nstationary (HPLC) afterwards. \nFurthermore, two moving localized clusters (MLC) of unknown origin \npropagated through nearly the whole displayed section \nand also through a 10 km long downstream section (not shown\nhere) giving a total of at least 30 km.\nRemarkably, as they crossed the PLC at D13,\nneither of the congested states seemed to be affected.\nThis complements the observations of Ref. \\cite{Kerner-rehb96},\ndesribing MLC states that propagated unaffected through \nintersections {in the absence of} PLCs.\n{As soon as the first MLC state reached\nthe location of the on-ramp of intersection I1 ($x=488.8$ km, \n$t\\approx$ 15:10 h), it triggered\nan additional pinned localized cluster, which dissolved at\n$t\\approx$ 16:00 h. The second MLC dissolved as soon as it reached\nthe on-ramp of I1 at $t\\approx 16:40$ h.}\n\nFigure \\ref{PLCMLCemp}(c) demonstrates that\nthe MLC and PLC states have characteristic signatures also in the\nempirical flow-density diagram.\nAs is the case for HCT and OCT, the PLC state is characterized\nby a two-dimensional flow-density regime (grey\nsquares). In contrast to the former states, however, there is no flow\nreduction (capacity drop) with respect to \nfree traffic (black bullets). As is the case for flow-density diagrams\nof HCT {compared to} OCT, it is expected that HPLCs are\ncharacterized by an isolated region, while the points\nof OPLC lie in a region which is connected to the region\nfor free traffic.\nDuring the periods were the MLCs crossed the PLC at D13,\nthe high traffic flow of the PLC state dropped drastically,\nand the traffic flow had essentially the property of the MLC,\n{see} also the velocity plot \\ref{PLCMLCemp}(e).\nTherefore, we omitted in the PLC data the \npoints corresponding to these intervals.\n\nThe black bullets for densities $\\rho>30$ vehicles/km\nindicate the region of the MLC (or TSG) states.\nDue to the aforementioned difficulties in determining the traffic\ndensity for very low velocities,\nthe theoretical line $J$ given by \n$Q_{\\rm J}(\\rho)= Q_{\\rm out} \n - (Q_{\\rm out}-Q_{\\rm jam})(\\rho-\\rho_{\\rm out})/\n (\\rho_{\\rm jam}-\\rho_{\\rm out})$\n(see Ref.~\\cite{Kerner-wide} and Fig.~\\ref{PLCMLCemp}(b))\nis hard to {\\em find empirically.} In any case, {the data suggest that \nthe line $J$ would lie} below the PLC region.\n\n\n\n%############### MLC+PLC(mic), TSG+PLC(mac) theo ####################\n\nTo simulate this scenario it is important that the PLC \nstates occurred in or near the freeway intersections.\nBecause at both intersections, the off-ramp is upstream of the on-ramp \n[Fig. \\ref{PLCMLCemp}(a)], the local flow at these locations is lower.\nIn the following, we will investigate the region around I2.\n{During the considered time interval, the average traffic flow\nof both, the on-ramp and the off-ramp was about 300 vehicles per hour\nand lane. With the exception of the time intervals, during which\nthe two MLCs pass by, we have about\n1200 vehicles per hour and lane at I2 (D15), \nand 1500 vehicles per hour and lane upstram (D16) and\ndownstream (D13) of I2.\nThis corresponds to an {\\it increase} of the effective \ncapacity {by} $\\delta Q\\approx -300$ vehicles per hour and lane\nin the region between the off-ramp and the subsequent on-ramp.}\n\nIn the simulation, we captured this \nqualitatively by {\\it decreasing}\nthe parameter $T$ in a section \n$x\\in[x_1,x_2]$ upstream of the empirically observed PLC\nstate. The hypothetical bottleneck located at D13, i.e., \nabout 1 km upstream of the on-ramp, {was neglected}. \n{Using real traffic flows as upstream and downstram\nboundary conditions and varying only the model parameter\n$T$ within and outside of the intersection,\nwe could not obtain\nsatisfactory simulation results.\nThis is probably because of the relatively high and fluctuating \ntraffic flow on this highway. It remains to be shown if \nsimulations with other model parameters can successfully reproduce the \nempirical data when applying real boundary conditions. \nNow, we show that the\nmain {\\it qualitative} feature on this highway, namely, the coexistence \nof pinned and moving localized clusters can, nevertheless,\nbe captured by our model.\nFor this purpose, we assume a constant inflow $Q_{\\rm in}=1390$\nvehicles per hour and lane\nto the freeway, with the corresponding equilibrium velocity. \nWe initialize the PLC by a triangular-shaped density peak in the\ninitial conditions,\nand initialize the MLCs by reducing the velocity at the downstream\nboundary to $V=12$ km/h during two five-minute intervals (see caption \nof Fig. \\ref{PLCMLCmic}). \n}\n%\n%(ALTE BESCHREIBUNG DER SIM. MIT EMP. BC)\n%Figure \\ref{PLCMLCmic}\n%shows the result of the microscosimulation of the 13 km long\n%section between cross sections D20 and D7 {\\bf [nicht D5?]} including \n%intersection I2 with the bottleneck at D13.\n%The off-ramp at D16 with an average outflow of 400 vehicles/h/lane\n%{Since} the on-ramp flow was comparable to that of the \n%off-ramp, we expect $\\delta Q\\approx 0$ downstream of the merging\n%zone. The overall \n%effect is a local {\\it increase} of the traffic capacity, which we captured\n%qualitatively by {\\it decreasing}\n%the parameter $T$ in a section upstream of the empirically observed PLC\n%state. {The hypothetical bottleneck} located at D13, i.e., \n%about 1 km upstream of the on-ramp, {was not modelled}. \n%As in the simulation of the A8 East, we\n%used empirical data for both upstream\n%and downstram boundary conditions. Furthermore, we \n%initialized the simulation at\n%$t=$\\,13:20 h with {linearly} interpolated traffic data.\n\nAgain, we obtained a qualitative agreement with the observed dynamics.\nIn particular, the simulation showed that also an \n{\\it increase} of the local capacity\nin a bounded region\ncan lead to pinned localized clusters.\nFurthermore, the regions of the MLC and PLC\nstates in the flow-density diagram were reproduced\nqualitatively, in particular, the coexistence of\npinned and moving localized clusters.\nWe did not observe such a coexistence\nin the simpler system \n{underlying the phase diagram in Fig.~\\ref{phasediag}, which \ndid not include a second low-capacity stretch \nupstream of the high-capacity stretch}.\n\n%######################################\n{To explain the coexistence of PLC and MLC in the more complex\nsystem consisting of one high-capacity stretch in the middle of\ntwo low-capacity stretches, it is useful to interpret the inhomogeneity not \nin terms of a\nlocal capacity {\\it increase} in the region $x\\in[x_1,x_2]$, but \nas a capacity {\\it decrease} for $x<x_1$ and $x>x_2$.\n(For simplicity, we will not explicitely include the 400 m long\ntransition regions of capacity increase at $x_1$ and decrease at $x_2$\nin the following discussion.) \nThen, the location $x=x_2$ can be considered as the beginning\nof a bottleneck, as in the system underlying the phase\ndiagram.\nIf the width\n$(x_2-x_1)$ of the region with locally increased capacity\nis {\\it larger than the width of PLCs}, such clusters are\npossible under the same conditions as in the standard phase diagram.\nIn particular, traffic in the standard system is stable in regions upstream of\na PLC, which is the reason why any additional\nMLC, triggered somewhere in the downstream region $x>x_2$\nand propagating upstream, will vanish as soon as it crosses \nthe PLC at $x=x_2$.\nHowever, this disappearance is not instantaneous, but the MLC will continue to\npropagate upstream for an additional \nflow-dependent ``dissipation distance'' or ``penetration depths''.\nIf the width $(x_2-x_1)$ is\n{\\it smaller than the dissipation distance for MLCs},\ncrossing MLCs\nwill not fully disappear before they reach the upstream \nregion $x<x_1$. There,\ntraffic is metastable again, so that the MLCs can persist.\nSince, in the metastable regime, the outflow of MLCs is equal to their\ninflow (in this regime, MLCs are equivalent to\n``narrow'' clusters, cf. Ref. \\cite{KK-94}), the passage of the MLC\ndoes not change the traffic flow at the position of the PLC,\nwhich can, therefore, persist as well.\n\nWe performed several simulations varying the inflow within the range\nwhere PLCs are possible.\nFor smaller inflows, the dissipation distance became smaller than\n$(x_2-x_1)$, and the moving localized cluster was absorbed within\nthe inhomogeneity.\nAn example for this can be seen in Fig. \\ref{PLCMLCemp}(b) at $t\\approx$\n16:40 h and $x\\approx$ 489 km.\nLarger inflows lead to an extended OCT state\nupstream of the capacity-increasing defect, which is\nalso in accordance with the observations.\n}\n\n%###################################################################\n\\section{Conclusion}\n%###################################################################\n\nIn this paper, we investigated, {to what extent}\nthe phase diagram Fig,\n\\ref{phasediag} can serve as a general description of \ncollective traffic dynamics in open, inhomogeneous systems.\nThe original phase diagram was formulated for on-ramps \nand resulted\nfrom simulations with macroscopic models \\cite{Phase,Lee}.\nBy simulations with a new car-following model we showed that\none can obtain the same phase diagram from microsimulations.\nThis includes even such subtle details as the small region of\ntristability. \nThe proposed intelligent-driver model (IDM) \nis simple, has only a few intuitive\nparameters with realistic values,\nreproduces a realistic collective dynamics,\nand also leads to a plausible ``microscopic''\nacceleration and deceleration behaviour of single drivers.\n%We also performed simulations with the optimal velocity model\n%\\cite{Bando}.\n%It turned out that the results depend\n%very sensitively on the model parameters, and we could not reproduce\n%most of the experimental findings\nAn interesting open question is whether the phase diagram\ncan be reproduced also with cellular automata.\n\nWe generalized the phase diagram from on-ramps\nto {other kinds of} inhomogeneities. Microsimulations\nof a flow-conserving\nbottleneck realized by\na locally increased safe time headway suggest that, with respect to\ncollective effects outside of the immediate neighbourhood of the\ninhomogeneity, all types of bottlenecks can be characterized by a single\nparameter, the bottleneck strength. This means, that the type of\ntraffic breakdown depends {essentially} on \nthe two control parameters\nof the phase diagram {only,} namely the traffic flow, \nand the bottleneck strength. {However, in some multistable regions,\nthe history (i.e., the previous traffic dynamics) matters as well.}\nWe checked this also by macroscopic simulations with \nthe same type of flow-conserving inhomogeneity and with\nmicrosimulations using a locally decreased desired velocity as\nbottleneck \\cite{TGF99-Treiber}. In all cases, we\nobtained qualitatively the same phase diagram.\nWhat remains to be done is to confirm the phase diagram also\nfor microsimulations of on-ramps. These can be implemented\neither by explicit multi-lane car-following\nmodels \\cite{Howe,Applet-engl},\nor, in the framework of single-lane\nmodels, by placing additional\nvehicles in suitable gaps between vehicles in the ``ramp'' region.\n\nBy presenting empirical data of congested traffic, we showed that\nall congested states proposed {by} the phase diagram were observed in\nreality, among them localized and extended states which can be\nstationary as well as oscillatory, furthermore, \nmoving or pinned localized clusters (MLCs or PLCs, respectively). \nThe data suggest that the typical {kind} of traffic\ncongestion depends on the specific freeway. This is in accordance\nwith other observations, for example, moving localized clusters on the\nA5 North \\cite{Kerner-rehb96}, or homogeneous congested traffic\n(HCT) on the A5-South\n\\cite{Kerner-sync}. {In contrast to another empirical study\n\\cite{Kerner-wide}, the frequent oscillating states (OCT) \nin our empirical data\ndid not show mergings of density clusters, \nalthough these can be reproduced with our model with other\nparameter values \\cite{coexist}.}\n%{\\bf (Furthermore, there is only one published \n%observation of PLCs \\cite{Lee-emp}.)}\n\nThe relative positions of the various observed\ncongested states in the phase diagram were\nconsistent with the theoretical predictions.\n{In particular, when increasing the {\\it traffic flow} on the freeway,\nthe phase diagram predicts (hysteretic) transitions from free\ntraffic to PLCs, and then\nto extended congested states. \nBy ordering the various forms of congestion on the A5-North with respect to\nthe average\ntraffic flow, the observations agree with these predictions. Moreover,\ngiven an extended congested state\nand increasing the {\\it bottleneck strength}, the phase\ndiagram predicts\n(non-hysteretic) transitions from triggered stop-and-go waves\n(TSG) to OCT, and then to HCT.\nTo show the qualitative agreement with the data, we had to estimate \nthe bottleneck strength $\\delta Q$. This was done directly by identifying\nthe bottleneck strength with ramp flows, e.g.,\non the A5-North, \nor indirectly, by comparing\nthe outflows from congested traffic with and without a\nbottleneck, e.g., for\nthe incident on the A5-South. With OCT and TSG on the A5-North, but\nHCT on the A5-South, where the bottleneck strength was much higher,\nwe obtained again the right behavior.}\nHowever, one needs a larger base of data to determine an empirical phase \ndiagram, in particular with its boundaries between the different\ntraffic states. Such a phase diagram has been {proposed} \nfor a Japanese highway \\cite{Lee-emp}. \nBesides PLC states, many breakdowns on \nthis freeway lead to extended congestions with fixed\n{downstream {\\it and} upstream} fronts. We did not observe such states \non German freeways\nand believe that the fixed upstream fronts were the result of a further\ninhomogeneity, but this remains to be investigated.\n\nOur traffic data indicate that the majority of traffic\nbreakdowns is triggered by some kind of {\\it stationary}\ninhomogeneity, so that the\nphase diagram is applicable. \n{Such inhomogeneities can be of a very general nature.\nThey include not only ramps, gradients, lane narrowings or -closings, \nbut also incidents in the {\\it oppositely} flowing traffic.\nIn the latter case, the bottleneck is constituted by a\ntemporary loss of concentration and by braking maneuvers \nof curious drivers {\\it at a fixed location}.}\nFrom the more \nthan 100 breakdowns on various German and\nDutch freeways investigated by us, there were only four {cases}\n(among them the two moving localized clusters\n{in} Fig. \\ref{PLCMLCemp}), where we could not explain the breakdowns \nby some sort of stationary bottleneck within the \nroad sections for which data were available to us.\nPossible explanations for the breakdowns in the four remaining cases\nare not only spontaneous breakdowns \\cite{sync-Letter}, but also\nbreakdowns triggered by a {\\it nonstationary} perturbation,\ne.g., moving ``phantom bottlenecks'' caused by two \ntrucks overtaking each other \\cite{Gazis-Herman}, or\ninhomogeneities outside the considered sections. \nOur simulations showed that stationary downstream fronts\nare a signature of non-moving\nbottlenecks. \n\n\nFinally, we could qualitatively reproduce the collective dynamics of\n{several} rather complex traffic breakdowns by microsimulations\nwith the IDM, using empirical data for the boundary conditions.\nWe varied only a {\\it single} model parameter, the safe time headway,\nto adapt the model to the individual capacities of the different\nroads, {\\it and} to implement the bottlenecks.\nWe also performed separate macrosimulations with \nthe GKT model and could reproduce the observations\nas well. Because both models are effective-single lane models, this\nsuggests that lane changes are not\nrelevant {to reproduce} \nthe collective dynamics causing the different types of \ncongested traffic. \nFurthermore, we assumed identical vehicles and therefore conclude\nthat also the heterogeneity of real traffic\nis not necessary for the basic mechanism of traffic instability.\nWe expect, however, that other yet unexplained aspects of congested\ntraffic require\na microscopic treatment of both, multi-lane traffic and heterogeneous \ntraffic. These aspects include\nthe wide scattering of flow-density data\n\\cite{Kerner-wide,GKT-scatter} {(see Fig.~\\ref{mixfund})},\nthe description of platoon formation \\cite{platoon-multilane},\n{and the realistic} simulation of speed limits %{\\it vs.} uphill gradients \n\\cite{note-phase2-speedlimit},\n%and the controversial determination of the density \n%from detector data.\nfor which a multi-lane generalization of the\nIDM seems to be promising \\cite{Applet-engl}.\\\\[4mm]\n{\\bf Acknowledgments:}\nThe authors want to thank for financial support by the BMBF (research\nproject SANDY, grant No.~13N7092) and by the DFG (grant No.~He 2789). \nWe {are also grateful to} the {\\it Autobahndirektion S\\\"udbayern}\nand the {\\it Hessisches Landesamt f\\\"ur Stra{\\ss}en und Verkehrswesen}\nfor providing {the} freeway data.\n\n% ##############################################################\n% References \n% ##############################################################\n\n\n% mit bibtex\n%\\bibliographystyle{/home/TeTeX/inputs/revtex/prsty}\n%\\bibliography{/home/treiber/tex/bibtex/traffic}\n\n% oder direkt\n% \\input{antrag.bbl}\n\\begin{thebibliography}{10}\n\n\\bibitem{Hall1}\nF.~L. 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E {\\bf 58}, 133 (1998).\n\n\\bibitem{MITSIM}\n{\\sloppy For an overview, see the internet page\\\\ {\\tt\n http://hippo.mit.edu/products/mitsim/main.html}.}\n\n\\bibitem{Howe}\nL.~D. Howe, Physica A {\\bf 246}, 157 (1997).\n\n\\bibitem{coexist}\nM. Treiber and D. Helbing, \n``Explanation of observed features of self-organization in traffic flow''.\n Preprint cond-mat/9901239.\n\n\\bibitem{TGF99-Treiber}\nM. Treiber, A. Hennecke, and D. Helbing, in {\\em Traffic and Granular Flow\n '99}, edited by D. Helbing, H.~J. Herrmann, M. Schreckenberg, and D.~E. Wolf\n (Springer, Berlin, 2000), in print. \n\n\\bibitem{Biham}\nO. Biham, A.~A. Middleton, and D. Levine, Phys. Rev. A {\\bf 46}, R6124\n (1992).\n\n\\bibitem{Cremer-multi}\nM. Cremer and J. Ludwig, Math. Comput. Simulation {\\bf 28}, 297 (1986).\n\n\\bibitem{Nagel-S}\nK. Nagel and M. Schreckenberg, J. Phys. I France {\\bf 2}, 2221 (1992).\n\n\\bibitem{Barlovic}\nR. Barlovic, L. Santen, A. Schadschneider, and M. Schreckenberg, {\\em Traffic\n and Granular Flow '97} (Springer, Singapore, 1998), p.\\ 335.\n\n\\bibitem{Helb-sblock}\nD. Helbing and B.~A. Huberman, Nature {\\bf 396}, 738 (1998).\n\n\\bibitem{Wolf-Galilei}\nD.~E. Wolf, Physica A {\\bf 263}, 438 (1999).\n\n\\bibitem{Kuehne}\nR.~D. K\\\"uhne, in {\\em Proceedings of the \n 9th International Symposium on Transportation and Traffic Theory},\n edited by I. Volmuller and R. Hamerslag\n (VNU Science Press, Utrecht, 1984), p. 21\n\n\\bibitem{Treiterer}\nJ. Treiterer and J.~A. Myers, in {\\em Proc. 6th Int. Symp. on Transportation\n and Traffic Theory}, edited by D.~J. Buckley (Elsevier, New York, 1974), p.\\\n 13, empirical observation of trajectories.\n\n\\bibitem{TGF99-Hennecke}\nA. Hennecke, M. Treiber, and D. Helbing, in {\\em Traffic and Granular Flow\n '99}, edited by D. Helbing, H.~J. Herrmann, M. Schreckenberg, and D.~E. Wolf\n (Springer, Berlin, 2000), p.\\ to be published.\n\n\\bibitem{Reuschel}\nA. Reuschel, {\\\"O}sterreichisches Ingenieur-Archiv {\\bf 4}, 193 (1950).\n\n\\bibitem{Chandler}\nR.~E. Chandler, R. Herman, and E.~W. Montroll, Operations Research {\\bf 6},\n 165 (1958).\n\n\\bibitem{Herman59}\nR. Herman, E.~W. Montroll, R.~B. Potts, and R.~W. Rothery, Operations Research\n {\\bf 7}, 86 (1959).\n\n\\bibitem{Gazis61}\nD.~C. Gazis, R. Herman, and R.~W. Rothery, Operations Research {\\bf 9}, 545\n (1961).\n\n\\bibitem{Edie}\nL.~C. Edie, Transp. Res. B {\\bf 28}, 66 (1961).\n\n\\bibitem{Bleile-Diss}\nT. Bleile, Modellierung des Fahrzeugfolgeverhaltens im innerst\\\"adtischen\nPKW-Verkehr,\n(Phd thesis, University of Stuttgart, Germany, 1999).\n\n\\bibitem{Gipps86}\nP.~G. Gipps, Transp. Res. {\\bf 20 B}, 403 (1986).\n\n\\bibitem{Nagatani-kink}\nM. Muramatsu and T. Nagatani, Phys. Rev. E {\\bf 60}, 180 (1999).\n\n\\bibitem{note-phase2-micperf}\nThe simulations were performed on a personal workstation with a 433 Mhz Alpha\n processor. \n%The timestep of the explicit numerical scheme was $\\Delta t=0.4$ s.\n\n\\bibitem{Kerner-ramp}\nB.~S. Kerner, P. Konh{\\\"a}user, and M. Schilke, Phys. Rev. E {\\bf 51}, 6243\n (1995).\n\n\\bibitem{note-phase2-speedlimit}\n{Since} speed limits lead to reduced desired velocities for many drivers as\n well, one might conclude that speed limits {also cause} undesirable local\n bottlenecks. The main effect of speed limits is, however, to reduce\n the velocity variance which leads to less braking {maneuvers\n } and {\\it de\n facto} to a higher capacity. In contrast, gradients {mainly}\n influence \n slow vehicles (trucks), leading to higher values for the velocity variance.\n\n\\bibitem{Gazis-Herman}\n D.~C. Gazis and R. Herman, \n Transportation Science {\\bf 26}(3),223--229 (1992).\n%\\bibitem{note-phase2-A5}\n%There is an uphill section and a relatively sharp curve at this location of the\n% A5-North. Nevertheless, the gradient is lower than that on the A8-East and it\n% is unclear, if it constitutes the bottleneck.\n\n\\bibitem{platoon-multilane}\nE. Ben-Naim and P.~L. Krapivsky, Phys. Rev. E {\\bf 56}, 6680 (1997).\n\n\\bibitem{Applet-engl}\nInteractive simulations of the multi-lane IDM are available at {\\tt\n www.theo2.physik.uni-stuttgart.de/treiber/MicroApplet/}.\n\n\\end{thebibliography}\n\n\n% ##############################################################\n% ######################## Tables #############################\n% ##############################################################\n\n%############ Table tab:param #####################################\n\n\\vspace{0mm}\n\n\\newcommand{\\entry}[2]{\\parbox{50mm}{#1} &\n \\parbox{30mm}{#2} }\n\n\\begin{table}\n\n\\begin{tabular}{l|l}\n%\n\\entry{Parameter} {Typical\\\\ value} \\\\[3mm] \\hline \n\\entry{}{}{} \\\\[-2mm]\n\\entry{Desired velocity $v_0$}\n {120 km/h}\n \\\\[0mm]\n\\entry{Safe time headway $T$}\n {1.6 s}\n \\\\[0mm]\n\\entry{Maximum acceleration $a$}\n {0.73 m/s$^2$}\n% {Corresponds to an accelerating time $v_0/a_0=35$ s}\n \\\\[0mm]\n\\entry{Desired deceleration $b$}\n {1.67 m/s$^2$}\n% {The deceleration does not exceed b=$b\\max/\\beta$\n% under regular conditions } \n \\\\[0mm]\n\\entry{Acceleration exponent $\\delta$}\n {4}\n% {Corresponds to the accelerating time from 0 to 100 km/h,\n \\\\[0mm]\n\\entry{Jam distance $s_0$}\n {2 m}\n% {Distance in queues where drivers do no accelerate} \n \\\\[0mm]\n\\entry{Jam Distance $s_1$}\n {0 m}\n \\\\[0mm]\n\\entry{Vehicle length $l = 1/\\rho_{\\rm max}$}\n {5 m}\n\\end{tabular}\n%\n\\vspace*{5mm}\n\\caption{\\label{tab:param} Model parameters of the IDM model\nused throughout this paper. {Changes of the freeway capacity were\ndescribed by a variation of the safe time headway $T$.}\n}\n\\end{table}\n\n\n% ##############################################################\n% ######################## Figures #############################\n% ##############################################################\n\n%##################################################################\n\\begin{figure}[ht]\n%\\vspace{25mm}\n \\begin{center}\n \\includegraphics[width=85mm]{fig1.eps}\n% \\includegraphics[width=45mm]{fig1a.eps}\n% \\hspace*{-5mm}\n% \\includegraphics[width=37.3mm]{fig1b.eps}\n% \\vspace{0mm}\n \\end{center}\n \\caption[]{\\label{fig:fund}\\protect \nEquilibrium flow-density relation of identical IDM \nvehicles with (a) variable\nacceleration exponent $\\delta$, and (b) variable\nsafe time headway $T$ and desired velocity $v_0$.\nOnly one parameter is varied at a time; the others\n{correspond to the ones in} Table \\ref{tab:param}.\n}\n\\end{figure}\n\n%##################################################################\n\\begin{figure}[ht]\n%\\vspace{25mm}\n \\begin{center}\n% \\includegraphics[width=85mm]{x_accbrake.eps}\\\\[-3mm]\n \\includegraphics[width=85mm]{fig2a.eps}\\\\[-3mm]\n \\includegraphics[width=85mm]{fig2b.eps}\n \\end{center}\n \\caption[]{\\label{fig:accbrake}\\protect \nTemporal evolution of velocity and \nacceleration of a single driver-vehicle unit\nwhich accelerates on a 2.5 km long stretch of free road\nbefore it decelerates when approaching \na standing obstacle at $x=2.5$ km.\nThe dynamics for the IDM parameters of Table \\ref{tab:param}\n(solid) is compared with the result for an increased\nacceleration $a_0=2$ m/s (dotted), or an increased braking\ndeceleration $b=5$ m/s$^2$ (dashed).\n}\n\\end{figure}\n\\newpage\n\n%##################################################################\n\\begin{figure}[ht]\n%\\vspace{25mm}\n \\begin{center}\n \\includegraphics[width=85mm]{fig3a.eps}\\\\[-1mm]\n \\includegraphics[width=85mm]{fig3b.eps}\n \\end{center}\n \\caption[]{\\label{fig:distance}\\protect \nAdaptation of a single vehicle \nto the equilibrium distance in the car-following regime.\nShown is (a) the net distance $s$, and (b) the velocity of a \nvehicle following\na queue of vehicles which all drive at $v^*=40.5$ km/h corresponding\nto an equilibrium distance $s_{e}=20$ m.\nThe initial conditions are $v(0)=v^*$ and $s(0)=s_{e}/2=10$ m.\nThe solid line is for the IDM standard parameters,\nand the dashed line for the deceleration parameter $b$ increased \nfrom 1.67 m/s$^2$ to 10 m/s$^2$.\n%{\\bf [Bitte ``Net Distance'' statt ``net distance'' und ``Velocity v''\n%statt ``v'' an den y-Achsen schreiben. Ausserdem fehlen die Labels\n%(a) und (b).]}\n}\n\\end{figure}\n\\newpage\n%##################################################################\n\\begin{figure}[ht]\n%\\vspace{25mm}\n \\begin{center}\n \\includegraphics[width=85mm]{fig4a.eps}\\\\\n \\includegraphics[width=85mm]{fig4b.eps}\n \\end{center}\n \\caption[]{\\label{fig:stab}\\protect \nStability diagram of homogeneous traffic \n{on a circular road}\nas a function of the homogeneous density $\\rho_h$\nfor small (grey) and large (black) initial\nperturbations of the density. In plot (a),\nthe upper two lines display the density inside\nof density clusters after a stationary state has been reached. \nThe lower two lines represent the density between the clusters.\nPlot (b) shows the corresponding flows and the\nequilibrium flow-density relation (thin curve).\nThe critical densities $\\rho_{{\\rm c}i}$ and flows\n$Q_{{\\rm c}i}$ are discussed in the \nmain text. For $\\rho_{\\rm c2}\\le\\rho_h\\le 45$ vehicles/km, the outflow\n$Q_{\\rm out}\\approx Q_{\\rm c2}$ and the corresponding density\n$\\rho_{\\rm out}$ are constant. Here, we have\n$Q_{\\rm max}\\approx Q_{\\rm out}$ \nfor the maximum equilibrium flow, but there are other parameter sets\n(especially if $s_1>0$) where $Q_{\\rm max}$ is clearly\nlarger than $Q_{\\rm out}$ \\protect\\cite{TGF99-Treiber}.\n%{\\bf [Man sollte erwaehnen, dass der Unterschied zwischen maximalem\n%Fluss $Q_{\\rm max}$ und Ausfluss $Q_{\\rm out}$ aus dem Stau\n%mit endlichem $s_1$ viel groesser ausfallen kann.]}\n}\n\\end{figure}\n\\newpage\n\n%########################### Qout(T) ################################\n\\begin{figure}[ht]\n%\\vspace{25mm}\n \\begin{center}\n \\includegraphics[width=85mm]{fig5.eps}\n \\end{center}\n \\caption[]{\\label{fig:Qout}\\protect \n Outflow $Q_{\\rm out}$ from congested traffic (solid curve), \nminimum flow $Q_{\\rm c1}$ for nonlinear instabilities (dashed),\nand maximum equilibrium flow $Q_{\\rm max}$ (dotted)\nas a function of the safe time headway.\nAn approximation for the bottleneck strength \n$\\delta Q$ of the phase diagram\nis given by the difference between the value \n$Q_{\\rm out}=1689$ vehicles/h for $T=1.6$ s (horizontal thin line),\nand $Q_{\\rm out}(T)$. For decerasing values of $T$ \ntraffic becomes more unstable which is indicated by increasing\ndifferences ($Q_{\\rm max} - Q_{\\rm out}$) or \n ($Q_{\\rm max} - Q_{\\rm c1}$).\nFor $T\\le 1.5$ s this even leads to\n$\\partial Q_{\\rm out}(T)/\\partial T > 0$. Furthermore,\n$Q_{\\rm out} \\approx Q_{\\rm c1}$ for\n$T\\le 1.4$ s.\n%{\\bf [Ueberarbeiten in Uebereinstimmung mit dem\n%Haupttext. Insbesondere fehlt $Q_{\\rm max}$, und die Auftragung soll\n%bei $T=1$ s beginnen. Man muss unbedingt\n%$\\delta Q$ aus $\\delta T$ bestimmen koennen, das ist fuer die\n%Diskussion unwahrscheinlich wichtig. Abbildungsunterschrift muss\n%danach aktualisiert werden.]}\n%{\\bf [Schoen waers. Gezeigt sind aber $Q_{\\rm\n%out}$ und $Q_{\\rm c1}$. Ich schlage vor, Du ergaenzt die Kurve\n%fuer $Q_{\\rm max}$ noch und waehltst gestrichelte Kurven. Abbildung 5\n%ist sehr wichtig, weil sie die Ermittlung der Bottleneckstaerke aus\n%dem Unterschied von $T$ und $T'$ erlaubt, die bei allen Simulationen\n%empirischer Ergebnisse abgeschaetzt werden sollte. Die Kurven sollten\n%etwa bei $T=1$ starten, um auch Effekte von negativen Bottlenecks\n%abschaetzen zu koennen, s. unten.]}\n}\n\\end{figure}\n\\newpage\n\n%##################################################################\n\n\n\n% ##############################################################\n% ######### Phase diagrams\n% ##############################################################\n\n% ######### Phase diagram Qin-delta Q ############################\n\n\\begin{figure}\n%\n\\begin{center}\n%\\includegraphics[width=85mm]{phase_T_dQ.eps}\n\\includegraphics[width=85mm]{fig6.eps}\n\\end{center}\n%\n\\caption[]{\n\\label{phasediag}\nPhase diagram resulting from IDM simulations of an open \nsystem with a flow-conserving bottleneck.\nThe bottleneck is realized by an increased IDM parameter $T$\nin the downstream region, cf. Eq. (\\protect\\ref{Tx}).\nThe control parameters are the traffic flow $Q_{\\rm in}$ and\nthe bottleneck strength $\\delta Q$, see Eq. (\\protect\\ref{deltaQ}).\nThe solid thick lines separate the congested\ntraffic states TSG, OCT, HCT,\nPLC, and MLC (cf. the main text and Fig. \\protect\\ref{3dphases}) \nand free traffic (FT)\nas they appear after adiabatically increasing the inflow to the value\n$Q_{\\rm in}$ and applying a large perturbation afterwards\n(history ``C'' in the main text).\nAlso shown is\nthe critical downstream flow $Q'_{\\rm c2}(\\delta Q)$ (thin solid line), \nbelow which free\ntraffic (FT) is (meta-)stable, and the maximum downstream flow\n$Q'_{\\rm max}(\\delta Q)$ (thin dotted) below which (possibly unstable)\nequilibrium traffic exists. \nTraffic is bistable for inflows above the lines FT-PLC or\nFT-MLC (whichever is lower), and below $Q'_{\\rm c2}(\\delta Q)$.\nIn the smaller shaded region, traffic is tristable and\nthe possible states FT, PLC, or OCT depend on the previous history (see \nthe main text). For history ``C'' we obtain OCT in this region.\n}\n\\end{figure}\n\\newpage\n\n\n% ######### IDM 3d traffic phases ###################################\n\n\\begin{figure}\n%\n\\begin{center}\n\\includegraphics[width=85mm]{fig7a.eps}\\\\[0mm]\n\\includegraphics[width=85mm]{fig7b.eps}\\\\[0mm]\n\\includegraphics[width=85mm]{fig7c.eps}\\\\[0mm]\n\\includegraphics[width=85mm]{fig7d.eps}\\\\[0mm]\n\\includegraphics[width=85mm]{fig7e.eps}\n\\end{center}\n%\n\\caption[]{\n\\label{3dphases}\nSpatio-temporal density plots of the traffic states appearing in the\nphase diagram of Fig.~\\protect\\ref{phasediag}.\n(a) Homogeneous congested traffic (HCT), \n(b) oscillating congested traffic (OCT),\n(c) triggered stop-and-go waves (TSG), \n(d) (stationary) pinned localized cluster (SPLC), and\n(e) oscillatory pinned localized cluster (OPLC).\n{The latter two states are summarized as pinned localized clusters \n(PLC).}\nAfter a stationary state of free traffic has developed,\na {density} wave is introduced through the downstream\nboundary ({or initial conditions})\nwhich eventually triggers the breakdown \n(History ``C'', cf. the main text).\n}\n\\end{figure}\n\\newpage\n\n\n% ######### IDM tristable #############################\n\n\\begin{figure}\n\n\\begin{center}\n\\includegraphics[width=85mm]{fig8a.eps}\\\\[0mm]\n\\includegraphics[width=85mm]{fig8b.eps}\\\\[0mm]\n\\includegraphics[width=85mm]{fig8c.eps}\n\\end{center}\n%\n\\caption[]{\n\\label{tristab}\nSpatio-temporal density plots {for the same phase point in}\nthe tristable traffic regime, {but different histories}.\n(a) Metastable PLC and stable OCT.\nThe system is the same as in Fig. \\protect\\ref{3dphases} with\n$Q_{\\rm in}=1440$ vehicles/h and $T'=1.95$ s ($\\delta Q=270$ \nvehicles/h). \nThe {metastable} PLC is triggered by a triangular-shaped density peak \nin the initial conditions ({of} total width 600 m,\ncentered at $x=0$), {in which} the \ndensity rises from 14 vehicles/km to 45 vehicles km.\nThe OCT is triggered by a {density} wave introduced by \nthe downstream boundary conditions.\n(b) Same system as in (a), but {starting with} metastable free traffic. \nHere, {the transition to OCT is triggered by a density wave coming \nfrom the downstream boundary.}\n(c) Similar behavior as in (a), but for the GKT model\n(with model parameters $v_0=120$ km/h,\n$T=1.8$ s, $\\tau=50$ s, $\\rho_{\\rm max}=130$ vehicles/km,\n$\\gamma=1.2$, $A_0=0.008$, and\n$\\Delta A=0.008$, cf. Ref. \\protect\\cite{GKT}).\n}\n\\end{figure}\n\\newpage\n\n% ##############################################################\n\n\n\n\n% ##############################################################\n% ###################### HCTemp ##################################\n% ##############################################################\n\n% ############## sketch + show lane blockage #########\n\\begin{figure}\n\n\\begin{center}\n\n\\includegraphics[width=85mm]\n {fig9a.eps} \\\\\n\\includegraphics[width=85mm]\n {fig9b.eps}\n\\end{center}\n\n\\vspace*{3mm}\n\n\\caption[]{\\label{HCTemp}\nTraffic breakdown to nearly homogeneous congested\ntraffic on the freeway A5-South near Frankfurt\ntriggered by a temporary incident between 16:20 and 17:30\n{on} Aug. 6, 1998 between the cross sections D11 and D12.\n(a) Sketch of the freeway. (b) Flows at cross section D11\non the right lane \n(solid black), middle \nlane (grey), and left lane (dotted).\n}\n\\end{figure}\n\n\\newpage\n\n% ############## timeseries emp+theo v,Q #########\n\n\\begin{figure}\n\n\\begin{center}\n\\includegraphics[width=160mm]{fig10.eps} \n%\\unitlength1mm\n%\\begin{picture}(160,130)\n%%\n%\\put(45,65){\n%\\includegraphics[width=65mm]{fig10a.eps} \n%}\n%% downstream acceleration\n%\\put(0, 40){\n%\\includegraphics[width=75mm]{fig10b.eps}\n%\\hspace*{5mm}\n%\\includegraphics[width=75mm]{fig10c.eps}\n%}\n%% at downstream front\n%\\put(0, 20){\n%\\includegraphics[width=75mm]{fig10d.eps}\n%\\hspace*{5mm}\n%\\includegraphics[width=75mm]{fig10e.eps}\n%}\n%% HCT\n%\\put(0, 0){\n%\\includegraphics[width=75mm]{fig10f.eps}\n%\\hspace*{5mm}\n%\\includegraphics[width=75mm]{fig10g.eps}\n%}\n%% upstream BC\n%\\put(0, -22){\n%\\includegraphics[width=75mm]{fig10h.eps}\n%\\hspace*{5mm}\n%\\includegraphics[width=75mm]{fig10i.eps}\n%}\n%\\end{picture}\n\\end{center}\n\n\\vspace*{0mm}\n\n\\caption[]{\\label{HCTemptheo}\n{\\bf [Please display in two columns as indicated]}\nDetails of the traffic breakdown\ndepicted in\nFig. \\protect\\ref{HCTemp}.\n(a) Flow-density diagram of the traffic breakdown on the A5 South, \n(b)-(g) temporal evolution of velocity and flow\nat three locations near the perturbation, and (h), (i) inflow boundary\nconditions taken from cross section D6.\nBesides the data (black lines), the results of the microsimulation are\nshown (grey).\nAll empirical quantities are averages over one minute and over\n{all} three lanes.\n}\n\\end{figure}\n\\newpage\n\n\n% ##############################################################\n% ################ HCT theoretical #################################\n% ##############################################################\n\n\n\\begin{figure}\n\n\n\\begin{center}\n\\includegraphics[width=160mm]{fig11.eps} \n%\\includegraphics[width=80mm]{fig11a.eps} \n%\\hspace*{-10mm}\n%\\includegraphics[width=80mm]{fig11b.eps} \\\\[-10mm]\n%\\includegraphics[width=80mm]{fig11c.eps} \n%\\hspace*{-10mm}\n%\\includegraphics[width=80mm]{fig11d.eps} \\\\\n%\\vspace*{0mm}\n\n\\end{center}\n\n\\caption{\n\\label{HCT3d}\n{\\bf [Please display in two columns as indicated]}\n(a), (b) Smoothed spatio-temporal velocity \nand flow from the data\nof the traffic breakdown depicted in\nFigs.~\\protect\\ref{HCTemp} and \\protect\\ref{HCTemptheo}. \n(c), (d) Corresponding IDM microsimulation with the parameter set from\nTable \\ref{tab:param}.\nThe upstream boundary conditions for velocity and traffic flow were\ntaken from cross section D6. Because of the fast relaxation of the\nvelocity to the model's equilibrium value, \nthe upstream boundary conditions for the empirical velocity\nplots (a) seem to be\ndifferent from the simulation (c) (see main text).\nHomogeneous von Neumann boundary conditions\nwere assumed downstream.\nThe temporary lane closing is modelled by\nlocally increasing the IDM parameter $T$ \nin a 1000 m long section centered at $x=478$ km during the time\ninterval 16:20 h $\\le t\\le$ 17:30 h of the incident.\n}\n\\end{figure}\n\\newpage\n\n% ##############################################################\n% ####################### OCTemp #########################\n% ##############################################################\n\n\\begin{figure}\n\n\\begin{center}\n\\vspace*{-5mm}\n\\includegraphics[width=160mm]{fig12.eps} \n%\n%\\vspace*{-5mm}\n%\\includegraphics[width=85mm]\n% {fig12a.eps} \\\\\n%\\hspace*{0mm}\n%\\includegraphics[width=90mm]{fig12b.eps}\n%\\hspace*{-5mm}\n%\\includegraphics[width=60mm]{fig12c.eps}\\\\\n%\\vspace*{0mm}\n%\\includegraphics[width=75mm]{fig12d.eps}\n%\\includegraphics[width=75mm]{fig12e.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{fig12f.eps}\n%\\includegraphics[width=75mm]{fig12g.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{fig12h.eps}\n%\\includegraphics[width=75mm]{fig12i.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{\\pathfigs/OCTemp_v19.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/OCTemp_Q19.eps}\\\\\n%\\vspace*{-5mm}\n\\end{center}\n\n\\caption[]{\\label{OCTemp}\n{\\bf [Please display in two columns as indicated]}\nTraffic breakdown to oscillating congested traffic in the evening rush \nhour of October 29, 1998 on the freeway A9-South near Munich.\n(a) Sketch of the considered section with the cross sections D16 to\nD30\nand their positions in kilometers.\nThe small on- and off-ramps between\nI1 and I2, which have been neglected in our simulation, are indicated\nby diagonal lines.\n(b) Spatio-temporal plot of the smoothed \nlane-averaged velocity. (c) Flow-density diagram obtained from\ntwo detectors in\nthe congested region. \n(d)-(i) Lane-averaged 1-minute data of\nvelocities and flows.\n}\n\\end{figure}\n\\newpage\n\n% upstream OCT and inverse velocity of full region at the end!\n\n% ##############################################################\n% OCT theo\n% ##############################################################\n\n\\begin{figure}\n\n\\begin{center}\n\\includegraphics[width=95mm]{fig13.eps} \\\\\n%\n%\\includegraphics[width=70mm]{\\pathfigs/OCTmic_v14200.eps}\n%\\includegraphics[width=70mm]{\\pathfigs/OCTmic_Q14200.eps}\\\\ \n%\\includegraphics[width=70mm]{\\pathfigs/OCTmic_v12000.eps}\n%\\includegraphics[width=70mm]{\\pathfigs/OCTmic_Q12000.eps}\\\\\n%\\includegraphics[width=70mm]{\\pathfigs/OCTmic_v9000.eps}\n%\\includegraphics[width=70mm]{\\pathfigs/OCTmic_Q9000.eps}\n\\end{center}\n\n\\caption[]{\\label{OCTtheo}\n{Smoothed spatio-temporal velocity plot from a qualitative \nIDM microsimulation of the situation depicted in \nFig. \\protect\\ref{OCTemp}. As inflow boundary conditions, we used the \ntraffic flow data of\ncross section D20. \nHomogeneous von Neumann\nboundary conditions were assumed at the downstream boundary.\nThe inhomogeneity was implemented by a local, but time-independent\nincrease of $T$ \nin the region $x\\ge 0$ km from\n$T=2.2$ s to $T'=2.5$ s.}\n}\n\\end{figure}\n\\newpage\n\n\n% ##############################################################\n% ####################### Flugzeug emp und theo ##############\n% ##############################################################\n\n\\newpage\n\\begin{figure}\n\n\\begin{center}\n\\includegraphics[width=85mm]\n {fig14a.eps} \\\\\n\\vspace*{-3mm}\n\\hspace*{0mm}\n\\includegraphics[width=90mm]{fig14b.eps}\\\\\n\\vspace*{0mm}\n\\includegraphics[width=90mm]{fig14c.eps}\\\\\n\\hspace*{-10mm}\n%\\includegraphics[width=60mm]{\\pathfigs/flugzeugemp_fundlower.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_v24.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_Q24.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_v23.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_Q23.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_v22.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_Q22.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_v20.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_Q20.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_v18.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_Q18.eps}\\\\\n%\\vspace*{-5mm}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_v16.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/flugzeugemp_Q16.eps}\\\\\n%\\vspace*{-5mm}\n\\end{center}\n\n\\caption[]{\\label{flugzeug}\nOscillating congested traffic (OCT) on an uphill\nsection of the freeway A8\nEast (near Munich). (a) Sketch of the section\nwith the cross sections D15 to D24 and\ntheir positions in kilometers. (b) Smoothed\nlane-averaged empirical velocity. An incident\nleading to a temporary lane closing between D23 and D24\n(near the downstream boundaries of the plots) induces even\ndenser congested traffic that propagates through the OCT region.\n(c) Microsimulation using the data of cross sections D15 and D23 as\nupstream and downstream boundary conditions, respectively.\nThe uphill section is modelled by linearly increasing the safe time headway\nfrom $T=1.6$ s (for $x<39.3$ km) to $T'=1.9$ s (for $x>40.0$ km).\n{As in the previous microsimulations, the {\\em velocity} near the upstram \nboundary relaxes quickly and,\ntherefore, seems to be inconsistent with the empirical values\n(see main text).}\n}\n\\end{figure}\n\\newpage\n\n\n% ##############################################################\n% ####################### TSG emp #################################\n% ##############################################################\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=85mm]\n {fig15a.eps}\n\\vspace*{0mm}\\\\\n\n\\hspace*{-5mm}\n\\includegraphics[width=95mm]{fig15b.eps} \\\\\n% fundamental diagram\n\\vspace*{-5mm}\n\\includegraphics[width=80mm]{fig15c.eps} \\\\ \n% TSG\n\\includegraphics[width=80mm]{fig15d.eps}\\\\\n%\\includegraphics[width=80mm]{\\pathfigs/TSGemp_Q06.eps}\\\\\n% TSG\n%\\includegraphics[width=80mm]{\\pathfigs/TSGemp_v09.eps}\\\\\n%\\includegraphics[width=80mm]{\\pathfigs/TSGemp_Q09.eps}\\\\\n\\vspace*{-2mm}\n% PLC\n\\includegraphics[width=80mm]{fig15e.eps}\\\\\n%\\includegraphics[width=80mm]{\\pathfigs/TSGemp_Q13.eps}\\\\\n\\vspace*{-2mm}\n% inflow\n\\includegraphics[width=80mm]{fig15f.eps}\n%\\includegraphics[width=80mm]{\\pathfigs/TSGemp_Q18.eps}\n\\end{center}\n\\vspace*{0mm}\n\n\\caption{\\label{PLCMLCemp}\nData of two moving localized\nclusters (MLCs), and \npinned localized clusters (PLC) on the freeway A5-North near Frankfurt.\n(a) Sketch of the infrastructure with the positions of the cross\nsections D5 - D16 in kilometers. (b) Spatio-temporal plot of the\ndensity. (c)\nFlow-density diagram {for} detector D6, where there are only stop-and-go \nwaves ($\\Box$), \nand {for detector} D13 ($\\bullet$), {where we\nhave omitted data points during the time intervals when\nthe MLCs passed by.}\n(d)-(f) Temporal evolution of the velocity (one-minute data)\nat the location of the PLC at D13, and upstream (D18)\nand downstream (D6) of it.\n}\n\\end{figure}\n\\newpage\n\n\n% ##############################################################\n% TSG theo\n% ##############################################################\n\n\\begin{figure}\n\n\\begin{center}\n\\includegraphics[width=105mm]{fig16a.eps} \\\\[0mm]\n%\n\\includegraphics[width=85mm]{fig16b.eps}\n\\end{center}\n%\n\\caption[]{\\label{PLCMLCmic}\n{Smoothed spatio-temporal plot of the traffic density\nshowing the coexistence of a pinned localized cluster (PLC)\nwith moving localized clusters (MLCs) in an IDM simulation.\nThe PLC is positioned at a road section with locally {\\it increased}\ncapacity corresponding to a bottleneck strength\n$\\delta Q=-300$ vehicles/h, which can be identified with the region\nbetween the off- and on-ramps of intersection I2 \nin Fig. \\protect\\ref{PLCMLCemp}(a). It was produced by \nlocally {\\it decreasing} the IDM parameter\n$T$ from 1.9 s to $T'=1.6$ s in a 400 m wide section \ncentered at $x=480.8$ km, and increasing it again from $T'$ to $T$ in\na 400 m wide section centered at $x=480.2$ km.\nThe initial conditions correspond to equilibrium traffic of flow \n$Q_{\\rm in}=1390$ vehicles/h, to which a triangular-shaped density peak\n(with maximum density 60 vehicles/km at $x=480.5$ km and total width 1 km)\nwas superposed to initialize the PLC.\nAs upstram boundary conditions, \nwe assumed free equilibrium traffic with\na constant inflow $Q_{\\rm in}$ of 1390 vehicles/h.\nAs downstream boundary conditions for the velocity, \nwe used the value for equilibrium free traffic most of\nthe time. However,\nfor two five-minute intervals at 14:20 h and 15:40 h, we\nreduced the velocity to $v=12$ km/h to initialize the\nMLCs.}\n%\n%of the part of the road section of Fig. \\protect\\ref{PLCMLCemp}\n%containing intersection I2 with the main pinned localized cluster.\n%The upstream and downstream boundary conditions {\\bf were} taken from cross\n%sections D20 and D7 {\\bf [nicht D5?]}, respectively. The initial conditions\n%at $t=$\\,13:30 h {\\bf were obtained by interpolation between the\n%measured data.} \n%{\\bf (All flows have been increased by a constant amount of 100\n%vehicles/h.) [Das gefaellt mir wieder nicht. Lieber sollten wir den\n%Grundwert von $T$ in dieser Simulation von 1.6 s verschieden waehlen.]}\n%(a) Spatio-temporal plot of the simulated traffic density\n%with a PLC coexisting with stop-and-go waves.\n%The {\\bf [which?, genauer beschreiben]}\n%intersection and the {\\bf inhomogeneity} downstream of it \n%is simulated by locally {\\it decreasing} the IDM parameter\n%$T$ from 1.6 s to $T'=1.2$ s in a 1000 m wide section \n%centered at $x=480.5$ km.\n(b) Flow-density diagram of virtual detectors located at the position\n$x=480.2$ km (PLC), and 2.2 km downstream of it (MLC).\nAgain, the time intervals, where {density} \nwaves passed through the PLC, were\nomitted in the {(``virtual'')} data points {belonging to} PLC.\nThe thin solid curve indicates the equilibrium flow density relation\nand \nthe thick solid line $J$ characterizes\nthe outflow from \nfully developed density clusters to free traffic, \ncf. Ref. \n\\protect\\cite{Kerner-wide}, both for $T=1.9$ s.\n}\n\\end{figure}\n\n% ##############################################################\n% Figure ``scatter'': emp (a9Dutch) + Multilane sim\n% ##############################################################\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=75mm]{fig17.eps}\n \\end{center}\n\\caption{\\label{mixfund}\nFlow-density diagram of a HCT state of single-lane\nheterogeneous traffic consisting of 70\\%\n``cars'' and 30\\% ``trucks''.\nTrucks are\ncharacterized by lower IDM parameters $v_0$ and $a$, and a larger\n$T$ compared to cars. The solid and dashed curves give\nthe equilibrium flow-density relations for traffic consisting only of\ncars and trucks, respectively. \nFor details, see Ref. \\protect\\cite{TGF99-Treiber}.\n}\n\\end{figure}\n%##################################################################\n\n\n\n% ##############################################################\n\\end{document}\n% ##############################################################\n\n\n\n\n\n\n\nIn all cases, the phase boundaries are given by the analytic\nexpressions (\\ref{HCT_OCT}) - (\\ref{MLC_ECT}).\nThey depend only on\nthe charateristic flows \n$Q_{\\rm c1}\\cdots Q_{\\rm c3}$, $Q_{\\rm cv}$, $Q_{\\rm out}$\ndetermined in closed homogeneous systems, and\non the effective bottleneck strengths\n$\\delta Q\\sup{ECT}(\\delta Q)$ and\n$\\delta Q\\sup{PLC}(\\delta Q)$ as a function of \nthe control parameter $\\delta Q$, cf. \nFig. \\ref{dQ}.\nThe flows characterize the stability properties,\nand the bottleneck strengths the specific inhomogeneity.\n\n\n\nwe explore hysteresis effects and multistability \nby determining the\nregions in control-parameter space, where each state is\nat least metastable. To determine the boundaries,\nwe adiabatically change the control parameters until a transition to\nanother phase occurs.\n\n\nThe transitions between the extended and pinned states\nshow distinct hysteretic effects and there exist regions,\nwhere the pinned states (PLC) are metastable and one of the extended state\nis stable (OCT+PLC, TSG+PLC). The transitions between pinned and extended \nstates\nstates depend both on the inflow and the\nbottleneck strength. Furthermore, also the pinned states can be\noscillatory (oscillatory pinned localized clusters, OPLC),\nor stationary (SPLC). We do not have analytic relations for the\n(nonhysteretic) transitions OPLC-SPLC.\n\nFree traffic in equilibrium is possible everywhere\nbelow the dotted???\nline $Q_{\\rm in}=Q_{\\rm max}-Q_{\\rm rmp}$,\nbut for inflows above\n$Q_{\\rm c2}-Q_{\\rm rmp}$ it is unstable in infinite systems.\nFor finite downstream regions, however, the downstream travelling\nperturbations of free traffic can be absorbed by the downstream boundary\n(by the next off-ramp in real traffic) before growing\nto nonlinear amplitude, so that in finite systems a stationary\nstate of free traffic is also possible in the lineraly unstable\nregime. Notice, however, that free traffic is at least\nmetastable at the phase boundaries for the transitions\nextended to pinned localized\nstates, so even the infinite system includes a true tristable \nregion in control-parameter space.\n\n\n\n%##################################################\n\\subsection{Analytic Expressions for the Phase Boundaries}\n{\\bf diesen Abschnitt hier weglassen?}\n%##################################################\n\nWe propose that dynamical effects caused by flow-conserving or\nnonconserving local defects depend essentially\non the induced local capacity drop,\nor ``bottleneck strength''.\nTo define a generalized bottleneck strength $\\delta Q$, we observe\nthat a bottleneck implies\neither additional traffic from on-ramps \nor a reduced local road capacity in the downstream region.\nConsequently, we define the bottleneck strength by\n%\n\\be\n\\label{deltaQ}\n\\delta Q := Q\\rmp + Q_{\\rm max}-Q'_{\\rm max}. \n%\\delta Q := Q\\rmp + Q_{\\rm out}-Q'_{\\rm out}. \n%\\delta Q := Q\\rmp + Q_{\\rm out}-\\tilQout.\n\\end{equation}\n%\nHere, $Q_{\\rm rmp}$ is the ramp flow,\nand $Q_{\\rm max}$ and $Q'_{\\rm max}$ are\nthe maximum possible\nequilibrium flows (static capacities) in the homogeneous upstream \nand downstream regions.\n\nIf the inhomogeneity is just an on-ramp to an otherwise\nhomogeneous road, we have $\\delta Q = Q\\rmp$, which is the\ncontrol parameter in the already published\nempirical \\cite{Lee-emp} and theoretical \\cite{Phase}\nphase diagrams.\nFor flow-conserving inhomogeneities, \n$\\delta Q =Q_{\\rm out}-Q'_{\\rm out}$ is equal to the static capacity\ndrop.\n\nNotice that $\\delta Q$ can also defined using\nthe self-organized outflow from congested traffic or \nstop-and-go waves (dynamic capacity) instead of the static capacity\n\\cite{Kerner-rehb96?,Kerner-asympt?,Daganzo-ST,Persaud}.\nHowever, we used the static capacity for simplicity.\n\n\nWe propose that the qualitative dynamics of the basic system depends only\non two control parameters:\n(i) the amplitude of the local defect\nmeasured by a suitably defined ``bottleneck strength''\n$\\delta Q$, and\n(ii) the incoming traffic flow $Q_{\\rm in}$.\nIn metastable regions of this control-parameter space\n$(\\delta Q,Q_{\\rm in})$\nthe selected state depends also on the initial conditions.\n\n[Discuss plots of phase diagram]\n\n[Erklaere phase boundaries, Einleitung]\n%\n%##############################################################\n\\paragraph{Extended congested states}\n%##############################################################\n%\nWe will consider the generic case where\nthe downstream front of extended \ncongested states (ECT) is pinned at the local defect.\nOnly in the last stage of dissolution, the downstream front\ncan detach from the inhomogeneity and transform into\na travelling front,\ncf. Figs. \\ref{OCTemp} and \\ref{OCTtheo}.\nThe pinned downstream front can be oscillatory or stationary,\nbut in any case,\nthe continuity equation (\\ref{rhoGKT} implies that\nthe time-averaged outflow $\\tilQout$ from the ECT state\nis equal to the averaged traffic flow \n$Q\\sup{ECT}$ in the congested\nregion plus, in case of nonconserving inhomogeneities,\nthe ramp flow $Q_{\\rm rmp}$.\nFurthermore, once ECT has formed,\n$\\tilQout$ is independent \nfrom the inflow and depends only on the inhomogeneity. \nWe therefore have the interesting consequence \nthat, at least not too close to the downstream front,\nthe bottleneck is dynamically eqivalent to\nDirichlet downstream boundary conditions prescribing\nthe traffic flow $\\tilQout-Q_{\\rm rmp}$.\n\nFrom simulations we obtain the interesting result\nthat the outflow $\\tilQout$ from\n{\\it fixed} downstream fronts of ECT\nis lower than the self-organized outflow $Q'_{\\rm out}$ \nfrom {\\it travelling} stop-and-go waves \nin the homogeneous downstream region.\nFurthermore, $\\tilQout$ depends on the specific\ninhomogeneity such that \nthe flow difference \n%\\be\n%\\label{deltaECT}\n%\\Delta\\sup{ECT}(\\delta Q) := Q'_{\\rm out} - \\tilQout\n%\\end{equation}\n$Q'_{\\rm out} - \\tilQout$\nis an increasing function of the bottleneck strength with\n$Q'_{\\rm out} \\approx \\tilQout$ for sufficiently small strengths $\\delta Q$\nor sufficiently large lengths $L_{\\rm inh}$ of the inhomogeneity.\n\nWith Eqs. (\\ref{deltaQ} we obtain\n%\n\\be\n\\label{QECT}\nQ\\sup{ECT} = Q_{\\rm out} - \\delta Q\\sup{ECT},\n\\end{equation}\n%\nwhere the {\\it effective} bottleneck strength\n%\n\\be\n\\label{dQECT}\n\\delta Q\\sup{ECT} := \\delta Q + Q'_{\\rm out} - \\tilQout\n\\end{equation}\n%\nfor ECT is slightly higher than the control parameter $\\delta Q$,\ncf. Fig. \\ref{dQ}.\nSince the properties of congested traffic like stability,\naverage density, or wavelength of oscillatory\nstates, are uniquely determined by the average flow $Q\\sup{ECT}$\n(see, for example, Fig. 8 of Ref. \\cite{GKT}),\nthe kind of ECT is determined uniquely by the control parameter\n$\\delta Q$.\nSpecifically, homogeneous equilibrium traffic \n(subscript e) is linearly stable\nfor very high densities $\\rho>\\rho_{\\rm c3}$ corresponding to\nflows $Q\\sup{ECT}<Q_{\\rm c3} := Q_e(\\rho_{\\rm c3})$.\nIt is linearly unstable but convectively stable \nin a density range $\\rho_{\\rm conv}<\\rho<\\rho_{\\rm c3}$\ncorresponding to a flow range\n$Q_{\\rm c3} < Q\\sup{ECT} < Q_{\\rm conv}:= Q_e(\\rho_{\\rm conv}) $\n\\cite{conv-instab?,Phase}, and it is unconditionally unstable\nfor $\\rho<\\rho_{\\rm conv}$ ($Q>Q_{\\rm c3}$).\nConvective stability is a new feature of open systems\n\\cite{Maneville}. \nIn the linearly unstable, convectively stable\nrange\n$Q_{\\rm c3} < Q\\sup{ECT} < Q_{\\rm conv} $ any\nperturbation increases with time, but\neventually propagates through the upstream boundary\nresulting in an\nhomogeneous asymptotic state.\nBecause real traffic always contains small perturbations which\ngrow in the upstream direction, \nthe actual asymptotic state consists of nearly homogeneous, congested\ntraffic (HCT)\nin the vicinity of the downstream boundary, and of oscillating\ncongested traffic (OCT) further upstream. \nFigure. \\ref{HCTemp} shows an example of this state which we will\ndenote as\nHCT/OCT. In summary, \nthe boundaries in control-parameter space from\nthe coexisting state HCT/OCT to\na purely homogeneous state (HCT) are given by\n%\n\\be\n\\label{HCT_OCT}\n%\\mbox{HCT/OCT - HCT : } \\tilQout\\sup{ECT} = Q_{\\rm c3} - Q_{\\rm rmp},\n\\mbox{HCT/OCT - HCT : } \\delta Q\\sup{ECT} = Q_{\\rm out} - Q_{\\rm c3}.\n\\end{equation}\n%\nand to a purely oscillating state (OCT) by\n%\n\\be\n%\\mbox{HCT/OCT - OCT : } \\tilQout\\sup{ECT} = Q_{\\rm cv} - Q_{\\rm rmp},\n\\mbox{HCT/OCT - OCT : } \\delta Q\\sup{ECT} = Q_{\\rm out} - Q_{\\rm cv}.\n\\end{equation}\n%\nThe unique relation between congested traffic flow and stability type\nimplies that these transitions are nonhysteretic and depend neither on \nthe inflow nor on the initial conditions.\n\nWhen adiabatically\ndecreasing the strength $\\delta Q$ of a bottleneck causing OCT,\nthe wavelength of the oscillations increases until, at a certain\ncongested traffic flow $Q_{\\rm TSG}$, there \nappear regions of free\ntraffic between each wave. At the downstream front there is \nan abrupt {\\bf ??}\ntransition from permanently congested to intermittent free traffic\n%which is associated with a small\n%discontinuity in the outflow $\\tilQout$ . As soon as \nwhere each stop-and-go wave triggers the next one as described in\nRef. \\cite{Phase}. This defines the\nboundary between OCT and triggered stop-and-go\nwaves (TSG) by\n%\n\\be\n\\label{OCT_TSG}\n\\mbox{OCT - TSG : } \\delta Q\\sup{ECT} = Q_{\\rm out} - Q_{\\rm TSG}.\n%\\mbox{OCT - TSG : } \\tilQout\\sup{ECT} = Q_{\\rm TSG} - Q_{\\rm rmp},\n\\end{equation}\n%\nFinally, ECT can only persist, if\nthe inflow exceeds the average flow of the congested state.\nOtherwise, the congested region would shrink \nto pinned localized congested states (PLC) or dissolve\nto free traffic,\n%\n\\be\n\\label{ECT_PLC}\n\\mbox{ECT}\\to\\mbox{PLC : } Q_{\\rm in} = Q_{\\rm out} - \\delta Q\\sup{ECT},\n%\\mbox{ECT}\\to\\mbox{PLC : } Q_{\\rm in} = \\tilQout\\sup{ECT} + Q_{\\rm rmp},\n\\end{equation}\n%\nwhere ECT is one of HCT, HCT/OCT, OCT, or TSG.\n\n\n%##############################################################\n\\paragraph{Localized states}\n%##############################################################\n%\nIn contrast to the outflow from extended states, the outflow\nfrom {\\it pinned} localized\nstates depends on the inflow such that the flows are balanced.\nFirst, we investigate the transition from PLC to ECT.\nRemarkably, the maximum \noutflow $\\tilQout\\sup{PLC}$ from pinned localized\nstates often is larger than the outflow $\\tilQout$ from extended\nstates.\n%, and even larger than the outflow $Q'_{\\rm out}$ from stop-and-go waves.\n\nThis is especially true for long ramps\nor for flow-conserving defects with a long transition region.\nA phenomenological explanation can be inferred\nfrom Fig. \\ref{triGKT}\nshowing spatio-temporal density plots of both states.\nBoth downstream fronts are in the ramp \nregion where the traffic flow increases\n(i.e., the local dynamical capacity decreases) in the downstream direction.\nThe front of the pinned state, however, is located further upstream\nwhere the larger local capacity results in a larger outflow.\n\n\nThe higher outflow implies, that\nthe effective bottleneck strength\n%\n\\be\n\\label{dQTSG}\n\\delta Q\\sup{TSG} := \\delta Q + Q'_{\\rm out} - \\tilQout\\sup{PLC}\n\\end{equation}\n\n\n\n\n\n\n\\begin{figure}\n\n\\includegraphics[width=140mm]{\\pathfigs/OCTempfull.eps}\\\\\n\n\n\\caption[]{\\label{OCTfullemp}\nEmpirical data OCT: The whole congested region with three intersections.\n}\n\\end{figure}\n\n\n\n\\begin{figure}\n\n\\begin{center}\n\\includegraphics[width=100mm]\n {A9MsketchLower.eps} \\\\\n\\vspace*{-15mm}\n\\hspace*{0mm}\n\\includegraphics[width=100mm]{\\pathfigs/OCTemp.eps}\\\\\n\\vspace*{-5mm}\n% downstream acceleration\n%\\includegraphics[width=75mm]{\\pathfigs/OCTemp_v18.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/OCTemp_Q18.eps}\\\\\n%\\vspace*{-5mm}\n% downstream acceleration\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_v16.eps}\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_Q16.eps}\\\\\n\\vspace*{-5mm}\n% pinch\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_v15.eps}\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_Q15.eps}\\\\\n\\vspace*{-5mm}\n% OCT\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_v14.eps}\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_Q14.eps}\\\\\n\\vspace*{-5mm}\n% OCT\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_v13.eps}\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_Q13.eps}\\\\\n\\vspace*{-5mm}\n% OCT\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_v12.eps}\n\\includegraphics[width=75mm]{\\pathfigs/OCTemp_Q12.eps}\\\\\n\\vspace*{-5mm}\n% begin OCT near upper front\n%\\includegraphics[width=75mm]{\\pathfigs/OCTemp_v11.eps}\n%\\includegraphics[width=75mm]{\\pathfigs/OCTemp_Q11.eps}\n\\end{center}\n\\vspace*{3mm}\n\n\n\\caption[]{\\label{OCTemp}\nOscillatory congested traffic on the freeway A9-South near Munich.\n(a) Sketch of the considered section. Several small junctions between\nI 1 and I 2 have been left out.\n(b) Spatio-Temporal plot of the lane-averaged density in the congested\nregion upstream of I 1, \n(c)-(g) lane-averaged velocities at several detectors,\nand (h)-(l) the lane-averaged flows at the same locations. \n}\n\\end{figure}\n\n\n\n\n\n\n\n%#################################################################\n%#################################################################\n%#################################################################\n\n\n\n\n% ##############################################################\n\n\\begin{figure}\n\n\\includegraphics[width=140mm]{\\pathfigs/OCTfull.eps}\n\n\\caption[]{\\label{OCTtheo}\nFull section with three intersections simulated with the IDM.\n\n}\n\\end{figure}\n\\newpage\n% ##############################################################\n\n\\begin{figure}\n\n\\includegraphics[width=140mm]{\\pathfigs/OCTmac1.eps} \\\\\n%\n\\includegraphics[width=70mm]{\\pathfigs/OCTmac1_v14200.eps}\n\\includegraphics[width=70mm]{\\pathfigs/OCTmac1_Q14200.eps}\\\\ \n\\includegraphics[width=70mm]{\\pathfigs/OCTmac1_v12000.eps}\n\\includegraphics[width=70mm]{\\pathfigs/OCTmac1_Q12000.eps}\\\\\n\\includegraphics[width=70mm]{\\pathfigs/OCTmac1_v9000.eps}\n\\includegraphics[width=70mm]{\\pathfigs/OCTmac1_Q9000.eps}\n\n\\caption[]{\\label{OCTmac1}\nSimulation of oscillating congested traffic with the GKT model\nusing a set of on- and off-ramps \n($L_{\\rm on}=L_{\\rm off}=400$ m) to model the inhomogeneity\nThe on-ramp is centered at $x=509.2$ km, and the off-ramp at\n510.2 km.\nThe ramp flow of the on-ramp is 110 vehicles/h/lane for\n$t\\le 2.25$ h, and 60 vehicles/h/lane afterwards.\nIn addition, we assumed a flow peak of 600 vehicles/h/lane\nfor 55 min $\\le t\\le$ 65 min which triggers the breakdown.\nThe flow of the off-ramp is -110 vehicles/km/lane, and the\ninflow imposed at the upstrema boundary is \n1450 vehicles/h/lane.\n}\n\\end{figure}\n% ##############################################################\n"
}
] |
[
{
"name": "cond-mat0002177.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{Hall1}\nF.~L. Hall and K. Agyemang-Duah, Transp. Res. Rec. {\\bf 1320}, 91 (1991).\n\n\\bibitem{Daganzo-ramp}\nC.~F. Daganzo, in {\\em Proceedings of the 13rd International Symbosium on\n Transportation and Traffic theory}, edited by J.~B. Lesort (Pergamon,\n Tarrytown, 1996), p.\\ 629.\n\n\\bibitem{Kerner-rehb96-2}\nB.~S. Kerner and H. Rehborn, Phys. Rev. E {\\bf 53}, R4275 (1996).\n\n\\bibitem{Kerner-sync}\nB.~S. Kerner and H. Rehborn, Phys. Rev. Lett. {\\bf 79}, 4030 (1997).\n\n\\bibitem{Nagatani-sync}\nT. Nagatani, J. Phys. Soc. Japan {\\bf 66}, L1928 (1997).\n\n\\bibitem{Persaud}\nB. Persaud, S. Yagar, and R. Brownlee, Transportation Research Record {\\bf\n 1634}, 64 (1998).\n\n\\bibitem{Kerner-wide}\nB.~S. Kerner, Phys. Rev. Lett. {\\bf 81}, 3797 (1998).\n\n\\bibitem{sync-Letter}\nD. Helbing and M. Treiber, Phys. Rev. Lett. {\\bf 81}, 3042 (1998).\n\n\\bibitem{TSG-science}\nD. Helbing and M. Treiber, Science {\\bf 282}, 2001 (1998).\n\n\\bibitem{Lee}\nH.~Y. Lee, H.~W. Lee, and D. Kim, Phys. Rev. Lett. {\\bf 81}, 1130 (1998).\n\n\\bibitem{Lee99}\nH.~Y. Lee, H.~W. Lee, and D. Kim, Phys. Rev. E {\\bf 59}, 5101 (1999).\n\n\\bibitem{Lee-emp}\nH.~Y. Lee and H.~W. Lee and D. Kim, {\\it Empirical Phase Diagram of Congested\n Traffic Flow}, Preprint cond-mat/9905292.\n\n\\bibitem{Daganzo-ST}\nC.~F. Daganzo, M.~J. Cassidy, and R.~L. Bertini, Transportation Research B {\\bf\n 33}, 25 (1999).\n\n\\bibitem{Phase}\nD. Helbing, A. Hennecke, and M. Treiber, Phys. Rev. Lett. {\\bf 82}, 4360\n (1999).\n\n\\bibitem{Kerner-Transp}\nB. S. Kerner, Congested traffic flow: Observations and theory, Transportation\n Research Record 1999, in print.\n\n\\bibitem{Kerner-rehb98}\nB.~S. Kerner and H. Rehborn, Internationales Verkehrswesen {\\bf 5}, 196\n (1998).\n\n\\bibitem{numerics}\nD. Helbing and M. Treiber, Computing in Science and Engineering (CiSE) {\\bf 5},\n 89 (1999).\n\n\\bibitem{Helb-book}\nD. Helbing, {\\em Verkehrsdynamik} (Springer, Berlin, 1997).\n\n\\bibitem{Helb-emp97}\nD. Helbing, Phys. Rev. E {\\bf 55}, R25 (1997).\n\n\\bibitem{GKT-scatter}\nM. Treiber and D. Helbing, J. Phys. A {\\bf 32}, L17 (1999).\n\n\\bibitem{Smulders1}\nS. Smulders, {\\em 7th International Conference on Road Traffic Monitoring and\n Control} (IEE, London, 1994), pp.\\ 145--149.\n\n\\bibitem{Vladi-98}\nV. Shvetsov and D. Helbing, Phys. Rev. E {\\bf 59}, 6328 (1999).\n\n\\bibitem{Kerner-rehb96}\nB.~S. Kerner and H. Rehborn, Phys. Rev. E {\\bf 53}, R1297 (1996).\n\n\\bibitem{Lighthill-W}\nM.~J. Lighthill and G.~B. Whitham, Proc. Roy. Soc. of London A {\\bf 229}, 317\n (1955).\n\n\\bibitem{Cremer-93}\nM. Cremer and F. Mei{\\ss}ner, {\\it Traffic Prediction and Optimization Using an\n Efficient Macroscopic Simulation Tool}. In: A. Pave, ed. {\\it Modelling and\n Simulation 1993}, pp. 513-519. Society for Computer Simulation International,\n Ghent, Belgium.\n\n\\bibitem{KK-94}\nB.~S. Kerner and P. Konh{\\\"a}user, Phys. Rev. E {\\bf 50}, 54 (1994).\n\n\\bibitem{GKT}\nM. Treiber, A. Hennecke, and D. Helbing, Phys. Rev. E {\\bf 59}, 239 (1999).\n\n\\bibitem{Newell}\nG.~F. Newell, Operations Research {\\bf 9}, 209 (1961).\n\n\\bibitem{Gipps81}\nP.~G. Gipps, Transportation Research B {\\bf 15}, 105 (1981).\n\n\\bibitem{Wiedemann}\nR. Wiedemann, {\\it Simulation des Stra{\\ss}enverkehrsflusses}. \n Schriftenreihe des IfV, Vol. {\\bf 8}\n(Institut f\\\"ur Verkehrswesen, University of\n Karlsruhe, Germany, 1974).\n\n\\bibitem{Bando}\nM. Bando {\\it et~al.}, Phys. Rev. E {\\bf 51}, 1035 (1995).\n\n\\bibitem{Krauss-traff98}\nS. Krau{\\ss}, in {\\em Traffic and Granular Flow '97}, edited by \nM. Schreckenberg and D.~E.~Wolf\n(Springer, Singapore, 1998), pp.\\ 269--283.\n\n\n%\\bibitem{Krauss-PhD}\n%S. Krau{\\ss}, {\\bf [Titel der Arbeit]} (Phd thesis, DLR, Cologne,\n%Germany, 1998).\n\n\\bibitem{Tilch-GFM}\nB. Tilch and D. Helbing, Phys. Rev. E {\\bf 58}, 133 (1998).\n\n\\bibitem{MITSIM}\n{\\sloppy For an overview, see the internet page\\\\ {\\tt\n http://hippo.mit.edu/products/mitsim/main.html}.}\n\n\\bibitem{Howe}\nL.~D. Howe, Physica A {\\bf 246}, 157 (1997).\n\n\\bibitem{coexist}\nM. Treiber and D. Helbing, \n``Explanation of observed features of self-organization in traffic flow''.\n Preprint cond-mat/9901239.\n\n\\bibitem{TGF99-Treiber}\nM. Treiber, A. Hennecke, and D. Helbing, in {\\em Traffic and Granular Flow\n '99}, edited by D. Helbing, H.~J. Herrmann, M. Schreckenberg, and D.~E. Wolf\n (Springer, Berlin, 2000), in print. \n\n\\bibitem{Biham}\nO. Biham, A.~A. Middleton, and D. Levine, Phys. Rev. A {\\bf 46}, R6124\n (1992).\n\n\\bibitem{Cremer-multi}\nM. Cremer and J. Ludwig, Math. Comput. Simulation {\\bf 28}, 297 (1986).\n\n\\bibitem{Nagel-S}\nK. Nagel and M. Schreckenberg, J. Phys. I France {\\bf 2}, 2221 (1992).\n\n\\bibitem{Barlovic}\nR. Barlovic, L. Santen, A. Schadschneider, and M. Schreckenberg, {\\em Traffic\n and Granular Flow '97} (Springer, Singapore, 1998), p.\\ 335.\n\n\\bibitem{Helb-sblock}\nD. Helbing and B.~A. Huberman, Nature {\\bf 396}, 738 (1998).\n\n\\bibitem{Wolf-Galilei}\nD.~E. Wolf, Physica A {\\bf 263}, 438 (1999).\n\n\\bibitem{Kuehne}\nR.~D. K\\\"uhne, in {\\em Proceedings of the \n 9th International Symposium on Transportation and Traffic Theory},\n edited by I. Volmuller and R. Hamerslag\n (VNU Science Press, Utrecht, 1984), p. 21\n\n\\bibitem{Treiterer}\nJ. Treiterer and J.~A. Myers, in {\\em Proc. 6th Int. Symp. on Transportation\n and Traffic Theory}, edited by D.~J. Buckley (Elsevier, New York, 1974), p.\\\n 13, empirical observation of trajectories.\n\n\\bibitem{TGF99-Hennecke}\nA. Hennecke, M. Treiber, and D. Helbing, in {\\em Traffic and Granular Flow\n '99}, edited by D. Helbing, H.~J. Herrmann, M. Schreckenberg, and D.~E. Wolf\n (Springer, Berlin, 2000), p.\\ to be published.\n\n\\bibitem{Reuschel}\nA. Reuschel, {\\\"O}sterreichisches Ingenieur-Archiv {\\bf 4}, 193 (1950).\n\n\\bibitem{Chandler}\nR.~E. Chandler, R. Herman, and E.~W. Montroll, Operations Research {\\bf 6},\n 165 (1958).\n\n\\bibitem{Herman59}\nR. Herman, E.~W. Montroll, R.~B. Potts, and R.~W. Rothery, Operations Research\n {\\bf 7}, 86 (1959).\n\n\\bibitem{Gazis61}\nD.~C. Gazis, R. Herman, and R.~W. Rothery, Operations Research {\\bf 9}, 545\n (1961).\n\n\\bibitem{Edie}\nL.~C. Edie, Transp. Res. B {\\bf 28}, 66 (1961).\n\n\\bibitem{Bleile-Diss}\nT. Bleile, Modellierung des Fahrzeugfolgeverhaltens im innerst\\\"adtischen\nPKW-Verkehr,\n(Phd thesis, University of Stuttgart, Germany, 1999).\n\n\\bibitem{Gipps86}\nP.~G. Gipps, Transp. Res. {\\bf 20 B}, 403 (1986).\n\n\\bibitem{Nagatani-kink}\nM. Muramatsu and T. Nagatani, Phys. Rev. E {\\bf 60}, 180 (1999).\n\n\\bibitem{note-phase2-micperf}\nThe simulations were performed on a personal workstation with a 433 Mhz Alpha\n processor. \n%The timestep of the explicit numerical scheme was $\\Delta t=0.4$ s.\n\n\\bibitem{Kerner-ramp}\nB.~S. Kerner, P. Konh{\\\"a}user, and M. Schilke, Phys. Rev. E {\\bf 51}, 6243\n (1995).\n\n\\bibitem{note-phase2-speedlimit}\n{Since} speed limits lead to reduced desired velocities for many drivers as\n well, one might conclude that speed limits {also cause} undesirable local\n bottlenecks. The main effect of speed limits is, however, to reduce\n the velocity variance which leads to less braking {maneuvers\n } and {\\it de\n facto} to a higher capacity. In contrast, gradients {mainly}\n influence \n slow vehicles (trucks), leading to higher values for the velocity variance.\n\n\\bibitem{Gazis-Herman}\n D.~C. Gazis and R. Herman, \n Transportation Science {\\bf 26}(3),223--229 (1992).\n%\\bibitem{note-phase2-A5}\n%There is an uphill section and a relatively sharp curve at this location of the\n% A5-North. Nevertheless, the gradient is lower than that on the A8-East and it\n% is unclear, if it constitutes the bottleneck.\n\n\\bibitem{platoon-multilane}\nE. Ben-Naim and P.~L. Krapivsky, Phys. Rev. E {\\bf 56}, 6680 (1997).\n\n\\bibitem{Applet-engl}\nInteractive simulations of the multi-lane IDM are available at {\\tt\n www.theo2.physik.uni-stuttgart.de/treiber/MicroApplet/}.\n\n\\end{thebibliography}"
}
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cond-mat0002178
|
Anomalous Diffusion in Quasi One Dimensional Systems
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[
{
"author": "F. M. Cucchietti"
}
] |
In order to perform quantum Hamiltonian dynamics minimizing localization effects, we introduce a quasi-one dimensional tight-binding model whose mean free path is smaller than the size of the sample. This size, in turn, is smaller than the localization length. We study the return probability to the starting layer using direct diagonalization of the Hamiltonian. We create a one dimensional excitation and observe sub-diffusive behavior for times larger than the Debye time but shorter than the Heisenberg time. The exponent corresponds to the fractal dimension $d^{*} \sim 0.72$ which is compared to that calculated from the eigenstates by means of the inverse participation number. PACS numbers: 72.10.Bg, 73.20.Dx, 73.20.Fz \begin{keyword} Anomalous Diffusion. Weak Localization. Fractal Dimension. \end{keyword}
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[
{
"name": "CP.tex",
"string": "\n\\documentclass{elsart}\n\n\\usepackage{epsfig}\n\\usepackage{epsf}\n\n\\begin{document}\n\n\n\\begin{frontmatter}\n\\title{Anomalous Diffusion in Quasi One Dimensional Systems}\n\\author{F. M. Cucchietti},\n\\author{H. M. Pastawski}\n\\address{LANAIS de RMS, Facultad de Matem\\'{a}tica, Astronom\\'{i}a y F\\'{i}sica, Universidad\nNacional de C\\'{o}rdoba, C\\'{o}rdoba, Argentina} \n\\begin{abstract}\nIn order to perform quantum Hamiltonian dynamics minimizing localization effects, \nwe introduce a quasi-one dimensional tight-binding model whose mean free path is \nsmaller than the size of the sample. This size, in turn, is smaller than the localization \nlength. We study the return probability to the starting layer using direct diagonalization of \nthe Hamiltonian. We create a one dimensional excitation \nand observe sub-diffusive behavior for times larger than the Debye time but shorter \nthan the Heisenberg time. The exponent corresponds to the fractal dimension $d^{*} \\sim 0.72$ \nwhich is compared to that calculated from the eigenstates by means of the \ninverse participation number. \nPACS numbers: 72.10.Bg, 73.20.Dx, 73.20.Fz\n\\begin{keyword}\nAnomalous Diffusion.\nWeak Localization.\nFractal Dimension.\n\\end{keyword}\n\\end{abstract}\n\\end{frontmatter}\n\nThe classical kinetic theory predicts that, in a disordered system, the\nreturn probability of an excitation decays with a diffusive law $P(t)\\sim\n(4\\pi Dt)^{-d/2}$, where $D$ is the diffusion coefficient and $d$ the\ndimension of the system. This fact has been extensively used in many areas\nof physics, in particular in electronic transport. However, the theory of\nquantum localization\\cite{AALR}, on the basis of steady state transport\nproperties, made clear that different regimes arise as a function of the\ndisorder $W$. For $W\\rightarrow \\infty $, the eigenstates are completely\nlocalized and wave packets do not move through the system (insulating\nphase). If $W\\rightarrow 0$, the motion is completely ballistic with a\nvelocity $Ja/\\hbar $. Between these two limits the diffusive behavior\n(metallic phase) is observed. The particles move around freely between\ncollisions for a certain average length $\\ell $, called the mean free path.\n\nMore recently, theoretical studies have considered the regime where the\ncritical amount of disorder $W_{C}$ is such that the system is at the\nmetal-insulator transition ($W_{C}$ $\\sim zJ$ is the typical exchange\nenergies with $z$ neighbors at distance $a$). In that case, the dynamics of\nthe system is diffusive, but with a smaller exponent implying $d^{*}<d$ \\cite\n{Huckestein}. This reduction in the effective dimension is attributed to the\n(multi)fractality of the eigenstates at the transition. On the other hand,\nthe diffusion of a spin excitation was directly observed \\cite{McDowell} in\nNMR\\ experiments. When the spin network has a cubic structure filling the space, \nthe intensity of the excitation decays as $(t-t_{0})^{-1.5}$,\nwhich is the expected value for diffusion in a three dimensional system.\nHowever, the same experiment performed in a chain-like structure (powdered\npolyethylene) shows anomalous exponents in the diffusion of the excitation;\nnamely $0.9$ and $0.7$ for the crystalline and amorphous parts of the sample\nrespectively. In these systems, the role of disorder is played by typical\nenergy differences $W$ between states, which are smaller than exchange\nenergies $zJ.$ This assures\\cite{Ander_Nobel} a diffusive dynamics with\\cite\n{QZeno} $D\\sim Ja^{2}/\\hbar $. Therefore, the effective dimensions $d^{*}=1.8\n$ and $1.4$ corresponding to these values reflect the spatial structure of\nthe spin network.\n\nIn this work we study the conditions to reach the diffusive regime from\nactual quantum dynamics, namely the numerical study of model Hamiltonians.\nIn particular, we developed a quasi one dimensional tight binding model\\textit{\\ }system,\nwhich we called \\textit{the Stars necklace model, }whose basic unit (layer)\nis a highly connected cluster (see inset in figure 1) with $N$ sites and intralayer \nhopping $J=V/N^{0.5}$.\nDisorder is introduced through on site energies characterized by a random\ndistribution of width $W$ . The initial wave function is a packet defined in\none layer of the system. We are interested in the probability of return to\nthe layer, $P$, which is the sum of probabilities of finding the particle in\nevery site of the initial layer. To calculate the dynamics of the system we\nperform an exact diagonalization of the Hamiltonian. One key aspect of the\nnumerical calculation is that the mean free path $\\ell $, the size of the\nsystem $L$ and the localization length $\\xi $ must obey some restrictive\nrelationships. The condition for a diffusive regime is $\\ell $ much smaller\nthan $L$, hence assuring that the particle will collide many times before\nreaching the boundaries of the system. In turn, to stay away from the\nlocalized regime, $L$ must be smaller than $\\xi $. For a strictly one\ndimensional wire $\\xi =2\\ell $, while for strips and bars with a given\nnumber of transverse modes (channels) $M$ the localization length is\nexpected to go\\cite{MacKinnon} as $\\xi =2M\\ell $. In our model the $M=N-1$\nchannels available for transport have the \\textit{same} group velocity $v=2Va/\\hbar$.\nThis striking feature of the model allows to reach a one dimensional\ndiffusive dynamics when $N\\rightarrow \\infty .$ For finite $N$ it provides\nan optimal representation for one-dimensional excitations. We studied many\nsystem sizes and amounts of disorder, a typical evolution is shown in\nfigure 1. The particular system for this figure has $N=12$ and a perimeter of $%\n100a$, $W=3V$, $\\ell \\sim 6a$ and $\\xi \\sim 140a$ . We see that after a\nballistic time $\\ell /v$, the evolution follows a power law which indicates\na diffusive behavior. Nevertheless, the exponent of this power law is somewhat different\n from the expected one dimensional value $0.5$. As in the examples\nmentioned above, this anomalous exponent could be due to a fractal effective\ndimension of the system. The fitting of the evolution to a power law with\na free exponent resulted in an effective dimension of the system $%\nd^{*}\\sim 0.7$. In our model, a possible cause for a fractality in\nthe eigenfunctions of the system is disorder. Strongly localized stated in\nthe band tails are confined around some random points. This means that they\nrepresent ``holes'' in the real space allowed to extended wave functions,\nthus making the effective dimension of the system smaller than the real one.\nFor times longer than the ones showed in the figure, the autocorrelation\nfunction saturates, this is a finite size effect (the saturation value\ndepends linearly on the system size). We also observed that a magnetic field\ndoes not change the exponent in the power law noticeable, but reduces the\nvalue of the saturation, meaning that there are fewer localized states.\nAnother way to study how the eigenstates of energy $\\varepsilon $ are\noccupying some fraction of the volume of space is by means of the inverse\nparticipation number $p^{-1}=\\sum_{i}\\left| \\varphi _{i}(\\varepsilon\n)\\right| ^{4}$ \\cite{MacKinnon}. For plane waves one obtains that $%\np=L^{d}$, i.e. equals the volume of the system. For a localized state $%\np$ is proportional to the volume in which the state has a non-vanishing\namplitude. However if the states are extended but fractal, in the thermodynamic\nlimit it diverges as $p=L^{d^{*}}$, with $d^{*}$ an effective dimension that may be\ndifferent from the dimension $d$ of the ordered system. We calculated the\ninverse participation number for each of the eigenstates of the system and\nthrough it the effective dimension ratio $d^{*}/d$ of everyone of them. The\nresults (shown in figure 2) are in very good agreement with the effective\ndimension calculated through the fitting of the autocorrelation function\ndepicted in Fig. 1. Summarizing, we have introduced a numerical Hamiltonian\nmodel whose exact solution shows a regime with sub-diffusive behavior.\nMoreover, we presented hints of a fractal dimension of the extended\neigenstates induced by the presence of disorder. By hindering particles from\na fraction of the available real space, disorder induces a weak breaking of\nthe ergodicity that anticipates the non-ergodicity associated with full\nlocalization.\n\n\\begin{thebibliography}{9}\n\\bibitem{AALR} E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V.\nRamakrishnan, Phys. Rev. Lett. \\textbf{42} (1979) 673\n\n\\bibitem{Huckestein} B. Huckestein and L. Schweitzer, Phys. Rev. Lett. \n\\textbf{72} (1994) 713\n\n\\bibitem{McDowell} S. Ding and C. A. McDowell, J. Phys.: Condens. Matter \n\\textbf{11} (1999) L199\n\n\\bibitem{Ander_Nobel} P. W. Anderson, Rev. Mod. Phys. \\textbf{50} (1978) 191\n\n\\bibitem{QZeno} H. M. Pastawski and G. Usaj, Phys. Rev. B \\textbf{57}\n(1998)5017\n\n\\bibitem{MacKinnon} B. Kramer and A. Mackinnon, Rep. Prog. Phys. \\textbf{12}\n(1993) 1469, and references therein.\n\\end{thebibliography}\n\n\\begin{figure}[tb]\n\\centering \\leavevmode\n\\center{\\epsfig{file=CP_figure_1.ps, ,width=16cm,angle=270}}\n\\caption{Evolution of the layer autocorrelation function (in thick solid\nline). The thin line is a best fit to $(4\\pi Dt)^{d^{*}/2}$, with the\nresults $d^{*}=0.72\\pm 0.005$ and $D=(4.84\\pm 0.08)a^{2}V/\\hbar $. In the\ninset is shown a schematics of the system. As shown, the layers are fully\nconnected, and periodic boundary conditions are applied to the longitudinal\ndimension.}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\centering \\leavevmode\n\\center{\\epsfig{file=CP_figure_2.ps, ,width=16cm,angle=270}}\n\\caption{Calculation of the effective dimension of every eigenstate of the\nsystem by means of the inverse participation number (dots) compared to the\neffective dimension obtained from the fitting of the dynamics of the system.\nExcept from very localized states in the band edges, agreement is quite\ngood. }\n\\end{figure}\n\n\\end{document}"
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[
{
"name": "cond-mat0002178.extracted_bib",
"string": "\\begin{thebibliography}{9}\n\\bibitem{AALR} E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V.\nRamakrishnan, Phys. Rev. Lett. \\textbf{42} (1979) 673\n\n\\bibitem{Huckestein} B. Huckestein and L. Schweitzer, Phys. Rev. Lett. \n\\textbf{72} (1994) 713\n\n\\bibitem{McDowell} S. Ding and C. A. McDowell, J. Phys.: Condens. Matter \n\\textbf{11} (1999) L199\n\n\\bibitem{Ander_Nobel} P. W. Anderson, Rev. Mod. Phys. \\textbf{50} (1978) 191\n\n\\bibitem{QZeno} H. M. Pastawski and G. Usaj, Phys. Rev. B \\textbf{57}\n(1998)5017\n\n\\bibitem{MacKinnon} B. Kramer and A. Mackinnon, Rep. Prog. Phys. \\textbf{12}\n(1993) 1469, and references therein.\n\\end{thebibliography}"
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cond-mat0002180
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cond-mat0002182
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Anomalous metallicity and electronic phase separation in the CsC$_{60}$ polymerized fulleride
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[
{
"author": "Barbara Simovi\\v{c}$^{(1)}$\\cite{Auth1}"
},
{
"author": "Denis J\\'{e}rome$^{(1)}$"
},
{
"author": "and L\\'{a}szl\\'{o} Forr\\'{o}$^{(2)}$"
}
] |
$^{133}\mathrm{Cs}$ and $^{13}\mathrm{C}$-NMR have been used to study the electronic properties of the polymerized phase of $\mathrm{CsC}_{60}$ at ambient and under hydrostatic pressure. The salient result of this study is the finding of fluctuations in the local field at $^{133}\mathrm{Cs}$ site which are independent of the applied pressure and due to thermally activated changes in the local electronic environment of $^{133}\mathrm{Cs}$ nuclei. We establish that the phase separation between magnetic and nonmagnetic domains observed in the low temperature state at ambient pressure is the result of a slowing down of these fluctuations likely related to polaronic charge excitations on the polymers.
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"name": "LP7370.tex",
"string": "\n\\documentstyle[aps,preprint,graphicx]{revtex}\n%\\documentstyle[prl,aps,graphicx]{revtex}\n\n\\begin{document}\n\\title{Anomalous metallicity and electronic phase separation in the CsC$_{60}$\npolymerized fulleride}\n%\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname\n\\author{Barbara Simovi\\v{c}$^{(1)}$\\cite{Auth1}, Denis\nJ\\'{e}rome$^{(1)}$, and L\\'{a}szl\\'{o} Forr\\'{o}$^{(2)}$}\n\\address{$^{(1)}$ Laboratoire de Physique des Solides (associ\\'{e} au CNRS),\nUniversit\\'{e} Paris-Sud, 91405 Orsay, France.\\\\$^{(2)}$\nLaboratoire de Physique des Solides Semicristallins, EPFL,\nSwitzerland.}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\n$^{133}\\mathrm{Cs}$ and $^{13}\\mathrm{C}$-NMR have been used to\nstudy the electronic properties of the polymerized phase of\n$\\mathrm{CsC}_{60}$ at ambient and under hydrostatic pressure. The\nsalient result of this study is the finding of fluctuations in the\nlocal field at $^{133}\\mathrm{Cs}$ site which are independent of\nthe applied pressure and due to thermally activated changes in the\nlocal electronic environment of $^{133}\\mathrm{Cs}$ nuclei. We\nestablish that the phase separation between magnetic and\nnonmagnetic domains observed in the low temperature state at\nambient pressure is the result of a slowing down of these\nfluctuations likely related to polaronic charge excitations on the\npolymers.\n\\end{abstract}\n\n\\pacs{PACS.61.48.+c, 76.60.-k, 71.38.+i}%]\n\n\\section{INTRODUCTION}\n When the face centered cubic (f.c.c) phase of ${\\rm A}%\n_{1}{\\rm C}_{60}$(A=K, Rb, Cs) compounds is slowly cooled from\n400K, one-dimensional polymerization of C$_{60}$ molecules\nspontaneously occurs along the (110) cubic direction and leads to\nan orthorhombic phase\\cite {Stephens94}. A drastic change in the\nelectronic properties is observed at the structural\ntransition\\cite{Tycko93,Chauvet94}. Indeed, the f.c.c. phase was\nshown to be a Mott insulator\\cite{Tycko93,Chauvet94}, whereas in\nthe orthorhombic phase a plasma frequency was measured in optical\nexperiments \\cite{Bommeli95}. However, the density of carriers is\nlikely to be rather low or their effective mass very large since\nthe plasma frequency is equal to 0.1eV in ${\\rm KC}_{60}$ polymer\nand even lower for ${\\rm RbC}_{60}$ and ${\\rm\nCsC}_{60}$\\cite{Bommeli95}. In addition, the low frequency\nconductivity of both ${\\rm RbC}_{60}$ and ${\\rm CsC}_{60}$\ndecreases\nsmoothly over a broad temperature range, being at variance with ${\\rm KC}%\n_{60}$ which remains conducting down to 4.2K\\cite{Bommeli95}.\nFurthermore, the temperature dependence of the $^{13}{\\rm C}$\nspin-lattice relaxation rate shows that strong magnetic\nfluctuations are present up to room temperature in ${\\rm\nRbC}_{60}$ and ${\\rm CsC}_{60}$\\cite{Brouet96} and the sharp\ndecrease of the uniform static susceptibility (measured from EPR\nline intensity) below 50K for ${\\rm RbC}_{60}$ and 40K for ${\\rm\nCsC}_{60}$\\cite{Bommeli95}, suggests that both compounds undergo\nmagnetic transitions at these respective temperatures. The\noccurrence of spin ordering is also evident from NMR\nexperiments\\cite {Brouet96,Senzier96}: the slowing down of\nmagnetic fluctuations gives rise to a divergent relaxation rate\nbelow 40K. However, the nature of the spin order is less obvious.\nOn one hand, EPR experiments \\cite{Janossy97,Bennati98} suggest\nthe onset of a spin density wave ground state as a result of a\npossible one-dimensional (1D) character of the band structure. On\nthe other hand, $\\mu {\\rm SR}$ studies\\cite{MacFarl95,Uemura95}\nshow a gradual transition towards a highly disordered magnetic\nphase and do not rule out the possibility of a random spin\nfreezing below 40K. \\newline In a recent NMR work\\cite\n{Barbara99}, we have shown that some of the $^{133}{\\rm Cs}$\nsites remain unaffected by the onset of the spin-ordering in the\nlow temperature state, magnetic and nonmagnetic domains being\nspatially distributed. At the temperature of 13.8K the occurrence\nof a charge redistribution and a concomitant decrease of the\nlocal electronic susceptibility inside these nonmagnetic domains\nhave been observed \\cite{Barbara99}. In agreement with this\nlatter result, detailed analysis of the EPR linewidth at ambient\npressure also suggest that two distinct magnetic environments\ncoexist in the low temperature state of ${\\rm RbC}_{60}$ and ${\\rm\nCsC}_{60}$ polymers \\cite{Atsarkin97,Coulon00} and insofar as a\ncharge redistribution occurs in the nonmagnetic domains at\n13.8K\\cite{Barbara99}, the spontaneous thermal contraction\nrecently observed at 14K by X-Ray diffraction in ${\\rm\nCsC}_{60}$\\cite{Rouziere00} strongly supports the fact that these\ninhomogeneities are intrinsic.\\newline In this manuscript, we\ngive experimental evidence showing that the ``conducting'' state\nof the ${\\rm CsC}_{60}$ polymerized phase cannot be understood\nwithin the framework of an electronic band conductor as claimed\nearlier\\cite{Chauvet94,Bommeli95,Brouet96,Janossy97,Bennati98,Erwin95}\n. We first report the temperature dependence of the spin lattice\nrelaxation rate $(T_{1})^{-1}$ for both $^{13}{\\rm C}$ and\n$^{133}{\\rm Cs}$ nuclei at different pressures up to 9 kbar,\nindicating that in the temperature domain above 80K two\ndifferent mechanisms govern the relaxation of $^{13}{\\rm C}$ and $^{133}{\\rm %\nCs}$ nuclei respectively. As far as $^{133}{\\rm Cs}$ is concerned,\n$^{133}(T_{1})^{-1}$ decreases linearly down to about 80K though\nremaining pressure independent up to 9kbar. This behavior is in\nsharp contrast with the $^{13}{\\rm C}$ nuclei for which\n$^{13}(T_{1})^{-1}$ strongly decreases under pressure up to 9kbar\nwhile remaining almost temperature independent. The difference\nbetween $^{133}{\\rm Cs}$ and $^{13}{\\rm C}$ nuclei exists {\\em\nindependently} of the nature of the ground state of the system.\nMore insight into these peculiar\nproperties is then obtained using quadrupolar echo experiments performed on the $%\n^{133}{\\rm Cs}$ nucleus which enable us to analyze with great\naccuracy the temperature dependence of the NMR spectrum at 1bar.\nWe show that the NMR spectrum of the two phases (magnetic and\nnonmagnetic) is motional narrowed\nabove 100K because of the fast motion of the local environment around the $%\n^{133}{\\rm Cs}$ sites. The evolution of the lineshape with\ntemperature reveals that the static coexistence of two different\n$^{133}{\\rm Cs}$ sites below 15K arises from a gradual freezing of\nthese fluctuations in the local environment.\n\n\\section{EXPERIMENTAL DETAILS}\n The measurements have been conducted on two powdered\nsamples with entirely consistent results, one of them (10\\%)\n$^{13}{\\rm {C}}$ enriched. The pressure set up is a homemade\ndouble-stage copper-beryllium cell using fluor-inert as the\npressure medium. This enables us to correct for each temperature\nthe loss of pressure within the sample chamber due to the gradual\nfreezing of the fluor-inert.\\newline The spin-lattice relaxation\nwere measured by monitoring the recovery of the magnetization\nafter saturation with a series of $\\pi/2$ pulses. The recovery\ncurve is exponential for $^{133}{\\rm Cs}$ and $^{13}{\\rm C}$\n\\cite{comment1} at room temperature. At ambient pressure, the\nrecovery curve gradually becomes bi-exponential for both nuclei\nbelow 40K. A large distribution of short relaxation rates is\nobserved giving raise to a recovery curve of the following shape\n$1-e^{{(-t/T_{1})}^\\beta}$ with a value of $\\beta$ of the order\nof 0.5 at the lowest temperature investigated i.e 4K. At 5kbar\nthe recovery curve is for $^{133}{\\rm Cs}$ exponential down to\n4K. Not so for $^{13}{\\rm C}$ since a nonexponential recovery is\nobserved below 20K. Different fit procedures did not help us to\ndetermine without ambiguity the shape of the recovery but no\nsignificant change were observed on the qualitative temperature\ndependence of $^{13}(T_{1})^{-1}$. The relaxation rates\n$^{13}(T_{1})^{-1}$ shown on Fig.1b at 5kbar are therefore\ndeduced below 20K from a fit of the recovery curve assuming it to\nbe exponential as above 20K. At 9kbar, the recovery curve is\nexponential for $^{133}{\\rm Cs}$ and $^{13}{\\rm C}$ in the all\ntemperature range investigated. \\newline Finally we should point\nout that despite the presence of a static quadrupole\nsplitting\\cite{comment2} for the NMR line of $^{133}{\\rm Cs}$ in\nthe orthorhombic phase, the smallness of the quadrupole frequency\nwhich is of the order of 5kHz enables us to saturate all the\ntransitions at once. Therefore the nuclear levels are initially\nequally populated establishing a well-defined spin temperature\n(equal to infinite). In that case no deviation from an exponential\nbehavior is expected for the relaxation of the\nmagnetization\\cite{Suter98} which\nperfect exponential recovery at room temperature is a proof of the homogeneity of the samples.\n\n\\section{$^{13}\\rm C$ and $^{133}\\rm Cs$-NMR UNDER PRESSURE}\nWe report on Fig.1a and Fig.1b the temperature dependence of the\nrelaxation rate for $^{133}{\\rm Cs}$ and $^{13}{\\rm C}$ nuclei at\nambient pressure, 5kbar and 9kbar. The large enhancement of\n$^{133}(T_{1})^{-1}$ and $^{13}(T_{1})^{-1}$ below 40K is due to\na slowing down of magnetic fluctuations which is completely\nsuppressed at 5kbar. At this pressure, both $^{133}(T_{1})^{-1}$ and $%\n^{13}(T_{1})^{-1}$ decrease exponentially below 20K revealing the\nopening of a spin-gap at $T_{C}\\approx 20K$, the ground state\nbeing homogeneous and nonmagnetic. The effect of an applied\npressure on this long range order has been carefully investigated\nby $^{133}{\\rm Cs}$-NMR. The temperature dependence of\n$^{133}(TT_{1})^{-1}$ at 5, 5.5, 5.7 and 9kbar is shown in\nFig.2.The well-defined\ninstability at 5kbar gives rise to a sharp peak on $^{133}(TT_{1})^{-1}$ at $%\nT_{C}$. Quite remarkably, a slight increase of the applied\npressure strongly\nreduces the amplitude of the spin-gap, without any significant change in $%\nT_{C}$ itself (as given by the position of the\n$^{133}(TT_{1})^{-1}$ peak, see on Fig.2). However, a smooth\ndecrease of the temperature ${\\rm T_{Mag}}$ at which the slowing\ndown of magnetic fluctuations occurs has been observed by EPR\nexperiments under pressure up to 4kbar\\cite{Forro96}. This fact is\nalso evident from the temperature dependence of the linewidth of\nthe $^{133}{\\rm Cs}$ NMR line shown at different pressures on\nFig.3. Thus, as the pressure increases ${\\rm T_{Mag}}$ drops\ncontinuously along a transition line which does not exist for the\ncase of the spin-singlet ground state. Henceforth we can infer\nthat the sharp suppression of the spin-gap below 20K which in\nturn gives rise to a metallic state, is not due to continuous\nchanges in the magnitude of the electronic interactions but may\nreflect some structural changes above 5kbar as suggested by DC\nconductivity measurements performed under\npressure\\cite{Khazeni97,Zhou00}.\n\n\n\\noindent Another striking feature in the response of the polymerized phase $%\n{\\rm CsC}_{60}$ to high pressure appears at glance in Fig.1a and\nFig.1b. Indeed, a clear distinction has to be made between the two\ntemperature\ndomains 4.2-80K and 80K-300K. Below 80K, $^{133}(T_{1})^{-1}$ and $%\n^{13}(T_{1})^{-1}$ exhibit a similar pressure and temperature\ndependence.\nThis is, however, no longer true above 80K, where $^{133}(T_{1})^{-1}$ and $%\n^{13}(T_{1})^{-1}$ behave in complete different ways. In\nparticular, we can see in Fig.1a that above 80K,\n$^{133}(T_{1})^{-1}$ shows no pressure dependence up to 9kbar\nunlike $^{13}(T_{1})^{-1}$, which is shown on Fig.1b. Within the\nfirst five kilobars, the relaxation of $^{13}{\\rm C}$ nuclei is\nstrongly affected by pressure in two manners:(i) an overall\ndepression is observed under pressure following the depression of\nthe uniform spin susceptibility ($\\chi $) measured by EPR\n\\cite{Forro96} which drops at a rate of about 10\\% per kbar, (ii)\na weakly temperature dependent contribution to $^{13}(T_{1})^{-1}$\n(20\\% decrease from 300 to 40K) is suppressed at 5 kbar.\\newline Broadly\nspeaking, the spin-lattice relaxation rate for a given nuclei and\nthe static electronic spin susceptibility are linked together by\nthe following relation : $(T_{1}T)^{-1}\\propto\n\\sum_{q}|A(\\vec{q})|^{2}\\chi{_{\\perp}} ^{^{\\prime \\prime\n}}(\\vec{q})$ where\n$A(\\vec{q})=\\sum_{i}A_{i}e^{i\\vec{q}.\\vec{r}_{i}}$ is the form\nfactor of the hyperfine interaction between a given nuclei and\nthe electronic spins located at the neighboring sites. Unlike\n$^{133}{\\rm Cs}$ which environment is octahedral, there is no\nparticular symmetry for $^{13}{\\rm C}$ sites. If both nuclei are\ncoupled to the same electronic spins then, that\n$^{13}(T_{1})^{-1}$ and $^{133}(T_{1})^{-1}$ display a different\npressure and temperature dependence above 80K, might be\nattributed to the presence of a spatially dependent electronic\nspin susceptibility which dominates the relaxation of $^{13}{\\rm\nC}$ nuclei. However, as previously shown for ${\\rm\nRbC}_{60}$ \\cite{Senzier96},\n the decrease of $^{13}(T_{1})^{-1}$ follows the decrease of\n the uniform spin susceptibility deduced from EPR\\cite{Forro96} within at least the first five\n kilobars. This reveals that in the low pressure regime, the dominant\ncontribution to the relaxation of $^{13}\\rm C$ above 80K is due to enhanced\nmagnetic fluctuations at the wave vector $\\vec{q}=0$ and therefore,\nthe differences described above between $^{13}\\rm C$ and\n$^{133}\\rm Cs$ cannot be ascribed to the form factor of the alkali\nsite in the polymerized phase. \\newline As it is, one can draw the\nfollowing conclusions. First, the absence of pressure dependence\nobserved for $^{133}(T_{1})^{-1}$ above 80K shows that the\ndominant contribution to the fluctuating field at $^{133}{\\rm\nCs}$ site in this temperature range is unrelated to the\nelectronic spins involved in the relaxation of $^{13}{\\rm C}$\nnuclei. Secondly, the fact that above 80K, $^{13}(T_{1})^{-1}$ is\nweakly temperature dependent at ambient pressure and constant at\n5kbar suggests that the electronic spins are localized. This\nlatter conclusion is in agreement with the calculated band\nstructure of the polymer $({\\rm C}_{60}^{-})^{n}$\\cite{Stafstrom95}\nwhich displays a\n dispersionless 1D half-filled band at the Fermi level but in apparent contradiction\n with\n transport measurements\\cite{Khazeni97,Zhou00} performed in the similar compound ${\\rm\n RbC}_{60}$.\\newline\n One can therefore conclude that a model based on a single electron specie is inadequate for\n describing the electronic properties of the polymerized phases ${\\rm\nRbC}_{60}$ and ${\\rm CsC}_{60}$. \\newline\nIn a previous work\\cite{Barbara99}, we have shown that the use of quadrupolar\nspin echoes of $^{133}{\\rm Cs}$ nuclei enables to reveal the\npresence of nonmagnetic domains within a magnetic background.\nHowever, whether this inhomogeneous state results from the\nexistence of static structural defects along the chains or is\npurely electronically driven e.g. as proposed for underdoped\ncuprates\\cite{Emery} and spin-ladders compounds\\cite{Scalapino},\nremained an open question. In what follows, we address this\nproblem again with the aid of quadrupolar spin echoes in order to\ndetermine how the inhomogeneous state at low temperature arises\nfrom the high temperature one.\n\n\\section{$^{133}\\rm Cs$-NMR AT AMBIENT PRESSURE}\n\nIn a similar way than in the reference\\cite{Barbara99}, the spin echoes of $^{133}{\\rm Cs}$ have been obtained after a ($%\n\\pi /2-\\tau -\\pi /8$) in-phase RF pulse sequence\\cite{comment3},\nmaintaining fixed the echo delay $\\tau $ at 40$\\mu s$. Half of\nthe spin-echo at 3$\\tau $ is then Fourier transformed. This\nprocedure gives rise to a spectrum containing two lines\n$5/2\\rightarrow 3/2$ and $-3/2\\rightarrow -5/2$ split\nby an amount $4\\nu _{Q}$, where $\\nu _{Q}$ is the quadrupole frequency of $%\n^{133}{\\rm Cs}$ nuclei in the polymerized phase\n\\cite{Barbara99,Barbara}. \\noindent The evolution of the\n$^{133}{\\rm Cs}$ spectrum is displayed on Fig.4 at different\ntemperatures between 100 and 4.2K. The expected doublet spectrum\ncorresponding to a single $^{133}{\\rm Cs}$ site is observed at\n100K, but as $T$ approaches 40K, the shape becomes asymmetric and\na fine\nstructure gradually develops. At 25K, the coexistence of two different $%\n^{133}{\\rm Cs}$ sites is evident in Fig.4, with a frequency\ndifference in the local field of the order of $4\\nu _{Q}$. This\nmeans therefore that two distinct magnetic environments are\nspatially distributed at this temperature. As the temperature is\nfurther lowered, the situation with a single quadrupolar split is\nrecovered and thus only one $^{133}{\\rm Cs}$ site contributes to\nthe spin echo signal below 15K. The amplitude of the spin echo\nrefocused at 3$\\tau $ is proportional to $e^{-3\\tau \\gamma \\Delta\nH(T)}$ where $\\Delta H(T)$ is the width of the local field\ndistribution due to the static electronic moments at a given\ntemperature~$T$. Considering two\ndistinct populations of $^{133}{\\rm Cs}$ nuclei below 30K, $N_{m}$ and $%\nN_{nm}$ which are coupled to the local field inhomogeneity $\\Delta\nH(T)$ and\nlocated inside the nonmagnetic domains respectively, the total number of $%\n^{133}{\\rm Cs}$ sites contributing to the spin echo signal at\n3$\\tau $ can be expressed as:~$N(T)=N_{m}/(1+(3\\tau \\gamma \\Delta\nH(T))^{2})+N_{nm}$. If the condition $3\\tau\n\\gamma \\Delta H(T)\\gg\n1$ is fulfilled, only a fraction $N_{nm}$ of the nuclei contribute\nto a spin echo at $3\\tau $ since this experiment selects those\n${\\rm Cs}$ sites which are entirely decoupled from the onset of\nlocal magnetism. Let $I(T)$ be the integrated intensity of the\nFourier transform performed on this spin echo. The temperature\ndependence of $N(T)$\n(equal to $I(T).T$) is reported on Fig.5. We observe that a majority of the $%\n^{133}{\\rm Cs}$ nuclei is gradually wiped out of the signal below\n40K. A minimal value for $N_{nm}$ is reached at 15K and amounts to\nabout 10\\% of the total number of nuclei at $40K$. However, the\nestimated ratio between the two phases from the $^{13}\\rm C$\nspectrum suggests that approximately half of the $^{13}\\rm C$ sites\ndo not see the magnetic moment distribution in the low temperature\nstate\\cite{Brouet96}. We may solve this puzzle by considering that\nthe $^{13}\\rm C$ spins probe the very local properties within each\n$\\rm C_{60}$ chains carrying the electronic spins whereas only\n${^{133}\\rm Cs}$ sites far from\nany magnetic domain will contribute to the echo signal refocused at $3\\tau $%\n. This would mean that the boundary surface is large compare to\nthe domains\nsize suggesting that the phase separation sets on a microscopic scale.%\n\\newline\nTo gain insight into the driving force of this process more\nattention must be paid to what happens above the spin ordering\ntemperature. In particular, we see on Fig.4 that the splitting of\nthe $^{133}{\\rm Cs}$ spectrum displays a fine structure near 40K\nalthough the NMR spectrum corresponding to two $^{133}{\\rm Cs}$\nsites is not yet resolved. This can be understood if we assume\nthat the local field of a $^{133}{\\rm Cs}$ nucleus jumps randomly\nfrom one value to the other in the ``conducting'' state. Indeed,\nusing only the difference between the resonance frequencies\n$\\delta \\omega $ and the hopping time $\\tau _{h}$, we can propose\nthe following scenario. At high temperature, $\\delta \\omega \\tau\n_{h}\\ll 1$ and the spectrum is motional narrowed, which means that\nonly one doublet is visible. When the temperature\nis lowered, the jump frequency ($1/\\tau _{h}$) decreases and the condition $%\n\\delta \\omega \\tau _{h}\\approx 1$ becomes fulfilled with a fine\nstructure\ndeveloping in the NMR spectrum. Finally, when $\\delta \\omega \\tau _{h}\\gg 1$%\n, the quadrupolar splitting of the two sites are well resolved,\n{\\it i.e}.\none for $^{133}{\\rm Cs}$ sites in the magnetic domains and the other for $%\n^{133}{\\rm Cs}$ sites in the nonmagnetic ones. We simulate each of\nthe three cases and our simulations at fixed $\\delta \\omega $ are\nshown in Fig.6 for different correlation times $\\tau _{h}$ and\nsuperimposed (dotted line) on the experimental spectra on Fig.4.\nClearly, the calculated spectra bear a strong resemblance with\nthe experimental ones displayed on Fig.3 between 100 and 30K. We\ncan thus infer the existence of a thermally activated change in\nthe local environment of $^{133}{\\rm Cs}$ sites which may become\nthe dominant contribution to $^{133}(T_{1})^{-1}$ when the\nfrequency $1/\\tau\n_{h} $ is of the order of the Larmor frequency (43 MHz) of the $^{133}{\\rm Cs%\n}$ nuclei. Therefore, from the results exposed in this section we\ncan conclude to the existence of another degree of freedom aside\nfrom the fluctuations of the electronic spins located on the\n${\\rm C}_{60}$ molecules, and possibly related to spontaneous\nlocal structural changes in the polymerized phase.\n\n\\section{DISCUSSION}\n As emphasized above, one of the difficulty aroused by\nour work is to bring together the conducting nature of the\npolymerized phase established by optical and transport\nmeasurements\\cite{Bommeli95,Khazeni97,Zhou00} with the pressure\nand temperature dependence of $^{13}(T_{1})^{-1}$ which strongly\nsuggest that electrons are localized. It therefore turns out\nnatural to question ourselves about the possible relationship\nbetween the local structural change around $^{133}{\\rm Cs}$\nnuclei and the presence of charge degrees of freedom like\npolarons in the polymerized phase. On the basis of the above NMR\nresults and anticipating results described further on, we suppose\nthat the mobility of a charge carrier in the polymerized phase\nmainly depends upon the occurrence of a local structural\ndistortion in its vicinity. From a point of view which is somewhat\nnaive, one may consider that at thermal equilibrium, the charge\ncarriers diffuse through the lattice under the action of a random\nforce $F(t)$ which takes on only two discrete values $\\pm f_{0}$.\nFor our particular purpose, the relevant physical quantity to be\nconsider is the spectral density $F(\\omega)$ defined as the\nFourier transform of the correlation function $\\langle\nF(t)F(t+\\tau)\\rangle$, the brackets indicating an ensemble\naverage. In our case, $\\langle F(t)F(t+\\tau)\\rangle$ can be\nassumed to be of the form\\cite{Slichter}: $f_{0}^{2}\ne^{-|t|/{\\tau_{h}}}$, which leads to the following\nspectral density : $F(\\omega)=\\tau_{h}/(1+(\\omega \\tau_{h})^2)$.\nBecause any excited state of the charge carriers is to relax due\nto the random force $F(t)$, the spectral density $F(\\omega)$ will\nlead to a strong frequency dependence in the response function of\nthe carriers to external oscillating fields. It is therefore of a\ngreat interest to focus on AC resistivity\nmeasurements\\cite{Zhou00} performed at ambient pressure in both\n${\\rm KC}_{60}$ and ${\\rm RbC}_{60}$. For ${\\rm KC}_{60}$ which\ndoes not exhibit a slowing down of spin fluctuations, AC and DC\nresistivities display a similar temperature dependence. This is\nhowever not true for ${\\rm RbC}_{60}$ since a frequency\ndependent peak is clearly observed on AC resistivity. The peak\nshifts from 35K at 1.1kHz down to 25K at 43Hz, the order of\nmagnitude of these frequencies being in good agreement with the\nvalue we deduced from our simulate spectra in the same\ntemperature range for ${\\rm CsC}_{60}$ (c.f.Fig.6). The fact that\nthe electronic properties of ${\\rm RbC}_{60}$ and ${\\rm\nCsC}_{60}$ display similar electronic and structural features as\nopposed to ${\\rm KC}_{60}$ allows us to extrapolate the results\nobtained by Zhou {\\it et al} for ${\\rm RbC}_{60}$ to the case of\n${\\rm CsC}_{60}$. Thus experiments show that in the\ntwo polymerized phases ${\\rm RbC}_{60}$ and ${\\rm CsC}_{60}$, the\ndissipation reaches a maximum when the hopping frequency of the\nlocal environment of the alkali ion becomes equal to the AC\nfrequency. Such a coincidence can be hardly fortuitous and\nsuggests that the mobility of the charge carriers in the\npolymerized phase is strongly coupled to the\nenvironment of the alkali ion. In this context, it is worthwhile to\nmention that polaron-like distortions such as ${\\rm\nC}_{60}^{-1-x}-{\\rm C}_{60}^{-1+x}$ have been predicted to be\nenergetically favorable in the charged polymer $({\\rm\nC}_{60}^{-})^{n}$ which exhibits a tendency to undergo a charge\ndensity wave transition \\cite{Springborg95}.\n In that particular case, the conduction mechanism would be due to an\n intramolecular property of the polymer itself and that would\ndrastically change our expectations regarding the pressure effect\non the electronic properties of the polymerized phase. However,\non the sole basis of the NMR experiment above described we cannot\naddress the microscopic mechanism at the origin of the\nspontaneous formation of polarons in the polymerized phase.\n\\newline In the light of the above considerations, it is\ninteresting to shortly reconsider the pressure effect on the\nspin-lattice relaxation rate $^{13}(T_{1})^{-1}$ of $^{13}{\\rm\nC}$ nuclei in the low pressure regime. As mentioned above,\n$^{13}(T_{1})^{-1}$ shows at room temperature a similar pressure\ndecrease than the electronic spin susceptibility deduced from EPR\n\\cite{Senzier96} which suggests that magnetic fluctuations\nat the wave vector $\\vec{q}=0$ dominate $^{13}(T_{1})^{-1}$ at\nambient pressure. One possible explanation for the origin of\nthese enhanced uniform fluctuations might be that polarons acting\nas local defects, induce disorder in the AF exchange coupling $J$\nalong the chain leading to the formation of spin\nclusters\\cite{Theodorou77}. It was indeed shown\ntheoretically\\cite{Theodorou77} that the low energy magnetic\nfluctuations (i.e when $T\\ll J$) of a disordered AF spins chain\nare merely governed by clusters with an odd number of spins, each\none acting as a nearly free localized (1/2) spin. In such a case\nthe {\\em reversible} suppression at 5kbar of a weak temperature\ndependent term in $^{13}(T_{1})^{-1}$ could be ascribed to the\nsuppression with applied pressure of disorder in the magnetic\ncoupling along the chain which presence would be henceforth\nclosely related to the slowing down of spin fluctuations in the\nlow temperature state. Much more experimental inputs are however\nrequired to go beyond this statement.\n\\newline As it is, the phase\n separation occurring in the low temperature state at ambient pressure appears\n to be the logical outcome of the twofold nature of the polymerized phase ${\\rm CsC}_{60}$ that is :\n mobile polarons spontaneously form aside from localized electrons and compete with a 3D magnetic order imposed by the transverse\n dipolar coupling between the chains. Note that the presence of nonmagnetic domains is in itself a strong hint that polarons are not randomly\nspatially distributed within the magnetic background but may form\ncollective structures developing a long range order below 14K as\nsuggested by NMR\\cite{Barbara99} and X-ray\nexperiments\\cite{Rouziere00}.\n\n\\section{Conclusion}\nThe work described in this manuscript deals with the electronic\n properties of the polymerized phase ${\\rm CsC}_{60}$ extensively studied by NMR of $^{13}{\\rm C}$\n and $^{133}{\\rm Cs}$ nuclei. The salient result is that the electronic properties of the\n polymerized phase ${\\rm CsC}_{60}$\ninvolve two degrees of freedom : one related to localized spins,\nthe other related to mobile charges which mobility is strongly\nentangled to the local environment of the Cs ion. The\npolymerized phase ${\\rm CsC}_{60}$ is therefore dynamically\ninhomogeneous and as shown by NMR under pressure, this\nfeature persists up to 9kbar. \\newline At ambient pressure static\ninhomogeneities gradually develop below 40K concomitantly with a\nslowing down of spin fluctuations. At 5kbar, the polymerized\nphase ${\\rm CsC}_{60}$ undergoes a nonmagnetic transition at\n$T_{c}$ equal to 20K. The ground state is homogeneous and a spin\ngap opened below 20K. Finally, a dramatic decrease of the\namplitude of the spin gap is observed above 5kbar without any\nsignificant decrease of $T_{c}$. The presence of magnetism therefore\nappears to be closely related to the occurrence of\nstatic inhomogeneities. How does the applied pressure suppress these inhomogeneities and\nstabilize a homogeneous nonmagnetic ground state? That cannot be\naddressed by the present work but remains an important issue to\nbe solved.\n\n\\section{ACKNOWLEDGMENT}\n It is a pleasure to thank F. Rachdi for the $ ^{13}\\rm\nC$ enriched ${\\rm C}_{60}$. We are also very grateful to C.\nBerthier, S. Brasovski, P. Carretta, P. Sotta and P. Wzietek\nfor illuminating discussions and to J. P. Cromi\\`{e}res and M.\nNardone for technical assistance.\\newline\nOne of the authors (L.F) is grateful for the support of the Swiss National Science Foundation.\n\n\n\\begin{references}\n\\bibitem[{*}]{Auth1} Present address: Condensed Matter and Thermal Physics,\nMST-10 MS K764, Los Alamos National Laboratory, Los Alamos, NM\n87545, USA.\n\n\n\n\\bibitem{Stephens94} P. W Stephens {\\it et al}, Nature {\\bf 370}, 636\n(1994).\n\n\\bibitem{Tycko93} R. Tycko, G. Dabbagh, D. W. Murphy, Q. Zhu, and J. E.\nFischer, Phys.~Rev.~B {\\bf 48}, 9097 (1993).\n\n\\bibitem{Chauvet94} O. Chauvet {\\it et al}., Phys.~Rev.~Lett. {\\bf 72},\n2721 (1990).\n\n\\bibitem{Bommeli95} F. Bommeli {\\it et al}., Phys.~Rev.~B {\\bf 51}, 14794\n(1995).\n\n\\bibitem{Brouet96} V. Brouet, H. Alloul, Y. Yoshinari, and L. Forr\\'{o},\nPhys.~Rev.~Lett. {\\bf 76}, 3638 (1996).\n\n\\bibitem{Senzier96} P. Auban-Senzier, D. J\\'{e}rome, F. Rachdi, G.\nBaumgartner, and L. Forr\\'{o}, J.~Phys.I~France {\\bf 6}, 1 (1996).\n\n\\bibitem{Janossy97} A. J\\'{a}nossy {\\it et al}., Phys.~Rev.~Lett. {\\bf 79},\n2718 (1997).\n\n\\bibitem{Bennati98} M. Bennati, R. G. Griffin, S. Knorr, A. Grupp, and M.\nMehring, Phys.~Rev.~B {\\bf 58}, 15603 (1998).\n\n\\bibitem{MacFarl95} W. A. MacFarlane, R. F. Kiefl, S. Dunsiger, J. E.\nSonier, and J. E. Fischer, Phys.~Rev.~B {\\bf 52}, 6695 (1995).\n\n\\bibitem{Uemura95} Y. J Uemura {\\it et al}., Phys.~Rev.~B {\\bf 52}, 6691\n(1995).\n\n\\bibitem{Barbara99} B. Simovi\\v{c}, D. J\\'{e}rome, F. Rachdi, G.\nBaumgartner, and L. Forr\\'{o}, Phys.~Rev.~Lett. {\\bf 82}, 2298\n(1999).\n\n\\bibitem{Atsarkin97} V.A. Atsarkin,V.V. Demidov and G.A. Vasneva, Phys.~Rev.~B {\\bf 56}, 9448 (1997).\n\n\\bibitem{Coulon00} C. Coulon, J. Duval, C. Lavergne, A.L. Barra,\nand A. P\\'{e}nicaud, J.~Phys.IV~France {\\bf 10}, 205 (2000).\n\n\\bibitem{Rouziere00} S. Rouzi\\`{e}re, S. Margadonna, K. Prassides,\nand A.N. Fitch, Cond-Mat/0002419\n\n\\bibitem{comment1} We can always fit the recovery curve with the expression\n$M(t)=M_{eq}(1-e^{(-t/T_{1})^{\\beta}})$, $M_{eq}$ being the\nequilibrium magnetization and $\\beta$ a dimensionless coefficient\ncomprised between 0.5 and 1. At room temperature and at any\npressure investigated $\\beta$ is equal to 1 for $^{133}$Cs\nwhereas a substantial deviation from an exponential recovery is\nobserved i.e $\\beta \\approx 0.72$ for $^{13}$C. Because of the\npossible presence of a small amount of pure amorphous C$_{60}$,\nsuch deviation might be extrinsic. Therefore, we talk about\n\"exponential recovery\" for $^{13}$C as opposed to the dramatic\nchanges in the recovery curve observed at low temperature and\nwhich occurrence is unambiguously related to some intrinsic\nproperties of the polymerized phase.\n\n\\bibitem{comment2} The presence of a quadrupole splitting does not\nimply a quadrupole relaxation but is consequent to the noncubic\nsymmetry of the local environment of $^{133}$Cs site. For a cubic symmetry\nthe three components of the electric field gradient\n are strictly equal to zero which implies no quadrupole\nsplitting. However, any fluctuations leading to a substantial\ndeviation from the cubic symmetry may lead to a quadrupole\nrelaxation depending upon the magnitude of the quadrupolar coupling compare to others.\nFor the particular case of $^{133}$Cs, the\npossibility of a quadrupole contribution in the relaxation of the\nmagnetization can be easily ruled out because of the remarkably\nsmall value of its quadrupole moment.\n\n\\bibitem{Suter98} A detail analysis of the magnetization recovery for nuclear spin I$>$1/2\n can be found in the following reference : A. Suter, M. Mali , J. Roos, and D. Brinkmann,\nJ. Phys. Condens. Matter {\\bf 10}, 5977 (1998)\n\n\\bibitem{Erwin95} S.C. Erwin, G.V. Krishna, and E.J. Mele, Phys.~Rev.~B {\\bf %\n51}, 7345 (1995).\n\n\\bibitem{Khazeni97} K. Khazeni, V. H. Crespi, J. Hone, A. Zettl, and M. L. Cohen, Phys.~Rev.~B {\\bf %\n56}, 6627 (1997).\n\n\\bibitem{Zhou00} W.Y. Zhou,S.S. Xie,L. Lu,E.S. Liu, and Z. Peng,\nJournal of Physics and Chemistry of Solids {\\bf 61}, 1159 (2000).\n\n\\bibitem{Stafstrom95} S. Stafstr\\\"{o}m, M. Boman, and J. Fagerstr\\\"{o}m,\nEurophysics.~Lett. {\\bf 30}, 295 (1995).\n\n\\bibitem{Forro96} L. Forr\\'{o} {\\it et al}., {\\it in Progress in Fullerene\nResearch}, edited by H. Kuzmany {\\it et al} (World Scientific,\nSingapore 1996)\n\n\n\\bibitem{Emery} O. Zachar, S.A. Kivelson, and V. J. Emery, Phys.~Rev.~B {\\bf %\n57}, 1422 (1998).\n\n\\bibitem{Scalapino} S.R. White, and D.J. Scalapino, Phys.~Rev.~B {\\bf 55},\n14701 (1997).\n\n\\bibitem{comment3} For a given nuclei, the amplitude of the spin echoes strongly depends upon\n the angle of the rotation applied by the second pulse to the magnetization as well as the relative phase between\n the two pulses. For the in-phase pulse sequence ($\\pi /2-\\tau -\\pi\n /8$) applied on $^{133}{\\rm Cs}$ nuclei, the calculated\\cite{Barbara} amplitude of the spin echo is maximal at $3\\tau$\n and strongly reduced at $5\\tau/2$ and $4\\tau$.\n\n\\bibitem{Barbara} B. Simovi\\v{c}, Thesis, Universit\\'{e} Paris-Sud, Orsay\n(1999).\n\n\\bibitem{Springborg95} M. Springborg, Phys.~Rev.~B {\\bf 52}, 2935 (1995).\n\n\\bibitem{Slichter} C.P.Slichter, Appendix C in Principles of Magnetic Resonance,\nThird edition, Springer-Verlag (1990).\n\n\n\\bibitem{Theodorou77} G. Theodorou, Phys.~Rev.~B {\\bf 16}, 2264 (1977).\n\n\\end{references}\n\n\\clearpage\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.48\\linewidth]{Fig1a.eps}\n \\includegraphics[width=0.48\\linewidth]{Fig1b.eps}\n \\caption{Temperature dependence of $T_{1}^{-1}$ for (a) $^{133}{\\rm\n Cs}$ (at 8 Tesla) and (b) $^{13}{\\rm C}$ (at 9 Tesla), at 1bar\n (empty triangles), 5kbar (black circles) and 9kbar (black\n triangles). At 1bar and below 40K, the magnetization recovery\n curves are bi-exponential for both $^{133}{\\rm Cs}$ and $^{13}{\\rm\n C}$ but only the rapid component is reported versus temperature.}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{Fig2.eps}\n \\caption{Temperature dependence of $^{133}(TT_{1})^{-1}$ at 5kbar, 5.5kbar, 5.7 and 9kbar.}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{Fig3.eps}\n \\caption{Temperature dependence of the linewidth of the $^{133}{\\rm\nCs}$-NMR line at 1bar from ref~\\protect\\cite{Brouet96} (empty hexagons), at\n3kbar (present study, empty triangles) and at 5kbar (present study,\nfull circles).}\n\\end{figure}\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.9\\linewidth]{Fig4.eps}\n \\caption{Evolution of the lineshape of the $^{133}{\\rm Cs}$\nquadrupolar normalized splitting from 100K down to 4.2K. Because\nwe take the Fourier transform of half of the spin echo at $3\\tau$, the\nother spin echoes induce distortion of the base line.}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{Fig5.eps}\n \\caption{Temperature dependence of $N(T)=I(T).T$ where $I(T)$ is the\nintegrated intensity of the Fourier transform of half of the spin echo\nrefocused at $3\\tau$.}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{Fig6.eps}\n \\caption{Simulation of the shape of the $^{133}{\\rm Cs}$ quadrupolar\nspectrum.(a) In the static case : for each of the two\nconfigurations, both the quadrupole frequency $\\nu_{Q}$ and the\nfrequency shift $K$ compare to the Larmor frequency are\ndetermined from the spectrum at 100K {\\it i.e.} in the fast motion\nlimit.This gives the respective values : $\\nu_{Q}$=4.17kHz and\n$K$=$-$350ppm (dashed line), $\\nu_{Q}$=3.57kHz,\n$K$=$-$209ppm (dotted line).(b) In the fast motion limit : $\\delta \\omega \\tau_h \\ll 1$%\n.(c) In the slow motion limit : $\\delta \\omega \\tau_h \\approx 1$.(d)\nFor a {\\em quasistatic} distribution : $\\delta \\omega \\tau_h \\gg 1$.\n}\n\\end{figure}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002182.extracted_bib",
"string": "\\bibitem[{*}]{Auth1} Present address: Condensed Matter and Thermal Physics,\nMST-10 MS K764, Los Alamos National Laboratory, Los Alamos, NM\n87545, USA.\n\n\n\n\n\\bibitem{Stephens94} P. W Stephens {\\it et al}, Nature {\\bf 370}, 636\n(1994).\n\n\n\\bibitem{Tycko93} R. Tycko, G. Dabbagh, D. W. Murphy, Q. Zhu, and J. E.\nFischer, Phys.~Rev.~B {\\bf 48}, 9097 (1993).\n\n\n\\bibitem{Chauvet94} O. Chauvet {\\it et al}., Phys.~Rev.~Lett. {\\bf 72},\n2721 (1990).\n\n\n\\bibitem{Bommeli95} F. Bommeli {\\it et al}., Phys.~Rev.~B {\\bf 51}, 14794\n(1995).\n\n\n\\bibitem{Brouet96} V. Brouet, H. Alloul, Y. Yoshinari, and L. Forr\\'{o},\nPhys.~Rev.~Lett. {\\bf 76}, 3638 (1996).\n\n\n\\bibitem{Senzier96} P. Auban-Senzier, D. J\\'{e}rome, F. Rachdi, G.\nBaumgartner, and L. Forr\\'{o}, J.~Phys.I~France {\\bf 6}, 1 (1996).\n\n\n\\bibitem{Janossy97} A. J\\'{a}nossy {\\it et al}., Phys.~Rev.~Lett. {\\bf 79},\n2718 (1997).\n\n\n\\bibitem{Bennati98} M. Bennati, R. G. Griffin, S. Knorr, A. Grupp, and M.\nMehring, Phys.~Rev.~B {\\bf 58}, 15603 (1998).\n\n\n\\bibitem{MacFarl95} W. A. MacFarlane, R. F. Kiefl, S. Dunsiger, J. E.\nSonier, and J. E. Fischer, Phys.~Rev.~B {\\bf 52}, 6695 (1995).\n\n\n\\bibitem{Uemura95} Y. J Uemura {\\it et al}., Phys.~Rev.~B {\\bf 52}, 6691\n(1995).\n\n\n\\bibitem{Barbara99} B. Simovi\\v{c}, D. J\\'{e}rome, F. Rachdi, G.\nBaumgartner, and L. Forr\\'{o}, Phys.~Rev.~Lett. {\\bf 82}, 2298\n(1999).\n\n\n\\bibitem{Atsarkin97} V.A. Atsarkin,V.V. Demidov and G.A. Vasneva, Phys.~Rev.~B {\\bf 56}, 9448 (1997).\n\n\n\\bibitem{Coulon00} C. Coulon, J. Duval, C. Lavergne, A.L. Barra,\nand A. P\\'{e}nicaud, J.~Phys.IV~France {\\bf 10}, 205 (2000).\n\n\n\\bibitem{Rouziere00} S. Rouzi\\`{e}re, S. Margadonna, K. Prassides,\nand A.N. Fitch, Cond-Mat/0002419\n\n\n\\bibitem{comment1} We can always fit the recovery curve with the expression\n$M(t)=M_{eq}(1-e^{(-t/T_{1})^{\\beta}})$, $M_{eq}$ being the\nequilibrium magnetization and $\\beta$ a dimensionless coefficient\ncomprised between 0.5 and 1. At room temperature and at any\npressure investigated $\\beta$ is equal to 1 for $^{133}$Cs\nwhereas a substantial deviation from an exponential recovery is\nobserved i.e $\\beta \\approx 0.72$ for $^{13}$C. Because of the\npossible presence of a small amount of pure amorphous C$_{60}$,\nsuch deviation might be extrinsic. Therefore, we talk about\n\"exponential recovery\" for $^{13}$C as opposed to the dramatic\nchanges in the recovery curve observed at low temperature and\nwhich occurrence is unambiguously related to some intrinsic\nproperties of the polymerized phase.\n\n\n\\bibitem{comment2} The presence of a quadrupole splitting does not\nimply a quadrupole relaxation but is consequent to the noncubic\nsymmetry of the local environment of $^{133}$Cs site. For a cubic symmetry\nthe three components of the electric field gradient\n are strictly equal to zero which implies no quadrupole\nsplitting. However, any fluctuations leading to a substantial\ndeviation from the cubic symmetry may lead to a quadrupole\nrelaxation depending upon the magnitude of the quadrupolar coupling compare to others.\nFor the particular case of $^{133}$Cs, the\npossibility of a quadrupole contribution in the relaxation of the\nmagnetization can be easily ruled out because of the remarkably\nsmall value of its quadrupole moment.\n\n\n\\bibitem{Suter98} A detail analysis of the magnetization recovery for nuclear spin I$>$1/2\n can be found in the following reference : A. Suter, M. Mali , J. Roos, and D. Brinkmann,\nJ. Phys. Condens. Matter {\\bf 10}, 5977 (1998)\n\n\n\\bibitem{Erwin95} S.C. Erwin, G.V. Krishna, and E.J. Mele, Phys.~Rev.~B {\\bf %\n51}, 7345 (1995).\n\n\n\\bibitem{Khazeni97} K. Khazeni, V. H. Crespi, J. Hone, A. Zettl, and M. L. Cohen, Phys.~Rev.~B {\\bf %\n56}, 6627 (1997).\n\n\n\\bibitem{Zhou00} W.Y. Zhou,S.S. Xie,L. Lu,E.S. Liu, and Z. Peng,\nJournal of Physics and Chemistry of Solids {\\bf 61}, 1159 (2000).\n\n\n\\bibitem{Stafstrom95} S. Stafstr\\\"{o}m, M. Boman, and J. Fagerstr\\\"{o}m,\nEurophysics.~Lett. {\\bf 30}, 295 (1995).\n\n\n\\bibitem{Forro96} L. Forr\\'{o} {\\it et al}., {\\it in Progress in Fullerene\nResearch}, edited by H. Kuzmany {\\it et al} (World Scientific,\nSingapore 1996)\n\n\n\n\\bibitem{Emery} O. Zachar, S.A. Kivelson, and V. J. Emery, Phys.~Rev.~B {\\bf %\n57}, 1422 (1998).\n\n\n\\bibitem{Scalapino} S.R. White, and D.J. Scalapino, Phys.~Rev.~B {\\bf 55},\n14701 (1997).\n\n\n\\bibitem{comment3} For a given nuclei, the amplitude of the spin echoes strongly depends upon\n the angle of the rotation applied by the second pulse to the magnetization as well as the relative phase between\n the two pulses. For the in-phase pulse sequence ($\\pi /2-\\tau -\\pi\n /8$) applied on $^{133}{\\rm Cs}$ nuclei, the calculated\\cite{Barbara} amplitude of the spin echo is maximal at $3\\tau$\n and strongly reduced at $5\\tau/2$ and $4\\tau$.\n\n\n\\bibitem{Barbara} B. Simovi\\v{c}, Thesis, Universit\\'{e} Paris-Sud, Orsay\n(1999).\n\n\n\\bibitem{Springborg95} M. Springborg, Phys.~Rev.~B {\\bf 52}, 2935 (1995).\n\n\n\\bibitem{Slichter} C.P.Slichter, Appendix C in Principles of Magnetic Resonance,\nThird edition, Springer-Verlag (1990).\n\n\n\n\\bibitem{Theodorou77} G. Theodorou, Phys.~Rev.~B {\\bf 16}, 2264 (1977).\n\n"
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cond-mat0002183
|
Casimir Force between two Half Spaces of Vortex Matter in Anisotropic Superconductors
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[
{
"author": "H.~P.\\ B\\\"uchler"
}
] |
We present a new approach to calculate the attractive long-range vortex-vortex interaction of the van der Waals type present in anisotropic and layered superconductors. The mapping of the statistical mechanics of two-dimensional charged bosons allows us to define a Casimir problem: Two half spaces of vortex matter separated by a gap of width $R$ are mapped to two dielectric half planes of charged bosons interacting via a massive gauge field. We determine the attractive Casimir force between the two half planes and show that it agrees with the pairwise summation of the van der Waals force between vortices.
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[
{
"name": "casimirphysicaC.tex",
"string": "\\documentstyle{elsart}\n\n \\begin{document}\n\n \\begin{frontmatter}\n \\title{Casimir Force between two Half Spaces of Vortex Matter in\n Anisotropic Superconductors} \n \n \\author[zuerich]{H.~P.\\ B\\\"uchler},\n \\author[zuerich,santacruz]{H.~G.\\ Katzgraber},\n \\author[zuerich]{and G.\\ Blatter}\n \\address[zuerich]{Theoretische Physik, Eidgen\\\"ossiche Technische\n Hochschule-H\\\"onggerberg, CH-8093 Z\\\"urich, Switzerland}\n \\address[santacruz]{\n Department of Physics, University of California, Santa Cruz,\n California 95064 }\n \n \\begin{abstract}\n We present a new approach to calculate the attractive long-range\n vortex-vortex interaction of the van der Waals type present in anisotropic \n and layered superconductors. The mapping of the statistical mechanics of\n two-dimensional charged bosons allows us to define a Casimir problem: Two\n half spaces of vortex matter separated by a gap of width $R$ are mapped to \n two dielectric half planes of charged bosons interacting via a massive\n gauge field. We determine the attractive Casimir force between the two\n half planes and show that it agrees with the pairwise summation of the van\n der Waals force between vortices.\n \\end{abstract}\n \n \\end{frontmatter}\n\n \\section{Introduction}\n\n\n Vortices in isotropic type-II superconductors repel each other with\n the interaction strength decaying exponentially on the scale $R > \\lambda$\n ($R$ is the distance between the vortices, while $\\lambda$ denotes the\n London penetration length). However, it\n has recently been shown \\cite{blatter96,katzgraber99} that in layered\n superconductors \n the thermal fluctuations of the flux lines give rise to a long-range\n {\\em attraction} $V_{{\\rm vdW}} \\sim - (T/d)(\\lambda/R)^{4}$ of the van der\n Waals type on the scale\n $R > \\lambda$, where $T$ denotes the temperature and $d$ is the interlayer\n distance. The equivalence of three dimensional statistical mechanics with the\n $2+1$ dimensional imaginary time quantum mechanics allows us to describe\n the flux lines as two dimensional charged bosons with an interaction mediated \n by a gauge field ${\\bf a}$. In the case of vortex matter \n the material properties are mapped to a dielectric\n permittivity for the gauge field. Geometric boundary conditions and\n dielectric permittivities then\n cause a shift in the zero-point energy of the gauge field\n known as the Casimir effect. It is subject of the present work to calculate\n the Casimir force between two half spaces of vortex matter separated by a\n vortex free region as shown in Fig.~\\ref{fig}.\n It is well known \\cite{lifshitz56} that in special geometries (e.~g.\\ two\n dielectric half spaces) van der Waals\n forces can be related to the Casimir force via pairwise summation. This\n interpretation of the Casimir force allows us to derive the van der Waals\n force between flux lines in \n anisotropic superconductors via the Casimir approach. \n \\input{psfig}\n \\begin{figure}[ht]\n \\makebox[12cm][c]{\n \\psfig{figure=cas_1.eps,height=4.02cm,width=10.34cm}}\n \\caption{Geometry used for the calculation of the Casimir force. The $ab$\n plane of superconductor is parallel to the $xy$ plane, while the $c$ axis \n is along the applied magnetic field in $z$ direction. The distance\n between the two half spaces is $R$. } \\label{fig}\n \\end{figure}\n\n \\section{Casimir Force between two half Spaces of Vortex Matter} \n \n {\\em Vortices as 2D Bosons}: Within the London theory the free energy of vortices in an isotropic\n superconductor takes the form \n \\begin{equation}\n {\\mathcal F} = \\frac{\\epsilon_{0}}{2} \\int d^{3}{\\bf x} \\: d^{3}{\\bf y} \\:\n {\\bf j}({\\bf x}) \n \\: \n \\frac{e^{- \n \\lambda |{\\bf x}-{\\bf y}|}}{|{\\bf x}-{\\bf y}|} \\: {\\bf j}({\\bf y}) \n \\end{equation}\n with the current ${\\bf j}= ({\\bf J}, \\rho) = \\sum_{\\mu} (\\partial_{z} {\\bf R} \n , 1) \\delta^{2}\\left({\\bf R} - {\\bf R}_{\\mu} \\right)$ describing the\n vortex lines. Following a suggestion of Nelson \\cite{nelson}, the statistical \n mechanics of vortices can be mapped to the imaginary time quantum mechanics\n of two-dimensional (2D) bosons. The $c$ axis of the superconductor is mapped \n to the imaginary time of the bosons $z \\rightarrow \\tau$, the temperature $T$\n becomes the Planck constant $\\hbar^{B}$, while\n the interaction between the vortices is\n mediated by a fake gauge field ${\\bf a}$ with a coupling $g^{2} = 4 \\pi\n \\varepsilon_{0}$, where $\\epsilon_{0}=(\\Phi_{0}/4 \\pi \\lambda)^{2}$ is the\n line energy \\cite{popov}. The action of the 2D bosons becomes\n \\begin{equation}\n \\begin{array}{l}\n {\\displaystyle {\\mathcal S}\\left[ {\\bf j}, {\\bf a} \\right] = \\int d\\tau\n \\sum_{\\mu} \\left\\{ \n \\frac{m}{2} \\left[ \\partial_{\\tau} {\\bf R}_{\\mu}(\\tau)\n \\right]^{2} - \n \\mu^{B} \\right\\} + \\int d\\tau\n d^{2}R {\\Bigg \\{ } i \\: {\\bf a} \\cdot {\\bf j} } \\\\\n \\hspace{80pt} \n {\\displaystyle \\left. + \n \\frac{1}{2 g^{2} \\lambda^{2}} {\\bf a}^{2} +\n \\frac{1}{2 g^{2}} \\left( \\nabla\n \\times {\\bf a} \n \\right)^{2}_{xy} + \n \\frac{c^{2}}{2 g^{2}} \\left( \\nabla \\times {\\bf a} \\right)_{\\tau}\n \\right\\}} \n \\end{array} \\label{2dbosons}\n \\end{equation}\n with the bare boson mass $m=\\varepsilon_{0}$. The\n self-interaction of the vortices via the gauge field leads to a mass\n renormalization \n $m\\rightarrow m^{B}= \\varepsilon_{l}$ where $\\varepsilon_{l}$ is the\n dispersive line tension \\cite{blatter94} of the vortex line. The\n anisotropy $\\epsilon$ between the $ab$ plane and the\n $c$ axis is introduced \n by $c=1/\\epsilon$ (the light velocity in the boson\n system). \n \n {\\em Material Properties}: We can map the material properties of the vortex matter to a dielectric\n permittivity $\\epsilon_{V}$ by integrating \n over the currents ${\\bf J}$ of the 2D bosons, leading to a term $\n (\\rho/m){\\bf a}_{xy}^{2}$ in the action. Performing functional\n derivatives leads to the dispersion relation for\n the gauge field ${\\bf a_{xy}}$\n \\begin{equation}\n \\left[ \\frac{\\omega^{2}}{c^{2}} \\epsilon_{V}(\\omega) + k^{2} +\n \\frac{\\epsilon^{2}}{\\lambda^{2}}\\right] {\\bf a}_{xy} = 0 \\hspace{20pt}\n \\mbox{with} \\hspace{10pt}\n \\epsilon_{V}(\\omega) = 1 + \\frac{g^{2} \\rho}{m^{B} \\omega^{2}} \\: .\n \\end{equation} \n The boundary conditions at the interface between the vortex matter and the\n vortex free region together with the dispersion relation and the\n dielectric permittivity define a Casimir problem \\cite{lifshitz56}.\n\n {\\em Casimir Force}: The Casimir force of the 2D dielectric system becomes\n \\begin{equation}\n \\begin{array}{c}\n {\\displaystyle f= -\\frac{\\hbar^{B}}{\\pi^{2}} \\int_{1}^{\\infty} dp\n \\int_{0}^{\\infty} d\\omega \\frac{s p \\omega^{2}}{c^{2} \\sqrt{p^{2}-1}}} \n \\hspace{180pt} \\\\ \\hspace{50pt}\n {\\displaystyle \\times \\left[ \\left( \\frac{\\left[ 1+\n (\\lambda \\omega)^{-2} \\right] s_{\\epsilon} + \\left[ \\epsilon_{V}\n +(\\lambda \n \\omega)^{-2} \\right] s}{\\left[ 1+\n (\\lambda \\omega)^{-2} \\right] s_{\\epsilon} - \\left[ \\epsilon_{V}\n +(\\lambda \\omega)^{-2} \\right] s} \\right)^{2}e^{2 R\n \\omega/c} -1 \\right]^{-1} \n }\\:, \n \\end{array}\n \\end{equation} \n where $p^{2} = 1+c^{2} k^{2} \\omega^{-2}$, $s_{\\epsilon}^{2} =\n \\epsilon_{V} -1 + p^{2} +(\\lambda \\omega)^{-2}$ and $ s^{2} = p^{2}+ (\\lambda\n \\omega)^{-2}$. This expression is determined by three frequencies:\n $\\omega_{d} = \\pi/d$ is an \n upper cut-off due to the layered structure of the superconductor, while\n $\\omega_{R}=1/\\epsilon R$ derives from the geometric length, and \n $\\omega_{\\lambda} = 1/\\lambda$ describes the mass of the gauge field. \n \n {\\em Pairwise Summation}: In the following we present the\n Casimir force for the three different length scales. The pairwise summation\n (i.~e.\\ the summation of the van der Waals forces \n between the vortex lines in each half space) then provides the van der Waals\n force between flux lines. For intermediate\n distances $R< d \\epsilon^{-1},\\lambda \\epsilon^{-1}$ the cut-off $\\omega_{d}$\n is relevant and we \n obtain the result\n \\begin{equation}\n \\begin{array}{rcl} \n {\\displaystyle f= - \\frac{\\left(\\epsilon_{V} -1 \\right)^{2}}{16 \\pi^{2}}\n \\frac{\\hbar^{B} \\omega_{d}}{R^{2}} } &\n \\hspace{10pt} \\Longrightarrow \\hspace{10pt} & \\displaystyle{ V_{{\\rm\n vdW}} = \n - \\frac{4 \n \\epsilon_{0}}{\\ln^{2}(\\pi \\lambda \n d^{-1})} \\frac{T}{d \\epsilon_{0}}\\left(\\frac{\\lambda}{R}\\right)^{4}}\n \\end{array} \\: .\n \\end{equation}\n At larger distances $d \\epsilon^{-1} < R < \\lambda \\epsilon^{-1}$ retardation\n of the gauge field becomes important and the frequency $\\omega_{d}$ is\n replaced by $ c/R$\n \\begin{equation}\n \\begin{array}{rcl}\n {\\displaystyle f= - \\frac{19 \\left(\\epsilon_{V} -1 \\right)^{2}}{1024 \\pi}\n \\frac{\\hbar^{B} \n c}{R^{3}} }& \\hspace{10pt} \\Longrightarrow \\hspace{10pt}\n & {\\displaystyle V_{{\\rm vdW}}^{{\\rm ret}} = - \\frac{(171\n \\pi / 256) \n \\epsilon_{0}}{\\ln^{2}(\\pi \\lambda \n (\\epsilon R)^{-1})} \\frac{T}{\\lambda \\epsilon \n \\epsilon_{0}}\\left(\\frac{\\lambda}{R}\\right)^{5} }\n \\end{array} \\: .\n \\end{equation} \n At very large distances the mass of the gauge field leads to an exponential\n decay \n \\begin{equation}\n \\begin{array}{rcl}\n {\\displaystyle f = - \\frac{\\hbar^{B} c}{R\\: \\lambda^{2}} \\: \n \\frac{8 \\pi \\lambda^{4} \\rho^{2}}{ c^{2}} e^{- \\frac{R}{\\lambda c}}}&\n \\hspace{10pt} \\Longrightarrow \\hspace{10pt} & \n {\\displaystyle V_{{\\rm vdW}} = - 4 \\sqrt{\\pi} \\varepsilon_{0} \\frac{T\n \\epsilon^{4}}{\\varepsilon_{0} \\lambda} \\left(\\frac{\\lambda}{\\epsilon\n R}\\right)^{\\frac{3}{2}} e^{- 2 \\frac{\\epsilon \n R}{\\lambda}}} \n \\end{array} \\: .\n \\end{equation}\n The van der Waals forces at intermediate and large distances agree with the\n derivation by Blatter and Geshkenbein \\cite{blatter96}.\n\n \n \\input{psfig}\n \\begin{figure}[ht]\n \\makebox[12cm][c]{\n \\psfig{figure=phasediag.eps,height=3.56cm,width=12.3cm}}\n \\caption{Phase diagrams: $T_{ce}^{\\rm vdW}$ denotes a critical endpoint,\n $T_{ce}^{\\rm vdW}$ is a \n critical point, and $T_{T}^{\\rm vdW}$ is a triple point } \\label{fig2}\n \\end{figure}\n {\\em Phase Diagram}: The attraction\n between flux lines leads to interesting modifications of the $B-T$ phase\n diagram of anisotropic type-II superconductors, see Fig.~\\ref{fig2} for a\n schematic drawing. At low\n temperature $T<T_{ce}^{\\rm vdW}$ a first order phase transition takes the\n Meissner state into the vortex solid ( with the critical field $H_{c_{1}}$\n lowered by the attraction), while at higher\n temperature $T_{c} > T >T_{ce}^{\\rm vdW}$ the Meissner state goes into a\n vortex gas through a second order phase transition. In the temperature \n range $T_{T}^{\\rm vdW}< T <T_{c}^{\\rm vdW}$ we can distinguish a low\n density vortex gas from a high density vortex liquid. Concurrent to the first \n order jump in the magnetization a phase separation is observed and the \n domains of vortex matter separated by vortex free regions interact via the\n Casimir force described in this paper.\n\n \\begin{thebibliography}{1} \n \\bibitem{blatter96}\n G. Blatter and V.~B. Geshkenbein, \\newblock Phys.\\ Rev.\\ Lett.\\ {\\bf 77},\n 4958 (1996). \n \\bibitem{katzgraber99}\n H.~G.\\ Katzgraber, H.~P.\\ B\\\"uchler, and G. Blatter, \\newblock Phys.\\ Rev.\\ \n B {\\bf 59}, 11990 (1999). \n \\bibitem{lifshitz56}\n E.~M. Lifshitz, \\newblock Zh.\\ \\'{E}ksp.\\ Teor.\\ Fiz.\\ {\\bf 29}, 94 (1955)\n [Sov.\\ Phys.\\ JETP {\\bf 2}, 73 (1956)].\n \\bibitem{nelson}\n D.~R. Nelson, \\newblock Phys.\\ Rev.\\ Lett.\\ {\\bf 60}, 1973 (1988).\n \\bibitem{popov}\n V.~N. Popov, {\\em Functional Integrals and Collective Excitations}\n (Cambridge University Press, Cambridge, 1987).\n \\bibitem{blatter94}\n G. Blatter, M.~V. Feigel'man, V.~B. Geshkenbein, A.~I. Larkin, and\n V.~M. Vinokur, \\newblock Rev.\\ Mod.\\ Phys.\\ {\\bf 66}, 1125 (1994).\n \\end{thebibliography}\n\\end{document}"
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"string": "\n% Psfig/TeX Release 1.8\n% dvips version\n%\n% All psfig/tex software, documentation, and related files\n% in this distribution of psfig/tex are \n% Copyright 1987, 1988, 1991 Trevor J. Darrell\n%\n% Permission is granted for use and non-profit distribution of psfig/tex \n% providing that this notice is clearly maintained. The right to\n% distribute any portion of psfig/tex for profit or as part of any commercial\n% product is specifically reserved for the author(s) of that portion.\n%\n% *** Feel free to make local modifications of psfig as you wish,\n% *** but DO NOT post any changed or modified versions of ``psfig''\n% *** directly to the net. Send them to me and I'll try to incorporate\n% *** them into future versions. If you want to take the psfig code \n% *** and make a new program (subject to the copyright above), distribute it, \n% *** (and maintain it) that's fine, just don't call it psfig.\n%\n% Bugs and improvements to trevor@media.mit.edu.\n%\n% Thanks to Greg Hager (GDH) and Ned Batchelder for their contributions\n% to the original version of this project.\n%\n% Modified by J. Daniel Smith on 9 October 1990 to accept the\n% %%BoundingBox: comment with or without a space after the colon. Stole\n% file reading code from Tom Rokicki's EPSF.TEX file (see below).\n%\n% More modifications by J. Daniel Smith on 29 March 1991 to allow the\n% the included PostScript figure to be rotated. The amount of\n% rotation is specified by the \"angle=\" parameter of the \\psfig command.\n%\n% Modified by Robert Russell on June 25, 1991 to allow users to specify\n% .ps filenames which don't yet exist, provided they explicitly provide\n% boundingbox information via the \\psfig command. Note: This will only work\n% if the \"file=\" parameter follows all four \"bb???=\" parameters in the\n% command. This is due to the order in which psfig interprets these params.\n%\n% 3 Jul 1991\tJDS\tcheck if file already read in once\n% 4 Sep 1991\tJDS\tfixed incorrect computation of rotated\n%\t\t\tbounding box\n% 25 Sep 1991\tGVR\texpanded synopsis of \\psfig\n% 14 Oct 1991\tJDS\t\\fbox code from LaTeX so \\psdraft works with TeX\n%\t\t\tchanged \\typeout to \\ps@typeout\n% 17 Oct 1991\tJDS\tadded \\psscalefirst and \\psrotatefirst\n%\n\n% From: gvr@cs.brown.edu (George V. Reilly)\n%\n% \\psdraft\tdraws an outline box, but doesn't include the figure\n%\t\tin the DVI file. Useful for previewing.\n%\n% \\psfull\tincludes the figure in the DVI file (default).\n%\n% \\psscalefirst width= or height= specifies the size of the figure\n% \t\tbefore rotation.\n% \\psrotatefirst (default) width= or height= specifies the size of the\n% \t\t figure after rotation. 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We continue searching if `(at end)'\n% was found after the `%BoundingBox:'.\n%\n\\long\\def\\epsf@aux#1#2:#3\\\\{\\ifx#1\\epsf@percent\n \\def\\epsf@testit{#2}\\ifx\\epsf@testit\\epsf@bblit\n\t\\@atendfalse\n \\epsf@atend #3 . \\\\%\n\t\\if@atend\t\n\t \\if@verbose{\n\t\t\\ps@typeout{psfig: found `(atend)'; continuing search}\n\t }\\fi\n \\else\n \\epsf@grab #3 . . . \\\\%\n \\not@eoffalse\n \\global\\no@bbfalse\n \\fi\n \\fi\\fi}%\n%\n% Here we grab the values and stuff them in the appropriate definitions.\n%\n\\def\\epsf@grab #1 #2 #3 #4 #5\\\\{%\n \\global\\def\\epsf@llx{#1}\\ifx\\epsf@llx\\empty\n \\epsf@grab #2 #3 #4 #5 .\\\\\\else\n \\global\\def\\epsf@lly{#2}%\n \\global\\def\\epsf@urx{#3}\\global\\def\\epsf@ury{#4}\\fi}%\n%\n% Determine if the stuff following the %%BoundingBox is `(atend)'\n% J. Daniel Smith. Copied from \\epsf@grab above.\n%\n\\def\\epsf@atendlit{(atend)} \n\\def\\epsf@atend #1 #2 #3\\\\{%\n \\def\\epsf@tmp{#1}\\ifx\\epsf@tmp\\empty\n \\epsf@atend #2 #3 .\\\\\\else\n \\ifx\\epsf@tmp\\epsf@atendlit\\@atendtrue\\fi\\fi}\n\n\n% End of file reading stuff from epsf.tex\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% trigonometry stuff from \"trig.tex\"\n\\chardef\\letter = 11\n\\chardef\\other = 12\n\n\\newif \\ifdebug %%% turn me on to see TeX hard at work ...\n\\newif\\ifc@mpute %%% don't need to compute some values\n\\c@mputetrue % but assume that we do\n\n\\let\\then = \\relax\n\\def\\r@dian{pt }\n\\let\\r@dians = \\r@dian\n\\let\\dimensionless@nit = \\r@dian\n\\let\\dimensionless@nits = \\dimensionless@nit\n\\def\\internal@nit{sp }\n\\let\\internal@nits = \\internal@nit\n\\newif\\ifstillc@nverging\n\\def \\Mess@ge #1{\\ifdebug \\then \\message {#1} \\fi}\n\n{ %%% Things that need abnormal catcodes %%%\n\t\\catcode `\\@ = \\letter\n\t\\gdef \\nodimen {\\expandafter \\n@dimen \\the \\dimen}\n\t\\gdef \\term #1 #2 #3%\n\t {\\edef \\t@ {\\the #1}%%% freeze parameter 1 (count, by value)\n\t\t\\edef \\t@@ {\\expandafter \\n@dimen \\the #2\\r@dian}%\n\t\t\t\t %%% freeze parameter 2 (dimen, by value)\n\t\t\\t@rm {\\t@} {\\t@@} {#3}%\n\t }\n\t\\gdef \\t@rm #1 #2 #3%\n\t {{%\n\t\t\\count 0 = 0\n\t\t\\dimen 0 = 1 \\dimensionless@nit\n\t\t\\dimen 2 = #2\\relax\n\t\t\\Mess@ge {Calculating term #1 of \\nodimen 2}%\n\t\t\\loop\n\t\t\\ifnum\t\\count 0 < #1\n\t\t\\then\t\\advance \\count 0 by 1\n\t\t\t\\Mess@ge {Iteration \\the \\count 0 \\space}%\n\t\t\t\\Multiply \\dimen 0 by {\\dimen 2}%\n\t\t\t\\Mess@ge {After multiplication, term = \\nodimen 0}%\n\t\t\t\\Divide \\dimen 0 by {\\count 0}%\n\t\t\t\\Mess@ge {After division, term = \\nodimen 0}%\n\t\t\\repeat\n\t\t\\Mess@ge {Final value for term #1 of \n\t\t\t\t\\nodimen 2 \\space is \\nodimen 0}%\n\t\t\\xdef \\Term {#3 = \\nodimen 0 \\r@dians}%\n\t\t\\aftergroup \\Term\n\t }}\n\t\\catcode `\\p = \\other\n\t\\catcode `\\t = \\other\n\t\\gdef \\n@dimen #1pt{#1} %%% throw away the ``pt''\n}\n\n\\def \\Divide #1by #2{\\divide #1 by #2} %%% just a synonym\n\n\\def \\Multiply #1by #2%%% allows division of a dimen by a dimen\n {{%%% should really freeze parameter 2 (dimen, passed by value)\n\t\\count 0 = #1\\relax\n\t\\count 2 = #2\\relax\n\t\\count 4 = 65536\n\t\\Mess@ge {Before scaling, count 0 = \\the \\count 0 \\space and\n\t\t\tcount 2 = \\the \\count 2}%\n\t\\ifnum\t\\count 0 > 32767 %%% do our best to avoid overflow\n\t\\then\t\\divide \\count 0 by 4\n\t\t\\divide \\count 4 by 4\n\t\\else\t\\ifnum\t\\count 0 < -32767\n\t\t\\then\t\\divide \\count 0 by 4\n\t\t\t\\divide \\count 4 by 4\n\t\t\\else\n\t\t\\fi\n\t\\fi\n\t\\ifnum\t\\count 2 > 32767 %%% while retaining reasonable accuracy\n\t\\then\t\\divide \\count 2 by 4\n\t\t\\divide \\count 4 by 4\n\t\\else\t\\ifnum\t\\count 2 < -32767\n\t\t\\then\t\\divide \\count 2 by 4\n\t\t\t\\divide \\count 4 by 4\n\t\t\\else\n\t\t\\fi\n\t\\fi\n\t\\multiply \\count 0 by \\count 2\n\t\\divide \\count 0 by \\count 4\n\t\\xdef \\product {#1 = \\the \\count 0 \\internal@nits}%\n\t\\aftergroup \\product\n }}\n\n\\def\\r@duce{\\ifdim\\dimen0 > 90\\r@dian \\then % sin(x+90) = sin(180-x)\n\t\t\\multiply\\dimen0 by -1\n\t\t\\advance\\dimen0 by 180\\r@dian\n\t\t\\r@duce\n\t \\else \\ifdim\\dimen0 < -90\\r@dian \\then % sin(-x) = sin(360+x)\n\t\t\\advance\\dimen0 by 360\\r@dian\n\t\t\\r@duce\n\t\t\\fi\n\t \\fi}\n\n\\def\\Sine#1%\n {{%\n\t\\dimen 0 = #1 \\r@dian\n\t\\r@duce\n\t\\ifdim\\dimen0 = -90\\r@dian \\then\n\t \\dimen4 = -1\\r@dian\n\t \\c@mputefalse\n\t\\fi\n\t\\ifdim\\dimen0 = 90\\r@dian \\then\n\t \\dimen4 = 1\\r@dian\n\t \\c@mputefalse\n\t\\fi\n\t\\ifdim\\dimen0 = 0\\r@dian \\then\n\t \\dimen4 = 0\\r@dian\n\t \\c@mputefalse\n\t\\fi\n%\n\t\\ifc@mpute \\then\n \t% convert degrees to radians\n\t\t\\divide\\dimen0 by 180\n\t\t\\dimen0=3.141592654\\dimen0\n%\n\t\t\\dimen 2 = 3.1415926535897963\\r@dian %%% a well-known constant\n\t\t\\divide\\dimen 2 by 2 %%% we only deal with -pi/2 : pi/2\n\t\t\\Mess@ge {Sin: calculating Sin of \\nodimen 0}%\n\t\t\\count 0 = 1 %%% see power-series expansion for sine\n\t\t\\dimen 2 = 1 \\r@dian %%% ditto\n\t\t\\dimen 4 = 0 \\r@dian %%% ditto\n\t\t\\loop\n\t\t\t\\ifnum\t\\dimen 2 = 0 %%% then we've done\n\t\t\t\\then\t\\stillc@nvergingfalse \n\t\t\t\\else\t\\stillc@nvergingtrue\n\t\t\t\\fi\n\t\t\t\\ifstillc@nverging %%% then calculate next term\n\t\t\t\\then\t\\term {\\count 0} {\\dimen 0} {\\dimen 2}%\n\t\t\t\t\\advance \\count 0 by 2\n\t\t\t\t\\count 2 = \\count 0\n\t\t\t\t\\divide \\count 2 by 2\n\t\t\t\t\\ifodd\t\\count 2 %%% signs alternate\n\t\t\t\t\\then\t\\advance \\dimen 4 by \\dimen 2\n\t\t\t\t\\else\t\\advance \\dimen 4 by -\\dimen 2\n\t\t\t\t\\fi\n\t\t\\repeat\n\t\\fi\t\t\n\t\t\t\\xdef \\sine {\\nodimen 4}%\n }}\n\n% Now the Cosine can be calculated easily by calling \\Sine\n\\def\\Cosine#1{\\ifx\\sine\\UnDefined\\edef\\Savesine{\\relax}\\else\n\t\t \\edef\\Savesine{\\sine}\\fi\n\t{\\dimen0=#1\\r@dian\\advance\\dimen0 by 90\\r@dian\n\t \\Sine{\\nodimen 0}\n\t \\xdef\\cosine{\\sine}\n\t \\xdef\\sine{\\Savesine}}}\t \n% end of trig stuff\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\def\\psdraft{\n\t\\def\\@psdraft{0}\n\t%\\ps@typeout{draft level now is \\@psdraft \\space . }\n}\n\\def\\psfull{\n\t\\def\\@psdraft{100}\n\t%\\ps@typeout{draft level now is \\@psdraft \\space . }\n}\n\n\\psfull\n\n\\newif\\if@scalefirst\n\\def\\psscalefirst{\\@scalefirsttrue}\n\\def\\psrotatefirst{\\@scalefirstfalse}\n\\psrotatefirst\n\n\\newif\\if@draftbox\n\\def\\psnodraftbox{\n\t\\@draftboxfalse\n}\n\\def\\psdraftbox{\n\t\\@draftboxtrue\n}\n\\@draftboxtrue\n\n\\newif\\if@prologfile\n\\newif\\if@postlogfile\n\\def\\pssilent{\n\t\\@noisyfalse\n}\n\\def\\psnoisy{\n\t\\@noisytrue\n}\n\\psnoisy\n%%% These are for the option list.\n%%% A specification of the form a = b maps to calling \\@p@@sa{b}\n\\newif\\if@bbllx\n\\newif\\if@bblly\n\\newif\\if@bburx\n\\newif\\if@bbury\n\\newif\\if@height\n\\newif\\if@width\n\\newif\\if@rheight\n\\newif\\if@rwidth\n\\newif\\if@angle\n\\newif\\if@clip\n\\newif\\if@verbose\n\\def\\@p@@sclip#1{\\@cliptrue}\n\n\n\\newif\\if@decmpr\n\n%%% GDH 7/26/87 -- changed so that it first looks in the local directory,\n%%% then in a specified global directory for the ps file.\n%%% RPR 6/25/91 -- changed so that it defaults to user-supplied name if\n%%% boundingbox info is specified, assuming graphic will be created by\n%%% print time.\n%%% TJD 10/19/91 -- added bbfile vs. file distinction, and @decmpr flag\n\n\\def\\@p@@sfigure#1{\\def\\@p@sfile{null}\\def\\@p@sbbfile{null}\n\t \\openin1=#1.bb\n\t\t\\ifeof1\\closein1\n\t \t\\openin1=\\figurepath#1.bb\n\t\t\t\\ifeof1\\closein1\n\t\t\t \\openin1=#1\n\t\t\t\t\\ifeof1\\closein1%\n\t\t\t\t \\openin1=\\figurepath#1\n\t\t\t\t\t\\ifeof1\n\t\t\t\t\t \\ps@typeout{Error, File #1 not found}\n\t\t\t\t\t\t\\if@bbllx\\if@bblly\n\t\t\t\t \t\t\\if@bburx\\if@bbury\n\t\t\t \t\t\t\t\\def\\@p@sfile{#1}%\n\t\t\t \t\t\t\t\\def\\@p@sbbfile{#1}%\n\t\t\t\t\t\t\t\\@decmprfalse\n\t\t\t\t \t \t\\fi\\fi\\fi\\fi\n\t\t\t\t\t\\else\\closein1\n\t\t\t\t \t\t\\def\\@p@sfile{\\figurepath#1}%\n\t\t\t\t \t\t\\def\\@p@sbbfile{\\figurepath#1}%\n\t\t\t\t\t\t\\@decmprfalse\n\t \t\t\\fi%\n\t\t\t \t\\else\\closein1%\n\t\t\t\t\t\\def\\@p@sfile{#1}\n\t\t\t\t\t\\def\\@p@sbbfile{#1}\n\t\t\t\t\t\\@decmprfalse\n\t\t\t \t\\fi\n\t\t\t\\else\n\t\t\t\t\\def\\@p@sfile{\\figurepath#1}\n\t\t\t\t\\def\\@p@sbbfile{\\figurepath#1.bb}\n\t\t\t\t\\@decmprtrue\n\t\t\t\\fi\n\t\t\\else\n\t\t\t\\def\\@p@sfile{#1}\n\t\t\t\\def\\@p@sbbfile{#1.bb}\n\t\t\t\\@decmprtrue\n\t\t\\fi}\n\n\\def\\@p@@sfile#1{\\@p@@sfigure{#1}}\n\n\\def\\@p@@sbbllx#1{\n\t\t%\\ps@typeout{bbllx is #1}\n\t\t\\@bbllxtrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@sbbllx{\\number\\dimen100}\n}\n\\def\\@p@@sbblly#1{\n\t\t%\\ps@typeout{bblly is #1}\n\t\t\\@bbllytrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@sbblly{\\number\\dimen100}\n}\n\\def\\@p@@sbburx#1{\n\t\t%\\ps@typeout{bburx is #1}\n\t\t\\@bburxtrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@sbburx{\\number\\dimen100}\n}\n\\def\\@p@@sbbury#1{\n\t\t%\\ps@typeout{bbury is #1}\n\t\t\\@bburytrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@sbbury{\\number\\dimen100}\n}\n\\def\\@p@@sheight#1{\n\t\t\\@heighttrue\n\t\t\\dimen100=#1\n \t\t\\edef\\@p@sheight{\\number\\dimen100}\n\t\t%\\ps@typeout{Height is \\@p@sheight}\n}\n\\def\\@p@@swidth#1{\n\t\t%\\ps@typeout{Width is #1}\n\t\t\\@widthtrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@swidth{\\number\\dimen100}\n}\n\\def\\@p@@srheight#1{\n\t\t%\\ps@typeout{Reserved height is #1}\n\t\t\\@rheighttrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@srheight{\\number\\dimen100}\n}\n\\def\\@p@@srwidth#1{\n\t\t%\\ps@typeout{Reserved width is #1}\n\t\t\\@rwidthtrue\n\t\t\\dimen100=#1\n\t\t\\edef\\@p@srwidth{\\number\\dimen100}\n}\n\\def\\@p@@sangle#1{\n\t\t%\\ps@typeout{Rotation is #1}\n\t\t\\@angletrue\n%\t\t\\dimen100=#1\n\t\t\\edef\\@p@sangle{#1} %\\number\\dimen100}\n}\n\\def\\@p@@ssilent#1{ \n\t\t\\@verbosefalse\n}\n\\def\\@p@@sprolog#1{\\@prologfiletrue\\def\\@prologfileval{#1}}\n\\def\\@p@@spostlog#1{\\@postlogfiletrue\\def\\@postlogfileval{#1}}\n\\def\\@cs@name#1{\\csname #1\\endcsname}\n\\def\\@setparms#1=#2,{\\@cs@name{@p@@s#1}{#2}}\n%\n% initialize the defaults (size the size of the figure)\n%\n\\def\\ps@init@parms{\n\t\t\\@bbllxfalse \\@bbllyfalse\n\t\t\\@bburxfalse \\@bburyfalse\n\t\t\\@heightfalse \\@widthfalse\n\t\t\\@rheightfalse \\@rwidthfalse\n\t\t\\def\\@p@sbbllx{}\\def\\@p@sbblly{}\n\t\t\\def\\@p@sbburx{}\\def\\@p@sbbury{}\n\t\t\\def\\@p@sheight{}\\def\\@p@swidth{}\n\t\t\\def\\@p@srheight{}\\def\\@p@srwidth{}\n\t\t\\def\\@p@sangle{0}\n\t\t\\def\\@p@sfile{} \\def\\@p@sbbfile{}\n\t\t\\def\\@p@scost{10}\n\t\t\\def\\@sc{}\n\t\t\\@prologfilefalse\n\t\t\\@postlogfilefalse\n\t\t\\@clipfalse\n\t\t\\if@noisy\n\t\t\t\\@verbosetrue\n\t\t\\else\n\t\t\t\\@verbosefalse\n\t\t\\fi\n}\n%\n% Go through the options setting things up.\n%\n\\def\\parse@ps@parms#1{\n\t \t\\@psdo\\@psfiga:=#1\\do\n\t\t {\\expandafter\\@setparms\\@psfiga,}}\n%\n% Compute bb height and width\n%\n\\newif\\ifno@bb\n\\def\\bb@missing{\n\t\\if@verbose{\n\t\t\\ps@typeout{psfig: searching \\@p@sbbfile \\space for bounding box}\n\t}\\fi\n\t\\no@bbtrue\n\t\\epsf@getbb{\\@p@sbbfile}\n \\ifno@bb \\else \\bb@cull\\epsf@llx\\epsf@lly\\epsf@urx\\epsf@ury\\fi\n}\t\n\\def\\bb@cull#1#2#3#4{\n\t\\dimen100=#1 bp\\edef\\@p@sbbllx{\\number\\dimen100}\n\t\\dimen100=#2 bp\\edef\\@p@sbblly{\\number\\dimen100}\n\t\\dimen100=#3 bp\\edef\\@p@sbburx{\\number\\dimen100}\n\t\\dimen100=#4 bp\\edef\\@p@sbbury{\\number\\dimen100}\n\t\\no@bbfalse\n}\n% rotate point (#1,#2) about (0,0).\n% The sine and cosine of the angle are already stored in \\sine and\n% \\cosine. The result is placed in (\\p@intvaluex, \\p@intvaluey).\n\\newdimen\\p@intvaluex\n\\newdimen\\p@intvaluey\n\\def\\rotate@#1#2{{\\dimen0=#1 sp\\dimen1=#2 sp\n% \tcalculate x' = x \\cos\\theta - y \\sin\\theta\n\t\t \\global\\p@intvaluex=\\cosine\\dimen0\n\t\t \\dimen3=\\sine\\dimen1\n\t\t \\global\\advance\\p@intvaluex by -\\dimen3\n% \t\tcalculate y' = x \\sin\\theta + y \\cos\\theta\n\t\t \\global\\p@intvaluey=\\sine\\dimen0\n\t\t \\dimen3=\\cosine\\dimen1\n\t\t \\global\\advance\\p@intvaluey by \\dimen3\n\t\t }}\n\\def\\compute@bb{\n\t\t\\no@bbfalse\n\t\t\\if@bbllx \\else \\no@bbtrue \\fi\n\t\t\\if@bblly \\else \\no@bbtrue \\fi\n\t\t\\if@bburx \\else \\no@bbtrue \\fi\n\t\t\\if@bbury \\else \\no@bbtrue \\fi\n\t\t\\ifno@bb \\bb@missing \\fi\n\t\t\\ifno@bb \\ps@typeout{FATAL ERROR: no bb supplied or found}\n\t\t\t\\no-bb-error\n\t\t\\fi\n\t\t%\n%\\ps@typeout{BB: \\@p@sbbllx, \\@p@sbblly, \\@p@sbburx, \\@p@sbbury} \n%\n% store height/width of original (unrotated) bounding box\n\t\t\\count203=\\@p@sbburx\n\t\t\\count204=\\@p@sbbury\n\t\t\\advance\\count203 by -\\@p@sbbllx\n\t\t\\advance\\count204 by -\\@p@sbblly\n\t\t\\edef\\ps@bbw{\\number\\count203}\n\t\t\\edef\\ps@bbh{\\number\\count204}\n\t\t%\\ps@typeout{ psbbh = \\ps@bbh, psbbw = \\ps@bbw }\n\t\t\\if@angle \n\t\t\t\\Sine{\\@p@sangle}\\Cosine{\\@p@sangle}\n\t \t{\\dimen100=\\maxdimen\\xdef\\r@p@sbbllx{\\number\\dimen100}\n\t\t\t\t\t \\xdef\\r@p@sbblly{\\number\\dimen100}\n\t\t\t \\xdef\\r@p@sbburx{-\\number\\dimen100}\n\t\t\t\t\t \\xdef\\r@p@sbbury{-\\number\\dimen100}}\n%\n% Need to rotate all four points and take the X-Y extremes of the new\n% points as the new bounding box.\n \\def\\minmaxtest{\n\t\t\t \\ifnum\\number\\p@intvaluex<\\r@p@sbbllx\n\t\t\t \\xdef\\r@p@sbbllx{\\number\\p@intvaluex}\\fi\n\t\t\t \\ifnum\\number\\p@intvaluex>\\r@p@sbburx\n\t\t\t \\xdef\\r@p@sbburx{\\number\\p@intvaluex}\\fi\n\t\t\t \\ifnum\\number\\p@intvaluey<\\r@p@sbblly\n\t\t\t \\xdef\\r@p@sbblly{\\number\\p@intvaluey}\\fi\n\t\t\t \\ifnum\\number\\p@intvaluey>\\r@p@sbbury\n\t\t\t \\xdef\\r@p@sbbury{\\number\\p@intvaluey}\\fi\n\t\t\t }\n%\t\t\tlower left\n\t\t\t\\rotate@{\\@p@sbbllx}{\\@p@sbblly}\n\t\t\t\\minmaxtest\n%\t\t\tupper left\n\t\t\t\\rotate@{\\@p@sbbllx}{\\@p@sbbury}\n\t\t\t\\minmaxtest\n%\t\t\tlower right\n\t\t\t\\rotate@{\\@p@sbburx}{\\@p@sbblly}\n\t\t\t\\minmaxtest\n%\t\t\tupper right\n\t\t\t\\rotate@{\\@p@sbburx}{\\@p@sbbury}\n\t\t\t\\minmaxtest\n\t\t\t\\edef\\@p@sbbllx{\\r@p@sbbllx}\\edef\\@p@sbblly{\\r@p@sbblly}\n\t\t\t\\edef\\@p@sbburx{\\r@p@sbburx}\\edef\\@p@sbbury{\\r@p@sbbury}\n%\\ps@typeout{rotated BB: \\r@p@sbbllx, \\r@p@sbblly, \\r@p@sbburx, \\r@p@sbbury}\n\t\t\\fi\n\t\t\\count203=\\@p@sbburx\n\t\t\\count204=\\@p@sbbury\n\t\t\\advance\\count203 by -\\@p@sbbllx\n\t\t\\advance\\count204 by -\\@p@sbblly\n\t\t\\edef\\@bbw{\\number\\count203}\n\t\t\\edef\\@bbh{\\number\\count204}\n\t\t%\\ps@typeout{ bbh = \\@bbh, bbw = \\@bbw }\n}\n%\n% \\in@hundreds performs #1 * (#2 / #3) correct to the hundreds,\n%\tthen leaves the result in @result\n%\n\\def\\in@hundreds#1#2#3{\\count240=#2 \\count241=#3\n\t\t \\count100=\\count240\t% 100 is first digit #2/#3\n\t\t \\divide\\count100 by \\count241\n\t\t \\count101=\\count100\n\t\t \\multiply\\count101 by \\count241\n\t\t \\advance\\count240 by -\\count101\n\t\t \\multiply\\count240 by 10\n\t\t \\count101=\\count240\t%101 is second digit of #2/#3\n\t\t \\divide\\count101 by \\count241\n\t\t \\count102=\\count101\n\t\t \\multiply\\count102 by \\count241\n\t\t \\advance\\count240 by -\\count102\n\t\t \\multiply\\count240 by 10\n\t\t \\count102=\\count240\t% 102 is the third digit\n\t\t \\divide\\count102 by \\count241\n\t\t \\count200=#1\\count205=0\n\t\t \\count201=\\count200\n\t\t\t\\multiply\\count201 by \\count100\n\t\t \t\\advance\\count205 by \\count201\n\t\t \\count201=\\count200\n\t\t\t\\divide\\count201 by 10\n\t\t\t\\multiply\\count201 by \\count101\n\t\t\t\\advance\\count205 by \\count201\n\t\t\t%\n\t\t \\count201=\\count200\n\t\t\t\\divide\\count201 by 100\n\t\t\t\\multiply\\count201 by \\count102\n\t\t\t\\advance\\count205 by \\count201\n\t\t\t%\n\t\t \\edef\\@result{\\number\\count205}\n}\n\\def\\compute@wfromh{\n\t\t% computing : width = height * (bbw / bbh)\n\t\t\\in@hundreds{\\@p@sheight}{\\@bbw}{\\@bbh}\n\t\t%\\ps@typeout{ \\@p@sheight * \\@bbw / \\@bbh, = \\@result }\n\t\t\\edef\\@p@swidth{\\@result}\n\t\t%\\ps@typeout{w from h: width is \\@p@swidth}\n}\n\\def\\compute@hfromw{\n\t\t% computing : height = width * (bbh / bbw)\n\t \\in@hundreds{\\@p@swidth}{\\@bbh}{\\@bbw}\n\t\t%\\ps@typeout{ \\@p@swidth * \\@bbh / \\@bbw = \\@result }\n\t\t\\edef\\@p@sheight{\\@result}\n\t\t%\\ps@typeout{h from w : height is \\@p@sheight}\n}\n\\def\\compute@handw{\n\t\t\\if@height \n\t\t\t\\if@width\n\t\t\t\\else\n\t\t\t\t\\compute@wfromh\n\t\t\t\\fi\n\t\t\\else \n\t\t\t\\if@width\n\t\t\t\t\\compute@hfromw\n\t\t\t\\else\n\t\t\t\t\\edef\\@p@sheight{\\@bbh}\n\t\t\t\t\\edef\\@p@swidth{\\@bbw}\n\t\t\t\\fi\n\t\t\\fi\n}\n\\def\\compute@resv{\n\t\t\\if@rheight \\else \\edef\\@p@srheight{\\@p@sheight} \\fi\n\t\t\\if@rwidth \\else \\edef\\@p@srwidth{\\@p@swidth} \\fi\n\t\t%\\ps@typeout{rheight = \\@p@srheight, rwidth = \\@p@srwidth}\n}\n%\t\t\n% Compute any missing values\n\\def\\compute@sizes{\n\t\\compute@bb\n\t\\if@scalefirst\\if@angle\n% at this point the bounding box has been adjsuted correctly for\n% rotation. PSFIG does all of its scaling using \\@bbh and \\@bbw. If\n% a width= or height= was specified along with \\psscalefirst, then the\n% width=/height= value needs to be adjusted to match the new (rotated)\n% bounding box size (specifed in \\@bbw and \\@bbh).\n% \\ps@bbw width=\n% ------- = ---------- \n% \\@bbw new width=\n% so `new width=' = (width= * \\@bbw) / \\ps@bbw; where \\ps@bbw is the\n% width of the original (unrotated) bounding box.\n\t\\if@width\n\t \\in@hundreds{\\@p@swidth}{\\@bbw}{\\ps@bbw}\n\t \\edef\\@p@swidth{\\@result}\n\t\\fi\n\t\\if@height\n\t \\in@hundreds{\\@p@sheight}{\\@bbh}{\\ps@bbh}\n\t \\edef\\@p@sheight{\\@result}\n\t\\fi\n\t\\fi\\fi\n\t\\compute@handw\n\t\\compute@resv}\n\n%\n% \\psfig\n% usage : \\psfig{file=, height=, width=, bbllx=, bblly=, bburx=, bbury=,\n%\t\t\trheight=, rwidth=, clip=}\n%\n% \"clip=\" is a switch and takes no value, but the `=' must be present.\n\\def\\psfig#1{\\vbox {\n\t% do a zero width hard space so that a single\n\t% \\psfig in a centering enviornment will behave nicely\n\t%{\\setbox0=\\hbox{\\ }\\ \\hskip-\\wd0}\n\t%\n\t\\ps@init@parms\n\t\\parse@ps@parms{#1}\n\t\\compute@sizes\n\t%\n\t\\ifnum\\@p@scost<\\@psdraft{\n\t\t%\n\t\t\\special{ps::[begin] \t\\@p@swidth \\space \\@p@sheight \\space\n\t\t\t\t\\@p@sbbllx \\space \\@p@sbblly \\space\n\t\t\t\t\\@p@sbburx \\space \\@p@sbbury \\space\n\t\t\t\tstartTexFig \\space }\n\t\t\\if@angle\n\t\t\t\\special {ps:: \\@p@sangle \\space rotate \\space} \n\t\t\\fi\n\t\t\\if@clip{\n\t\t\t\\if@verbose{\n\t\t\t\t\\ps@typeout{(clip)}\n\t\t\t}\\fi\n\t\t\t\\special{ps:: doclip \\space }\n\t\t}\\fi\n\t\t\\if@prologfile\n\t\t \\special{ps: plotfile \\@prologfileval \\space } \\fi\n\t\t\\if@decmpr{\n\t\t\t\\if@verbose{\n\t\t\t\t\\ps@typeout{psfig: including \\@p@sfile.Z \\space }\n\t\t\t}\\fi\n\t\t\t\\special{ps: plotfile \"`zcat \\@p@sfile.Z\" \\space }\n\t\t}\\else{\n\t\t\t\\if@verbose{\n\t\t\t\t\\ps@typeout{psfig: including \\@p@sfile \\space }\n\t\t\t}\\fi\n\t\t\t\\special{ps: plotfile \\@p@sfile \\space }\n\t\t}\\fi\n\t\t\\if@postlogfile\n\t\t \\special{ps: plotfile \\@postlogfileval \\space } \\fi\n\t\t\\special{ps::[end] endTexFig \\space }\n\t\t% Create the vbox to reserve the space for the figure\n\t\t\\vbox to \\@p@srheight true sp{\n\t\t\t\\hbox to \\@p@srwidth true sp{\n\t\t\t\t\\hss\n\t\t\t}\n\t\t\\vss\n\t\t}\n\t}\\else{\n\t\t% draft figure, just reserve the space and print the\n\t\t% path name.\n\t\t\\if@draftbox{\t\t\n\t\t\t% Verbose draft: print file name in box\n\t\t\t\\hbox{\\frame{\\vbox to \\@p@srheight true sp{\n\t\t\t\\vss\n\t\t\t\\hbox to \\@p@srwidth true sp{ \\hss \\@p@sfile \\hss }\n\t\t\t\\vss\n\t\t\t}}}\n\t\t}\\else{\n\t\t\t% Non-verbose draft\n\t\t\t\\vbox to \\@p@srheight true sp{\n\t\t\t\\vss\n\t\t\t\\hbox to \\@p@srwidth true sp{\\hss}\n\t\t\t\\vss\n\t\t\t}\n\t\t}\\fi\t\n\n\n\n\t}\\fi\n}}\n\\psfigRestoreAt\n\n\n\n\n\n"
}
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[
{
"name": "cond-mat0002183.extracted_bib",
"string": "\\begin{thebibliography}{1} \n \\bibitem{blatter96}\n G. Blatter and V.~B. Geshkenbein, \\newblock Phys.\\ Rev.\\ Lett.\\ {\\bf 77},\n 4958 (1996). \n \\bibitem{katzgraber99}\n H.~G.\\ Katzgraber, H.~P.\\ B\\\"uchler, and G. Blatter, \\newblock Phys.\\ Rev.\\ \n B {\\bf 59}, 11990 (1999). \n \\bibitem{lifshitz56}\n E.~M. Lifshitz, \\newblock Zh.\\ \\'{E}ksp.\\ Teor.\\ Fiz.\\ {\\bf 29}, 94 (1955)\n [Sov.\\ Phys.\\ JETP {\\bf 2}, 73 (1956)].\n \\bibitem{nelson}\n D.~R. Nelson, \\newblock Phys.\\ Rev.\\ Lett.\\ {\\bf 60}, 1973 (1988).\n \\bibitem{popov}\n V.~N. Popov, {\\em Functional Integrals and Collective Excitations}\n (Cambridge University Press, Cambridge, 1987).\n \\bibitem{blatter94}\n G. Blatter, M.~V. Feigel'man, V.~B. Geshkenbein, A.~I. Larkin, and\n V.~M. Vinokur, \\newblock Rev.\\ Mod.\\ Phys.\\ {\\bf 66}, 1125 (1994).\n \\end{thebibliography}"
}
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cond-mat0002185
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Relative energetics and structural properties of zirconia using a self-consistent tight-binding model
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[
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"author": "Relative energetics and structural properties of zirconia using a self-consistent tight-binding model"
}
] |
We describe an empirical, self-consistent, orthogonal tight-binding model for zirconia, which allows for the polarizability of the anions at dipole and quadrupole levels and for crystal field splitting of the cation $d$ orbitals. This is achieved by mixing the orbitals of different symmetry on a site with coupling coefficients driven by the Coulomb potentials up to octapole level. The additional forces on atoms due to the self-consistency and polarizabilities are exactly obtained by straightforward electrostatics, by analogy with the Hellmann-Feynman theorem as applied in first-principles calculations. The model correctly orders the zero temperature energies of all zirconia polymorphs. The Zr-O matrix elements of the Hamiltonian, which measure covalency, make a greater contribution than the polarizability to the energy differences between phases. Results for elastic constants of the cubic and tetragonal phases and phonon frequencies of the cubic phase are also presented and compared with some experimental data and first-principles calculations. We suggest that the model will be useful for studying finite temperature effects by means of molecular dynamics.
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{
"name": "amstruc.tex",
"string": "&latex209\n\\documentstyle[12pt,aps,prb]{revtex}\n%\\documentstyle[twocolumn,aps,prb]{revtex}\n\n\\input psfig.tex\n\n\\begin{document}\n% \\draft command makes pacs numbers print\n\\draft\n% repeat the \\author\\address pair as needed\n\\title{Relative energetics and structural properties of zirconia\n using a self-consistent tight-binding model}\n\\author{Stefano Fabris, Anthony T. Paxton and Michael W. Finnis}\n\\address{Atomistic Simulation Group, Department of Pure and Applied\nPhysics, Queen's University, \\\\ Belfast BT7 1NN, United Kingdom}\n\\date{\\today} \n\\maketitle\n\\begin{abstract}\n\nWe describe an empirical, self-consistent, orthogonal tight-binding model\nfor zirconia, which allows for the polarizability of the anions at dipole\nand quadrupole levels and for crystal field splitting of the cation $d$\norbitals. This is achieved by mixing the orbitals of different symmetry\non a site with coupling coefficients driven by the Coulomb potentials up\nto octapole level. The additional forces on atoms due to the\nself-consistency and polarizabilities are exactly obtained by\nstraightforward electrostatics, by analogy with the Hellmann-Feynman\ntheorem as applied in first-principles calculations. The model correctly\norders the zero temperature energies of all zirconia polymorphs. The Zr-O\nmatrix elements of the Hamiltonian, which measure covalency, make a\ngreater contribution than the polarizability to the energy differences\nbetween phases. Results for elastic constants of the cubic and\ntetragonal phases and phonon frequencies of the cubic phase are also\npresented and compared with some experimental data and first-principles\ncalculations. We suggest that the model will be useful for studying\nfinite temperature effects by means of molecular dynamics.\n\n\\end{abstract}\n\\pacs{31.15.Ar,71.15.Fv,81.30-t}\n\n\\section{Introduction}\n\nSolid solutions of zirconia (ZrO$_2$) containing other oxides are among\nthe major representatives of modern ceramic materials. The wide range of\napplications, including traditional structural refractories, fuel cells\nand electronic devices such as oxygen sensors,~\\cite{Heuer81,Claussen84}\ntestifies to the technological importance of zirconias. Different\ndivalent and trivalent oxides are added to ZrO$_2$ in order to improve\nits thermomechanical properties, and charge-compensating vacancies are\nthereby introduced on the anion sublattice. The macroscopic effects\nassociated with the impurities are very well\nknown,~\\cite{Grain67,Scott75,Stubican81} but a microscopic model which\ngives a theoretical interpretation is still missing. As a preliminary\nstep, this paper provides a physical picture of the crystal\nthermodynamics of pure zirconia, combining the results of first\nprinciples density functional and semiempirical Tight Binding (TB)\ncalculations.\n\nZirconia has three zero-pressure polymorphs; these have cubic\n($c$), tetragonal ($t$) and monoclinic ($m$) symmetry. The high\ntemperature $c$ phase \\cite{Aldebert85,Ackermann77} ({\\it Fm}3{\\it m}) is\nstable between 2570 K and the melting temperature of 2980 K. The $t$\nstructure \\cite{Howard88,Teufer62} ($P4_2/nmc$), which is stable between\n1400 and 2570 K, is closely related to the $c$ one: the internal degree\nof freedom $\\delta$ shifts the oxygen ions away from the centrosymmetric\npositions along the $X_2^-$ mode of vibration (Figure~\\ref{cellct}) and\nforces the $c/a$ ratio of the unit cell to adjust. Below 1400 K the\nlow-symmetry $m$ phase \\cite{Adam59,McCullough59,Smith65} ($P2_1/c$) is\nthermodynamically stable.\n\nBesides its technological implications, the relationship between these\nstructures is of fundamental interest. The mechanisms of the phase\ntransformations, the effects of impurities and vacancies on\nthem, and their relationship to the nature of the bonding still require\nexplanation, and this may shed light on the properties of other, more\ncomplex oxides.\n\nThe crystal structure of purely ionic bonded materials can be determined\non the basis of radius-ratio rules,~\\cite{Kingery60} based purely on\nelectrostatic arguments. Because of the small size of the Zr$^{4+}$\nions, these rules place ZrO$_2$ on the border between the 8-fold\ncoordinated fluorite structure and the 6-fold coordinated rutile one\n($P4_2/mnm$). The radius-ratio is too blunt a tool to account for the\nabsolute stability of the unique 7-fold coordinated $m$ structure.\n\n%Following this approach, the small size of the Zr$^{4+}$ ions explains\n%the relative instability of the $c$ phase compared to the $t$ one. We\n%will show that the $c \\leftrightarrow t$ energetics can be understood\n%in terms of minimization of electrostatic interactions, which underlie\n%the radius-ratio rules. \n\nThe classical empirical models of zirconia are based on the {\\it a\npriori} assumption of its {\\it ionicity}. Empirical approaches like the\nShell Model (SM) or the Rigid Ion Model (RIM) described the\nstructural,~\\cite{Dwivedi90} dynamical \\cite{Mirgorodsky97,Cormack90}\nand transport \\cite{Li95,Shimojo92-I,Shimojo92-II} properties of the\nphases on which they were parameterized, but failed to predict the\nabsolute stability of the $m$ structure. The most detailed of such\nmodels was developed by Wilson {\\it et al.},~\\cite{Wilson96zirc} whose\nenvironment-dependent Compressible and Polarizable Ion Model (CIM-DQ)\ndemonstrated the importance of the anion polarizabilities at both dipole\nand quadrupole levels on the energetics of zirconia. However, further\ncalculations~\\cite{Laurea97} carried out with this model revealed that\neven though it predicted the correct energy ordering of the $c$, $t$ and\n$m$ phases, it predicted that the rutile structure should be even more\nstable, and this phase is never observed experimentally in zirconia.\n\nThe experience gained with the CIM-DQ model suggests that a successful\nempirical model of zirconia should describe the effects of the atomic\npolarization, but should also go beyond a purely ionic description of the\nbonding. The partial covalent character of zirconia has already been\npostulated \\cite{Ho82} and is evident from electronic structure\ncalculations based on density functional theory. In this paper we further\ninvestigate the recently proposed polarizable self-consistent tight\nbinding (SC-TB) model \\cite{Finnis97,Finnis98,Schelling98} which combines\nthe physical concepts of covalency, ionicity and polarizability. Using\nthe SC-TB model we are drawn to the conclusion that the covalent\ncharacter of the Zr-O bond makes a significant contribution to the\nrelative energetics of different structures, which would explain the\nlimited predictive power of the previous ionic models.\n\nThere have been several previous approaches to analyzing the structural\nand electronic properties of zirconia. Boyer and Klein \\cite{Boyer85b}\nused the APW method to derive pair potentials with which to investigate\nthe equation of state of the $c$ phase. Cohen {\\it et al.} \\cite{Cohen88}\ncalculated the relative energetics and the elasticity using the Potential\nInduced Breathing (PIB) method based on the Gordon-Kim\napproach. Zandiehnadem {\\it et al.} \\cite{Zandiehnadem88} studied the\nelectronic structure with a first principles LCAO method. The FLAPW\ncalculations of Jansen \\cite{Jansen91} predicted for the first time the\ncorrect energetic ordering between the $c$ and $t$ structures at zero\nabsolute temperature, identifying the double well in the potential\nenergy that governs their relative stability. The double well was subsequently\nconfirmed by {\\it ab initio} Hartree-Fock (HF)\ncalculations,~\\cite{Orlando92,Stefanovich94} but these did not predict\nthe stability of the $m$ structure over the $t$ one. Only the very recent\nDensity Functional Theory (DFT) calculations\n\\cite{Kralik98,Stapper99,Jomard99} consistently reproduce the relative\nenergetics of the three zirconia polymorphs at 0 K.\n\nThe plan of the present paper is as follow. In Section \\ref{TB model} we\ndescribe the model used in the calculations, the inclusion of the atomic\npolarizability in the TB framework and the parameterization procedure. A\npreliminary account of this work has been published.~\\cite{Finnis98} We\nhave made DFT calculations of band structures of the simple structures\nfor this purpose, using a new full-potential, linear muffin tin orbital\nmethod (NFP-LMTO). The predictive power of the new model is tested\nagainst the DFT calculations in Section \\ref{en-vol}, where we study the\nrelative energetics of zirconia. Section \\ref{c-t dist} focuses on the\nrelationship between the $c$ and $t$ structures: the Landau theory of\nphase transformation is used to interpret the results of the static\ncalculations. In Section \\ref{distortions}, we explore the elastic and\nthe vibrational properties of the high symmetry phases. The results are\nsummarized in the concluding Section.\n\n\\section{The Tight Binding Model}\n\\label{TB model}\n\n\\subsection{Including polarizabilities in TB}\n\nIn the TB approximation the crystal wave function can be expressed as a\nlinear combination of atom-centered orbitals which we denote $\\left. \\mid\n{\\bf R} L \\right>$: \n\n\\begin{equation}\n\\left. \\mid \\Psi^{n {\\bf k}} \\right> = \\sum c_{{\\bf\nR} L}^{n {\\bf k}} \\left. \\mid {\\bf R} L \\right>.\n\\end{equation}\n$L$ is a composite angular momentum index $L=\\left(\\ell,m \\right)$ of the\natomic orbital centered on the site whose position is {\\bf R}, $n$ and\n${\\bf k}$ are the band and ${\\bf k}-$vector indices of the single\nparticle wave function. For the purpose of derivation, we express the\nlocal orbitals as a product of a radial function and a real spherical\nharmonic\n\\begin{equation}\n\\left< {\\bf r} \\! \\mid\\! {\\bf R} L \\right> = f_{{\\bf R}\\ell}(\\mid \\!\n{\\bf r} - {\\bf R} \\! \\mid ) \\, Y_{L}({\\bf r}- {\\bf R}), \\label{loc-orb}\n\\end{equation} \nalthough in our {\\it empirical} TB scheme the explicit functional forms\nof the radial wave functions are not required. To simplify the notation,\nwe will frequently suppress the site index ${\\bf R}$, in which case one\ncan take it we are referring to an atom at the origin and ${\\bf r}$ is a\nsmall vector in its neighborhood.\n\n%\\begin{equation}\n%\\left. \\mid \\Psi^{n {\\bf k}} \\right> = \\sum c_{{\\bf R} L}^{n {\\bf k}}\n%\\left. \\mid {\\bf R} L \\right> \n%\\end{equation}\n\nThe total Hamiltonian ${\\cal H}$ can be expressed as a sum of two terms,\n${\\cal H} = {\\cal H}^0 + {\\cal H}^\\prime$. In traditional Self-Consistent\n(SC) TB, ${\\cal H}^0$ contains both on-site and inter-site terms. The\non-site terms are diagonal in $L$, and are often taken as Hartree-Fock\nterm values of the isolated atoms. The inter-site terms are the bonding\nintegrals. The additional part of the Hamiltonian, ${\\cal H}^\\prime$, is\ndiagonal in {\\bf R} and $L$ in the traditional approach (Majewski and\nVogl \\cite{Majewski86,Majewski87}). It controls the charge redistribution\nbetween neighboring sites which results from the balance between the\nopposite effects due to the on-site Coulomb repulsion (Hubbard $U$) and\nMadelung potentials.\n\n% and controls, via the Hubbard potentials $U_{{\\bf R}L}$,\n% the amount of charge transfer between a site and it's neighbors,\n% introducing the physical concept of {\\it ionicity} in the representation\n% of the crystal.\n\nWhat is missing in the previous model is the effect of the crystal\nfields on the valence electrons, {\\it i.e.} the atomic {\\it\npolarizability}. In a preliminary account of this work \\cite{Finnis98}\nwe indicated how to include the polarization effects in a\nSC-TB formalism by adding off-diagonal terms ${\\cal H}^\\prime_{{\\bf\nR}L{\\bf R}L^\\prime}$ to the on-site blocks of the Hamiltonian. Here we\ndescribe how we make that extension.\n\nIf we assume the on-site charge distribution to be localized, then its\ntotal multipole moment $Q_L$ has a monopole contribution from the ionic\ncore charge and a multipole (including monopole) contribution from the\nvalence charge:\n\n\\begin{equation}\nQ_{L} = Q^i \\, \\delta_{L0} + Q^e_{L}. \n\\label{tmpol}\n\\end{equation}\n\nAs Stone \\cite{Stone96} points out, the electronic multipole moment on a\nsite is the expectation value of the operator \n\\begin{equation}\n \\hat{Q}_L^e=e \\, \\hat{r}^\\ell Y_L({\\bf \\hat{r}}), \\label{stone}\n\\end{equation}\nwhere $e$ is the charge of the electron. Neglecting inter-site terms like\n$\\left<{\\bf R^\\prime} L^\\prime \\left| \\hat{Q}^e_{{\\bf R} L} \\right| {\\bf\nR^{\\prime\\prime}} L^{\\prime\\prime} \\right>$ for ${\\bf R^\\prime},{\\bf\nR^{\\prime\\prime}} \\ne {\\bf R}$, the definition of the on-site multipole\nmoment is therefore:\n\n\\begin{equation}\nQ_L^e \\equiv \\sum_{L^\\prime L^{\\prime\\prime}} \\sum_{n {\\bf k}}^{\\rm occ.}\nc_{L^\\prime}^{n {\\bf k}} c_{L^{\\prime\\prime}}^{n {\\bf k}} \\left<L^\\prime\n\\left| \\hat{Q}^e_L \\right| L^{\\prime\\prime} \\right>. \\label{mpol}\n\\end{equation}\nBy invoking equations (\\ref{loc-orb}) and (\\ref{stone}), the last factor\nof Eq.(\\ref{mpol}) can be expressed as a product of two quantities, the\nGaunt coefficients $C_{L^\\prime L^{\\prime\\prime}L}$, which dictate the\nselection rules, and the integrals $\\Delta_{\\ell^\\prime\n\\ell^{\\prime\\prime} \\ell}$, which will be new parameters of the model:\n\n% It is possible to show that the last factor of (\\ref{mpol}) is\n% proportional to the Gaunt coefficients $C_{L^\\prime\n% L^{\\prime\\prime}L}$. The proportionality constants are new parameters of\n% the model which we will define as $\\Delta_{l^\\prime l^{\\prime\\prime}\n% l}$.\n\n\\begin{eqnarray}\n\\left< L^\\prime \\left| \\hat{Q}^e_L \\right| L^{\\prime\\prime} \\right> & = &\ne \\,\\, {\\Delta_{\\ell^\\prime \\ell^{\\prime\\prime} \\ell}} \\,\\, {C_{L^\\prime\nL^{\\prime\\prime} L}} \\label{product} \\\\\n{C_{L^\\prime L^{\\prime\\prime} L}} & = & \\int \\!\\! Y_{L^{\\prime}}\nY_{L^{\\prime \\prime}} Y_L \\, \\, d{\\Omega} \\label{gaunt} \\\\ \n{\\Delta_{\\ell^{\\prime} \\ell^{\\prime\\prime} \\ell}} & = & \\int \\!\\!\nf_{\\ell^{\\prime}}(r) f_{\\ell^{\\prime\\prime}}(r) r^{\\ell+2} dr \\, , \\label{delta} \n%\\left< L^\\prime \\left| \\hat{Q}^e_L \\right| L^{\\prime\\prime} \\right> & = &\n%e \\,\\, \\Delta_{l^\\prime l^{\\prime\\prime} l} \\,\\, C_{L^\\prime\n%L^{\\prime\\prime} L} \\nonumber \\\\ & = & e \\,\\, \\Delta_{l^\\prime\n%l^{\\prime\\prime} l} \\int \\!\\! Y_{L^{\\prime\\prime}} Y_{L^{\\prime}} Y_L \\,\n%\\, d\\Omega.\\label{gaunt}\n\\end{eqnarray} \nwhere $d \\Omega$ stands for the element of solid angle $\\sin \\theta \\,\nd\\theta \\, d\\phi$. The r$\\hat{\\rm o}$le of the Gaunt coefficients, which\ndepend on the angular part of the wave function only, is to select the\nterm with symmetry $L$ arising from the coupling of the on-site orbitals\n$L^\\prime$ and $L^{\\prime\\prime}$. The $\\Delta$ parameters, depending on\nthe radial part of the wave function, determine the magnitude of the\ncoupling. The substitution of Eq.(\\ref{product}) in Eq.(\\ref{mpol})\ndefines the multipole moment of symmetry $L$ on the site {\\bf R}.\n\n\nHaving defined the on-site multipole moments, we can calculate the fields\nwhich they generate on all the lattice sites. The derivation uses\nstandard results from classical electrostatics. The electrostatic\npotential is expanded in partial waves about the site:\n\\begin{equation}\nV({\\bf r}) = \\sum_{L} V_L \\, r^\\ell Y_L({\\bf r}),\n\\end{equation}\nwhere, using the Poisson equation,\n\\begin{equation}\nV_{L} = 4\\pi \\sum_{{\\bf R^\\prime} \\neq {\\bf 0}} \\sum_{L^\\prime}\n\\tilde{B}_{L L^\\prime} \\left({\\bf R^\\prime} \\right) \nQ_{{\\bf R^\\prime}L^\\prime}, \\label{pot}\n\\end{equation}\nand\n\\begin{eqnarray}\n\\tilde{B}_{L L^\\prime} \\left({\\bf R} \\right) \n& = & \\frac{4\\pi }{(2\\ell+1)!! \\, (2\\ell^\\prime\n+1)!!} \\\\ \n& & \\times \\sum_{L^{\\prime \\prime}} \n \\,\n\\frac{ (-1)^{\\ell^\\prime} \\, (2\\ell^{\\prime \\prime}-1)!!}{\n\\left| {\\bf R} \\right|^{\\ell^{\\prime \\prime} +1}} \n\\, Y_{L^{\\prime \\prime}}({\\bf R}) \\, C_{L^{\\prime \\prime} L^\\prime L } . \\nonumber\n\\end{eqnarray}\nThe sum over $L^{\\prime\\prime}$ is restricted to the values for which\n$\\ell^{\\prime\\prime} = \\ell + \\ell^\\prime$; $\\tilde{B}_{L L^\\prime}$ are\nproportional to the well known LMTO-ASA structure\nconstants.~\\cite{Andersen84} The component of electrostatic potential\n$V_L$ couples different orbitals on a site giving the matrix elements:\n\\begin{equation}\n\\left< L^\\prime \\! \\mid \\! {\\cal H}^\\prime \\! \\mid \\! L^{\\prime\\prime}\n\\right> = \\sum_L \\, V_L \\, \\, \\Delta_{\\ell^{\\prime} \\ell^{\\prime\\prime}\n\\ell} \\, \\, C_{L^\\prime L^{\\prime\\prime}L} .\n\\end{equation}\n\nThe diagonal elements of the Hamiltonian are adjusted by using a single\nHubbard $U$ in the standard way, which adds a term $U \\,\\delta N_{{\\bf\nR}\\ell}$ to each diagonal matrix element. The quantities $\\delta N_{{\\bf\nR}\\ell}$ are the changes in the electronic charge projected onto a site and\norbital compared to the input, non-self-consistent charge. We use the\nstandard Mulliken projection. Finally the Schr\\\"odinger equation is\nsolved using a self-consistent iterative procedure with charge mixing to\nobtain the coefficients $c_{{\\bf R} L}^{n {\\bf k}}$ and hence the\nmultipoles.\n\nIt is useful to step back at this point and compare the above model with\nthe Hohenberg-Kohn-Sham (HKS) one, whose exchange and correlation energy\nfunctional $U^{xc}[n]$ has been expanded to second order in the electron\ndensity n({\\bf r}):~\\cite{Foulkes89}\n\n\\begin{eqnarray}\n U^{\\rm HKS} & = & \\sum_{n, {\\bf k}} ^{\\rm occ} \\left< \\Psi^{n\n{\\bf k}} \\mid {\\cal T}_S + V_0^{xc} + V_0^{\\rm H} + V_0^i \\mid \\Psi^{n {\\bf k}} \\right>\n\\label{HKS} \\\\\n& & + U^{xc}[n_0] - \\int \\!\\! V^{xc}_0 \\, n_0 \\, d {\\bf r}\n - U^{\\rm H}[n_0] + U^{ii} \\nonumber \\\\\n%\\frac{1}{2} \\int \\!\\!\\! \\frac{n_0 n_0^\\prime}{\n% \\mid \\! {\\bf r-r^{\\rm \\prime}} \\!\\! \\mid} \\, d{\\bf r}\\, d{\\bf r^{\\rm\n% \\prime}} \\nonumber \\\\\n& & + \\frac{1}{2} \\int \\!\\!\\!\\! \\int \\!\\!\\left(\n\\frac{e^2}{\\mid \\! {\\bf r-r^{\\rm \\prime}} \\!\\! \\mid} +\n\\left. \\frac{\\delta^2 U^{xc}}{\\delta n \\, \\delta n^\\prime}\n\\right|_{n=n_0} \\right) \\, \\delta n \\, \\delta n^\\prime \\, d{\\bf r}\\, d{\\bf\nr^{\\rm \\prime}}. \\nonumber\n\\end{eqnarray}\n$n_0$ denotes a reference electron density, which we will consider as a\nsuperposition of spherical ionic charges; ${\\cal T}_S$ is the kinetic\nenergy operator of the non-interacting electron gas, $V_0^{xc}$,\n$V_0^{\\rm H}$ and $V_0^i$ are the exchange and correlation, Hartree and\nionic potentials calculated at the reference charge $n_0$; $\\delta n$\ndenotes the deviation from that reference ($\\delta n = n - n_0$) and\n${n^\\prime}$ refers to the electron density at ${\\bf r^\\prime}$. $U^{\\rm\nH}$ and $U^{ii}$ are respectively the Hartree and the ion-ion\nelectrostatic energies.\n\nWithout the last term, this is simply the Harris-Foulkes functional. It\ngenerates a non-self-consistent TB model in which the first term is the\nsum of the eigenvalues while the second is a sum of pair\npotentials.~\\cite{Sutton88} If the last term is included, the total energy\nmust be minimized iteratively, and the last term now provides the\nself-consistency correction to the Kohn-Sham Hamiltonian.\n\nThe last line of Eq.(\\ref{HKS}) represent the Hartree energy of the\ndeviation from the reference charge, $U^{\\rm H}[\\delta n]$, and the\nsecond order term of the $U^{xc}$ Taylor expansion. We can identify this\nterm in our SC-TB model as follows:\n\\begin{eqnarray}\n\\frac{1}{2} \\int \\!\\!\\!\\! \\int \\!\\!\\left(\n\\frac{e^2}{\\mid \\! {\\bf r-r^{\\rm \\prime}} \\!\\! \\mid} +\n\\left. \\frac{\\delta^2 U^{xc}}{\\delta n \\, \\delta n^\\prime}\n\\right|_{n=n_0} \\right) \\, \\delta n \\, \\delta n^\\prime \\, d{\\bf r}\\, d{\\bf\nr^{\\rm \\prime}}\n\\equiv \\\\\n\\equiv \\frac{1}{2} \\sum_{{\\bf R}L} \\left( U \\, \\delta N^2_{{\\bf R}\\ell} + \n Q_{{\\bf R}L}\\, V_{{\\bf R}L} \\right) \\, . \\nonumber\n\\end{eqnarray}\n\n\nOur total energy in the SC-TB model is therefore\n\\begin{eqnarray}\nU^{\\rm TB} & = & \\sum_{n, {\\bf k}}^{\\rm occ} \\left< \\Psi^{n {\\bf k}} \\mid\n{\\cal H}^0 \\mid \\Psi^{n {\\bf k}} \\right> + U^{\\rm pair} \\nonumber \\\\ & &\n+ \\frac{1}{2} \\sum_{{\\bf R}L} \\left( U \\, \\delta N^2_{{\\bf R}\\ell} + Q_{{\\bf\nR}L} \\, V_{{\\bf R}L} \\right) \\label{UTB}\n\\end{eqnarray}\n\nIt can be verified that, by minimizing the above expression\nwith respect to the expansion coefficients in the wave functions,\nwe recover the Schr\\\"odinger equation with the SC-TB Hamiltonian. \n\nCalculation of the forces on the ions is very straightforward once we\nhave the self-consistent wave functions and multipoles. For if an ion is\nmoved a small distance $\\delta {\\bf R}$, there is no change in total\nelectronic energy to first order in the $\\delta c_{{\\bf R} L}^{n {\\bf\nk}}$. Therefore we can calculate the force due to the change in the\nfirst term of (\\ref{UTB}) by the conventional formulae, using the\nderivatives of the non-self-consistent Hamiltonian matrix elements (see\nfollowing section). In calculating the forces due to the last term of\n(\\ref{UTB}) we can hold the multipoles fixed and use standard\nelectrostatics. There is no contribution to the forces from the on-site\nenergy containing $U$. The simple form of these results for the forces in\nTB is a direct analogy with the application of the Hellmann-Feynman\ntheorem in DFT.\n\n\n\n\n\\subsection{Parameterization}\n\nEach parameter of the model has been adjusted to the results of NFP-LMTO\ncalculations, details of which are specified in the previous work on\nzirconia.~\\cite{Finnis98} Our TB description of zirconia uses a minimal\nbasis of atomic orbitals. The oxygen atoms are modelled with $2p$ and\n$3s$ orbitals and with a fixed core charge of +4, while on the zirconium\natoms there are $4d$ orbitals and a core charge of +4. The purpose of the\n$3s$ orbital on the oxygen is twofold: to allow an extra degree of\nfreedom for polarization, which is otherwise restricted to charge\ntransfer between its $2p$ orbitals, and to better reproduce the structure\nof the conduction bands.\n\nA repulsive Born-Mayer pair potential $U^{\\rm pair}$ has been chosen\nin order to reproduce the lattice parameter and the bulk modulus of the\n$c$ phase. Only the first Zr-O coordination shell has been included in\nthis interaction.\n\nThe Hamiltonian ${\\cal H}^0$ has been adjusted to the {\\it ab initio}\nelectronic structure of the $c$ phase shown in Figure~\\ref{bndstr} (c).\nWe chose the Goodwin-Skinner-Pettifor \\cite{Goodwin89} distance\ndependence of the 10 hopping integrals involved. The Hubbard $U$ have\nbeen fixed to 1 Ry. The parameters of the SC-TB model are collected in\nTable~\\ref{param}.\n\nThe basis set chosen reduces the number of symmetry-allowed $\\Delta$\nparameters to 4: $\\Delta_{spp}$, $\\Delta_{ppd}$, $\\Delta_{ddd}$ and\n$\\Delta_{ddg}$. The first two refer to the $s$ and $p$ orbitals of oxygen\nions, the last two to the $d$ orbitals on the zirconium.\n\nIn the highly symmetric $c$ structure the first non spherical terms of\nthe potential $V_L$ on the cation and anion sites have $g$ and $f$\nsymmetry respectively. The latter cannot interact with the oxygen\norbitals, the former splits the energetic levels of the zirconium $d$\norbitals and $\\Delta_{ddg}$ determines the magnitude of the energy\nsplitting $\\delta \\epsilon$. Cubic crystal field theory\n\\cite{Stoneham75} predicts the proportionality between $\\delta\n\\epsilon$ and the radial distribution of charge $< \\! \\! r^4 \\! \\! >$\nwhich is the definition of $\\Delta_{ddg}$ given in Eq.(\\ref{delta}).\nFigure~\\ref{bndstr} (a) and (b) shows the effect of the $\\Delta_{ddg}$\npolarization term on the band structure of the $c$ phase: the\nsplitting of the $d$ bands could not be captured with the SC-TB without\nthe polarizability parameters. Reasonable values of the\n$\\Delta_{ddd}$ parameter have no significant effect on any physical\nproperties studied here, therefore we set it to zero.\n\nLess symmetric structures are necessary to parameterize the remaining\n$\\Delta$'s. In the rutile phase, the $\\ell=3$ component of the crystal\nfield acting on the oxygen ions splits the $p$ levels. Consequently, it\ncontributes to the width of the $2p$ band: this effect is controlled by\n$\\Delta_{ppd}$ which we adjust to match the {\\it ab initio} band\nstructure of the rutile phase. The last term $\\Delta_{spp}$ has been\nchosen in order to reproduce the depth of the double well in the\npotential energy of the $t$ structure.\n\n\\section{Energetics of Bulk Phases}\n\n\\subsection{Energy-Volume curves}\n\\label{en-vol}\n\n\\subsubsection{Zero-pressure phases}\n\nThe predictive power of the polarizable TB model has been investigated\nby comparing its results with NFP-LMTO calculations. The Energy-Volume\ncurves calculated with the two methods are shown in\nFigure~\\ref{envol}. Each energy value involved the full relaxation of\nall the degrees of freedom of the structures.\n\nThe $c$ and the $t$ phases were used in the parameterization\nprocedure, therefore there is automatic agreement of the two methods\nfor these crystal structures. The true prediction of the model is the\nabsolute stability of the monoclinic phase. This indicates the\ntransferability of the parameters between the phases. \n\nThe rutile phase, which is not experimentally observed, has been included\nin the study because further calculations with the CIM-DQ\n\\cite{Laurea97,MRS99} model predicted the rutile phase to be more stable\nthan the monoclinic one. Figure~\\ref{envol} shows that the SC-TB model\ndoes not suffer from this problem, although the relative energy of the\nrutile phase is less than with the DFT. To our knowledge, the SC-TB is\nthe first semi-empirical model which reproduces the correct ordering of\nthese polymorphs at zero temperature, including the stability of the $m$\nphase.\n\nTable~\\ref{strucpar} summarizes the structural properties calculated with\nthe NFP-LMTO method and with the polarizable SC-TB model, comparing them\nwith other theoretical and experimental works. The $c$ and $m$ lattice\nparameters are referred to the 12-atoms unit cell, while the $t$ ones\nare given in terms of the 6-atoms unit cell. A comparison of the energy\ndifferences between the phases of zirconia calculated with different\nmethods is given in Table~\\ref{energy}.\n\n\\subsubsection{High-Pressure phases}\n\nUnder pressure, the low temperature $m$ phase transforms to an\northorhombic structure, known as ortho I ($o_I$), whose crystallography is\nstill controversial. X-ray diffraction analysis\\cite{Kudoh86,Suyama85}\nsuggests it belongs to the $Pbcm$ space group while neutron diffraction\nstudies\\cite{Ohtaka90,Howard91} propose the $Pbca$ space group. We\ncarried out the calculations using the latter structure. The phase\ntransition pressure strongly depends on the state of the sample and is\nbelieved to be between 3 and 6 GPa.~\\cite{Liu80,Block85,Ohtaka91} A\nsecond pressure-induced phase transition is observed around 15\nGPa,~\\cite{Ohtaka91} where the $o_I$ transforms to the orthorhombic phase\ntermed ortho II ($o_{II}$). The latter is isostructural to cotunnite\n(PbCl$_2$) and belongs to the $Pnam$ space group.~\\cite{Ming85} The\npressure increases the coordination number of the zirconium atoms from 7\nto 9. \n\nA comprehensive first-principles study of the two orthorhombic phases has apparently not\nyet been made: Stapper {\\it et al.}\\cite{Stapper99} studied the $o_I$\nstructure only, while Jomard {\\it et al.}\\cite{Jomard99} focused on the\n$o_{II}$ phase. \n\nThe atomic environment of the high pressure phases is completely\ndifferent to that of the $c$ and $t$ phases used in the\nparameterization of the TB model, therefore these orthorhombic structures\nprovides a severe benchmark for the transferability of the TB\nparameters.\n\nThe energy ordering of the phases predicted by the TB model is\n\n\\[ U^m < U^{o_I} < U^t < U^c < U^{o_{II}}, \\] \nwhich is the same as we obtain by combining the results of\nRefs. \\onlinecite{Stapper99} and \\onlinecite{Jomard99}. The numerical\nvalues of the energy differences are summarized in Table~\\ref{energy} and\ncompare reasonably well with the {\\it ab initio} results. The\nEnergy-Volume curves of the orthorhombic phases are shown in\nFigure~\\ref{ortenvol}: all the degrees of freedom were fully relaxed and\ntheir values are collected in Table~\\ref{parort}.\n\nAlthough the TB model predicts the correct relative energetics\nof the phases, it is not capable of describing the subtle\npressure-induced phase transformation $m \\leftrightarrow\no_I$. Figure~\\ref{ortenvol} shows the common-tangent between the\n$m$ and the $o_{II}$ phases. As the pressure is increased,\nthe model misses the correct sequence of the phases,\npredicting a $m \\leftrightarrow o_{II}$ pressure-induced phase\ntransformation at 5 GPa. \n\n\\subsection{Cubic versus Tetragonal Phases}\n\\label{c-t dist}\n\n\\subsubsection{Static calculations}\n\nThe relationship between the cubic and the tetragonal phases is governed\nby a volume dependent double well in the potential energy. Since the FLAPW\ncalculation of Jansen \\cite{Jansen88,Jansen91} who predicted it first,\nthe double well has been confirmed by several other {\\it ab initio} calculations\nand it is now well established.\n\nIn this section we analyze the nature of the 0 K energy surface by\ncombining the information gained using two very different approaches: the\nNFP-LMTO method and the polarizable TB model. The qualitative and\nquantitative agreement between the results of the two calculations, shown\nin the previous section, entitles us to use the physical picture provided\nby the simpler model to interpret the {\\it ab initio} results.\n\nStarting from the $c$ phase, the $t$ structure can be obtained by\ncontinuously stretching the unit cell along the $c$ crystallographic\ndirection and by displacing the oxygen columns by $\\delta$ along the\ntetragonal axis according to the $X_2^-$ mode of vibration\n(Figure~\\ref{cellct}). We calculated the total energy of the crystal\nusing the two methods, for different values of ($\\delta$, $c/a$) at\nseveral volumes.\n\nThe energy curve exhibits a single well or a double well structure\ndepending on the specific volume. At small volumes, V$_1$, the tetragonal\ndistortion is energetically unfavored and the equilibrium structure is\ncubic (Figure~\\ref{well}). When the cubic phase is stable, there is no\ndistinct metastable tetragonal phase with which to compare its energy, so\nthe energies of the two phases merge. At larger volumes, V$_2$, a\nstructural instability appears and the $c$ structure spontaneously\ndistorts to the $t$ one (Figure~\\ref{well}).\n\nThe curvature of the energy surfaces is related to the phase transition\nmechanism. It is clear from Figure~\\ref{well} that $\\frac{\\partial^2E}{\\partial \\eta^2}$ is positive, while $\\frac{\\partial^2 E}{\\partial\n\\delta^2}$ is negative: this suggests that the phase transition is\ndriven by the $\\delta$ instability and that the adjustment of the $c/a$\nratio is a secondary effect. The coupling between these two order\nparameters will be further discussed when we interpret the double\nwell using Landau Theory.\n\nOur LDA and TB results for the depth of the double well at the $t$ phase\nequilibrium volume, V$_2$, are consistent with the recent LDA values of\n$\\approx$ 7 mRy.~\\cite{Stapper99,Jomard99} This energy barrier for the\n6-atom unit cell corresponds to a temperature of $\\approx$ 1100 K. The\nsame result was obtained by Jansen \\cite{Jansen88} with the FLAPW method\nwho proposed a value of $\\approx$ 1200 K. It is natural that these\ntemperatures, extrapolated from the 0 K potential energy, underestimate\nthe experimental phase transition temperature of 2570 K.~\\cite{Aldebert85} The\nexperimentally observed phase transition temperature can be considered as the sum of\nthe kinetic contributions of all the activated eigenmodes, while the\ncalculated energy barrier refers to the kinetic contribution of the\n$X_2^-$ eigenmode only. Even though it is reasonable to expect that at\nthe phase transition the soft mode in the phonon spectra (Figure \\ref{phon}) will be\nhighly weighted in the total density of states, the kinetic energy $kT$\nassociated with all the other modes of vibrations will still contribute\nto the measured phase transition temperature.\n\n\\subsubsection{Physical interpretation of the double well}\n\n What causes the $c \\leftrightarrow t$ symmetry breaking? The tetragonal\n distortion of the oxygen sublattice implies the following geometrical\n changes: (i) Two Zr-O bond lengths get smaller and two get longer but\n the average Zr-O distance increases. (ii) entire columns of oxygen atoms\n shift one with respect to each other (see Fig.~\\ref{cellct}) therefore\n the nearest neighbor O-O distances along the column remain constant\n while the other 4 nearest neighbor O-O distances increase. (iii) All\n the Zr-Zr distances remain constant. The overall increase of both the\n Zr-O and the O-O bond lengths is the basis of our interpretation of the\n double well, founded mainly on electrostatic arguments.\n\n By adjusting the various parameters describing ionicity, covalency and\n polarizability of the TB model we can select and isolate the effects\n that induce the double well, but before doing so it is instructive to understand\n how a simple RIM answers to the same question. It has been shown\n \\cite{Wilson96zirc} that it is possible to reproduce the double well with a RIM\n in which there are two contributions: a repulsive short ranged pairwise\n interaction $U^{\\rm pair}$ and a long ranged electrostatic term\n $U^{ii}$.\n\n\\begin{equation} U^{\\rm RIM} = \\sum_{i<j} A \\, e^{ - \\, b \\, r_{ij} }\n\\, + \\, \\sum_{i<j}\n\\frac{z_i \\, z_j}{r_{ij}} = U^{\\rm pair} + U^{ii}, \\label{RIM}\n\\end{equation}\n$z$ is the ionic charge and $r_{ij}$ is the interatomic distance\nbetween the ions $i$ and $j$. \n\n The Zr-O bonds increase and decrease in length in a symmetric way. As\n a net result, the centrosymmetric position of the oxygen atoms is a\n relative maximum of the Coulomb energy $U^{ii}$. The change in the\n Madelung potential caused by the tetragonal distortion is shown in\n Figure \\ref{split1} (a). The overall increase of the O-Zr and O-O\n distances makes the oxygen sites much more sensitive to the change of\n the Madelung potential then the zirconium ones. The structural\n instability can therefore be interpreted as an effective way of\n minimizing the electrostatic energy of the oxygen sublattice. The\n repulsive Zr-O interaction counteracts the structural instability\n driven by the electrostatics, in a way which dominates at large\n displacements because of the exponential distance dependence of this\n repulsion. The double well shape of the energy profile is due to the\n different functional form of these opposing energy terms of\n Eq.(\\ref{RIM}). This argument clearly depends on the strength of the\n repulsion, and does not work if the repulsion is too weak.\n\n It can be noticed that analogous terms are present in the TB model and a\n similar interpretation is tempting. However, we now have the additional\n effects due to polarization, covalency, and charge\n redistribution. Figure \\ref{split1} (b) shows that the absolute value of\n the self-consistent equilibrium charge $Q$ decreases on both\n species. Consequently, in this approximation, the on-site energy\n\\begin{equation}\n\\frac{1}{2} \\sum_{{\\bf R}L} \\left[ U \\, \\delta N^2_{{\\bf R}\\ell} +\n Q_{{\\bf R}L} \\, V_{{\\bf R}L} \\right] \\, , \\label{elcst-en}\n\\end{equation}\nplotted in Figure \\ref{split1} (c), decreases not only because of the\n previous geometric arguments but also because the charge redistribution\n reduces the ionic charges and therefore both the O-O and Zr-Zr\n electrostatic interactions.\n\n It is interesting to note that, on the oxygen atoms, the self-consistent\n charge $|Q^e|$ decreases with $\\delta$ even though the total on-site\n potential [the sum of the Hubbard and electrostatic terms as in\n Eq.(\\ref{elcst-en})] increases. This non-intuitive behavior of the\n charge transfer is due to covalency. The charge transfer is controlled\n both by the on-site potential and by the bonding integrals, which depend\n on the Zr-O distance. For $\\delta \\ne 0$, the overall increase in the\n Zr-O distance results in a decrease in the magnitude of the hopping\n integrals, and this overcomes the opposing effect of change in the\n on-site potential, {\\em pushing back} some electrons from the oxygen to\n the zirconium sites.\n\n In the CIM-DQ, it was the quadrupole polarization of the O ions which\n stabilized the tetragonal structure, so it is of interest to see if it\n is also the development of a quadrupole moment in the tetragonal phase\n which stabilizes it within the SC-TB model.\n\n In fact it turns out that covalency is the main effect, although\n polarizability is still significant. The $t$ structure is stable with\n respect to the $c$ one even with a {\\it non} polarizable SC-TB model\n [Figure~\\ref{split2} (a)]: the small energy difference is due to both\n ionicity and covalency of the crystal. The addition of the oxygen\n polarizability enhances the energy difference between the two phases\n deepening and broadening the double well [Figure~\\ref{split2} (b)].\n\n We can be more specific about the nature of the polarization. In the $c$\n structure, the first non-zero components of the electrostatic potential\n are $V_0$ and $V_3$. The latter could, in principle, induce an octapole\n moment $Q_3$ on the anions. We truncated the multipolar expansion of the\n atomic multipole moments to the quadrupoles $Q_2$ therefore, within this\n approximation, the ions in the $c$ structure are not polarized. Higher\n order terms can be included in the expansion, but the overall agreement\n of the results with both experiments and first-principle calculations\n demonstrates that the model is already capturing the important physics\n of the system.\n\n As the anion sublattice is distorted, the symmetry lowering induces the\n $\\ell=1$ and $\\ell=2$ components of the potential which couple the $s$\n and $p$ oxygen atomic orbitals. The magnitude of the coupling, and\n therefore of the multipole moments, is controlled by the parameters\n $\\Delta_{spp}$ and $\\Delta_{ppd}$. The latter, fixed in order to\n reproduce the electronic structure of the rutile phase, produces very\n weak quadrupole moments, whose contribution to the double well is\n negligible. The former controls the size of the dipole moments whose\n symmetric distribution further minimize the electrostatic energy [Figure\n \\ref{split1} (d)]. The total effect on the double well is shown in\n Figure~\\ref{split2}.\n\n\\subsubsection{Landau theory} \n\nThe $c \\leftrightarrow t$ phase transition can be interpreted in terms of\nthe Landau Theory.~\\cite{Landau5} In a subsequent paper we plan to\nexplore the free energy surface at $T>0$ with this formalism, so it is\nconvenient to introduce it here to discuss the $T=0$\nresults. Experimentally, the mechanism of this phase transition has been\nvery controversial and a clear description is still\nmissing.~\\cite{Cohen63,Sakuma85,Heuer84,Heuer87,Stubican78,Sakuma87,Zhou91,Katamura97}\n\nTheoretically, Chan \\cite{Chan88} suggested that a partial softening of\nan elastic constant is the driving force of this phase transition and,\nafter symmetry considerations based on the elastic strains only,\nconcluded that the phase transition must be of first order. We show here\nthat the inclusion of the order parameter $\\delta$ gives a second order\nphase transition. A similar discussion has been given by Ishibashi and\nDvo\\'{r}\\u{a}k.~\\cite{Ishibashi89}\n\nAccording to the Landau Theory, the appropriate thermodynamic potential\nwhich describes the relationship between the two phases of interest, is\nexpanded in a Taylor series in one or more order parameters, in which the\nexpansion coefficients are temperature dependent. The order parameters\nare non-zero in the low symmetry phase and vanish in the high symmetry\none, providing therefore a unique way to differentiate the two phases.\nThe terms involved in the Taylor expansion are invariants under the\nsymmetry operations of the high symmetry phase and can be identified\nusing group theory.\n\nIn the case of zirconia, the $c$ structure is unstable along the three\ncrystallographic directions, therefore the distortions along $x,y$ and\n$z$ have to be explicitly treated in the energy expansion. This suggests\nthe following 9 order parameters, defined in terms of the strain tensor\n{\\boldmath $ \\epsilon $}, and grouped into 4 symmetry-adapted bases which\nspans the corresponding irreducible representations:\n\n\\begin{eqnarray*}\n\\delta_x , \\delta_y , \\delta_z & \\hspace{1cm} & {\\rm T_1} \\\\\n\\epsilon_{xx} + \\epsilon_{yy} + \\epsilon_{zz} & &{\\rm A_1} \\\\\n\\left(2\\epsilon_{zz} - \\epsilon_{xx} - \\epsilon_{yy}\n\\right), \\, \\sqrt{3} \\left(\\epsilon_{xx} - \\epsilon_{yy} \\right) \n& & {\\rm E} \\\\ \n\\epsilon_{xy}, \\epsilon_{yz}, \\epsilon_{zx} & & {\\rm T_2}\n\\end{eqnarray*}\n\nA complete analysis involving all the order parameters will be done in a\nseparate paper, here we simplify the total energy expansion selecting one\nof the three possible directions of the tetragonal axis. Under this\nhypothesis three order parameters are necessary to describe the $c\n\\leftrightarrow t$ phase transition of zirconia: $\\delta$, $\\eta$ and\n$\\eta_0$. The high temperature $c$ phase has the full cubic symmetry\n$m3m$ and the only degree of freedom is the hydrostatic strain $\\eta_0=\n\\epsilon_{xx} + \\epsilon_{yy} + \\epsilon_{zz}$. The low-symmetry $t$\nphase is defined by the distortion of the anionic sublattice $\\delta$,\nwhich we define as the amplitude of the $X_2^-$ mode of vibration, and by\nthe tetragonal strain $\\eta=\\left(2\\epsilon_{zz} - \\epsilon_{xx} -\n\\epsilon_{yy} \\right)$.\n\nThe three order parameters can be hierarchically classified according to\nthe amount of symmetry breaking that they involve. The hydrostatic strain\n$\\eta_0$ preserves the cubic symmetry of the crystal. The\ntetragonal strain $\\eta$ maintains the number of atoms in the primitive\ncell and lowers the symmetry to the point group $4/mmm$ which still has\nthe mirror symmetry operation perpendicular to the tetragonal axis. The\ntetragonal distortion $\\delta$ breaks this symmetry operation and\ninvolves cell doubling. Therefore, according to Landau theory, $\\delta$\nis the primary order parameter, $\\eta$ is the secondary and $\\eta_0$ is\nthe tertiary one.\n\nThe potential energy is expanded as a power series in these order\nparameters around the equilibrium volume of the cubic phase $V_0$\n(Figure \\ref{envol-ct}):\n\n\\begin{eqnarray} \\label{lanexp}\nU - U^{c}_{V_0} & = & \\frac{a_2}{2} \\: \\delta^2 +\n\\frac{a_4}{4} \\: \\delta^4 + b_0 \\: \\delta^2 \\eta_0 + b_1 \\: \\delta^2\n\\eta + \\\\ \\nonumber \n& & \\frac{c_0}{2} \\: \\eta_0^2 + \\frac{c_1}{2} \\: \\eta^2 + {\\cal O}\n(\\delta^6). \n\\end{eqnarray}\nThe elastic constants $c_0$ and $c_1$ are proportional respectively to\nthe bulk modulus and to $C^{\\prime}=\\frac{1}{2}(c_{11}-c_{12})$ in the\n$c$ phase described in the next section. The third order term\n$\\delta^3$ is forbidden by symmetry, therefore this transition is of\nsecond order if $a_2$ goes negative.\n\nThe volume dependence of the order parameters can be studied by setting\nto zero $\\nabla_\\eta U$ and $\\nabla_{\\eta_0} U$. Both the {\\it ab initio}\nand TB results (Figure~\\ref{landvol}) confirm the analytic expressions:\n\n\\begin{eqnarray}\n\\left\\{ \\begin{array}{lcl}\n\\eta & = & - \\frac{b_1}{c_1}\\, \\delta^2 \\\\\n& & \\\\\n\\eta_0 & = & - \\frac{b_0}{c_0} \\, \\delta^2 \n\\end{array} \\right. &\n\\hspace{0.5cm} \\Rightarrow \\hspace{0.5cm} & \n\\left\\{ \\begin{array}{lcl}\n\\delta & \\propto & \\sqrt{\\eta_0} \\\\\n& & \\\\\n\\eta & \\propto & \\eta_0\n\\end{array} \n\\right. \\label{land-orvol}\n\\end{eqnarray}\n\nThese expressions show that the second-order strain terms of\nEq.(\\ref{lanexp}) are already proportional to $\\delta^4$ and therefore,\nwithin the chosen order of approximation, it is not necessary to\ninclude third-order terms in $\\epsilon_{ij}$. Moreover, from the static\nresults it is clear that the description of the high temperature\nstability of the $c$ phase must go beyond the quasi-harmonic\napproximation. The higher the temperature, the larger the volume and,\naccording to Figure~\\ref{landvol}, the larger $\\delta$ and $\\eta$.\nTherefore, in a simple quasi-harmonic picture, a higher temperature\nseems to favor the $t$ phase with respect to $c$, in contradiction to\nthe experimental observation.\n\nThe parameters $c_1$ and $c_0$ are known from the elastic properties of\nthe crystal and have been calculated independently (see next section).\nThe coefficients $a_2$ and $a_4$ have been fitted to the double well of an\nundistorted stress-free cubic crystal (in the sense $\\eta=0$ and $\\eta_0=0$). In a\nsimilar way, $b_1$ and $b_0$ have been fitted to the double well of a tetragonal\ncrystal at $V_0$ ($\\eta_0=0$, $\\eta \\ne 0$) and of a cubic crystal near\n$V_0$ ($\\eta=0$, $\\eta_0 \\ne 0$) respectively. Figure~\\ref{lansurf} (a)\nshows the three curves used for the fitting procedure. The agreement is\nvery good even far away from the reference volume of the energy\nexpansion [Figure~\\ref{lansurf} (b)]. This demonstrate that the fourth\norder truncation in Eq.(\\ref{lanexp}) is sufficient to capture all the\nessential features of the 0 K energy surface. \n\nNardelli {\\it et al.} \\cite{Nardelli92,Nardelli95} have shown the crucial\nr$\\hat{\\rm o}$le played by the coupling between different order\nparameters and how it can affect the correct interpretation of the phase\ntransformation. To see this we substitute the relationships\n(\\ref{land-orvol}) back in Eq.(\\ref{lanexp}):\n\n\\begin{equation}\nU - U^{c}_{V_0} = \\frac{a_2}{2} \\delta^2 + \\left[ \\frac{a_4}{4} -\n\\frac{b_0^2}{2 \\, c_0} - \\frac{b_1^2}{2 \\, c_1} \\right] \\delta^4 + {\\cal O}\\left(\n\\delta^6 \\right).\n\\end{equation}\nThe above equation shows that the coupling term $\\left( \\frac{b_0^2}{2 \\,\nc_0} + \\frac{b_1^2}{2 \\, c_1} \\right)$ can renormalize the fourth order\ncoefficient, and could make it negative. In that case it would be\nnecessary to truncate Eq.(\\ref{lanexp}) at the sixth order term in\n$\\delta$, including therefore the third-order terms in the strain. These\nwould then drive the phase transition making it first order.~\\cite{Chan88,Anderson65}\nThe numerical values of the coefficients (Table~\\ref{landcfc}) allow us\nto estimate the amount of the coupling. We find that the coupling term is\n$\\approx 20\\%$ of $\\frac{a_4}{4}$, not big enough to affect the sign of\nthe fourth order coefficient and therefore the 0 K calculations suggest\nthat the phase transition is displacive of second order.\n\nThe temperature dependence of the elastic constants might change this\ndescription and the final answer will be given by high temperature MD\ncalculations which are in progress. \n\n\\section{Distortions}\n\\label{distortions}\n\n\\subsection{Elastic constants}\n\nThe elasticity of $c$ and $t$ zirconia has been explored with the TB\nmodel. The analysis involved the distortion of the crystal along high\nsymmetry directions, the calculation of the total energy for different\nvalues of the distortion parameter and the fit of the results to a\npolynomial. The rigidity of the crystal with respect the particular\ndistortion applied has been extracted from the quadratic coefficient of\nthe energy series expansion. For each strain of the $t$ structure, we\nconstrained the volume to the predicted equilibrium value and minimized\nthe energy with respect to the internal degrees of freedom.\n\nVolume conserving stretches along the high symmetry directions of the $c$\nunit cell $<\\!\\!100\\!\\!>$ and $<\\!\\!111\\!\\!>$ provide\n$C^\\prime=\\frac{1}{2}(c_{11}-c_{12})$ and $c_{44}$ respectively. Extra\ndistortions are necessary when the symmetry is lower: if $z$ is the\ntetragonal axis, an independent set of 5 shear moduli were obtained by\nstretching along $<\\!\\!100\\!\\!>$, $<\\!\\!001\\!\\!>$, $<\\!\\!111\\!\\!>$,\n$<\\!\\!110\\!\\!>$ and $<\\!\\!101\\!\\! >$. The bulk moduli have been obtained\nby fitting the Energy-Volume curves with a Birch-Murnaghan Equation of\nState.~\\cite{Murnaghan44,Birch78}\n\nLiu {\\it et al.}~\\cite{Liu87}used the slope of the acoustic branches\nat small wavelength of a ZrO$_2$-Y$_2$O$_3$ (15 $\\%$) system to\nestimate the elastic constants of the cubic phase. Kandil {\\it et\nal.}~\\cite{Kandil82} directly measured the elastic constants of Yttria\nStabilized Zirconia (YSZ) single crystals: the reference values included\nin Table~\\ref{elconst} are extrapolations to 0$\\%$ impurities. To our\nknowledge there is no equivalent experimental study of the elasticity\nof the $t$ phase. The most recent values \\cite{Kisi98b} are measured via\na powder diffraction technique on 12\\% Ce-doped $t$ zirconia. \n\n%Other\n%available data \\cite{TROVA} are extrapolation of the monoclinic elastic\n%constants at the $t \\leftrightarrow m$ phase transition composition. Clearly these\n%numbers have to be considered as a guide only.\n\nWe compare our predictions with theoretical and experimental data in\nTable~\\ref{elconst}. The results of two other theoretical approaches, the\nHartree-Fock and the PIB ones, are very different. As already\nmentioned in the Introduction, none of these calculations predicted the\ncorrect relative energetics of the crystal structures. Elasticity is a\nproperty of the energy second derivative: a good description of the\nenergy curves is a prerequisite for reliable elastic constant\ncalculations.\n\nThe fairly good agreement of our calculations with the experiments\nfurther indicates that the SC-TB model captures the main physics of the\nbonding. The bulk modulus, however, is seriously overestimated: this\nmay not be an intrinsic limitation of the TB model, because it was fit\nprecisely to the NFP-LMTO calculation, which similarly overestimates\nthis quantity.\n\n\\subsection{Phonon Spectra}\n\nIn order to test the model further, as well as to give further insight\ninto the spontaneous symmetry breaking of the $c$ phase, we studied its\nvibrational properties. First principle calculations\n\\cite{Detraux98,Parlinski97} predict an imaginary frequency at the\nboundary of the BZ: this reinforces the idea that the phase transition is\ndisplacive, and driven by the softening of an optic mode.\n\nOur calculations were carried out with the TB model on a 96-atom\nunit cell. The eigenvalues and eigenvectors of\nthe possible vibrational modes in that unit cell, were found by\ndiagonalising the dynamical matrix which we calculated using the direct\nmethod. The procedure was as follows.\n\nWithin the harmonic approximation, the potential energy $\\Phi$ is\nexpanded to second order in powers of the atomic displacements ${\\bf\nu}$:\n\n\\begin{equation}\n\\Phi= \\Phi_0 + \\frac{1}{2}\n\\sum_{ \\scriptsize \\begin{array}{c} l, \\kappa, \\alpha \\\\\n\t\t l^{\\prime} \\! \\! , \\kappa^{\\prime} \\! \\! , \\beta\n \\end{array}} \\Phi_{\\alpha\\beta} \\left( \\! \\! \\begin{array}{c} {l l^\\prime} \\\\ \\kappa \\kappa^\\prime\n\\end{array} \\! \\! \\! \\right) u_\\alpha\n\\left( \\! \\! \\begin{array}{c} {l} \\\\ \\kappa \\end{array} \\! \\!\n\\right) u_\\beta \\left( \\! \\! \n\\begin{array}{c} {l^\\prime} \\\\ \\kappa^\\prime \\end{array} \\! \\! \\! \\right) + \\ldots \n\\end{equation}\nWe use the notation of Maradudin {\\it et al.}~\\cite{Maradudin63}:\n$\\kappa$ and $l$ label respectively the atom in the primitive cell\nand the position of the primitive cell with respect to some origin. The\ndirect method consists in computing the force constants\n$\\Phi_{\\alpha\\beta}$ via total energy and force calculations. In general,\nthe atom $\\kappa$ in the $l$ cell is displaced by a small amount\nin direction $\\alpha$ and the Hellmann-Feynman forces on the other atoms\nare recorded. These give directly the quadratic terms in the total\nenergy expansion. The force constants $\\Phi_{\\alpha\\beta}$ \ncan be related to the\ncorresponding term of the dynamical matrix {\\bf D} via the usual\nrelation:\n\n\\begin{equation} \nD_{\\alpha\\beta} \\left( \\! \\! \\begin{array}{c} {\\bf k} \\\\ \\kappa \\kappa^\\prime\n\\end{array} \\! \\! \\! \\right) = \\frac{1}{\\sqrt{M_\\kappa\nM_{\\kappa^\\prime}}} \\sum_l \\Phi_{\\alpha\\beta} \\left( \\! \\! \\begin{array}{c} {l} \\\\ \\kappa \\kappa^\\prime\n\\end{array} \\! \\! \\! \\right) e^{-2 \\pi {\\bf k \\cdot x{\\it (l)}}} .\n\\end{equation}\n$M_\\kappa$ is the mass of the atom $\\kappa$ and $ \\bf k$ is a point in\nthe BZ. The crystal symmetry can considerably reduce the number of\nnecessary independent calculations.~\\cite{Maradudin68,Warren68}\n\nThe phonon spectra plotted along the high symmetry direction\n$<\\!\\!100\\!\\!>$ are shown in Figure~\\ref{phon}. The main feature of the\nspectra is the imaginary frequency of the $X_2^-$ mode of vibration which\ncorresponds to the tetragonal instability shown in Figure~\\ref{well}. As\nalready mentioned the tetragonal instability involves cell doubling\ntherefore the corresponding eigenvector appears at the BZ border of the\n$c$ phase. The soft mode at the $X$ point $\\nu_s=5.1i$ is the natural\nconsequence of the negative curvature of the energy surface at $\\delta=0$\n(Figure~\\ref{well}). Setting to zero the dipolar polarizability of the\nanions ($\\Delta_{spp}=0$), the $X_2^-$ mode is still soft,\n$\\nu_s=0.8i$, but the force constant corresponding to the instability\nis much smaller. This is consistent with Figure~\\ref{split2} where the\nsame effect is studied from the energetic point of view: the energy curve\nis concave at $\\delta=0$ even when the oxygens are not polarizable.\n\nThe effect of the oxygen polarizability is evident on the $T_{1u}$\nIR-active mode, which involves the rigid displacement of the two atomic\nsublattices. The calculated vibration frequency is 7.9 THz when the\nanions are not polarizable and 6.3 THz when the dipolar degree of freedom\nis allowed. The closer agreement of the non-polarizable result with the\nDFT frequencies of $8.1 - 8.5$ THz, together with the overestimation of\nthe bulk modulus suggests that the present model could slightly\noverestimate both the short-range repulsion between closed shells of\nelectrons, responsible for the high bulk modulus, and the long range\npolarization effects which make the $T_{1u}$ frequency lower than the\n{\\it ab initio} values. The results might be improved with a more\naccurate re-parameterization but the physical interpretation of the {\\it\nab-initio} results, which is the main objective of this analysis, is\nunlikely to change.\n\nTable~\\ref{phontab} shows the general agreement of the TB model with\nother calculations and with the experimental data. The latter are\nmeasured by Raman spectroscopy and inelastic neutron scattering at high\ntemperatures on YSZ.\n\nCertain non-analytical terms in the dynamical matrix have been neglected,\nnamely those relating to macroscopic polarization or the Berry phase.\nFor this reason our calculations cannot reproduce the LO-TO splitting of\n12 THz calculated by Detraux {\\it et al.}.~\\cite{Detraux98} The\nnon-analytical terms can be approximated by knowing the Born effective\ncharge and the dielectric tensor, both of which could in principle be\nobtained from our model. This has previously been done in a TB framework\n,~\\cite{Bennetto96} although not for ZrO$_2$, and we plan to investigate\nthe effect in the future.\n\n\\section{Conclusions}\n\nWe have explored the predictive power of a polarizable SC-TB model by\ninvestigating the crystal stability of pure zirconia. The results of this\nextended TB model are in overall good agreement with our own {\\it ab\ninitio} (NFP-LMTO) calculations and with previous experimental and\ntheoretical {\\it ab initio} studies. This semiempirical model has\ncaptured the basic physics of the relative phase stability of zirconia\nwith a set of parameters which are transferable between the crystal\nstructures. A noteworthy improvement over all previous models is the\nabsolute stability of the monoclinic structure at 0 K with respect to the\nusual set of alternatives. This demonstrates that the model is ready to\ndeal with more complicated crystalline environments such as solid\nsolutions, high temperature distortions, or interfaces.\n\nThe TB model predicts that the covalent character of the Zr-O bond\n plays a major r$\\hat{\\rm o}$le in the energetics of zirconia, more so\nthan the polarizability of the oxygen ions. For example, the double\nwell about the cubic structure, absent in a rigid ion model, exists\nwhen covalency is included; it is further enhanced by including also\npolarizability at the dipole level. We do not believe that the\nseparation between covalent effects and polarizability effects is\nunique, since it depends on the choice of basis functions. Quite\npossibly the previous polarizable ion models were capturing some\neffects of charge redistribution which could alternatively be described\nby covalency. It remains to be seen if a model for zirconia without\nexplicit covalency could satisfactorily reproduce all the structural\nenergies.\n\n The Landau Theory, used to interpret the TB and {\\it ab initio} results,\ntogether with the lattice dynamic analysis, shows that the $c\n\\leftrightarrow t$ phase transition is displacive of the second order and\nis driven by the softening of the $X_2^-$ mode of vibration. If it had\nbeen driven by a softening of the corresponding elastic constant\n$c_{11}-c_{12}$ it would have been a first order transition. The partial\nsoftening of the elastic constants due to the temperature could also in\nprinciple change the character of the phase transition. We are currently\napplying the molecular dynamics technique to understand the high\ntemperature thermodynamic stability of the $c$ phase and to explore the\ncharacter of the phase transition. To this end we can use thermodynamic\nintegration to go beyond the harmonic approximation. The preliminary\nresults of these calculations will appear in the near\nfuture.~\\cite{MDzirc} \n\nSince the valence electrons are treated explicitly within the\nSC-TB model we also hope to be able to study the effects of point\ndefects. This would be more difficult with a classical polarizable ion model\nbecause of the problems associated with charge conservation and\nredistribution.\n\n\\acknowledgements\n\nSF is grateful for support from the European Science Foundation, Forbairt\nand the British Council, and for discussions with John Corish and Nigel\nMarks. ATP and MWF are grateful to the EPSRC for funding under Grants\nNo. L66908 and No. L08380. 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Energy in Ry and\nlengths in atomic units.}\n\\label{param}\n\\begin{tabular}{cc}\n\\multicolumn{2}{c}{\\it On site parameters} \\\\\n ${\\cal H}^0_{s}$ = 0.35 & $U_s$ = 1 \\\\\n ${\\cal H}^0_{p}$ =-0.70 & $U_p$ = 1 \\\\\n ${\\cal H}^0_{d}$ =-0.10 & $U_d$ = 1 \\\\ \n\\hline\n\\multicolumn{2}{c}{\\it Bond integrals } \\\\\n\\multicolumn{2}{c}{$ V_{ll^\\prime}\n \\left( \\frac{d}{r} \\right)^n \n \\exp \\left\\{ n \\left[ - \\left( \\frac{r}{r_c} \\right)^{n_c} +\n \\left( \\frac{d}{r_c} \\right)^{n_c}\n \\right] \\right\\} $ } \\\\\n\\begin{tabular}{cccccc}\n & $V_{ll^\\prime}$ & $n$ & $n_c$ & $d$ & $r_c$ \\\\ \\hline\n$ss\\sigma$ & -0.060 & 2 & 0 & 4.90 & 6.24 \\\\ \n$sp\\sigma$ & ~0.070 & 2 & 0 & 4.90 & 6.24 \\\\\n$pp\\sigma$ & ~0.050 & 3 & 4 & 4.90 & 6.24 \\\\\n$pp\\pi$ & -0.008 & 3 & 4 & 4.90 & 6.24 \\\\\n$sd\\sigma$ & -0.050 & 3 & 0 & 4.24 & 4.90 \n\\end{tabular} &\n\\begin{tabular}{cccccc}\n & $V_{ll^\\prime}$ & $n$ & $n_c$ & $d$ & $r_c$ \\\\ \\hline\n$pd\\sigma$ & -0.100 & 4 & 0 & 4.24 & 4.90 \\\\\n$pd\\pi$ & ~0.058 & 4 & 0 & 4.24 & 4.90 \\\\\n$dd\\sigma$ & -0.050 & 5 & 0 & 6.02 & 6.93 \\\\\n$dd\\pi$ & ~0.033 & 5 & 0 & 6.02 & 6.93 \\\\\n$dd\\delta$ & ~0.008 & 5 & 0 & 6.02 & 6.93 \n\\end{tabular} \\\\\n\\hline\n\\multicolumn{2}{c}{\\it Polarization terms} \\\\\n$\\Delta_{spp}$ = 0.73 & $\\Delta_{ddd}$ = 0~~~~ \\\\\n$\\Delta_{ppd}$ = 1.89 & $\\Delta_{ddg}$ = 63.5 \\\\\n\\hline\n\\multicolumn{2}{c}{\\it Pair potential} \\\\\n\\multicolumn{2}{c}{$ U(r)= A \\, e^{\\left(-b \\, r \\right)} $} \\\\\n$A$ = 181.972 & $b$ = 1.652 \\\\\n\\end{tabular}\n\\end{table}\n\n%%%%%%% !! POSSIBLY ON TWO COLUMNS !!! %%%%%%%%%%%%%%%%%%%\n\\begin{table*}\n\\caption{Equilibrium structural parameters for the 0-pressure phases of\nZrO$_2$. The lattice parameters $a$, $b$, $c$ (a.u.), and the volumes\n(a.u./ZrO$_2$) of the $c$, $t$, and $m$ structure are referred to the\n12-atoms, 6-atoms, and 12-atoms unit cells respectively. $\\delta$ denotes\nthe internal degree of freedom of the $t$ phase (see Fig.~\\ref{cellct}),\n$\\beta$ is the angle of the $m$ cell in degrees, and $x$, $y$, $z$ are\nthe fractional coordinates of the non-equivalent sites in the $m$\nstructure.}\n\\label{strucpar}\n\\begin{tabular}{c|cccccc}\n & Expt.\\tablenote[1]{The experimental values of the cubic and\n tetragonal structures have been extrapolated at 0 K using the\n thermal expansion data from Ref.~\\onlinecite{Aldebert85}} \n & SC-TB & NFP-LMTO & PW-PP & PW-PP & FLAPW \\\\ \n & Refs.~\\onlinecite{Aldebert85,Howard88} & this work & this work & Ref.~\\onlinecite{Kralik98} &\nRef.~\\onlinecite{Stapper99} & Ref.~\\onlinecite{Jansen91} \\\\ \\hline\n & \\multicolumn{6}{c}{$Cubic$} \\\\\n{Volume } & 222.50 & 213.40 & 210.33 & 215.29 & 220.84 & 217.81 \\\\ \n$a$ & 9.619 & 9.486 & 9.442 & 9.514 & 9.595 & 9.551 \\\\ \\hline\n & \\multicolumn{6}{c}{$Tetragonal$} \\\\\n{Volume } & 229.93 & 217.73 & 215.16 & 218.69 & 225.31 & 218.84 \\\\ \n$a$ & 6.748 & 6.709 & 6.695 & 6.734 & 6.797 & 6.747 \\\\ \n$c/a$ & 1.451 & 1.442 & 1.434 & 1.432 & 1.435 & 1.425 \\\\ \n$\\delta/c$ & 0.057\\tablenote[2]{At 1568 K} \n & 0.047 & 0.051 & 0.042 & 0.042 & 0.029 \\\\ \\hline\n & \\multicolumn{6}{c}{$Monoclinic$} \\\\\n{Volume } & 237.67 & 222.89 & 226.13 & 230.51 & 236.46 & \\\\ \n$a$ & 9.733 & 9.592 & 9.417 & 9.611 & 9.733\\tablenote[3]{Fixed\n to the experimental values of Ref.~\\onlinecite{Howard88}} & \\\\ \n$b/a$ & 1.012 & 1.001 & 1.036 & 1.024 & 1.012$^{\\rm c}$ & \\\\ \n$c/a$ & 1.032 & 1.019 & 1.057 & 1.028 & 1.032$^{\\rm c}$ & \\\\ \n$\\beta$ & 99.23 & 98.00 & 98.57 & 99.21 & 99.23$^{\\rm c}$ & \\\\ \n & & & & & & \\\\ \n$x_{\\rm Zr}$ & 0.275 & 0.272 & 0.274 & 0.278 & 0.277 & \\\\ \n$y_{\\rm Zr}$ & 0.040 & 0.027 & 0.040 & 0.042 & 0.043 & \\\\ \n$z_{\\rm Zr}$ & 0.208 & 0.217 & 0.212 & 0.210 & 0.210 & \\\\ \n$x_{\\rm O_1}$ & 0.070 & 0.078 & 0.069 & 0.077 & 0.064 & \\\\ \n$y_{\\rm O_1}$ & 0.332 & 0.336 & 0.339 & 0.349 & 0.324 & \\\\ \n$z_{\\rm O_1}$ & 0.345 & 0.342 & 0.338 & 0.331 & 0.352 & \\\\ \n$x_{\\rm O_2}$ & 0.450 & 0.452 & 0.448 & 0.447 & 0.450 & \\\\ \n$y_{\\rm O_2}$ & 0.757 & 0.752 & 0.753 & 0.759 & 0.756 & \\\\ \n$z_{\\rm O_2}$ & 0.479 & 0.472 & 0.478 & 0.483 & 0.479 & \\\\ \n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table}[t]\n\\caption{Energy differences (mRy/ZrO$_2$) between the zirconia polymorphs\nand the $c$ phase calculated at the minimized structural parameters of\nTables~\\ref{strucpar} and~\\ref{parort}. The experimental values are\nderived from enthalpy differences at the phase transition temperature.}\n\\label{energy}\n\\begin{tabular}{ll|cc|cc}\n & & $\\Delta U^{t-c}$ & $\\Delta U^{m-c}$ & $\\Delta U^{O_I-c}$ &\n$\\Delta U^{O_{II}-c}$ \\\\ \\hline\nExpt. & Ref.~\\onlinecite{Ackermann75}& -4.2 & -8.8 & & \\\\\nSC-TB & & -3.0 & -7.4 & -3.6 & 2.8 \\\\\nNFP & & -3.6 & -7.7 & - & \\\\\nPW-PP & Ref.~\\onlinecite{Kralik98} & -3.3 & -7.5 & - & \\\\\nPW-PP & Ref.~\\onlinecite{Stapper99} & -3.5 & -8.2 & -5.3 & - \\\\\nPW-PP & Ref.~\\onlinecite{Jomard99} & -1.5\\tablenote[1]{LDA calculation}\n -5.9\\tablenote[2]{Perdew-Wang GGC calculation} \n & -4.4$^{\\rm a}$ \n -13.9$^{\\rm b}$\n & - & 0.7$^{\\rm a}$\n 7.3$^{\\rm b}$\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{External and internal degrees of freedom of the orthorhombic\nstructures. Lattice parameters $a$, $b$, $c$ in a.u., volumes in\na.u./ZrO$_2$. The fractional coordinates of the non-equivalent sites are\ndenoted with $x$, $y$, and $z$.}\n\\label{parort}\n\\begin{tabular}{c|ccc|ccc}\n & \\multicolumn{3}{c}{$Ortho \\, I$} & \\multicolumn{3}{c}{$Ortho \\, II$} \\\\\n & Expt. & SC-TB & PW-PP & Expt. & SC-TB & PW-PP \\\\ \n & Ref.~\\onlinecite{Ohtaka90proc} \n & this work & Ref.~\\onlinecite{Stapper99} \n & Ref.~\\onlinecite{Haines97} \n & this work \n & Ref.~\\onlinecite{Jomard99} \\\\ \\hline\nVol. & 228.159 & 218.69 & 226.7 & 203.54 & 196.08 & 212.44 \\\\\n$a$ & 19.060 & 18.737 & 19.060\\tablenote[1]{Fixed to the experimental\n values of Ref.\\onlinecite{Ohtaka90proc}} & 10.558 & 10.541 & 10.721 \\\\\n$b/a$ & 0.522 & 0.520 & 0.522$^{\\rm a}$ & 0.596 & 0.592 & 0.593 \\\\\n$c/a$ & 0.505 & 0.511 & 0.505$^{\\rm a}$ & 1.161 & 1.139 & 1.163 \\\\ \\hline\n$x_{\\rm Zr}$ & 0.884 & 0.880 & 0.884 & 0.246 & 0.255 & 0.253 \\\\\n$y_{\\rm Zr}$ & 0.033 & 0.002 & 0.036 & 0.250 & 0.250 & 0.250 \\\\\n$z_{\\rm Zr}$ & 0.256 & 0.256 & 0.253 & 0.110 & 0.099 & 0.111 \\\\\n$x_{\\rm O_1}$ & 0.978 & 0.978 & 0.978 & 0.360 & 0.354 & 0.360 \\\\\n$y_{\\rm O_1}$ & 0.748 & 0.745 & 0.739 & 0.250 & 0.251 & 0.250 \\\\\n$z_{\\rm O_1}$ & 0.495 & 0.509 & 0.499 & 0.424 & 0.421 & 0.425 \\\\\n$x_{\\rm O_2}$ & 0.791 & 0.784 & 0.790 & 0.025 & 0.022 & 0.023 \\\\\n$y_{\\rm O_2}$ & 0.371 & 0.371 & 0.374 & 0.750 & 0.749 & 0.750 \\\\\n$z_{\\rm O_2}$ & 0.131 & 0.130 & 0.127 & 0.339 & 0.338 & 0.340 \n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Coefficients (a.u.) of the energy Taylor expansion (\\ref{lanexp}).}\n\\label{landcfc}\n\\begin{tabular}{clllc}\n\\hspace{0.5cm} & $a_2$ = -0.053 & $b_0$ = -0.062 & $c_0$ = 0.621 &\n\\hspace{0.5cm} \\\\ \n & $a_4$ = ~0.347 & $b_1$ = -0.152 & $c_1$ = 0.818 & \n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Elastic constants (GPa) of the $c$ and $t$ structures.}\n\\label{elconst}\n\\begin{tabular}{c||ccccc}\n & SC-TB & Expt. & PIB & HF & DFT \\\\\n & this work & Refs.~\\onlinecite{Liu87},~\\onlinecite{Kandil82},~\\onlinecite{Kisi98b}&\n Ref.~\\onlinecite{Cohen88}& Ref.~\\onlinecite{Orlando92} & \n Ref.~\\onlinecite{Stapper99} \\\\ \\hline\n & \\multicolumn{5}{c}{\\it Cubic } \\\\\n K$_0$ & 310 & 194 254 & 288 & 222 & 268 \\\\\n $C^\\prime$ & 175 & 167 165 & 195 & 304 & - \\\\\n $c_{44}$ & 57 & 47 61 & 180 & 82 & - \\\\ \\hline\n & \\multicolumn{5}{c}{\\it Tetragonal} \\\\\n K$_0$ & 190 & 151 & 179 & & 197 \\\\\n $c_{11}$ & 366 & 327 & 465 & - & - \\\\\n $c_{33}$ & 286 & 264 & 326 & - & - \\\\\n $c_{12}$ & 180 & 100 & 83 & - & - \\\\\n $c_{13}$ & 80 & 62 & 49 & - & - \\\\\n $c_{44}$ & 78 & 59 & 101 & - & - \\\\\n $c_{66}$ & 88 & 64 & 156 & - & - \n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Phonon frequencies (THz) at the $\\Gamma$ and $X$ points of the BZ.}\n\\label{phontab}\n\\begin{tabular}{l|cccc}\n & SC-TB & DFT & DFT & Expt. \\\\ \nMode & this work & Ref.~\\onlinecite{Detraux98} &\n Ref.~\\onlinecite{Parlinski97} &\nRefs.~\\onlinecite{Liu87},~\\onlinecite{Liu84},~\\onlinecite{Cai95} \\\\ \\hline\n & \\multicolumn{4}{c}{$\\Gamma$ point} \\\\\n$T_{1_u}$ (TO) & { 6.3} & { 8.1} & { 8.5} & { 9.6} \\\\ \n%\\tablenote[3]{Reference~\\onlinecite{Liu84}} \\\\\n$T_{2_g}$ & 15.0 & 17.6 & 16.5 & 18.3 \\\\\n%\\tablenote[4]{Reference~\\onlinecite{Cai95}} \\\\\n$T_{1_u}$ (LO) & - & 20.1 & 19.7 & 21.1 \\\\ \\hline \n & \\multicolumn{4}{c}{$X$ point} \\\\\n$X_2^-$ & { ~5.1}$i$ &{ ~5.8}$i$ & { ~5.9}$i$ & \\\\\n$X_5^-$ & { 4.5} &{ 4.9} & { 3.5} & { 5.1} \\\\\n% \\tablenote[5]{Reference ~\\onlinecite{Liu87}} \\\\\n$X_5^+$ & { 5.0} &{ 8.9} & 11.7 & \\\\\n$X_4^-$ & 12.5 & 11.0 & 11.6 & \\\\\n$X_4^+$ & 18.1 & 17.0 & 16.0 & \\\\\n$X_2^-$ & 25.0 & 21.0 & 21.0 & \\\\ \n\\end{tabular}\n\\end{table}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% FIGURES\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{figure}\n\\caption{Cubic and tetragonal structures of ZrO$_2$. Light and dark\ncircles denote oxygen and zirconium atoms respectively. Arrows represent\nthe structural instability of the oxygen sublattice along the $X_2^-$\nmode of vibration.}\n\\label{cellct}\n\\centerline{\\psfig{file=cell.tet.ps,height=6cm,angle=-90} }\n\\end{figure} \n\n%%%%%%%%%%%%%%%%%%%% POSSIBLY ON TWO COLUMNS !! %%%%%%%%%%%%%%%%%%%%%\n\\begin{figure*}\n\\caption{Band structure of cubic zirconia. In all the panels, starting\nfrom the bottom it is possible to identify the oxygen 2$p$ valence bands\nand the unoccupied zirconium 4$d$ bands which are partly hybridized with\nthe oxygen 3$s$ one. The large crystal field splitting of the 4$d$ bands\npredicted by the LDA calculation (c) is reproduced with the SC-TB model,\n(a) and (b), when the $\\Delta_{ddg}$ parameter is included.}\n\\label{bndstr}\n\\centerline{\\psfig{file=bands2.ps,width=15cm,angle=0} }\n\\end{figure*}\n\n\n\\begin{figure}\n\\caption{SC-TB (top) and NFP-LMTO (bottom) Energy-Volume data for the cubic\n(c), tetragonal (t), and monoclinic (m) phases of zirconia fitted with\nMurnaghan equation of states.}\n\\label{envol}\n\\centerline{\\psfig{file=envol.all.ps,width=8.5cm,angle=-90}} \n\\end{figure}\n\n\\begin{figure}\n\\caption{Energy-Volume curves for the monoclinic (m) and orthorhombic\n($o_I$ and $o_{II}$) phases calculated with the TB model.}\n\\label{ortenvol}\n\\centerline{\\psfig{file=envol.ort.ps,width=8.5cm,angle=-90}}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Energy-Volume curves for the $c$ and $t$ structures:\nnote the convergence at small volumes V$_1$. V$_0$ and V$_2$ are\nthe equilibrium volumes of the $c$ and $t$ phases respectively.}\n\\label{envol-ct}\n\\centerline{\\psfig{file=envol-ct.ps,width=8cm,angle=-90}} \n\\end{figure}\n\n\\begin{figure*}\n\\caption{SC-TB cohesive energy vs. tetragonal distortion $\\delta$: volume and\n$c/a$ dependence. (a) Single well at V$_1$=198 a.u/ZrO$_2$; (b) Double\nwell at V$_2$=218 a.u/ZrO$_2$.}\n\\label{well}\n\\centerline{\\psfig{file=well.ps,width=8.5cm,angle=-90} }\n\\end{figure*}\n\n\\begin{figure}\n\\caption{$\\delta$ dependence of: (a) Madelung potential, (b) self-consistent charge\n$Q=Q^e+Q^i$, (c) Electrostatic and Hubbard energies as in\nEq.~(\\ref{elcst-en}), (d) the same including dipoles and quadrupoles. The\nzero of energy is the top of the double well at V$_2$, total energies are in\nRy/formula unit, other quantities in a.u./ion.}\n\\label{split1}\n\\centerline{\\psfig{file=split1.ps,width=8.5cm,angle=0} }\n\\end{figure} \n\n\\begin{figure}\n\\caption{Double well in the TB total energy at V$_2$ : ({\\boldmath\n$\\times$}) no coupling between the potential and the oxygen atomic\norbitals; ({\\boldmath $\\circ$}) with dipoles and quadrupoles on the\noxygen atoms.}\n\\label{split2}\n\\centerline{\\psfig{file=split2.ps,width=8.5cm,angle=-90}} \n\\end{figure} \n\n\n\\begin{figure}\n\\caption{Volume dependence of the order parameters calculated with the TB\nmodel: $\\eta_0$ is the hydrostatic strain of the cubic cell from the\nreference volume V$_0$, $\\eta$ is the tetragonal strain of the cell,\nand $\\delta$ (a.u.) is the tetragonal distortion of the oxygen sublattice.}\n\\label{landvol}\n\\centerline{\\psfig{file=landau1.ps,width=8.5cm,angle=-90} }\n\\end{figure}\n\n\\begin{figure}\n\\caption{SC-TB total energy versus tetragonal distortion $\\delta$. (a) Fit\nof the data with the Landau energy expansion Eq.(\\ref{lanexp}); (b)\ntransferability of the coefficients at values of hydrostatic ($\\eta_0$)\nand tetragonal ($\\eta$) strains different from the reference ones.}\n\\label{lansurf}\n\\centerline{\\psfig{file=lansurf.ps,width=8.5cm,angle=-90}} \n\\end{figure}\n\n\\begin{figure}[]\n\\caption{Phonon dispersion curves of cubic zirconia in the [100]\ndirection. Closed circles are TB calculations, dashed lines are guides to\nthe eye. Note the imaginary frequency of the $X_2^-$ mode of vibration.}\n\\label{phon}\n\\centerline{\\psfig{file=disp.ps,width=6cm,angle=0}} \n\\end{figure}\n\n\\end{document}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n"
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[
{
"name": "cond-mat0002185.extracted_bib",
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